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4th Quarter, Math Level 1 (9th grade math)

00.00 Start Here (Math Level 1)

Course Description

The fundamental purpose of Mathematics I is to formalize and extend the mathematics that students learned in 8th grade.  Students will focus on linear expressions, equations, and functions but also be introduced to exponential functions. They will solve systems of equations and inequalities. They will review learn the basic terms of geometry and solve problems involving angles, triangles, parallel lines, perimeter and area. In addition, they will use the Pythagorean Theorem to solve problems and find the distance and midpoint between two points. They will review transformations and use this knowledge to understand triangle congruence.  Finally, they will study the statitistical concepts of spread, frequency tables, histograms, box plots. scatter plots, correlation and causation, and standard deviation. Students should experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Class Overview

This integrated Secondary Mathematics I course is based on the New Utah State Standards Initiative.

Credit

This course is worth .25 credits, or nine weeks of Mathematics I. There are four Mathematics II quarter classes available. Taking all four will add up to one credit or one year of Mathematics I. In order to earn credit for each quarter, you must commit to following the EHS Honor Code: "As a student of the Electronic High School, I agree to turn in my assignments in a timely manner, do my own work, not share my work with others, and treat all students, teachers, and staff with respect." This course is for ninth grade students. After completing the work for the class, students must pass a proctored final exam to earn credit. There is not a paper-based textbook assigned for this course. If you find that having a textbook is useful, you can check out a textbook from most local libraries. You can also search for topics on the Internet to find many useful resources.

Prerequisites

You should have successfully completed 8th grade math.

Supplies needed

  • Graph paper (this can also be downloaded)
  • Scientific or Graphing Calculator (You can download scientific and graphing calculator simulators or find online versions, but as you will need a graphing calculator for the rest of your high school career, you may consider buying one now.)
  • Access to a printer to print the daily assignments is vital. Most assignments are NOT interactive and must be printed out to complete.
  • Ability to scan or photograph a completed assignment to submit electronically.

Organization of Secondary Math Level 1

Units: There are 10 units for the full credit of Secondary Math 2.

Quarter 1 has two units.
Quarter 2 has two unit.
Quarter 3 has three units.
​Quarter 4 has three units.

Schedule: When you enroll in a quarter class, you are given 10 weeks to finish all the requirements needed to earn the credit for a quarter. There are no "required" due dates for the assignments in this course. However, there is a pacing guide provided for you that will help you stay on track to being successful and finishing the course within the 10 week time frame. The pacing guide is located in the Syllabus in Module 1. Before you begin, go over the pacing guide to help you set up your own due dates for the assignments. Give your parents permission to nag you about it. You don't want to be one of those students who does a whole lot of work, but never finishes the course.

This Quarter Class

The units in this class have lessons, assignments, quizzes and a unit test.

Lessons: Each lesson provides instruction on a given topic. Many include instructional videos (hosted on YouTube) and one or more assignments for independent practice.

Assignments: Print, then complete the practice worksheet, showing how you arrived at your answer. In order to earn full points on each problem, you are required to show step by step the process needed to arrive at your answer. Then circle your answer so it is easily seen. I do give partial credit for showing your work--so ALWAYS show your work even if you think it is obvious. Assignments are submitted by uploading the digital assignment through the course website by following the instructions within each assignment. Under some circumstances, you may snail-mail a hard copy of the assignment to the instructor. If you choose this option, be sure to make a copy for yourself, as the instructor will NOT return your assignment. Also, please send the instructor an e-mail if you must mail an assignment.

Quizzes: Some quizzes are taken online. Others are more like the assignments. Print, then complete the quiz, showing all of your work as to how you arrived at your answer. In order to earn full points on each problem, you are required to show step by step the process needed to arrive at your answer. Then circle your answer so it is easily seen. I do give partial credit for showing your work--so ALWAYS show your work even if you think it is obvious. Quizzes are submitted by uploading the digital assignment through the course website by following the instructions within each quiz.

Proctored Final

Each quarter class has a proctored final exam and is worth 25% of the final grade.

Information about the Final Exam

  1. You must get approval from your teacher before you are allowed to take the final exam.
  2. You must complete every assignment and have an overall grade of C in the course to be approved to take the final exam.
  3. The final exam is a comprehensive exam that must be taken with an approved proctor.
  4. You are allowed to have a page of notes. You also will need a calculator and scratch paper.
  5. You must pass the final exam with a 60% in order to pass the class.
  6. The final exam is worth 25% of your final grade.
  7. The exam is timed. You will have 2.0 hours to complete the exam. You must finish it in one attempt.

Final Grade

Assignments and quizzes are worth 75% of the final grade. The proctored final test is worth 25% of the final grade.

Grading Scale

You earn a grade based on a modified total points percentage method. This means that the total number of points you earned is divided by the total number of points possible, times 100%. That will make up 75% of your final grade. The final exam is the remaining 25%. These scores are combined for a total percentage of the class. This percentage is translated into a grade based on this standard scale:

94-100% A
90-93% A-
87-89% B+
83-86% B
80-82% B-
77-79% C+
73-76% C
70-72% C-
67-69% D+
63-66% D
60-62% D-
0-59% no credit

00.01 Curriculum Standards (Math Level 1)

Overview information on the Utah Mathematics Level I Core is here.

00.01.01 Student Software Needs

 

Students need access to a robust internet connection and a modern web browser.

This class may also require the Apple QuickTime plug-in to view media.

For students using a school-issued Chromebook, ask your technical support folks to download the QuickTime plug-in and enable the plug-in for your Chromebook.

$0.00

00.02 About Me (Math Level 1)

teacher-scored 10 points possible 10 minutes

{\color{Red} CAREFULLY } {\color{Red} FOLLOW } {\color{Red} THE } {\color{Red} DIRECTIONS } {\color{Red} BELOW!! }

About Me Assignment: This assignment gives me, as your teacher, a chance to get to know you better! To complete and submit this assignment copy the material between the asterisks into a blank word-processing document. Answer the questions using complete sentences, appropriate punctuation and sentence structure. Please write your answers in either BOLD or a {\color{Magenta}DIFFERENT } {\color{Magenta}COLOR }. Save the document. Finally, select all, copy, then paste the entire document into the box that opens when you click to submit this assignment.

************************************************************************************************

1. What is your full name, what name do you prefer to go by, your parent's/guardian's names, and contact information for both you and your parents? (email addresses and phone numbers.)

2. What high school do you attend and what grade are you in? What is the name of the last math class you completed?

3. Why have you chosen to take this math class with EHS?

4. What is your counselor's full name and contact information?

5. Have you read the EHS Honor Code and do you commit to following it? EHS Honor Code "As a student of the Electronic High School, I agree to turn in my assignments in a timely manner, do my own work, not share my work with others, and treat all students, teachers, and staff with respect."

6. Are you committed to finishing the class within the 10 week time frame, completing your final exam in week 9?

7. Now tell me about you! What are your likes/dislikes etc. Please be sure to include anything you think I need to know as your teacher.

************************************************************************************************

I am excited to learn more about you!

Grading criteria:

1. All requested information is included.

2. Complete sentences, correct punctuation and correct grammar are used.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


08.00 Coordinate Geometry Overview (Math Level 1)

Connecting algebra and geometry through coordinates and building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. By the end of the unit, students will be able to:

  • Use the Pythagorean Theorem to find the missing side of a triangle.
  • Use the distance formulas to calculate lengths of line segments in a coordinate plane.
  • Use the midpoint to find the midpoint of a line segment in the coordinate plane.
  • Explain linear relationships dealing with parallel and perpendicular lines, distance and midpoint in a coordinate plane.
  • Classify shapes and compute perimeter and area in a coordinate plane.
  • Use the knowledge of geometric shapes to generate complex shapes.

08.01 Pythagorean Theorem (Math Level 1)

Find the mean, median, mode, and range of a set of data, compare my results to an expected distribution, and interpret results based on different sample sizes.

Something to Ponder

How would you describe the Pythagorean Theorem and how would you explain how and when to use it?

Mathematics Vocabulary

Pythagorean Theorem: in any right triangle the sum of the squared sides is equal to the squared length of the hypotenuse \fn_jvn a^{2} +b^{2}=c^{2}

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.01 Pythagorean Theorem - Explanation Video Link (Math Level 1)

08.01 Pythagorean Theorem - Explanation Videos (Math Level 1)

See video


08.01 Pythagorean Theorem - Extra Video (Math Level 1)

I highly recommend that you click on the links above and watch the videos before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

Find the value of x in each right triangle. 

a)

b)

Example 2:

Find the missing sides in each of the right triangles: 

a) 

b) 

c) 

Example 3:

A baseball diamond is a square with 90-ft. sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base? 

Answers

Example 1:

Find the value of x in each right triangle. 

a)

x2 + 62 = 102

x2 + 36 = 100

x2 = 64

x = 8

b)

152 + 82 = x2

225 + 64 = x2

289 = x2

17 = x

Example 2:

Find the missing sides in each of the right triangles: 

a)

x2 + 32 = 52

x2 + 9 = 25

x2 = 16

x = 4

b)

52 + 122 = x2

25 + 144 = x2

169 = x2

13 = x

c)

x2 + 222 = 252

x2 + 484 = 625

x2 = 141

x = 11.9 rounded to the nearest tenth.

Example 3:

A baseball diamond is a square with 90-ft. sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base? 

902 + 902 = x2

8100 + 8100 = x2

16,200 = x2

x = 127.3 rounded to the nearest tenth.

08.01 Pythagorean Theorem - Worksheet (Math Level 1)

teacher-scored 42 points possible 30 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


08.02 Distance Formula (Math Level 1)

Use the distance formula to calculate lengths of line segments in a coordinate plane.

Something to Ponder

How would you explain the distance formula and how to use it?

Mathematics Vocabulary

Distance Formula: the distance between any two points \fn_jvn (x1,y1) and (x2,y2) can be found by d= \sqrt{ (x2 - x1)^{2} + (y2 - y1)^{2}} 

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.02 Distance Formula - Explanation Video Link (Math Level 1)

08.02 Distance Formula - Explanation Videos (Math Level 1)

See video


08.02 Distance Formula - Extra Video (Math Level 1)

I highly recommend that you click on the links above and watch the video and work through the material before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1: 

Find the distance between R(-2,6) and S(6,-2) to the nearest tenth

Example 2:

AB has the endpoints A(1,-2) and B(-4,4). Find AB to the nearest tenth.

Example 3:

Find the length of the line segment XY with endpoints X(0,0) and Y(-5,-2).

Answers

Example 1: 

Find the distance between R(-2,6) and S(6,-2) to the nearest tenth.

\fn_phv d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(6--2)^{2}+(-2-6)^{2}}

d=\sqrt{(8)^{2}+(-8)^{2}}

d=\sqrt{64+64}

d=\sqrt{128}

d = 11.3 rounded to the nearest tenth.

Example 2:

AB has the endpoints A(1,-2) and B(-4,4). Find AB to the nearest tenth.

d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(-4-1)^{2}+(4--2)^{2}}

d=\sqrt{(-5)^{2}+(6)^{2}}

d=\sqrt{25+36}

d=\sqrt{61}

d = 7.8 rounded to the nearest tenth.

Example 3:

Find the length of the line segment XY with endpoints X(0,0) and Y(-5,-2).

d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(-5-0)^{2}+(-2-0)^{2}}

d=\sqrt{(-5)^{2}+(-2)^{2}}

d=\sqrt{25+4}

d=\sqrt{29}

d = 5.4 rounded to the nearest tenth.

08.02 Distance Formula - Worksheet (Math Level 1)

teacher-scored 20 points possible 40 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


08.03 Midpoint Formula (Math Level 1)

Use the midpoint to find the midpoint of a line segment in the coordinate plane.

Something to Ponder

How would you explain the Midpoint Formula and how to use it?

Mathematics Vocabulary

Midpoint Formula: the location of the midpoint of a segment with end points \fn_jvn (x1,y1) and (x2,y2) is found by \left (\frac{(x_{2}+x_{1})}{2},\frac{(y_{2}+y_{1})}{2} \right )

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.03 Midpoint Formula - Explanation Video Link (Math Level 1)

08.03 Midpoint Formula - Explanation Videos (Math Level 1)

See video


08.03 Midpoint Formula - Extra Video (Math Level 1)

I highly recommend that you click on the link above and work through the material before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1: 

AB has endpoints (8,9) and (-6,-3). Find the coordinates of its midpoint M.

Example 2:

Find the coordinates of the midpoint of XY with endpoints X(2,-5) and Y(6,13).

Example 3:

Find the coordinates of the midpoint of CD with endpoints C(6,-4) and D(2,7).

Answers

Example 1

AB has endpoints (8,9) and (-6,-3). Find the coordinates of its midpoint M.

 \fn_phv M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

M = \left ( \frac{8+-6}{2},\frac{-3+9}{2} \right )

M = \left ( \frac{2}{2},\frac{6}{2} \right )

M = \left (1,3)

Example 2:

Find the coordinates of the midpoint of XY with endpoints X(2,-5) and Y(6,13).

M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

\left ( \frac{2+6}{2}, \frac{-5+13}{2} \right )

\left ( \frac{8}{2}, \frac{8}{2} \right )

M = (4,4)

Example 3:

Find the coordinates of the midpoint of CD with endpoints C(6,-4) and D(2,7).

M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

M = \left ( \frac{6+2}{2},\frac{-4+7}{2} \right )

M = \left ( \frac{8}{2},\frac{3}{2} \right )

M = \left (4,\frac{3}{2} \right )

08.03.01 Midpoint Formula - Worksheet (Math Level 1)

teacher-scored 20 points possible 40 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


08.03.02 Checkpoint Quiz (Math Level 1)

teacher-scored 54 points possible 40 minutes

Quiz Check Point

  1. Print the quiz. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your corrected quiz.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


08.04 Verifying Linear Relationships (Math Level 1)

Explain linear relationships dealing with parallel and perpendicular lines, distance and midpoint in a coordinate plane.

Something to Ponder

How would you explain how to determine if lines are parallel or perpendicular in a coordinate plane?

Mathematics Vocabulary

Linear relationship: A linear relationship exists when you plot two values on a coordinate system and you get a straight line, or values that would "average out" to be a straight line.

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.04 Verifying Linear Relationships - Explanation Video Link (Math Level 1)

Optionally: use the link above to view the explanatory math video.

08.04 Verifying Linear Relationships - Explanation Videos (Math Level 1)

See video


08.04 Verifying Linear Relationships - Extra Link (Math Level 1)

I highly recommend that you click on the link above and work through the material.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

Determine if the following lines are parallel, perpendicular or neither. Verify your decision using slope: 

Example 2:

Determine the lengths of the following line segments using the distance formula. Then, find the coordinates of the midpoints. 

Answers

Example 1:

Determine if the following lines are parallel, perpendicular or neither. Verify your decision using slope: 

1st Graph:

Upper line: Find two points on the graph - (0, 2) and (3, 0)

Find the slope of the line given the two points:

\fn_phv m=\frac{y{_2-y{_1}}}{x{_2-x{_1}}}

m=\frac{0-2}{3-0}=-\frac{2}{}3

Lower line: Find two points on the graph - (0, -1) and (3, -3)

Find the slope of the line given the two points:

m=\frac{y{_2-y{_1}}}{x{_2-x{_1}}}

m=\frac{-3--1}{3-0}=\frac{-3+1}{3} = -\frac{2}{3}

The slopes are the same so the lines are parallel.

2nd Graph:

Line 1: Find two points on the graph - (0, 1) and (-1, -2)

m=\frac{y{_2-y{_1}}}{x{_2-x{_1}}}

m=\frac{-2-1}{-1-0} = \frac{-3}{-1} = 3

Line 2: Find two points on the graph - (-3, 0) and (0, -1)

m=\frac{y{_2-y{_1}}}{x{_2-x{_1}}}

m=\frac{-1-0}{0 - - 3}=-\frac{1}{}3

Since 3\cdot -\frac{1}{}3 = 1 (negative reciprocal), the lines are perpendicular.

Example 2:

Determine the lengths of the following line segments using the distance formula. Then, find the coordinates of the midpoints. 

Step 1: Find the endpoints of each of the lines:

Line A: (-3, 4) and (-2, -4)

Line B: (2, 4) and (-2, -2)

Line C: (3, 1) and (-3, 0)

Step 2: Find the distance between each set of points:

Line A:

d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(-2--3)^{2}+(-4-4)^{2}}

d=\sqrt{(1)^{2}+(-8)^{2}}

d=\sqrt{1+64}

d=\sqrt{65}

d = 8.1 rounded to the nearest tenth.

Line B:

d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(-2-2)^{2}+(-2-4)^{2}}

d=\sqrt{(-4)^{2}+(-6)^{2}}

d=\sqrt{16+36}=\sqrt{52}

d = 7.2 to the nearest tenth

Line C:

d=\sqrt{(x{_2}-x{_1})^{2}+(y{_2}-y{_1})^{2}}

d=\sqrt{(-3-3))^{2}+(0-1)^{2}}

d=\sqrt{(-6))^{2}+(-1)^{2}}

d=\sqrt{36+1}=\sqrt{37}

d = 6.1 to the nearest tenth.

Step 3: Find the midpoints of each line:

Line A:

M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

M = \left ( \frac{-3+-2}{2},\frac{4+-4}{2} \right )

M = \left ( \frac{-5}{2},\frac{0}{2} \right )=\left ( \frac{-5}{2},0 \right )

Line B:

M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

M = \left ( \frac{2+-2}{2},\frac{4+-2}{2} \right )

M = \left ( \frac{0}{2},\frac{2}{2} \right )=\left ( 0,1\right )

Line C:

M = \left ( \frac{x{_1}+x{_2}}{2},\frac{y{_1}+y{_2}}{2} \right )

M = \left ( \frac{3+-3}{2},\frac{1+0}{2} \right )

M = \left ( \frac{0}{2},\frac{1}{2} \right )=\left ( 0,\frac{1}{}2 \right )

08.04 Verifying Linear Relationships - Worksheet (Math Level 1)

teacher-scored 52 points possible 40 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


08.05 Geometric Figures in the Coordinate Plane (Math Level 1)

Classify shapes and compute perimeter and area in a coordinate plane.

Something to Ponder

How would you explain how to find the perimeter and area in a coordinate plane?

Mathematics Vocabulary

Perimeter: the total distance around the edge of the figure

Area: the size of a surface refered to by the number of square units the figure covers

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.05 Geometric Figures in the Coordinate Plane - Explanation Video Link (Math Level 1)

08.05 Geometric Figures in the Coordinate Plane - Explanation Videos (Math Level 1)

See video


08.05 Geometric Figures in the Coordinate Plane - Extra Links (Math Level 1)

I highly recommend that you click on the links above and work through the material.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

Find the perimeter of the following polygons. Also find the area of triangles, parallelograms and trapezoids.

Example 2:

Based on your knowledge of slope, distance and midpoint formulas, determine what shape each set of ordered pairs represents. 

a) {(1,-6),(4,-6),(5,-9),(0,-9)}

b) {(-8,-4),(-6,-2),(-4,-4),(-5,-6),(-7,-6)}

c) {(-4,7),(-2,4),(-6,1)}

d) {(3,-1),(7,-2),(7,-5),(3,-4)}

e) {(-1,2),(2,2),(3,0),(2,-2),(-1,-2),(-2,0)}

f) {(7,2),(7,-1),(10,-1)}

g) {(2,7),(6,7),(4,5),(8,5)}

Answers

Example 1:

Find the perimeter of the following polygons. Also find the area of triangles, parallelograms and trapezoids.

Triangle:

To find the perimeter, we need to know the lengths of all three sides. We can use the distance formula or the Pythagorean theorem to find the sloping sides. For this one, I’ll use the Pythagorean theorem.

Left side: bottom = 4, vertical side = 6, hypotenuse = \fn_phv \sqrt{4^{2}+6^{2}} = \sqrt{16+36} = \sqrt{52} \approx 7.2

Right side: bottom =2, vertical side = 6, hypotenuse =\sqrt{2^{2}+6^{2}} = \sqrt{4+36} = \sqrt{40} \approx 6.3

Bottom side: = 6

P = 7.2 + 6.3 + 6 = 19.5 units

A = \frac{1}{2}Bh = \frac{1}{2}x6x6 = 18 square units

b) Trapezoid:

To find the perimeter, we need to know the lengths of all four sides. We can use the distance formula or the Pythagorean theorem to find the sloping sides. Again, I’ll use the Pythagorean theorem as needed.

Top side = 6

Bottom side = 3

Left side: We will need to find the hypotenuse of the triangle

top = 1, vertical side = 4, hypotenuse = \sqrt{1^{2}+4^{2}}=\sqrt{1+16}=\sqrt{17}\approx 4.1

Right side: We will need to find the hypotenuse of the triangle

Top side = 2, vertical side = 4, hypotenuse = \sqrt{2^{2}+4^{2}}=\sqrt{4+16}=\sqrt{20}\approx 4.5

P = 6 + 3 + 4.1 + 4.5 = 17.6 units

To find the area, I’m going to find the area of the left triangle, the right triangle and the rectangle inbetween.

Left triangle = Bh = \frac{1}{2}x1x4 = 2 square units

Right trianlge = Bh = \frac{1}{2}x2x4 = 4 square units

Rectangle = l x w = 3 x 4 = 12 square units

A = 2 + 4 + 12 = 18 square units

c) Parallelogram:

To find the perimeter, we need to know the lengths of all four sides. We can use the distance formula or the Pythagorean theorem to find the sloping sides. Again, I’ll use the Pythagorean theorem as needed.

Each of the right triangles are congruent. I just need to find the hypotenuse of one.

Vertical side is \frac{1}{2} x 6 = 3, horizontal side is \frac{1}{2} x 6 = 3; hypotenuse =\sqrt{3^{2}+3^{2}}=\sqrt{9+9}}=\sqrt{18}\approx 4.24

P = 4.24 x 4 = 16.96 units

A = \frac{d{_1}\cdot d{_2}}{2}=\frac{6\cdot 6}{2}= 18 square units

d) Pentagon:

To find the perimeter, we need to know the lengths of all five sides. We can use the distance formula or the Pythagorean theorem to find the sloping sides. Again, I’ll use the Pythagorean theorem as needed.

I’m going to find the length of the sides of the top left triangle first.

Bottom side = 2, vertical side = 2, hypotenuse = \sqrt{2^{2}+2^{2}}=\sqrt{4+4}}=\sqrt{8}\approx 2.83

The top right triangle’s hypotenuse also \approx 2.83

Now for the bottom left triangle:

Top = 1, vertical side = 3, hypotenuse = \sqrt{1^{2}+3^{2}}=\sqrt{1+9}}=\sqrt{10}\approx 3.16

The bottom right triangle’s hypotenuse is also \approx 3.16

P = 2.83 + 3.16 + 2 + 3.16 + 2.83 = 13.98 or \approx 14

Example 2:

Based on your knowledge of slope, distance and midpoint formulas, determine what shape each set of ordered pairs represents. 

You can follow the directions and determine the answers – but here is a little math test hint. If you know an easier way – use it! The easiest way to do these problems is to graph them!!

a) {(1,-6),(4,-6),(5,-9),(0,-9)}

Isosceles Trapezoid

b) {(-8,-4),(-6,-2),(-4,-4),(-5,-6),(-7,-6)}

Pentagon

c) {(-4,7),(-2,4),(-6,1)}

Triangle

d) {(3,-1),(7,-2),(7,-5),(3,-4)}

Parallelogram

e) {(-1,2),(2,2),(3,0),(2,-2),(-1,-2),(-2,0)}

Hexagon

f) {(7,2),(7,-1),(10,-1)}

Right Triangle

g) {(2,7),(6,7),(4,5),(8,5)}

Parallelogram

08.05 Geometric Figures in the Coordinate Plane - Worksheet (Math Level 1)

teacher-scored 82 points possible 45 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps..
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


08.06 Complex Geometric Shapes (Math Level 1)

Use my knowledge of geometric shapes to generate complex shapes.

Something to Ponder

How would you create a illustrated story using tangram shapes?

Mathematics Vocabulary

Simple shape: described by basic geometry objects such as a set of two or more points, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere).

Complex shape: most shapes occurring in the physical world are complex such as plant structures and coastlines. Complex shapes may defy traditional mathematical description

Learning these concepts

Click the images or the links below to launch the videos to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

08.06 Complex Geometric Shapes - Explanation Video Links (Math Level 1)

08.06 Complex Geometric Shapes - Explanation Videos (Math Level 1)

See video
See video


Guided Practice
After watching the video try these problems. The worked solutions follow.

Tangrams are popular in China.  Find a square piece of cardboard. Draw grid lines on the cardboard to form an array of 4 by 4 squares.

Draw the tangram lines.  Cut the cardboard along the tangram lines to form the seven pieces of the tangram set.  You end up with five triangles, a square, and a parallelogram.

Answers

The five triangles are blue, pink, yellow, brown and green. The square is light purple. The parallelogram is red.

08.06 Complex Geometric Shapes - Worksheet (Math Level 1)

teacher-scored 56 points possible 80 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


08.06 Unit 08 Review Quiz (Math Level 1)

teacher-scored 46 points possible 40 minutes

Unit Review Quiz

  1. Print the quiz. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your corrected quiz.

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


09.00 Congruence Overview (Math Level 1)

In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop no- tions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. By the end of the unit, students will be able to:

  • Define, identify and make rigid transformations (rotations, reflections and translations), contrast them with non-rigid transformations and discuss symmetry.
  • Use mappings to transform geometric figures and write rules from transformations.
  • Justify congruence of triangles.
  • Construct congruent segments and angles, bisect segments and angles, and construct perpendicular and parallel lines.
  • Construct triangles, squares and hexagons.

09.01 Transformations (Math Level 1)

Define, identify and make rigid transformations (rotations, reflections and translations), contrast them with non-rigid transformations and discuss symmetry.

Something to Ponder

How would you define Rotations, Reflections, Translations and Compositions?

Mathematics Vocabulary

A transformation occurs when we take one object and change it into a new object. The constructions duplicating a line segment and duplicating an angle were examples of transformations. In both of these transformations the old object was congruent to the new object. Not all transformations produce a new object that is congruent to the old object.

In this lesson we will learn about transformations that preserve both the length of a line segment and the measure of an angle. If a transformation preserves both of these, we say it is a rigid motion. A rigid object is an object that holds its shape. In the real world, solids are rigid (remember, that solid/liquid/gas thing from science class). Many everyday objects are solid: your desk, your coffee mug, your sunglasses, most of the parts of your computer. Many objects are plastic (the adjective, not the noun). Plastic means that the object will hold a shape, but it can be changed to a different shape by applying force (and usually heat). Plastic things include plastic cups, rubber balls, modeling clay and some of the parts of your computer.

What is my point? A rigid motion is one that a rigid object can undergo.

If you decide to slide your desk across the wall to the other corner of the room, the desk will have the same shape in the new corner that it had in the old corner. (Of course, this is only true if you didn't break off a leg or something trying to get it to slide across the carpet--I've done that :) ) So sliding your desk is one example of a rigid motion.

Now, once you get the desk into the corner, you decide it should face the other wall. So you turn it. I bet it has the same shape as it did before! How awesome is that! So another rigid motion we can do is to turn an object.

The final rigid motion is one that you can't physically do to your desk, but is a real motion nonetheless. If you look at your desk in the mirror, the mirror desk has the same shape as the real desk. This is also an example of a rigid motion, because although the mirror desk is inverted, the lengths and angles are all the same. (Now this isn't true if you happen to have a fun house mirror in your room instead of a regular flat mirror. What kind of a room do you have anyway!) You can see physical examples of the mirror transformation by walking through a neighborhood where all the homes where built at the same time. Neighborhoods like this are usually built with a few floor plans. Each floor plan is expensive to design so developers take each floor plan and invert it. If it is easy to find two houses that are mirror images of one another, it is because they were built using mirror image floor plans.

Now consider a ball of clay as an example of a non-rigid object. This object can still undergo rigid motions, but it can be transformed in other ways as well. You can squish it, you can stretch it, you can warp in in all kinds of ways!

Let's talk about each rigid transfromation.

Translations

The example of moving your desk across the room is a translation. Translations are the simplest and most easily understood of the rigid motions we will consider. A translation is where you slide an object from one position in space to another. We can translate any of the objects we have considered so far in this class.

Consider the following objects. The lighter color represents the original object and the darker color is the transformed object.

Point H is translated to the right.
 

Line v is translated more or less upwards.
 

Why didn't I say that line v is translated upward and to the right? Remember lines are infinite. Although we only drew part of the line, it extends forever, so we don't actually know that it is any farther to the right. All we know is that it is translated in a direction perpendicular to the line. We would need to include at least one point to know whether the line is translated to the right or not.

is translated down and to the left.
 

∠P is translated down and to the right.
 

is translated to the left.
 

Triangle ABC (△ABC) is translated up and to the right.
 

Get the idea? Notice that the measure of the angle and the length of the line segment do not change.

We will mostly be transforming figures. There are a few other properties of figures that can be preserved. However, if the length of each segment is preserved and the measure of each angle is preserved, all the other properties we consider will also be preserved.
 

Rotations

If you turn your desk to face the other wall you have performed the next rigid motion we consider, a rotation. A rotation is when we turn something. To specify a rotation you need two pieces of information: the angle of the rotation and the center of the rotation.

When you rotate an object, it is standard to rotate it counter-clockwise. You are probably familiar with measuring rotations in terms of degrees. We will be using the commonly known degrees of 90o , 180o and 270o .

The other thing you need to know is the center of rotation. Rotating from the center of a figure will produce a different result than rotating from the edge.

Consider the following rotations. The lighter color represents the original object. The first image shows point H rotated a quarter circle around point G, and the second image shows point I rotated a quarter circle about point I. Notice that the point I hasn't changed. You probably suspected that.
 

This image shows line v rotated a quarter circle both about point U on line v (the turquoise line) and about point T not on line v (the green line). Notice that these rotations produce parallel lines, but not the same line.
 


Okay, let's see if you can predict what will happen when you rotate objects. What will happen if you rotate it 90o about point J? What about rotating it 90o about point K? Will you get the same result? How will the results differ?
 

Now rotate ∠P 90o about its vertex? What about a rotating it the same amount about a third point O as shown? How will these results differ? What will not change?
 

Take a second and draw your predictions. The answers are on the next page.

This image shows rotated 90o both about point K (the blue line) and about point J (the turquoise line). Notice that these rotations produce parallel line segments but not the same line.
 

The image below shows ∠P rotated 90o both about its vertex (dark red) and also ∠P rotated 90o about point O. Did you predict these results?
 

In these images the lengths of the rays have changed. Why is this irrelevant?

Okay, how about a recognition question. The image below shows △ ABC (light magenta) and four 90o rotations about different centers of rotation. Determine the centers of rotation for the blue triangle, the green triangle, the red triangle, and the magenta triangle. The answers are on the next page.
 

Answers:

First image:

  • The blue triangle is rotated 90o circle about point A.
  • The red triangle is rotated 90o about point B.
  • The magenta triangle is rotated 90o about point C.
  • The green triangle is rotated 90o about a point inside △ ABC.

 

Did you get these correct? Did you get close at least?

 

Reflections

The final rigid motion we consider is a reflection. A reflection produces a mirror image of the original object, just like your reflection in the mirror.

To specify a reflection you need to give the axis of reflection. By changing the axis of reflection the result can be very different. Consider a few reflections below. The lighter image is the original object and the darker color is the reflected object. The gray line is the axis of reflection. The key to a reflection is that the perpendicular distance between all points and the axis of reflection doesn't change.
 

Consider a few additional reflections. Again, the light image is the original ∠P. The gray line is the axis of reflection. The dark red image shows ∠P reflected across a line that is parallel to one of the rays. The magenta image shows ∠P reflected across a line that is oblique to both rays that compose ∠P. The dark blue image shows ∠P reflected across a line that is perpendicular to one of the rays. Notice how the reflection changes in each of these images; also notice the relationship between the reflected image and the original.
 

In the images below observe how the different axes of reflection change the reflected object. Also, observe how the relationship between the original and reflected object changes depending on the axis of reflection. The light magenta triangle is the original △ ABC. The gray line is the axis of reflection.
In the first image, the magenta triangle is reflected across an axis that is parallel to .
 

In the second image, the red triangle is reflected across an axis that is oblique to each segment  , and . This line does go through point C.
 

In the third image, the turquoise triangle is reflected across an axis that is perpendicular to  and that intersects .
 

Notice in each of these images the orientation of the vertices has changed. In the original image the vertices were labeled A, B and C going counter-clockwise from point A. In each of the reflected images the vertices are still labeled A, B and C, but the orientation is clockwise from point A.

It is now time to do a few practice problems. First, predict what the reflected △ ABC will look like with an axis parallel to  on the left of the triangle, as shown. What about the reflection across an oblique line that intersectsand , such as the one shown? The answers are on the next page.
 

The next question: given a reflection can you determine where the location of the axis of refection? Try it for the next images. Remember the perpendicular distance between all the points and the axis doesn't change.
 

 

Answers:
 

Did you get all of those?
 

Learning these concepts

Click the mathematician images or the links below to launch the videos to help you better understand this "mathematical language."

09.01 Transformations - Careers (Math Level 1)

09.01 Transformations - Explanation Video Links (Math Level 1)

09.01 Transformations - Explanation Videos (Math Level 1)

See video
See video


Constructing Rigid Transformations

Constructing Translations

You now know what transformations we will be doing in this class. I am certain you are entertaining a question: How do I construct these transformations for myself? Right? Okay, maybe not, but this is what we are going to learn next anyway. We will be constructing these both using transparent paper, a straight edge, and a compass, and on a computer drawing program.

The first transformation we considered was a translation, so this will be our first construction. Let's start with the paper and pencil method.

The first construction we will do is a translation to the right.

Using your straight edge, draw a figure on a sheet of transparent paper. It can be any shape you wish. I will use a trapezoid for my example. Draw a horizontal line under the figure, like this.
 

Copy the figure and horizontal line onto a different sheet of transparent paper. Place the first sheet on top of the second sheet so that the horizontal lines coincide. However, the figure on the bottom is shifted to the right. Copy the figure onto the first sheet of paper. Tah dah!

How could you use this method to translate the figure upwards? What about at an arbitrary oblique angle? How much fun is this anyway?

In the homework you are asked to translate the object a specific distance. Using your compass to measure the distance, how would you complete this task?

The next task is to perform a translation to the right using a drawing program. I mentioned previously that I use Open Office Draw but I suspect the features are similar for most drawing programs.

Start again by drawing any figure you like. I will use the same trapezoid as before. Copy and paste the drawn figure. At this point your drawing program may differ. Some programs will paste the image on top of the original. Others will paste the image in the center of the workspace. Either way you will need to move it to a new location.

Next, you will want to move the figure to the right of the original. Again, programs vary in how to do this. I have used drawing programs that if you push the shift button while moving an object it will only allow motion vertically, horizontally, or at an angle of 45 degrees. Open Office Draw does not have that feature. Instead I will repeat the process I used previously by drawing a guide line under the figure. Most drawing programs will draw a horizontal line if you hold the shift button while drawing the line. I can then reposition it so that the translated image is in the same location with respect to the line. I can even copy and paste the guide line which will make it more obvious when the objects are aligned. Now that I have a strategy for moving the object to the right, I will do so.

What if I wish to move it a set distance? For example, say I wish to translate it to the right by 1 unit, where I have defined 1 unit to be the base of the trapezoid? Since I was silly and drew my trapezoid so that the base is at an odd angle, I will have to be clever. I cannot make the measurement with a rectangle (one of my favorite ways to measure distance in a drawing program).

How did I measure distance on paper? With a compass! I can draw a circle. Open Office Draw will let me draw a circle from the center by using the shift and alt buttons while drawing. It doesn't tell me where the center is so I will draw a pair of intersecting lines with the center of the circle on the intersection. Next, I will select all three objects and move them together. The steps I will take are shown below. The gray lines are the guides.
 


 

Once this is done, I can delete the guide lines. In your homework, you may wish to leave them. Is it clear how and why I did each of the steps? Can you duplicate this translation downward instead of to the right?
 

Constructing Rotations on Paper

Constructing a rotation with pencil and paper is a bit more difficult. Because we are not using protractors, it is hard to estimate a third of a circle (120o), for example. On the other hand, it is fairly easy to estimate half a circle (180o), a quarter of a circle (90o), three quarters of a circle (270o), even eighths and sixteenths of circles. How, you ask? By folding a sheet of paper in half, then in half again, then in half again, etc. If we do this three times, the creases in the paper will look like this.
 

By drawing a guide line that we align with the creases on the paper, we can easily rotate an object an eight of a circle (45o), a quarter of a circle (90o), three eights of a circle (135o), a half of a circle (180o), etc. Two more careful folds will divide the circle into sixteenths and we can make even smaller rotational increments by continuing to fold. Also, this awesome image gives us a center of rotation.

We will be using two sheets of transparent paper in addition to the folded/unfolded sheet we will use as our angle guide. You may wish to draw lines through the creases to make them easier to see.

To start, draw a figure on one sheet of transparent paper. Draw the center of rotation and a horizontal line through the center of rotation. Say, for example, I wish to rotate the figure about the bottom left vertex. I would draw an image like the one below.
 

Trace this image onto the second piece of transparent paper. With the creased guide sheet on the bottom, place the second sheet on top of this so that the center of rotation is at the center of the guides, as shown here.
 

Rotate the transparent paper so that the center point is still at the center of the guide, but with the horizontal line aligned with the correct angle. For example, an eighth of a circle (45o), a half-circle (1800) and a three-quarters of a circle (270o) would look like this:
 

Finally, place the original sheet of transparent paper on top of the others (again with the center point at the center) and the horizontal line aligned with the horizontal guide line of the first image.
 

Now, copy the image onto the top sheet. Your top paper should look like the image below.
 

This will work for any arbitrary center of rotation such as the one shown here.
 

Make a copy of this image and use the guide to rotate it to the desired angle.
 

Place the original image on top of the rotated copy, aligning the horizontal guide and center of rotation, and copy the rotated copy.
 

Ready to do this on your own?

Constructing Rotations with a Computer Drawing Program

The next task is to perform a rotation using a computer drawing program. Your computer draw program has a rotate tool, so you will want to use this tool. However, it has some limits. For example, while you can move the center of rotation to an arbitrary location, I find it difficult to align the center. Also, you will still need to be able to determine how far you have rotated the shape; so again, it is worth drawing a set of guides.

Before you start, you will want to draw the angles of rotation. By using the shift button while drawing a line, you can draw lines that are horizontal, vertical and at 45 degree angles. This will give you the same set of guides we were able to produce for the pencil and paper rotation.

You may also be able to obtain guide lines for sixth of a circle rotations. One of the preset figures in your drawing program should be a hexagon. If this is a regular hexagon, you merely need to hold the shift key while drawing the hexagon. The Open Office hexagon is not quite regular but I can use this function to construct a regular hexagon. First, draw a circle (again, from the center with the center marked). Next draw a hexagon from the same center. Size the hexagon such that the vertices lie on the circle, as shown here.
 

In general, just because a hexagon is inscribed in a circle doesn't mean that it is a regular hexagon. This technique works with the drawing program because it creates a semi-regular hexagon. Inscribing the semi-regular hexagon in the circle forces it to become regular.

Next, draw lines connecting opposite vertices, as shown here. Now each line corresponds to one-sixth of a circle. If we only keep the lines, and the 4 we drew earlier, we will be able to make the following fractions of a circle rotations:and . That's not every fraction, but quite a few.
 

By drawing a regular octagon, we can add the sixteenth fractions. However, Open Office doesn't have a regular octagon, so I won't go there. Depending on your draw program you may be able to generate a much larger variety of fractions of a circle.

Back to the original problem. To begin, draw the figure you wish to rotate. Next draw the center of rotation, and a horizontal line through the center, such as the example shown.
 

Copy and paste these three objects. Move the guide lines you just drew so that the center of these lines coincides with your center of rotation, as seen here.
 

To rotate these three objects select them then choose the rotate tool. Move the center of the rotate tool so that it coincides with the center of rotation. Rotate the selected objects until the horizontal guide line aligns with the desired guide angle. You will definitely need to practice this.

I have not had good luck relocating the center of rotation. Instead, I figured out how to force the computer program into placing the center of rotation at my desired center of rotation. I draw a square that is larger than the three objects I plan to rotate, centered at my center of rotation, such as the one shown here.
 

Then I select these four objects, and rotate them. Because the square is larger than any of the three objects I wish to rotate, the drawing program places the center of rotation at the center of the square. However, I drew the square such that its center is my desired center of rotation. Problem solved.
 

Once I have completed the rotation I can delete the square. You may also wish to delete the guide lines (or move them elsewhere so you don't have to redraw them if you need them later). Of course, for the purpose of earning full credit on the homework, it isn't a bad idea to leave the guide lines. A finished homework problem might look like the image below.
 


 

Constructing Reflections

The third transformation we considered was a reflection. Constructing reflections with transparent paper is fairly straightforward.

As usual, start by drawing the figure you wish to transform on a sheet of transparent paper. Next, draw the line you wish to reflect the figure across. Now, as a guide, construct a second line that is perpendicular to the reflection line.

One example is shown here. In this case, the brown line is the axis of reflection and the gray line is the perpendicular guide. It doesn't matter where you construct the guide.
 

Fold the paper in half along the axis of reflection. Use the perpendicular guide to get the fold perfect.

Your paper should look like the image. Now copy the figure onto the other side of the transparent paper.
 

When you unfold the paper, the reflected image will be visible through the transparent paper. You may wish to redraw it so that it is easier to see.
 

If you wish to create a reflection across two lines, simply repeat the steps above.

To construct a reflection across an axis that intersects the figure, you would follow the same steps as above, but you will need to copy part of the figure on one side of the paper, then turn the folded paper over and finish copying the figure on the other side. Finally, unfold the paper.

To reflect an image using a computer drawing program may be very simple, or very complicated, depending on the drawing program. I have used a program that allows you to specify the axis of reflection with the reflection tool. You select the object you wish to reflect, then you select the reflection tool, use the shift button to prevent the axis from moving, and reflect the image. Easy as pie!

Unfortunately, this was a commercial drawing program. I suspect the free program you downloaded, or the program that came with whatever Office package you purchased for your computer, doesn't have this capability. Open Office only allows horizontal and vertical "flips", but we can use that reflection, coupled with carefully constructed perpendicular lines, and the rotation tool to reflect a figure across any axis.

Start by drawing the figure you wish to reflect, and the axis of reflection. Now, draw a line perpendicular to the axis of reflection. Do you remember how to do this? Great.
 

Hint:

Copy and paste the perpendicular line. You need two of these. Move the lines so that each line goes through one vertex. An example is shown here.
 

Next, draw a circle, centered at each intersection, so that the edge of the circle intersects the vertices. Remember, in a reflection, each point is the same perpendicular distance from the axis of reflection, so the lines and circles give us a perpendicular distance that we can use as a guide to align the reflected image.
 

This image shows the same figure moved to the other side and in a different color.
 

Now use the rotate tool and the moving tool to align the two vertices with the corresponding circle/line intersection. You may have to rotate and move the figure a few times to get it right. Your finished reflection should look like the image below. Tuh duh!
 

Can you do these on your own? If not, practice, practice, practice!

 

09.01 Transformations - Extra Links (Math Level 1)

I highly recommend that you click on the links above and work through the material.

09.01 Transformations - Worksheet (Math Level 1)

teacher-scored 20 points possible 45 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

 

Pacing: complete this by the end of Week 4 of your enrollment date for this class.


09.02 Mappings (Math Level 1)

Use mappings to transform geometric figures and write rules from transformations.

Something to Ponder

How would you describe what the concept of input/output has to do with mappings.?

Mathematics Vocabulary

Mapping: the function rule that is applied to the pre-image to get the image

Learning these concepts

Click the mathematician images or the linsk below to launch the videos to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

09.02 Mappings - Explanation Video Links (Math Level 1)

From Oswego City School District Regents Exam Prep Center: http://www.regentsprep.org/regents/math/geometry/gt5/reviewTranformation...

Translations

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction. 

The rule is (x, y) = (x \pm h, y \pm k) where “h” is the number of units of “slide” horizontally and “k” is the number of units of “slide” vertically.

Example:

In this case the triangle is translated 3 units up. The rule is: (x, y) = (x, y + 3)

Reflections

Across x-axis: When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. Rule: P(x, y) P’(x, –y) 

Example:

Across y-axis: When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. Rule: P(x, y) P’(–x, y) 

Example:

Across the line y = x: When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite.  Rule: P(x, y) P’(y, x) 

Example:

Rotations

A rotation turns a figure through an angle about a fixed point called the center.  A positive angle of rotation turns the figure counterclockwise, and a negative angle of rotation turns the figure in a clockwise direction. Therefore, every rotation needs two pieces of information: the point the figure rotates around and the degree of rotation.

Rotation around the origin: While any point in the coordinate plane may be used as a point of reflection, the most commonly used point is the origin.  

Rotation of 90º about the origin – Rule: P(x, y) P’(–y, x)

Rotation of 180º around the origin – Rule: P(x, y) P’(–x, –y)

You can also perform transformations on functions. We will only discuss vertical translations in this unit.

Let’s start by graphing the following linear functions:

f(x) = 2x

f(x) = 2x + 2

f(x) = 2x – 2

The + 2 and –2 translate the original function to the right and left. Notice that the three lines are parallel? You should know from your study of the slope-intercept form of equations, that the + 2 and –2 are the y-intercepts of the lines.

For vertical translations of all linear functions, we can state that f(x) = f(x) + k (realizing that k can be positive or negative as in our example).

Now, let’s consider vertical translations of exponential functions.

Let’s graph the following exponential functions:

f(x) = 2^x

f(x) = 2^x + 2

f(x) = 2^x – 2

You will remember that the general exponential functions is f(x) = b^x + c.  Therefore for vertical translations of all exponential functions, we can state that f(x) = f(x) + c (realizing that c can be positive or negative as in our example).

09.02 Mappings - Explanation Videos (Math Level 1)

See video
See video
See video


09.02 Mappings - Extra Links (Math Level 1)

I highly recommend that you click on the links above and work through the material.

These are the same links from lesson 1 - but this time pay attentions to the rules.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

For the coordinates of \bigtriangleupABC, write new coordinates according to the transformation mappings: 

a) A(1,4), B(-1,3), C(0,1)
   (x,y)\rightarrow(x,-y)

b) A(-2,-4), B(0,1), C(-1,4)
   (x,y)\rightarrow(x+2,y-3)

c) A (1,3), B(3,5), C(5,3)
   (x,y)\rightarrow(x-4,y+2)

d) A(2,2), B(4,5), C(5,3)
   (x,y)\rightarrow(-y,x)

e) reflection across the x-axis 

f) translation (x,y)\rightarrow(x-5,y+1)

g) rotation clockwise of 90°

Example 2:

Graph the following figures and draw the reflected images: 

a) \bigtriangleupABC with A(-3,4), B(0,1), C(-4,-2)
   (x,y)\rightarrow(-x,y)

b) \bigtriangleupABC with A(-2,-4), B(2,0), C(1,-3)
   (x,y)\rightarrow(x,-y)]

Example 3:

Graph the following triangle and the translation:

\bigtriangleupABC with A(-2,-4),B(-1,0),C(3,-3)
(x,y)\rightarrow(x+3,y+2)

Example 4:

Graph the following triangle and thengraphitsrotationof 90°, 180°and 270° on the same graph. \bigtriangleupABC with A(1,1), B(3,4), C(5,0)

Example 5:

Write a rule for the following transformations:

a) A(3,5), B(-1,3), C(0,1) to A'(5,2), B'(1,0), C'(2,-2)

b) A(1,3), B(4,5), C(2,1) to A'(-1,3), B'(-4,5), C'(-2,1)

c) A(-2,3), B(0,4), C(1,2) to A'(-3,-2), B'(-4,0), C'(-2,1)

d) (image is bold)

Example 6:

Draw the graph of f(x) = x and then the graph of the function with a vertical translation of -3.

Example 7:

Draw the graph of f(x) = \dpi{100} \fn_phv \frac{1}{2}^{x}and then the graph of the function with a vertical translation of 3.

Example 8:

Find the value of k or c on the following graphs.

a.

b.

Answers

Example 1:

For the coordinates of \bigtriangleupABC, write new coordinates according to the transformation mappings: 

a) A(1,4), B(-1,3), C(0,1)
   (x,y) \rightarrow (x,-y)

A’(1, -4), B’(-1, -3), C’(0, -1)

b) A(-2,-4), B(0,1), C(-1,4)
   (x,y) \rightarrow (x+2,y-3)

A’(0, -7), B’(2, -2), C’(2, 1)

c) A (1,3), B(3,5), C(5,3)
   (x,y) \rightarrow (x-4,y+2)

A’(-3, 5), B’(-1, 7), D’(1, 5)

d) A(2,2), B(4,5), C(5,3)
   (x,y) \rightarrow (-y,x)

A’(-2, 2), B’(-5, 4), C’(-3, 5)

e) reflection across the x-axis 

Rule: (x, y) \rightarrow (x, -y)

A’(1, -1), B’(4, -1), C’(3, -4)

f) translation (x,y) \rightarrow (x-5,y+1)

A’(-4, 6), B’(-2, 4), C’(0, 4)

g) rotation clockwise of 90°

Rule: (x, y) \rightarrow (y, -x)

A’(4, -1), B’(1, -4); C’(3, -5)

Example 2:

Graph the following figures and draw the reflected images: 

a) \bigtriangleup ABC with A(-3,4), B(0,1), C(-4,-2)
   (x,y) \rightarrow (-x,y)

b) \bigtriangleup ABC with A(-2,-4), B(2,0), C(1,-3)
   (x,y) \rightarrow (x,-y)]

Example 3:

Graph the following triangle and the translation:

\bigtriangleup ABC with A(-2,-4),B(-1,0),C(3,-3)
(x,y) \rightarrow (x+3,y+2)

Example 4:

Graph the following triangle and then graph its rotation of 90°, 180°and 270° on the same graph.

\bigtriangleup ABC with A(1,1), B(3,4), C(5,0)

90^{\circ} rule (x, y) \rightarrow (y, -x): A’(1, -1), B’(4, -3), C’(0, -5)

180^{\circ} rule (x, y) \rightarrow (-x, -y); A’(-1,-1), B’(-3,-4), C’(-5,0)

270^{\circ} rule (x, y) \rightarrow (-y, x); A’(-1,1); B’(-4,3); C’(0,5)

Example 5:

Write a rule for the following transformations:

a) A(3,5), B(-1,3), C(0,1) to A'(5,2), B'(1,0), C'(2,-2)

(x, y) \rightarrow (x + 2, y – 2)

b) A(1,3), B(4,5), C(2,1) to A'(-1,3), B'(-4,5), C'(-2,1)

(x, y) \rightarrow (-x, y)

c) A(-2,3), B(0,4), C(1,2) to A'(-3,-2), B'(-4,0), C'(-2,1)

(x, y) \rightarrow (-y, x)

d) (image is bold)

(x, y) \rightarrow (x+2, y+3)

Example 6:

Example 7:

Example 8:

a. k = 4

b. c = 2

09.02 Mappings - Worksheet (Math Level 1)

teacher-scored 86 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 4 of your enrollment date for this class.


09.03.01 Triangle Congruence, Part 1 (Math Level 1)

Justify congruence of triangles.

Something to Ponder

How would you why explain how ASA, SAS, and SSS are sufficient to show congruence and why SSA is not sufficient to show congruence?

Mathematics Vocabulary

Congruent triangles: One triangle is congruent to another when there is a rigid motion that maps the first onto the second. 

We are now ready to determine the requirements for triangle congruence. If two triangles are congruent, each side of one triangle is congruent to the corresponding side of the other triangle. It also means that each angle of one triangle is congruent to the corresponding angle of the other triangle.

However, to determine whether two triangles are congruent, do you really need to know all three sides and all three angles? Is it enough to know that one set of corresphttps://share.ehs.uen.org/node/28062/edit?destination=view-all%2F302onding sides are congruent? How about one set of corresponding angles? There are six measures you could know. How many do you really need?

One Parameter: One Side or One Angle

We will start at the very beginning with one parameter. I really hope you don't expect that we will have any luck with this--but to paraphrase Rodgers and Hammerstein, the very beginning is a very good place to start.

Consider the line segment , shown here. How many triangles can you construct using this segment as one side?

First of all, how would you construct a triangle? I recommend the following method. This may seem overkill at this point, but it will be necessary soon. Start by placing the stationary end of your compass on one endpoint. Open the compass to whatever span you like. Draw an arc above the segment. Move the compass to the other endpoint, change the span so that is is wide enough to intersect the first arc. Draw another arc above the segment. Label this point something.

Draw the lines connecting this point with the endpoints. It is important to use a straight edge to draw these segments. Repeat until you get bored.

It is fairly obvious that none of these triangles are congruent to any of the other triangles. Therefore, having one congruent side is not enough to declare that two triangles are congruent.

What about one angle? Consider the angle ∠U, shown here. How many triangles can you construct using this as one angle?

Again, for reasons that will become apparent later, we are going to construct triangles in the following way. Start by extending the sides to arbitrary lengths. You need to use a straight edge for this. Next place the stationary end of the compass on point U. Open it enough that the span is smaller than the total length of the one ray of the angle. Draw an arc that intersects this ray. Label the point something.

Now, without moving the stationary point, change the span to a distance that is not longer than the other ray. Draw another arc that intersects this ray. Label this point something else. Draw the segment connecting the two points. Use a straight edge to draw this segment.

Repeat until you get bored.

Bored yet?

Again, none of the triangles are congruent to any of the other triangles, so one angle is also not enough to declare congruence.

Two Parameters: Two Sides, One Side and One Angle, or Two Angles

This should be more promising. With two parameters the problem is more constrained, and the weird method we just used to construct triangles will now make sense.

Let's start with two sides. Consider the segments \overline{ST} and \overline{VW}, shown below. How many triangles can you construct with these segments as sides?

Start by measuring \overline{VW} by placing the stationary end of your compass on one endpoint, opening the span so that it is the same distance as \overline{VW} and drawing an arc that intersects the other endpoint. Next, without changing the span of the compass, place the stationary end on one endpoint of \overline{ST} and draw an arc above the segment.

Move the stationary end of your compass to the other endpoint. Change the span so that it is wide enough to intersect the first arc. Draw this arc. Label the point something. Draw the segments connecting the endpoints to this point. Again, you need to use a straight edge to draw these segments.

Repeat until you get bored.

Each of these triangles has two congruent sides, yet is is clear they are not congruent with one another. Two sides are not adequate.

The next possible choice is one side and one angle.

Consider \overline{ST}and \angle U.

How many triangles can you construct with these? It is easier to duplicate on one of the rays that compose ∠U than it is to construct ∠U at one endpoint of \overline{ST}. That is what we will do.

Therefore, to start, you want to extend one of the rays of ∠U so that it is longer than \overline{ST}. Extend the other ray an arbitrary length. Next, use your compass to measure \overline{ST}. You should know how to extend a segment and measure a length by now.

Now, without changing the span of your compass, place the stationary end on point U and use the other end to mark of the distance . Label this point something. Without moving your compass, change the span to a width less than the other ray. Mark off this distance and label the point.

Finally, draw the segment connecting these two points. You should know how to draw a segment connecting two points. Repeat until you get bored.

Again, it should be clear that one side and one angle is not enough to declare congruence.

The third two-parameter choice is two angles.

Okay, you know enough about triangles to realize that if you know two angles, the third angle is fully constrained. In other words, if you know two angles, you can calculate the third. This means we actually have three parameters here.

Again, consider the angles ∠U and ∠Z.

Since ∠U is more or less upright and ∠Z is at a less nice orientation, I will construct ∠Z on ∠U. Start with ∠U. Extend the rays that form ∠U to arbitrary lengths. Use your compass to mark a vertex along one ray. Label this something (notice I have gone through the alphabet, so I am labeling this A, even though it is not the same point A as before).

Use your compass to measure ∠Z. As a reminder, start by placing the stationary end of your compass at the vertex of ∠Z. Open it to a span that is not longer then the segments that makeup ∠Z. Draw an arc that intersects both segments, as shown. Now, without changing the span of the compass, move the stationary end to the vertex you just constructed on one ray of ∠U (point A in our example). Draw the same arc at this vertex. Be sure to draw it larger than you think you need.

Now, place the stationary end of your compass at one of the intersections on ∠Z and measure the distance to the other intersection. (Open the compass wide enough that when you draw an arc it intersects the previous point of intersection, as shown.) Next, without changing the span of your compass, move the stationary point to the point of intersection on \overline{UA} and mark off this distance on the arc you drew at vertex A.

Draw a line segment connecting points A and the intersection you just constructed. This duplicates ∠Z at point A. Extend this segment and the other ray of ∠U (if needed) until these segments intersect. Label this point something. Repeat until you get bored.

It should be clear that these triangles are not congruent; therefore, two angles are insufficient criteria to determine congruence. However, do you notice that there is something similar about these triangles?

Three Parameters

You may suspect that three is the magic number here. We will actually look at all the possible three parameter options and see that three almost always determines congruence.

What are the possible combinations? Three sides, two sides and one angle, two angles and one side, or three angles. There is one way to have three sides. There are two ways to have two sides and one angle: side-angle-side and side-side-angle. There are also two ways to have one side and two angles: angle-side-angle, and angle-angle-side. However, since if you know two angles, you know the third, we will see that these two combinations are really the same. Finally, there is only one way to have three angles. However, we already recognized that because the sum of all angles in a triangle is 180°, two angles is really the same thing as three angles, so we also already considered that case. (We will come back to it since it is an interesting case.) This means we have four more cases to consider.

1. Side-Angle-Side, SAS

The side-angle-side, or SAS, congruence criteria is the first congruence theorem in the Elements, so there is historical precedence for starting here. There is another reason we are starting here, which I will get to soon.

The order here is important. SAS means that you know the length of two sides, and the angle between them. Again, consider the segments and angles we used in lesson 3.

How many triangles can you construct with sides that are length ST and VW and have ∠U between these?

Where will you start? I would start by extending the rays that make up ∠U so that the segments are longer than \overline{ST} and \overline{VW}, respectively. Next, using your compass to measure \overline{ST}, mark off this length on one of the rays that make up ∠U, as shown. Label this intersection something.

Repeat these steps with \overline{VW} and the other ray.

Construct the segment connecting these two points.

I expect you can tell that this construction is fully constrained. This means that this is the ONLY triangle we can construct in this particular way.

However, we should still ask, are there any other possible triangles? What if you measured off \overline{VW} along the bottom ray and \overline{ST} along the side, instead?

Does this triangle have a different shape than the previous triangle?

What if we constructed ∠U along \overline{ST} and then measured off \overline{VW}? Or vice versa? There are many ways to change this construction. Can we prove that all these triangles are congruent?

Demonstration of SAS Congruence

Now that we have constructed several triangles in which two sides and the angle between these sides are all congruent, we need to demonstrate that these triangles are also congruent.

Consider ΔUKL and ΔUJI.

We can demonstrate the congruence of these triangles using rigid transformations. Remember the rigid transformations are translations (moving something from one place to another), rotations (turning something about a center point), and reflections (flipping something across an axis). Consider ΔUKL and ΔUJI, shown here.

Start by translating ΔUKL so that the points U align and that sides align.

Next, draw the angle bisector of ∠U. You bisect an angle by placing the stationary end of your compass at point U, open the compass and draw an arc that intersects both sides of the triangle. Next, move the stationary point of the compass to one intersection, and draw an arc. Without changing the span, move the stationary point of the compass to the other intersection, and draw an intersecting arc, as shown. Draw the line segment between the points U and the intersection you just drew. This is the angle bisector of ∠U.

Finally, reflect ΔUKL across this line. Since ΔUKL and ΔUJI align, these triangles are congruent.

It is possible to use other rigid motions to demonstrate that all of the triangles we constructed are in fact congruent.

2. Side-Side-Side, SSS

The side-side-side, or SSS, congruence criteria is the second triangle congruence theorem by Euclid. In a similar way as before, we will see how many triangles we can construct with three given sides. Consider the segments \overline{KL}, \overline{DE}, and \overline{UV} below.

Rather than starting with one of these as the base, I am going to draw a horizontal line, and use that as the base. Draw a horizontal line. Mark off a point to use as one vertex. Label this something. Measure the length KL and mark this off on the horizontal line. Label that point something else.

Measure the length DE. Without changing the span of your compass, place the stationary point on one of the vertices you just constructed on the horizontal line. Draw a wide arc above this line. Both possibilities are shown below.

Measure the distance UV. Without changing your compass span, place the stationary end of your compass on the other vertex and draw an arc that intersects with the previous arc. Label this intersection something. Again, both possibilities are shown below.

 

Finally, draw lines connecting the vertices.

 

You cannot construct a different triangle. Three sides is enough to constrain the triangle. Therefore, three sides are enough to determine triangle congruence.

3. Side-Side-Angle, SSA

Side-side-angle, or SSA, means that we have two sides, and an angle not between them. It would be fine to say angle-side-side, except that this acronym is considered an impolite word in the United States. In England, it is merely another name for a donkey.

Now, with that out of the way, let's see how many triangles we can create with two sides and an angle not between the sides. Let's use the sides \overline{KL} and \overline{DE} again, as well as ∠M, as shown.

 

Start by drawing a horizontal line and marking off a distance on this line with a length KL. Label both vertices.

 

At one vertex, duplicate ∠M. (You do remember how to duplicate an angle?)

 

Measure length DE. Without changing the compass span, move the stationary end of your compass to the other vertex and draw a sweeping arc above the horizontal line. Label the intersection something.

Oops, which intersection? Umm, pick one. Then pick the other one! Draw the lines connecting the vertices. How many triangles can you make doing it this way?

ΔNPO has sides \overline{NO}\overline{KL}, \overline{NP}\overline{DE}, and ∠O ≅ ∠M. ΔNQO also has sides \overline{NO}\overline{KL}, \overline{NQ}\overline{DE}, and ∠O ≅ ∠M. However, it should be clear that ΔNPO is not congruent to ΔNQO.Therefore, we potentially have two different triangles we can create with this combination of parameters. This is called the ambiguous case and it cannot be used to determine congruence. There is more than one triangle with these parameters.
 

4. Angle-Side-Angle, ASA

This is the case with two angles and a side between them. Earlier I claimed that this is the same case as the side-angle-angle case. However, the construction is a bit different so we will treat them differently.

Let's test this problem in the same way as before. Start with side \overline{DE}, ∠M, and ∠R, shown below.

    

Again, let's start with a horizontal line on which we have marked the distance DE. Label these points something.

 

Next, duplicate ∠M on one vertex and ∠R on the other.

   

Finally, extend the rays until they intersect. Label this point something.

There's only one possible triangle here. Therefore ASA is valid criteria for triangle congruence.

It is okay to also call this AAS. The order doesn't matter. But ASA is used most often.

Summary of Congruence Criteria

1. Each triangle is defined by six parameters: three sides and three angles.

2. We need at least three parameters to uniquely define a triangle

    a. Therefore, we need at least three parameters to determine if two triangles are congruent.

3. Not every combination of three parameters will uniquely define a triangle.

    a. Therefore, not every combination of parameters will prove that two triangles are congruent.

4. The sets of parameters that uniquely determine a triangle, and therefore prove congruence, are

     a. Side-Angle-Side, SAS

     b. Side-Side-Side, SSS

     c. Angle-Side-Angle,ASA (or SAA)

 

5. The ambiguous case

    a. Side-Side-Angle, SSA

    b. With these parameters, the triangles might be congruent, or they might not; we cannot tell.

I know this is A TON OF WORK to come to know the parameters for triangle congruence. Believe me - I understand!! Many, many years ago when I took Geometry, we were just given the information and we memorized it. Today, mathematicians want students to understand congruence through the lens of transformations and constructions. Lucky you!!

09.03.02 Triangle Congruence, Part 2 (Math Level 1)

Identify corresponding parts of congruent triangles.

This material is from Engage NY Geometry Module 1 Lesson 20 and 21 licensed under Creative Commons license 3.0 

Corresponding Parts

One figure is congruent to another when there is a rigid motion that maps the first onto the second. That rigid motion is called a congruence.

A rigid motion F always produces a one-to-one correspondence between the points in a figure (the pre-image) and points in its image. If P is a point in the figure, then the corresponding point in the image is F(P). A rigid motion also maps each part of the figure to a corresponding part of the image. As a result, corresponding parts of congruent figures are congruent since the very same rigid motion that makes a congruence between the figures also makes a congruence between each part of the figure and the corresponding part of the image.

In each figure below, the triangle on the left has been mapped to the one on the right by a 240^{\circ} rotation about P. Let’s identify all six pairs of corresponding parts (vertices and sides).

Corresponding Vertices Corresponding Sides
A \rightarrow X \overline{AB}\rightarrow \overline{XY}
B \rightarrow Y \overline{AC}\rightarrow \overline{XZ}
C \rightarrow Z \overline{BC}\rightarrow \overline{YZ}

 

Example 1:

Corresponding Angles Corresponding Sides
\angle BAC\rightarrow \angle DAC \overline{AB}\rightarrow \overline{AD}
\angle ABC\rightarrow \angle ADC \overline{BC}\rightarrow \overline{DC}
\angle BCA\rightarrow \angle DCA \overline{AC}\rightarrow \overline{AC}

 

  • Are the corresponding sides and angles congruent?   

Since the triangle is a rigid transformation, all angles and sides maintain their size.

  • Is △ABC ≅ △ADC? Justify your response.

Yes. Since ADC is a reflection of ABC, they must be congruent.

Each example below shows a sequence of rigid motions that map a pre-image onto a final image. We will identify each rigid motion in the sequence. Then, we will trace the congruence of each set of corresponding sides and angles through all steps in the sequence, proving that the pre-image is congruent to the final image by showing that every side and every angle in the pre-image maps onto its corresponding side and angle in the image. Finally, we will make a statement about the congruence of the pre-image and final image.

Example 2:

 

Sequence of rigid motions - there are two Rotation then translation
Sequence of corresponding sides

AB \rightarrow A"B"

BC \rightarrow B"C"

AC \rightarrow A"C"

Sequence of corresponding angles

A\rightarrow A"

B\rightarrow B"

C\rightarrow C"

Triangle congruence statement \Delta ABC \cong \Delta A"B"C"

 

Guided Practice:

Now try to fill in the table:

Sequence of rigid motions - there are three                                                         
Sequence of corresponding sides  
Sequence of corresponding angles  
Triangle congruence statement  

 

Answer:

Sequence of rigid motions - there are three

 

{\color{Red} Relfection, translation, rotation}

 

Sequence of corresponding sides

{\color{Red} AC \rightarrow A"C"}

{\color{Red} BC \rightarrow B"C"}

{\color{Red} AC \rightarrow A"C"}

Sequence of corresponding angles

{\color{Red} A \rightarrow A"}

{\color{Red} B \rightarrow B"}

{\color{Red} C \rightarrow C"}

Triangle congruence statement {\color{Red} \Delta ABC \cong \Delta A"B"C"}

 

Corresponding parts of congruent triangles are congruent (CPCTC) will be studied in greater depth in Secondary Math 2.

Good luck on your assignment!

09.03.03 Triangle Congruence - Extra Links (Math Level 1)

I highly recommend that you click on the links above before continuing.

  1. View the video
  2. Work through the material.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

\bigtriangleupABC \cong \bigtriangleupQTJ

List the congruent corresponding parts.

Example 2:

Decide whether \bigtriangleupABC \cong \bigtriangleupCED. Justify your answer.

Example 3:

Can you conclude \bigtriangleupJKL \cong \bigtriangleupMNL? Justify your answer. 

Answers

Example 1:

\bigtriangleup ABC \cong \bigtriangleup QTJ

List the congruent corresponding parts.

Congruent sides: \fn_phv \overline{AB}\cong \overline{QT}, \overline{AC}\cong \overline{QJ}, \overline{BC}\cong \overline{TJ}

Congruent angles: \angle A \cong \angle Q, \angle B \cong \angle T, \angle C \cong \angle J

09.03.04 Triangle Congruence (Math Level 1)

teacher-scored 35 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.


09.04 Constructions with Segments, Angles and Lines (Math Level 1)

Construct congruent segments and angles, bisect segments and angles, and construct perpendicular and parallel lines.

Something to Ponder

How would you describe the process of constructing perpendicular lines from points on the line and not on the line?

Mathematics Vocabulary

Perpendicular bisector: the line perpendicular to a line segment passing through the segment's midpoint. 

Angle bisector: a line or ray that divides an angle in half

Perpendicular lines: lines at right angles (90°) to each other. 

Parallel lines: lines on a plane that never meet and always the same distance apart. 

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

09.04 Constructions with Segments, Angles and Lines - Explanation Video Link (Math Level 1)

09.04 Constructions with Segments, Angles and Lines - Explanation Videos (Math Level 1)

See video


09.04 Constructions with Segments, Angles and Lines - Extra Video (Math Level 1)

I highly recommend that you click on the link above and watch the video before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

Construct a congruent segment to a given segment.

Example 2:

Construct a congruent angle to a given angle.

Example 3:

Construct the perpendicular bisector of a given segment.

Example 4:

Construct the angle bisector of a given angle.

Example 5:

Construct a perpendicular line from point not on a given line 

Example 6:

Construct a parallel line to a given line.

Example 1:

Construct a congruent segment to a given segment.

From: en.wikibooks.org/wiki/Geometry/Chapter_16

Example 2:

Construct a congruent angle to a given angle.

From: en.wikibooks.org/wiki/Geometry/Chapter_16

Example 3:

Construct the perpendicular bisector of a given segment.

From: en.wikibooks.org/wiki/Geometry/Chapter_16

Example 4:

Construct the angle bisector of a given angle.

From: en.wikibooks.org/wiki/Geometry/Chapter_16

Example 5:

Construct a perpendicular line from point not on a given line.

From: en.wikibooks.org/wiki/Geometry/Chapter_16

Example 6:

Construct a parallel line to a given line.

From: www.wikihow.com/Construct-a-Line-Parallel-to-a-Given-Line-Through-a-Given-Point

09.04 Constructions with Segments, Angles and Lines - Worksheet (Math Level 1)

teacher-scored 70 points possible 50 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps. You must use a straight edge, compass and the processes shown in the lesson on each construction.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.


09.05 Constructions with Triangles, Squares and Hexagons (Math Level 1)

Construct triangles, squares and hexagons.

Something to Ponder

How would you describe the process of inscribing a hexagon inside a circle?

Mathematics Vocabulary

Inscribed: a polygon in a circle with the vertices of the polygon on the circle itself

Circumscribed: a circle that passes through all vertices of a polygon and contains the entire polygon in the interior of the circle 

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

09.05 Constructions with Triangles, Squares and Hexagons - Explanation Video Link (Math Level 1)

09.05 Constructions with Triangles, Squares and Hexagons - Explanation Videos (Math Level 1)

See video


09.05 Constructions with Triangles, Squares and Hexagons - Extra Video (Math Level 1)

I highly recommend that you click on the links above and watch the videos before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Construction 1: Construct an Equilateral Triangle

Construction 2: Construct a Square

Construction 3: Construct a Hexagon

Construction 4: Construct an Equilateral triangles in a circle: 

Construction 5: Construct a Square in a circle: 

Answers

Construction 1: Construct an Equilateral Triangle:

Start with an arbitrary line segment. I will call it \overline{AB}

Place the stationary end of your compass at one endpoint, say point A. Measure the distance to point B. This means you need to draw the arc of the circle with center A and radius AB that intersects point B, as shown.


Place the stationary end of your compass at one endpoint, say point A. Measure the distance to point B. This means you need to draw the arc of the circle with center A and radius AB that intersects point B, as shown.

Next, swing the compass around so that you continue this arc above the segment in a region that is about halfway along the segment, as shown.

Repeat the steps above at the other endpoint, as shown.

Label the intersection of these arcs point C. Draw the segment connecting points C and A. Draw the segment connecting points C and B.

You have now constructed equilateral triangle ABC.

Construction 2: Construct a Square:

Start by constructing a line segment. On the segment, mark a point that will be the first vertex. Don't place this point near the end of the segment.

Next, construct a line perpendicular to the first line through this point. Again, this was something we learned in quarter 1, but we will review it here.

Start by placing the stationary end of your compass at the vertex. Open it so that the span is not greater than the distance from the vertex to the end of the segment. Mark this distance on either side of the vertex, as shown.

Move the stationary end of the compass to one of these intersections. Open the span a bit wider and construct an arc below the segment.

Without changing the span of the compass, move the stationary end to the other intersection, and construct the arc that intersects the first.

Now, using your straight edge, construct the line between this intersection and the vertex. This line is perpendicular to the first line.

Open the span of your compass to the length you want each side to be. Place the stationary end of your compass on the vertex and mark off this distance along both lines.

Now, through one of these points, construct another perpendicular. It shouldn't matter which point you use. I am going to use the bottom vertex since I didn't leave much space along the other line. Start by drawing arcs on either side of the vertex.

Then draw the arcs below the vertex.

Next connect the intersection with the vertex.

Now, you will need to remeasure the length of each side and measure off this distance between the vertex and the newly constructed perpendicular line.

Finally, connect the remaining two vertices. Label all vertices. Figure UVWX is a square.


 

Construction 3: Construct a Hexagon:

Regular Hexagon Inscribed in a Circle 240px

Construction 4: Construct an Equilateral Triangle in a circle:

Equilateral Triangle Inscribed in a Circle

 

Construction 5: Construct a Square in a circle

Straight Square Inscribed in a Circle 240px

The last three constructions are from:

09.05 Constructions with Triangles, Squares and Hexagons - Worksheet (Math Level 1)

teacher-scored 60 points possible 40 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps. You are required to use a straight edge, compass and the processes shown in the lessons on each construction.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


09.06 Rotations, Reflections, and Symmetry

teacher-scored 30 points possible 60 minutes

Activity for this lesson

  • Print the worksheet. Work all the problems showing ALL your steps.
  • Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


09.06 Rotations, Reflections, and Symmetry (Math Level 1)

Most of this lesson is from 6.5 and 6.6 of the Mathematics Vision Project licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Rotations, Reflections, and Symmetry

A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation is said to have
rotational symmetry.

Every four‐sided polygon is a quadrilateral. Some quadrilaterals have additional properties and are given special names like squares, parallelograms and rhombuses. A diagonal of a quadrilateral is formed when opposite vertices are connected by a line segment. In this task you will use rigid motion transformations to explore line symmetry and rotational symmetry in various types of
quadrilaterals.

Describe the rotations and/or reflections that carry the following quadrilaterals onto itself. (Be as specific as possible in your descriptions. Base your decisions about lines of symmetry or centers of rotation NOT simply on intuition and “it looks like it works” type of justification, but based on the definitions of the rigid‐motion transformations and the defining properties of the geometric figures).

1. A rectangle is a quadrilateral that contains four right angles. Is it possible to reflect or rotate a rectangle onto itself?

For the rectangle shown below, find

  • any lines of reflection, or
  • any centers and angles of rotation

that will carry the rectangle onto itself.

It is easy to find the "mirror image" lines of reflection.

The diagonals, however, are not lines of reflection!

What about centers and angles of rotation? Looking at the two lines of symmetry, the center of rotation is the point of intersection. What are the angles of rotation? A square can be rotated 90°, 180°, 270° or 360° about this center of rotation. Do they all work?

Let's try 90^{\circ}:

Nope! Do you need to check 270^{\circ}?

What about 180^{\circ}?

Yep! The vertices were labeled to help you see the rotation. Do we need to check 360^{\circ}

2. A parallelogram is a quadrilateral in which opposite sides are parallel. Is it possible to reflect or rotate a parallelogram onto itself?

For the parallelogram shown below, find

  • any lines of reflection, or
  • any centers and angles of rotation

that will carry the parallelogram onto itself.

The lines of reflection to check are the diagonals - as we did with the rectangle. Do they work?

 

Nope!

What about the center and angles of rotation? Looking at the two diagonals, the center of rotation is the point of intersection. What are the angles of rotation?

Let's try 90^{\circ}

Nope!

How about 180^{\circ}?

Yep! The vertices were labeled to help you see the rotation. Do we need to check 360^{\circ}?

3. A rhombus is a quadrilateral in which all sides are congruent. Is it possible to reflect or rotate a rhombus onto itself?

For the rhombus shown below, find

  • any lines of reflection, or
  • any centers and angles of rotation

that will carry the rhombus onto itself.

Again, we will check the diagonals to see if they are lines of reflection.

The horizontal diagonal worked!

The vertical diagonal worked!

What about the center and angles of rotation? Looking at the two diagonals, the center of rotation is the point of intersection. What are the angles of rotation?

Let's try 90^{\circ}:

Nope! It should be obvious that 270^{\circ} will not work as well.

Let's try 180^{\circ}

Yep! It should be obvious that 360^{\circ} will work as well.

4. A square is both a rectangle and a rhombus. Is it possible to reflect or rotate a square onto itself?

For the square shown below, find

  • any lines of reflection, or
  • any centers and angles of rotation

that will carry the square onto itself.

Again, we will check the diagonals to see if they are lines of reflection.

It is easy to find the "mirror image" lines of reflection.

What about the diagonals?

Obviously, the other diagonal would also be a line of reflection. So, a square has four lines of reflection.

What about the center and angles of rotation?

Let's try 45^{\circ}:

Nope! It should be obvious that all factors of 45 won't work either.

Let's try 90^{\circ}:

Yep! Clearly, 360^{\circ} would also work.

By now you should be able to find lines of symmetry and any rotational symmetries for a given figure.

5. A regular pentagon has five congruent sides. Is it possible to reflect or rotate a pentagon onto itself?

For the pentagon shown below, find

  • any lines of reflection, or
  • any centers and angles of rotation

that will carry the pentagon onto itself.

First, let's draw the lines of reflection.

You can see diagonal drawn is a line of reflection. What is true for one is true for all five vertices.

You should try to label each vertex with each of the five reflections!

Now, what about the center and angles of reflection?

The intersection of each of these diagonal lines is the center point. Now for the angles of reflection.

Let's try 45^{\circ}

Nope! What about 90^{\circ}?

Nope again. So, it looks like we need to try a different approach. Since there are five sides, each interior angle = \frac{360^{\circ}}{5}= 72^{\circ}. Let's try this.

Yep! This means that 144^{\circ} and 216^{\circ} and 288^{\circ} and 360^{\circ} will all work as well. Feel free to check!

You will be working with more regular polygons in your assignment. Have fun!

09.06.01 Student Project (Math Level 1)

teacher-scored 60 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. 
  2. Carefully follow the directions!!
  3. This project is worth 60 points and is in lieu of a unit review. It must show a depth of knowledge and evidence of significant time spent.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


10.00 Analyzing Statistical Models Overview (Math Level 1)

Experience with descriptive statistics began as early as Grade 6. Students were expected to display numerical data and summarize it using measures of center and variability. By the end of middle school they were creating scatterplots and recognizing linear trends in data. This unit builds upon that prior experience, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit. By the end of the unit, students will be able to:

  • Find the mean, median, mode, and range of a set of data, compare my results to an expected distribution, and interpret results based on different sample sizes.
  • Organize, display and analyze data in histograms.
  • Organize, display and analyze data in box plots.
  • Describe the strengths, weaknesses of and most appropriate use for dot plots, histograms and box plots.
  • Use two-way frequency tables of data to compare different categories of data.
  • Find mean and compute standard deviation and analyze data using mean and standard deviation.
  • Create scatter plots and use them to analyze data and estimate linear and exponential functions that fit the data.
  • Understand the difference between correlation and causation and explain that a strong correlation does not mean causation.
  • Choose the best method to display and analyze statistical data.

 


10.01 Measures of Center and Spread (Math Level 1)

Find the mean, median, mode, and range of a set of data, compare my results to an expected distribution, and interpret results based on different sample sizes.

Something to Ponder

How would you explain the differences between the mean, median and mode of a set of data?

Mathematics Vocabulary

Mean: a mathematical term for “average” which you may already know.

Median: the middle value when the data is ordered.

Mode: the data value that appears most often in a data set.

Learning these concepts

This part of the lesson is from Monterey Institute.org Developmental Math Unit 8 Lesson 2 Topic 1under a Creative Commons license.

Introduction

Mean, median, and mode are important tools in the statistician’s toolbox. These measures of center all use data points to approximate and understand a “middle value” or “average” of a given data set. Range is another measure of interest which uses the greatest and least values of the data set to help describe the spread of the data.

So why would you need to find out the middle of a data set? And why do you need three measures instead of just one? Let’s look closely at these measures of center and learn how they can help us understand sets of data.

Mean, Median, and Mode

“Mean” is a mathematical term for “average” which you may already know. Also referred to as the “arithmetic mean,” it is found by adding together all the data values in a set and dividing that sum by the number of data items.

You can often find the average of two familiar numbers, such as 10 and 16, in your head without much calculation. What number lies half way between them? 13. A mathematical way to solve this, though, is to add 10 and 16 (which gives you 26) and then divide by 2 (since there are 2 numbers in the data set). 26 ÷ 2 = 13

Knowing the process helps when you need to find the mean of more than two numbers. For example, if you are asked to find the mean of the numbers 2, 5, 3, 4, 5, and 5, first find the sum: 2 + 5 + 3 + 4 + 5 + 5 = 24. Then, divide this sum by the number of numbers in the set, which is 6. So the mean of the data is 24 ÷ 6, or 4.

In the previous data set, notice that the mean was 4 and that the set also contained a value of 4. This does not always occur. Look at the example that follows—the mean is 18, although 18 is not in the data set at all.

Next, let’s look at the “median.” The median is the middle value when the data is ordered. If there are two middle values, the median is the average of the two middle values.

To calculate the median, you first put your data into numerical order from least to greatest. Then identify the middle value(s).

For example, let’s look at the following values: 4, 5, 1, 3, 2, 7, 6. To find the median of this set, you would put it in order from least to greatest.

1  2   3  {\color{Red} 4}  5  6  7

Then identify the middle value. There are three values to the right of four and three values to the left of four. The middle value is 4, so 4 is the median.

If there is an even number of data items, however, the median will be the mean of the two center data items.

Finally, let’s consider the “mode.” The mode is found by looking for the data value that appears most often. If there is a two-way tie for most often, the data is bimodal and you use both data values as the modes. Sometimes there is no mode. This happens when there is no data value that occurs most often. In our example data set (2, 3, 4, {\color{Red} 5, 5, 5}), the number 5 appears 3 times and all other numbers appear once, so the mode is 5.

Let’s look at an example with some relevant data.

What can be learned from the mean, median, and mode of Carlos’ test scores? Notice that these values are not the same.

Both the mean and the median give us a picture of how Carlos is doing. Looking at these measures, you notice that the middle of the data set is in the mid-80s: the mean value is 86, and the median value is 84. That’s all you are really after when using median and mean—finding the center, or middle, of the data. Notice, also, that there is no mode, since Carlos did not score the same on two tests. In the case of test taking, the mode is often meaningless—unless there are a lot of 0s, which could mean that the student didn’t do his homework, or really doesn’t know what’s going on!

In this case, the mean, median, and mode are very close in value. This shows some consistency in the data, with a middle (average) value of about 11. If this data represented the ages of students on a chess team, for example, you would have a good idea that everyone on the team was about 11 years old, with a few older and younger members.

Question: During a seven-day period in July, a meteorologist recorded that the median daily high temperature was 91º.

Which of the following are true statements?

  1. The high temperature was exactly 91º on each of the seven days.
  2. The high temperature was never lower than 92º.
  3. Half the high temperatures were above 91º and half were below 91º.

A) ! only

B) 2 only

C) 3 only

D) 1, 2, and 3

Take time to think this through before looking at the answer below.

Range

There are other useful measures other than mean, median, and mode to help you analyze a data set. When looking at data, you often want to understand the spread of the data: the gap between the greatest number and the least number. This is the range of the data. To find the range, subtract the least value of the data set from the greatest value. For example, in the data of 2, 5, 3, 4, 5, and 5, the least value is 2 and the greatest value is 5, so the range is 5 – 2, or 3.

Let’s look at a couple of examples.

Answer:

A) 1 only

Incorrect. Just because the median high was 91º does not mean that the temperature reached 91º on each day. The correct answer is statement iii only.

B) 2 only

Incorrect. You know the median is 91º, so 91º is a member of the data set—meaning that the temperature had to have been lower than 92º at least one time during the week. The correct answer is statement iii only.

C) 3 only

Correct. Half the high temperatures were above 91º and half were below 91º since the median will always represent the value where half the data is higher and half the data is lower.

D) 1, 2, and 3

Incorrect. Statement 1 is incorrect because a median high of 91º does not necessarily mean that the temperature reached 91º on each day, and 2 is incorrect because you know the median is 91º, so 91º is a member of the data set—meaning that the temperature had to have been lower than 92º at least one time during the week. The correct answer is statement 3 only.

Now let’s look at the role outliers – data that is much higher or lower than the bulk of the data set.

This part of the lesson is from Oswego City School District Regents Exam Prep Center Algebra. The copyright for this material has expired.

Consider this set of test score values:

The two sets of scores above are identical except for the first score.  The set on the left shows the actual scores.  The set on the right shows what would happen if one of the scores was WAY out of range in regard to the other scores.  Such a term is called an outlier.

With the outlier, the mean changed.
With the outlier, the median did NOT change.

How do I know which measure of central tendency to use?

  MEAN MEDIAN MEDIAN
  Use the mean to describe the middle of a set of data that does not have an outlier. Use the median to describe the middle of a set of data that does have an outlier. Use the mode when the data is non-numeric or when asked to choose the most popular item.
Advantages
  • Most popular measure in fields such as business, engineering and computer science.
  • It is unique - there is only one answer.
  • Useful when comparing sets of data.
  • Extreme values (outliers) do not affect the median as strongly as they do the mean.
  • Useful when comparing sets of data.
  • It is unique - there is only one answer.
  • Extreme values (outliers) do not affect the mode.
Disadvantages
  • Affected by extreme values (outliers)
  • Not as popular as mean.
  • Not as popular as mean and median.
  • Not necessarily unique - may be more than one answer
  • When no values repeat in the data set, the mode is every value and is useless.
  • When there is more than one mode, it is difficult to interpret and/or compare.

 

Learning these concepts

Click the mathematician image OR click the links below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

10.01 Measures of Center and Spread - Explanation Video Link (Math Level 1)

10.01 Measures of Center and Spread - Explanation Videos (Math Level 1)

See video


10.01 Measures of Center and Spread - Extra Video (Math Level 1)

I highly recommend that you click on the links above and watch the videos before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1: 

Find the mean, median, mode, and range of the following data:

2 6 10 11 14 18 3 8 20 11 8 14

Answers

Example 1: 

Find the mean, median, mode, and range of the following data:

2 6 10 11 14 18 3 8 20 11 8 14

Step 1: Write the numbers is order from least to greatest: 2, 3, 6, 8, 8, 10, 11, 11, 14, 14, 18, 20 (12 numbers)

Step 2: Find the sum of the numbers: 125

Step 3: Find the mean = \fn_phv \frac{125}{12} = 10.4 rounded to the nearest hundredth

Step 4: Find the median. There are 12 numbers so there are two middle numbers. The median is the average of 6th and 7th numbers 10 and 11 = 10.5

Step 5: Find the mode: There are three numbers that occur twice: 8, 11, and 14.

Step 6: Find the range: 20 = 2 = 18

10.01 Measures of Center and Spread - Worksheet (Math Level 1)

teacher-scored 20 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


10.02 Dot Plot (Math Level 1)

Describe and give a simple interpretation of a graphical representation of data on dot plots.

Something to Ponder

Why is it important to organize, display and analyze a data set?

Mathematics Vocabulary

Dot plot: a representation of a distribution consisting of group of data points plotted on a simple scale.

Skew: when a distribution is has more data points to the right or to the left of center.

Symmetrical distribution: has a middle where an imaginary line can be drawn through the center, creating a fairly equal "look" on either side.

Bell shaped: a symmetrical distributions that is tall in the middle with the two sides thinning out. The sides are referred to as tails.

Outlier: data points much smaller than or much larger than the rest of the data.

Learning these concepts

Most of the following material is from:

under Creative Commons AttributionNonCommercial 3.0 Unported (CC BY-NC 3.0)

cK12.org: user:anoka/book/anoka-hennepin-probability-and-statistics/section/5.3/

In the next few lessons you will be constructing graphical representations of data using dot plots, frequency tables and histograms, and box plots. You will be asked to find the measures of center for each data set represented. You will then be asked to describe the representation.

Describing a Numerical Distribution

Once you have constructed a graphical representation of a data set, the next step is to describe what the graph shows. There are several characteristics that should be mentioned when describing a numerical distribution, and your description needs to explain what this specific data represents. Describe the shape of the graph, whether or not there are any outliers present in the data, the location of the center of the data and how spread out the data is. All of this should be done in the specific context of the individuals and variable being studied. We will use an acronym to help you remember what to include in your descriptions (S.O.C.C.S.) - shape, outliers, context, center and spread. An explanation of each of these characteristics follows.

Shape

Once a graphical display is constructed, we can describe the distribution. When describing the distribution, we should be sure to address its shape. Although many graphs will not have a clear or exact shape, we can usually identify the shape as symmetrical or skewed. A symmetrical distribution will have a middle where we can draw an imaginary line through the center, and a fairly equal "look" on either side of that imaginary line. If you were to fold along the imaginary center line, the two sides would almost match up. Many symmetrical distributions are bell shaped, they will be tall in the middle with the two sides thinning out. The sides are referred to as tails. A skewed distribution is one in which the bulk of the data is concentrated on one end, with the other side being a longer tail. The direction of the longer tail is the direction of the skew. Skewed right will have a longer tail to the right, or higher numbers. Skewed left will have a longer tail off to the left, or the lower values. Other shapes that you might see are uniform (almost consistent height all the way across) and bimodal (having two peaks in the distribution).

 

Outliers

If there are any outliers, gaps, groupings, or other unusual features in the distribution, we should be sure to mention them. An outlier is a value that does not fit with the rest of the data. Some distributions will have several outliers, while others will not have any. We should always look for outliers because they can affect many of our statistics. Also, sometimes an outlier is actually an error that needs to be corrected. If you have ever 'bombed' one test in a class, you probably discovered that it had a big impact on your overall average in that class. This is because the mean will be affected by an outlier-it will be pulled toward it. This is another reason why we should be sure to look at the data, not just look at the statistics about the data. When an outlier is part of the data and we do not realize it, we can be misled by the mean to believe that the numbers are higher or lower than they really are.

Context

Do not forget that the graph, the numbers and the descriptions are all about something--its context. All of these elements of the distribution should be described in the specific context of the situation in question.

Center

The center of the distribution should always be included in the verbal analysis as well. People often wonder what the 'average is'. The measure for center can be reported as the median, the mean, or the mode. Even better, give more than one of these in your description. Remember that outliers affect the mean, but do not affect the median. For example, the median of a list of data will stay in the center even when the largest value increases tremendously, but such a change would affect the mean quite a bit.

Spread

Another thing to include in the description is the spread of the data. The spread is the specific range of the data. When analyzing a distribution, we don't want to simply say that the range is equal to some number. It is much more informative to say that the data ranges from_____ to ______ (minimum value to maximum value). For example, if the news reports that the temperature in St. Paul had a range of 20o during a given week, this could mean very different temperatures depending on the time of year. It would be more informative to say something specific like, the temperature in St. Paul ranged from 68^{\circ} to 88^{\circ} last week.

Dot Plots

A dot plot consists of a horizontal scale (a number line) on which dots are placed to show the numerical values of the data. If a data value repeats the dots are piled up at that location – one dot for each repetition. We say that the dot plot displays the distribution of the data.

A dot plot is easier to look at compared to a long list of numbers. By examining the dot plot in the gray box, you can see that the numbers 25 and 40 are repeated the most. You can see that the number 50 is repeated the least. Not only that but you can count very easily how many times each number is repeated. A dot plot is a good way of organizing numbers and values.

How would you find the mean of this dot plot?

Value Number of Values Total
5 6 30
10 5 50
15 3 45
20 7 140
25 9 225
30 7 210
35 4 140
40 9 360
45 3 135
50 2 100
Sum 55 1435

 

Mean = 1435/55 = 26.1 rounded to the nearest tenth.

How would you find the median of this dot plot?

There are 55 total data points so the middle point is the 28th point – which falls in the list of 25’s.

Median = 25.

How would you find the mode of this dot plot?

This is the easiest to find. In the example there are two modes: 25 and 40. This means the data is bimodal.

Outliers are data points much smaller than or much larger than the rest of the data.  Here is an example:

Source: syllabus.bos.nsw.edu.au

How does the outlier affect the measures of center?

What is the mean WITH the outlier?

Sum of values: 24 + 65 + 28 + 24 = 110

Mean = 110/10 = 11

What is the mean WITHOUT the outlier?

Sum of values: 24 + 65 + 28 = 117

Mean = 117/9 = 13

What is the median WITH the outlier?

With 11 values, the median is 6th points = 13.

What is the median WITHOUT the outlier?

With 10 values, the median is the average of the 5th and 6th points = 13.

What is the mode WITH the outlier? 13

What is the mode WITHOUT the outlier? 13

Outliers heavily influence the mean. Not so with the median and mode.

Example 1: Here are two dot plots. Students were surveyed and asked what they thought their grade would be on a math exam. The results of the survey are in green. Then the students took the math test and the scores were plotted on the top graph in purple. The blue line is the halfway point between the scores. What conclusions can we make based on looking at the two dot plots?

Some thoughts:

  • The students tended to think that they would score lower than they actually did.
  • Counting the dots, more people scored above 15, where in the survey, more students thought they would score below 15.
  • In the survey, only 14 students thought they would score above 25. On the actual test, 24 people scored above a 25.
  • On the actual math test, scores were spread out a lot more than in the survey scores.

Example 2:

The following tables show the number of cars sold each month in one year for two separate dealers. Create a dot plot for each of the following tables of data using the same grid. Give three inferences based on the graphs and how they overlap.

First we look at the lowest and highest number of cars sold for both because the graph must be able to handle all data. The lowest number is 23 and the highest is 38. The horizontal axis is made to include 23 - 38. The vertical axis is left to increase by 1’s.

Then we plot the individual data points for each of the tables. 

TAKE TIME TO DO THIS BEFORE CONTINUING!!

 

Three inferences based on the dot plots of both sets of data are:

  1. Mac’s sold more cars than Sid’s.
  2. Mac and Sid had three months where they sold the same amount of cars.
  3. The maximum number of cars sold in one month by Sid was 35 and the maximum number of cars sold by Mac was 38.

There are more!

Good luck on your assignment!

teacher-scored 30 points possible 45 minutes

Activity for this lesson.

  1. Please print the worksheet.
  2. Complete each problem showing all your work and highlighting your answer.
  3. Digitize (scan or take digital photo) and upload your assignment.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


10.03 Frequency Tables and Histograms (Math Level 1)

Organize, display and analyze data in histograms.

Something to Ponder

How would you explain the difference between a Histogram and a Bar Graph?

Mathematics Vocabulary

Frequency Table: a table that lists data and uses tally marks to indicate frequency of occurrence 

Histogram: a graph of frequency distribution 

Learning these concepts

The content of the following is from CK-12-Basic-Probability-and-Statistics-Concepts/section/7.8/ with minor edits to the frequency tables.

An extension of the bar graph is the histogram. A histogram is a type of vertical bar graph in which the bars represent grouped continuous data. The shape of a histogram can tell you a lot about the distribution of the data, as well as provide you with information about the mean, median, and mode of the data set. The following are some typical histograms, with a caption below each one explaining the distribution of the data, as well as the characteristics of the mean, median, and mode. Distributions can have other shapes besides the ones shown below, but these represent the most common ones that you will see when analyzing data. In each of the graphs below, the distributions are not perfectly shaped, but are shaped enough to identify an overall pattern.

Figure a represents a bell-shaped distribution, which has a single peak and tapers off to both the left and to the right of the peak. The shape appears to be symmetric about the center of the histogram. The single peak indicates that the distribution is unimodal. The highest peak of the histogram represents the location of the mode of the data set. The mode is the data value that occurs the most often in a data set. For a symmetric histogram, the values of the mean, median, and mode are all the same and are all located at the center of the distribution.

 
Figure b represents a distribution that is approximately uniform and forms a rectangular, flat shape. The frequency of each class is approximately the same.

Figure c represents a right-skewed distribution, which has a peak to the left of the distribution and data values that taper off to the right. This distribution has a single peak and is also unimodal. For a histogram that is skewed to the right, the mean is located to the right on the distribution and is the largest value of the measures of central tendency. The mean has the largest value because it is strongly affected by the outliers on the right tail that pull the mean to the right. The mode is the smallest value, and it is located to the left on the distribution. The mode always occurs at the highest point of the peak. The median is located between the mode and the mean.

 
Figure d represents a left-skewed distribution, which has a peak to the right of the distribution and data values that taper off to the left. This distribution has a single peak and is also unimodal. For a histogram that is skewed to the left, the mean is located to the left on the distribution and is the smallest value of the measures of central tendency. The mean has the smallest value because it is strongly affected by the outliers on the left tail that pull the mean to the left. The median is located between the mode and the mean.

Figure e has no shape that can be defined. The only defining characteristic about this distribution is that it has 2 peaks of the same height. This means that the distribution is bimodal.

While there are similarities between a bar graph and a histogram, such as each bar being the same width, a histogram has no spaces between the bars. The quantitative data is grouped according to a determined bin size, or interval. The bin size refers to the width of each bar, and the data is placed in the appropriate bin.

The bins, or groups of data, are plotted on the x-axis, and the frequencies of the bins are plotted on the y-axis. A grouped frequency distribution is constructed for the numerical data, and this table is used to create the histogram. In most cases, the grouped frequency distribution is designed so there are no breaks in the intervals. The last value of one bin is actually the first value counted in the next bin. This means that if you had groups of data with a bin size of 10, the bins would be represented by the notation [0-10), [10-20), [20-30), etc. Each bin appears to contain 11 values, which is 1 more than the desired bin size of 10. Therefore, the last digit of each bin is counted as the first digit of the following bin.

The first bin includes the values 0 through 9, and the next bin includes the values 10 through 19. This makes the bins the proper size. Bin sizes are written in this manner to simplify the process of grouping the data. The first bin can begin with the smallest number of the data set and end with the value determined by adding the bin width to this value, or the bin can begin with a reasonable value that is smaller than the smallest data value.

Example A

Construct a frequency distribution table with a bin size of 10 for the following data, which represents the ages of 30 lottery winners: 38, 41, 29, 33, 40, 74, 66, 45, 60, 55, 25, 52, 54, 61, 46, 51, 59, 57, 66, 62, 32, 47, 65, 50, 39, 22, 35, 72, 77, 49

Step 1: Determine the range of the data by subtracting the smallest value from the largest value.

Range: 77−22 = 55

Step 2: Divide the range by the bin size to ensure that you have at least 5 groups of data. A histogram should have from 5 to 10 bins to make it meaningful: 55/10=5.5 ≈ 6. Since you cannot have 0.5 of a bin, the result indicates that you will have at least 6 bins.

Step 3: Construct the table.

Bin Frequency
[20–30) 3
[30–40) 5
[40–50) 6
[50–60) 7
[60–70) 6
[70–80) 3

 

Step 4: Determine the sum of the frequency column to ensure that all the data has been grouped.

 

3+5+6+8+5+3 = 30

When data is grouped in a frequency distribution table, the actual data values are lost. The table indicates how many values are in each group, but it doesn't show the actual values.

Step 5: Create the histogram.

From looking at the tops of the bars, you can see how many winners were in each category, and by adding these numbers you can determine the total number of winners. You can also determine how many winners were within a specific category. For example, you can see that 9 winners were 60 years of age or older. The graph can also be used to determine percentages. For example, it can answer the question, “What percentage of the winners were 50 years of age or older?” as follows:

16/30 = 0.533 or approximately 5.3%

Example B

The numbers of years of service for 75 teachers in a small town are listed below:

1, 6, 11, 26, 21, 18, 2, 5, 27, 33, 7, 15, 22, 30, 8,31, 5, 25, 20, 19, 4, 9, 19, 34, 3, 16, 23, 31, 10, 4, 2, 31, 26, 19, 3, 12, 14, 28, 32, 1, 17, 24, 34, 16, 1,18, 29, 10, 12, 30, 13, 7, 8, 27, 3, 11, 26, 33, 29, 20, 7, 21, 11, 19, 35, 16, 5, 2, 19, 24, 13, 14, 28, 10, 31

Using the above data, construct a frequency distribution table with a bin size of 5.

Range: 35−1 = 34

34/5 = 6.8 ≈ 7

You will have 7 bins.

When the number of data values is very large, another column is often inserted in the distribution table. This column is a tally column, and it is used to account for the number of values within a bin. A tally column facilitates the creation of the distribution table and usually allows the task to be completed more quickly. For each value that is in a bin, draw a stroke in the Tally column. To make counting the strokes easier, draw 4 strokes and cross them out with the fifth stroke. This process bundles the strokes in groups of 5, and the frequency can be readily determined.

14 + 10 + 10 + 13 + 7 + 11 + 10 = 75

Now that you have constructed the frequency table, the grouped data can be used to draw a histogram. Like a bar graph, a histogram requires a title and properly labeled x- and y-axes.

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

10.03 Frequency Tables and Histograms - Explanation Video Link (Math Level 1)

10.03 Frequency Tables and Histograms - Explanation Videos (Math Level 1)

See video


10.03 Frequency Tables and Histograms - Extra Video (Math Level 1)

I highly recommend that you click on the links above and watch the video before continuing.

Preparation
Sample items to consider while viewing the videos and before beginning the worksheet.

Example 1:

The results of testing 12 batteries of the same type are shown. Use the data to make a frequency table and histogram. 

Hours of Battery Life

9 12 14 10 15 10 18 23 10 14 22 11

Answers

Example 1:

The results of testing 12 batteries of the same type are shown. Use the data to make a frequency table and histogram. 

Hours of Battery Life

9 12 14 10 15 10 18 23 10 14 22 11

Step 1: Write the numbers is order from least to greatest: 9, 10, 10, 10, 11, 12, 14, 14, 15, 18, 22, 23

Step 2: Choose the range of numbers for the frequency table (answers may vary)

Range of Numbers Frequency
0 - 9 1
10 - 14 7
15 - 19 2
20 - 24 2

 

Step 3: Create the histogram

10.03 Frequency Tables and Histograms - Worksheet (Math Level 1)

teacher-scored 40 points possible 45 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


10.04 Box Plots (Math Level 1)

Organize, display and analyze data in box plots.

 

Something to Ponder

How would you describe the process for making a Box Plot?

Mathematics Vocabulary

Box Plot: a representation of data consisting of a five number summary (using median, quartiles and range) 

Quartiles: a division of data into 4 parts

Interquartile range: the difference between the first and third quartiles 

Learning these concepts

This part of the lesson is from NROC Developmental Math Unit 8 Lesson 2 Topic 1 under a Creative Commons Attribution 3.0 Unported License.

Another type of graph that you might see is called a box-and-whisker plot. These graphs provide a visual way of understanding both the range and the middle of a data set.

Here is a sample set of 15 numbers to get us started.

12, 5, 18, 20, 11, 9, 3, 5, 7, 18, 12, 15, 6, 10, 11

Creating a box-and-whisker plot from this data requires finding the median of the set. To do this, order the data.

3, 5, 5, 6, 7, 9, 10, {\color{Blue} 11} , 11, 12, 12, 15, 18, 18, 20

This data set has 15 numbers, so the median will be the 8th number in the set: [{\color{Blue} 11}] .

Finding the median of the data set essentially divides it into two—a set of numbers below the median, and a set of numbers above the median. A box-and-whisker plot requires you to find the median of these numbers as well!

Lower set: 3, 5, 5, {\color{Red} 6}, 7, 9, 10. Lower Median: {\color{Red} 6}

Upper set: 11, 12, 12, {\color{Red} 15}, 18, 18, 20. Upper Median: {\color{Red} 15}

So, the median of the set is {\color{Blue} 11}, the median of the lower half is {\color{Red} 6}, and the median of the upper half is {\color{Red} 15}.

3, 5, 5, [{\color{Red} 6}] , 7, 9, 10, {\color{Blue} 11}, 11, 12, 12, {\color{Red} 15}, 18, 18, 20

A box-and-whisker plot for this data set is shown here. Do you see any similarities between the numbers above and the location of the box?

Notice that one “box” (rectangle section) begins at {\color{Red} 6} (the median of the lower set) and goes to {\color{Blue} 11} (the median of the full set), and the other box goes from {\color{Blue} 11} to {\color{Red} 15} (the median of the upper set).

The “whiskers” are the line segments on either end. One stretches from 3 (the least value in the set) to 6, and the other goes from 15 to 20 (the greatest value in the set).

The box-and-whisker plot essentially divides the data set into four sections (or quartiles): whisker, box, box, whisker. The size of the quartiles may be different, but the number of data points in each quartile is the same.

You can use a box-and-whisker plot to analyze how data in a set are distributed. You can also the box-and-whisker plots to compare two sets of data.

Now let’s look at this same set of data with an outlier added:

3, 5, 5, {\color{Red} 6}, 7, 9, 10, {\color{Blue} 11}, 11, 12, 12, {\color{Red} 15}, 18, 18, 20, {\color{Green} 40}

I used a website to create this. Notice that the outlier – {\color{Green} 40} – is not included in the plot. It is marked with an “x” outside of the box plot! If you are asked to create a box plot, make certain that you exclude any outliers!

Five Number Summary

This part of the lesson is from Oswego City School District Regents Exam Prep Center. The copyright has expired.

When describing a set of data, without listing all of the values, we have seen that we can use measures of location such as the mean and median.  It is also possible to get a sense of the data's distribution by examining a five statistical summary (or five number summary), the (1) minimum, (2) maximum, (3) median (or second quartile), (4) the first quartile, and (5) the third quartile.  Such information will show the extent to which the data is located near the median or near the extremes.

The first quartile (25th percentile) is the middle (the median) of the lower half of the data.  One-fourth of the data lies below the first quartile and three-fourths lies above. The second quartile (50th percentile) is another name for the median of the entire set of data. The median of a data set equals the second quartile of the data set. The third quartile (75th percentile) is the middle (the median) of the upper half of the data.  Three-fourths of the data lies below the third quartile and one-fourth lies above.

 

A quartile is a number, it is not a range of values.  A value can be described as "above" or "below" the first quartile, but a value is never "in" the first quartile.

Consider:  Check out this five statistical summary for a set of tests scores.

Minimum 1st Quartile 2nd Quartile (Median) 3rd Quartile Maximum
65 70 80 90 100

 

While we do not know every test score, we do know that half of the scores are below 80 and half are above 80.  We also know that half of the scores are between 70 and 90.  (The difference between the third and first quartiles is called the interquartile range.)

A five statistical summary can be represented graphically as a box-and-whisker plot.

The first and third quartiles are at the ends of the box, the median is indicated with a vertical line in the interior of the box, and the maximum and minimum are at the ends of the whiskers.

http://practicalstats.labanca.net/index.php?title=File:Interquartilerange.gif#filelinks 

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

10.04 Box Plots - Explanation Video Link (Math Level 1)

10.04 Box Plots - Explanation Videos (Math Level 1)

See video


10.04 Box Plots - Extra Link (Math Level 1)

I highly recommend that you click on the link above and work through the material before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

The results of testing 12 batteries of the same type are shown. Find the quartile values, the minimum value, and the maximum value and then make a box plot:

Hours of Battery Life

9 12 14 10 15 10 18 23 10 14 22 11

Answers

Example 1:

The results of testing 12 batteries of the same type are shown. Find the quartile values, the minimum value, and the maximum value and then make a box plot:

Hours of Battery Life

9 12 14 10 15 10 18 23 10 14 22 11

Step 1: Write the numbers is order from least to greatest: 9, 10, 10, 10, 11, 12, 14, 14, 15, 18, 22, 23 (12 numbers)

Step 2: Find the median: Since there are 12 numbers, it is the average of the 6th and 7th numbers: 13

Step 3: Find the minimum and maximum

  • Minimum: 9
  • Maximum: 23

Step 4: Find the quartiles:

  • 2nd Quartile is the median = 13
  • 1st Quartile is the median of the lower half of the numbers which is the average of the 3rd and 4th numbers: 9, 10, 10, 10, 11, 12 = 10
  • 3rd Quartile is the median of the upper half of the numbers which is the average of the 3rd and 4th numbers: 14, 14, 15, 18, 22, 23 = (15+18)/2 = 16.5

Step 5: Create the box plot

10.04 Box Plots - Worksheet (Math Level 1)

teacher-scored 20 points possible 40 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


10.05 Strengths and Weaknesses of Various Data Representations (Math Level 1)

Describe the strengths, weaknesses of and most appropriate use for dot plots, histograms and box plots.

 

Something to Ponder

  • Which graphical representation is the best to use for a set of data?

Mathematics Vocabulary

Numerical Data* – Values or observations that can be measure and placed in order such as height and weight. 

Categorical Data* – Values or observations that can be sorted into groups or categories like eye color or gender. 

*Definitions from Australia Bureau of Statistics.

Learning these concepts

The following is from CPALMS – a trademark of Florida State University – under a Creative Commons Attribution 3.0 license.

Joshua, a sophomore at Hoover High School, usually goes to bed around 11:00 p.m. and gets up around 8:00 a.m. to get ready for school. That means that he gets about 9 hours of sleep on a school night. He decided to investigate this statistical question: How many hours per night do sophomores usually sleep when they have school the next day?

Joshua surveyed 20 sophomores.  The following data set represents the average number of hours each student sleeps on a school night: 7 8 5 9 9 9 7 7 10 10 11 9 8 8 8 12 6 11 9 10

Let’s make a dot plot, histogram, and box plot to display the data.

  • Which graph best illustrates the data given the research question?

Before answering the question let’s look at the advantages and disadvantages of each type of graph. 

Dot Plot

  1. A dot plot is a graphic display using dots and a simple scale to compare the frequency within categories or groups.
  2. A dot plot is useful for relatively small sets of data.
  3. Dot plots clearly display clusters/gaps of data and outliers.
  4. In dot plots, the frequency axis is not necessary but you need to count to find the frequency in each stack of dots, and they can be hard to construct and interpret for data sets with many points.
  5. They can be used with numerical and categorical data.

Histogram

  1. A histogram is a type of graph that shows the frequency distribution of data within equal intervals (thus, there are no spaces between the bars).
  2. It shows the number of values within an interval and not the actual values. 
  3. You can graph huge data sets easily with histograms. 
  4. They are used only for numerical data.
  5. You could change the intervals of the histogram to see which gives a better description of the data.

Box Plot

  1. The box plot is a standardized way of displaying the distribution of data based on the minimum, first quartile, median, third quartile, and maximum of the data set.
  2. A box plot is a good way to summarize large amounts of data.
  3. It displays the range and distribution of data along a number line. 
  4. Box plots provide some indication of the data’s symmetry and skew-ness.
  5. Box plots show outliers.
  6. Original data is not clearly shown in the box plot; also, mean and mode cannot be identified in a box plot.
  7. They can be used only with numerical data.

Remember

  1. Graphs must always be clearly labeled.
  2. Changing the scales in a graph can make the data look very different, ultimately changing the impression that the graph makes.
  3. When comparing two or more sets of data, the scales must be consistent; otherwise, it is difficult to compare the data.

Back to the question: Which graph best illustrates the data given the research question?

This is an informal survey. It isn’t a complex situation. That easiest display would work well. There is no need to know the median or quartiles. There is no need to take the time to create intervals. In this case, the dot plot best illustrates the data give this research question.

Good luck on your assignment!

teacher-scored 44 points possible 45 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


10.06 Two-Way Frequency Tables of Data (Math Level 1)

Use two-way frequency tables of data to compare different categories of data.

Two-Way Frequency Tables

Categorical data are often summarized in the media, research studies, or general discussions. However, categorical data are summarized differently than numerical data. There is no mean or median that answers the question, “What is your favorite soft drink?” Methods for analyzing categorical data are developed in this lesson (Engage NY Lesson 9: Summarizing Bivariate Categorical Data under a Creative Commons Attribution Share-Alike 3.0 License).

The material for this lesson is from DIGI 203 Algebra 1 (d203algebra.wikispaces.com/Guided+Learning+1B) under a Creative Commons Attribution Share-Alike 3.0 License.

A two-way frequency table is used to compare different categories of data. The table below was created from a survey where 50 students were asked what foreign language they were taking. In the table below, the different categories being compared are gender (boys and girls), and subject taken (Spanish, French, and German). The table can provide different pieces of information. For example, if you wanted to know how many girls are currently taking French, you would look at the "girls" row and move to the "French" column. You can see that it is 12. If you wanted to know the number of total boys who are taking a foreign language, you would look at the "boys" row, and go to the "total" column to find out that 20 boys are taking a foreign language.

Looking at the number of girls taking French is called a joint frequency, because you are combining two different categories (gender and foreign language). Looking at the total number of boys taking a foreign language in general is called a marginal frequency because you are only looking at one category (gender).

The marginal frequency (the number of times a certain response is given) is shown in red. The joint frequency (the number of times a certain response is given by a certain group) is shown in blue.

We can interpret the data even further by creating what is called a two-way relative frequency table. In this type of table, the percent of a joint frequency or marginal frequency is compared to a total. A two-way relative frequency table can be created with respect to the table total, the table rows, or the table columns.

When a two-way relative frequency table is created with respect to the table total, all of the entries in the table are divided by the overall table total. The entries are then replaced with the decimal that represents the percentage.

When a two-way relative frequency table is created with respect to the rows, the data entries in each row are divided by the total for that row. For example, in order to create the relative frequency for the Boys row, take each entry in that row (10, 2, 8, and 20), and divide by the total for the Boys row (20). This is where the 0.5, 0.1, 0.4, and 1.00 comes from in the table on the right.

When a two-way relative frequency table is created with respect to the columns, the data entries in each column are divided by the total for that column. For example, in order to create the relative frequency for the Spanish column, take each entry in that column (10, 15, and 25), and divide by the total for the Spanish column (25). This is where the 0.4, 0.6, and 1.00 comes from in the table on the right.

Example: Create a two-way frequency table and answer the questions that follow. Twenty five kids were asked what they like to do on a hot day. Ten of the 13 girls said they like to go to the pool. Eight of the boys said they like to play video games indoors. Create the two-way frequency table. Then create a two-way relative frequency table with respect to the table total.

a) How many boys prefer to go to the pool?

b) What percentage of girls out of all the kids prefer to watch video games?

c) What percentage of the kids are boys?

a) 4 boys
b) 12%
c) 48%

Guided Practice

1. You took a survey in your class of 30 students and the results are as follows:

a. How many boys are in the set?

b. How many girls are in the set?

c. How many students can bike?

d. How many girls can't bike?

e. Create a two-way frequency table (with respect to the table total).

f. With respect to the total what percent of girls could bike?

g. Create a two-way frequency table (with respect to the rows).

h. What percent of boys could bike?

i. Create a two-way frequency table (with respect to the columns).

j. What percent of people that could bike are girls?
 

2. There are 150 children at a local pool are signed up in the aquatic sports of which 61 signed up for swim team. There were a total of 52 children that signed for water polo and 28 of them also signed up for swim team. Use the two-way frequency table below to summarizing the data.

a. How many children did not sign up for water polo? Does this represent a marginal or joint frequenc

b. How many students did not sign up for either water polo or swim team? Does this represent a marginal or joint frequency?

3. You are in charge of boosting profits for the concession stand at the basketball games. You decide to do some research. On Friday night as each person entered the basketball game, you counted how many of the 400 people had a hotdog, a slice of pizza, soda and bottled water. You found that out of 182 people that had Pizza, 120 chose soda and only four did not have a drink. 50 people bought a hotdog and a soda. 158 people bought hotdogs. Ten people walked in without any food or a drink. The stand sold 200 sodas and 140 waters. Using this data, fill in the two way frequency table and determine the marginal frequencies of the data.

a. What was the joint frequency of people who ordered pizza AND a drink?

b. Based on the data, is it more likely that people will order pizza without a drink or a hot dog without a drink? Justify your answer using information from the frequency table.

Answers

1. Biking to School

a. 11

b. 19

c. 16

d. 10

e.

f. 30%

g.

h. 64%

i.

j. 56%

2. Aquatic Sports

a.  98

b. marginal

c.  65

d. joint

3. Concession Stand Sales

a. 98%

b.  Hot dog/no drink = 41%

Good luck on your assignment!

teacher-scored 60 points possible 45 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


10.07 Standard Deviation (Math Level 1)

Find mean and compute standard deviation and analyze data using mean and standard deviation.

Something to Ponder

How would you define Standard Deviation?

Mathematics Vocabulary

Variance: the measure of how far each value in the data set is from the mean.

Standard deviation: the measure of dispersion for a collection of data 

This material is from

Probability and Statistics Advanced 2

CK-12 Foundation is licensed under Creative Commons Attribution NonCommercial 3.0 Unported (CC BY-NC 3.0)

 

Most high schools have a set amount of time in between classes in which students must get to their next class. If you were to stand at the door of your statistics class and watch the students coming in, think about how the students would enter. Usually, one or two students enter early, then more students come in, then a large group of students enter, and then the number of students entering decreases again, with one or two students barely making it on time, or perhaps even coming in late! Try the same by watching students enter your school cafeteria at lunchtime. Spend some time in a fast food restaurant or café before, during, and after the lunch hour and you will most likely observe similar behavior. Have you ever popped popcorn in a microwave? Think about what happens in terms of the rate at which the kernels pop. Better yet, actually do it and listen to what happens! For the first few minutes nothing happens, then after a while a few kernels start popping. This rate increases to the point at which you hear most of the kernels popping and then it gradually decreases again until just a kernel or two pops. Try measuring the height, or shoe size, or the width of the hands of the students in your class. In most situations, you will probably find that there are a couple of very students with very low measurements and a couple with very high measurements with the majority of students centered around a particular value.

This shows a typical pattern that seems to be a part of many real life phenomena. In statistics, because this pattern is so pervasive, it seems to fit to call it “normal”, or more formally the normal distribution. The normal distribution is an extremely important concept because it occurs so often in the data we collect from the natural world, as well as many of the more theoretical ideas that are the foundation of statistics.

Characteristics of a Normal Distribution

Shape

If you think of graphing data from each of the examples in the introduction, the distributions from each of these situations would be mound-shaped and mostly symmetric. A normal distribution is a perfectly symmetric, mound-shaped distribution. It is commonly referred to the as a normal, or bell curve.

Because so many real data sets closely approximate a normal distribution, we can use the idealized normal curve to learn a great deal about such data. In practical data collection, the distribution will never be exactly symmetric, so just like situations involving probability, a true normal distribution results from an infinite collection of data, or from the probabilities of a continuous random variable.

Center

Due to this exact symmetry the center of the normal distribution, or a data set that approximates a normal distribution, is located at the highest point of the distribution, and all the statistical measures of center we have already studied, mean, median, and mode are equal.

It is also important to realize that this center peak divides the data into two equal parts.

Spread

Let’s go back to our popcorn example. The bag advertises a certain time, beyond which you risk burning the popcorn. From experience, the manufacturers know when most of the popcorn will stop popping, but there is still a chance that a rare kernel will pop after longer, or shorter periods of time. The directions usually tell you to stop when the time between popping is a few seconds, but aren’t you tempted to keep going so you don’t end up with a bag full of un-popped kernels? Because this is real, and not theoretical, there will be a time when it will stop popping and start burning, but there is always a chance, no matter how small, that one more kernel will pop if you keep the microwave going. In the idealized normal distribution of a continuous random variable, the distribution continues infinitely in both directions.

Because of this infinite spread, range would not be a possible statistical measure of spread. The most common way to measure the spread of a normal distribution then is using the standard deviation, or typical distance away from the mean.

The symbol for mean in statistics is \mu, the Greek letter mu if it is a population and \overline{x} if is it a sample. The symbol for standard deviation is \sigma, the Greek letter sigma.

Because of the symmetry of a normal distribution, the standard deviation indicates how far away from the maximum peak the data will be. Here are two normal distributions with the same center (mean):

 

The first distribution pictured above has a smaller standard deviation and so the bulk of the data is concentrated more heavily around the mean. There is less data at the extremes compared to the second distribution pictured above, which has a larger standard deviation and therefore the data is spread farther from the mean value with more of the data appearing in the tails.

Because of the similar shape of all normal distributions we can measure the percentage of data that is a certain distance from the mean no matter what the standard deviation of the set is. The following graph shows a normal distribution with μ = 0 and σ = 1. This curve is called a standard normal distribution. In this case, the values of x represent the number of standard deviations away from the mean.

Notice that vertical lines are drawn at points that are exactly one standard deviation to the left and right of the mean. We have consistently described standard deviation as a measure of the “typical” distance away from the mean. How much of the data is actually within one standard deviation of the mean? To answer this question, think about the space, or area under the curve. The entire data set, or 100% of it, is contained by the whole curve. What percentage would you estimate is between the two lines? It is a reasonable estimate to say it is about 2/3 of the total area.

The empirical rule states that the percentages of data in a normal distribution within 1, 2, and 3 standard deviations of the mean, are approximately 68, 95, and 99.7, respectively.

Variation

This part of the lesson is from Saylor Foundation's Introductory Statistics eBook under a Creative Commons Attribution 3.0 Unported License.

Look at the two dot plots below:

The two sets of ten measurements each center at the same value: they both have mean, median, and mode equal to 40. Nevertheless a glance at the figure shows that they are markedly different. In the plot on the left, the measurements vary only slightly from the center, while the plot on the left's  measurements vary greatly. Just as we have attached numbers to a data set to locate its center, we now wish to associate to each data set numbers that measure quantitatively how the data either scatter away from the center or cluster close to it. These new quantities are called measures of variability, and we will discuss three of them.

The Range

The first measure of variability that we discuss is the simplest. To find the range you simply subtract the smallest data point from the largest data point.

For the plot on the left the maximum is 43 and the minimum is 38, so the range is R = 43 – 38 = 5.

For the plot on the right the maximum is 47 and the minimum is 33, so the range is R = 47 – 33 = 14.

The range is a measure of variability because it indicates the size of the interval over which the data points are distributed. A smaller range indicates less variability (less dispersion) among the data, whereas a larger range indicates the opposite.

The Variance and the Standard Deviation

The other two measures of variability that we will consider are more elaborate and also depend on whether the data set is just a sample drawn from a much larger population or is the whole population itself (that is, a census).

To find the variance:

  1. Find the mean (\overline{x}): (40 + 38 + 42 + 40 + 39 + 39 + 43 + 40 + 39 + 40)/10 = 400/10 = 40
  2. Subtract each data point from the mean

Left Data Set:

x 40 38 42 40 39 39 43 40 39 40
x – \overline{x} 0 –2 2 0 1 1 –3 0 1 0

 

   3. Square each difference.

x – \overline{x} 0 –2 2 0 1 1 –3 0 1 0
(x-\overline{x})^{2} 0 4 4 0 1 1 9 0 1 0

 

   4. Find the average of the squared differences: (0 + 4 + 4 + 0 + 1 + 1 + 9 + 0 + 1 + 0)/10 = 20/10 = 2

 

This is the variance.

Your turn! Find the variance of Data Set 2.

Make certain you get the correct answer: 22.4

The standard deviation is the square root of the variance. For the first data set of data:

Standard deviation = \dpi{100} \fn_phv \sqrt{2}  = 1.41.

What is the standard deviation for the second data set?

Make certain you get the correct answer: 4.73

Of course, there is a formula to calculate the standard deviation without having to follow all these steps. In addition, if you enter the data into a spreadsheet, there is a standard deviation function that will also compute it.

Note: There are actually two formulas for variance and standard deviation. One is for a population – all values included from the complete population. This is what has been taught to this point.

If the values instead were a random sample drawn from some larger parent population (for example, they were 8 marks randomly chosen from a class of 20), then you must calculate the sample variance and deviation.

The only difference between the two is divisor. If you have eight values and they represent the entire population, you divide by 8. If you have a sample of the entire population you divide by one less or 7 instead.

For the purposes of this class we will find the variance and standard deviation of populations.

This material is from Engage NY Algebra 1 Module 2 licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.

Here is a dot plot of the lives of the Brand A batteries.

Let's calculate the standard deviation:

  • First, find each deviation from the mean.
  • Then, square the deviations from the mean. For example, when the deviation from the mean is −18 the squared deviation from the mean is (-18)^{2}=324.
Life (Hours) 83 94 96 106 113 114
Deviation from the Mean –18 –7 –5 5 12 13
Squared Deviations from the Mean 324 49 25 25 114 169
  • Add up the squared deviations:

324 + 49 + 25 + 25 + 144 + 169 = 736.

This result is the sum of the squared deviations.

The number of values in the data set is denoted by n. In this example, n is 6.

  • You divide the sum of the squared deviations by n if it is an entire population and n – 1 if it is a sample population. In this case, we have a sample population we use 5..

736/5 = 147.2

Finally, you take the square root of 147.2, which to the nearest hundredth is 12.13.

That is the standard deviation! It seems like a very complicated process at first, but you will soon get used to it.

We conclude that a typical deviation of a Brand A battery lifetime from the mean battery lifetime for Brand A is 12.13 hours. The unit of standard deviation is always the same as the unit of the original data set. So, the standard deviation to the nearest hundredth, with the unit, is 11.07 hours.

Now you can calculate the standard deviation of the lifetimes for the eight Brand B batteries. The mean is 100.5. We already have the deviations from the mean.

Life (Hours) 73 76 92 94 110 117 118 124
Deviation from the Mean –27.5 –24.5 87.5 –6.5 9.5 16.5 17.5 23.5
Squared Deviation from the Mean                

 

  1. Calculate the squared deviations in the table.
  2. Add up the squared deviations.
  3. What is the value of n for this data set?
  4. Divide the sum of the squared deviations by n – 1. Round your answer to the nearest thousandth.
  5. Take the square root to find the standard deviation. Record your answer to the nearest hundredth.
  6. Make a statement about the answer as it relates to the problem.

Seriously! Take time to do this before looking below.

ANSWERS

1.

Squared Deviation from the Mean 756.25 600.25 72.25 42.25 90.25 272.25 306.25 552.25

 

2. 2692

3. 8

4. 2792/7 = 384.571

5. \sqrt{384.571}=19.16

6. The standard deviation, 19.16 hours, is a typical deviation of a Brand B battery lifetime from the mean battery lifetime for Brand B.

So, now we have computed the standard deviation of the data on Brand A and of the data on Brand B. We can now compare the two and describe what you notice in the context of the problem.

The fact that the standard deviation for Brand B is greater than the standard deviation for Brand A tells us that the battery life of Brand B had a greater spread (or variability) than the battery life of Brand A. This means that the Brand B battery lifetimes tended to vary more from one battery to another than the battery lifetimes for Brand A.

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

10.07 Standard Deviation - Explanation Video Link (Math Level 1)

10.07 Standard Deviation - Explanation Videos (Math Level 1)

See video


10.07 Standard Deviation - Extra Link (Math Level 1)

I highly recommend that you click on the link above and work through the material before continuing.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1:

Find the mean and the standard deviation for the values: 48.0, 53.2, 52.3, 46.6, 49.9

Answers

Example 1:

Find the mean and the standard deviation for the values: 48.0, 53.2, 52.3, 46.6, 49.9

Step 1: Find the mean - the sum divided by 5 = 300\div6 = 50.

Step 2: Calculate the difference from the mean for each data point:

Mean Difference from the mean
48 50 – 48 = 2
53.2 50 – 53.2 = –3.2
52.3 50 – 52.3 = –2.3
46.6 50 – 46.6 = 3.4
49.9 50 – 49.9 = 0.1

 

Step 3: Square each difference

Difference Difference Squared
2 4
– 3.4 10.24
– 2.3 5.29
3.4 11.56
0.1 0.01

 

Step 4: Find the average of the square differences or the variance \sigma^{2}:

\fn_phv \frac{4+10.24+5.29+11.56+0.01}{5}=6.22

Step 5: Take the square root of the variance to find the standard deviation, \sigma:

\sqrt{6.22} \approx 2.49

10.07 Standard Deviation - Worksheet (Math Level 1)

teacher-scored 20 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


10.08 Scatter Plots, Correlation and Causation (Math Level 1)

Create scatter plots and use them to analyze data and estimate linear and exponential functions that fit the data.

Something to Ponder

How would you describe the process for estimating a trend line?

How would you explain the difference between correlation and causation?

Mathematics Vocabulary

Bivariate data: set of data on two variables

Causation: the direct effect of one set of data on the other 

Clusters: points in a scatter plot that form two or more distinct clouds of points. 

Correlation: the linear relationship between two sets of data 

Correlation Coefficient: a number between –1 and +1 that measures the strength and direction of a linear relationship; denoted by the letter r.

Outliers: An unusual point in a scatter plot that does not seem to fit the general pattern or that is far away from the other points in the scatter plot.

Scatter plot: a graph that relates two sets of data 

Statistical relationship: when one variable tends to vary in a predictable way as the values of the other variable change.

Learning these concepts

This material is from Engage NY Grade 8 Module 6 under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

A bivariate data set consists of observations on two variables. For example, you might collect data on 13 different car models. Each observation in the data set would consist of an (x, y) pair.

x = weight (in pounds, rounded to the nearest 50 pounds)

and

y = fuel efficiency (in miles per gallon, mpg)

The table below shows the weight and fuel efficiency for 13 car models with automatic transmissions manufactured in 2009 by Chevrolet.

Model Weight (pounds) Fuel Efficiency (mpg)
1 3200 23
2 2550 28
3 4050 19
4 4050 20
5 3750 20
6 3550 22
7 3550 19
8 3500 25
9 4600 16
10 5250 12
11 5600 16
12 4500 16
13 4800 15

 

In the table above, the observation corresponding to Model 1 is (3200, 23).

  • What is the fuel efficiency of this car? What is the weight of this car?

The fuel efficiency is 23 miles per gallon, and the weight is 3200 pounds.

One question of interest is whether there is a relationship between the car weight and fuel efficiency. The best way to begin to investigate is to construct a graph of the data. A scatter plot is a graph of the (x, y) pairs in the data set. Each (x, y) pair is plotted as a point in a rectangular coordinate system.

Here is the graph of these coordinate points:

  • Do you notice a pattern in the scatter plot?

There does seem to be a pattern in the plot. Higher weights tend to be paired with lesser fuel efficiencies, so it looks like heavier cars generally have lower fuel efficiency.

  • Is there a relationship between price and the quality of athletic shoes? The data in the table below are from the Consumer Reports website.

x = price (in dollars)

and

y = Consumer Reports quality rating

The quality rating is on a scale of 0 to 100, with 100 being the highest quality.

Shoe Price (dollars) Quality Rating
1 65 71
2 45 70
3 45 62
4 80 59
5 110 58
6 110 57
7 30 56
8 80 52
9 110 52
10 70 52

 

To construct a scatter plot of these data, you need to start by thinking about appropriate scales for the axes of the scatter plot. The prices in the data set range from $30 to $110, so one reasonable choice for the scale of the x-axis would range from $20 to $120. The smallest y-value is 51, and the y-largest value is 71. So, the y-axis could be scaled from 50 to 75.

This is the scatter plot.

  • Do you see any pattern in the scatter plot indicating that there is a relationship between price and quality rating for athletic shoes?

You may think there is a slight downward trend or you might not see a pattern at all.

Some people think that if shoes have a high price, they must be of high quality.

  • How would you respond?

The data do not support this. 

A pattern in a scatter plot indicates that the values of one variable tend to vary in a predictable way as the values of the other variable change. This is called a statistical relationship. In the fuel efficiency and car weight example, fuel efficiency tended to decrease as car weight increased.

This is useful information, but be careful not to jump to the conclusion that increasing the weight of a car causes the fuel efficiency to go down. There may be some other explanation for this. For example, heavier cars may also have bigger engines, and bigger engines may be less efficient. You cannot conclude that changes to one variable cause changes in the other variable just because there is a statistical relationship in a scatter plot

Data were collected on

x = shoe size

and

y = score on a reading-ability test

for 30 elementary school students. The scatter plot of these data is shown below. Does there appear to be a statistical relationship between shoe size and score on the reading test?

The pattern in the scatter plot appears to follow a line. As shoe sizes increase, the reading scores also seem to increase. There does appear to be a statistical relationship because there is a pattern in the scatter plot.

  • Is it reasonable to conclude that having big feet causes a high reading score? Can you think of a different explanation for why you might see a pattern like this?

You cannot conclude that just because there is a statistical relationship between shoe size and reading score that one causes the other. These data were for students completing a reading test designed for younger elementary school children. Older children, who would have bigger feet than younger children, would probably tend to score higher on a reading test for younger students.

Summary

  • A scatter plot is a graph of numerical data on two variables.
  • A pattern in a scatter plot suggests that there may be a relationship between the two variables used to construct the scatter plot.
  • If two variables tend to vary together in a predictable way, we can say that there is a statistical relationship between the two variables.
  • A statistical relationship between two variables does not imply that a change in one variable causes a change in the other variable (a cause-and-effect relationship).

When you look at a scatter plot, you should ask yourself the following questions:

  1. Does it look like there is a relationship between the two variables used to make the scatter plot?
  2. If there is a relationship, does it appear to be linear?
  3. If the relationship appears to be linear, is the relationship a positive linear relationship or a negative linear relationship?

To answer the first question, look for patterns in the scatter plot. Does there appear to be a general pattern to the points in the scatter plot, or do the points look as if they are scattered at random? If you see a pattern, you can answer the second question by thinking about whether the pattern would be well-described by a line. Answering the third question requires you to distinguish between a positive linear relationship and a negative linear relationship. A positive linear relationship is one that is described by a line with a positive slope. A negative linear relationship is one that is described by a line with a negative slope.

Look at the following scatter plots. 

1. Scatter Plot 1

  • Is there a relationship?

Yes.

  • If there is a relationship, does it appear to be linear?

Yes.

  • If the relationship appears to be linear, is it a positive or negative linear relationship?

The slope is downward so the slope is negative.

2. Scatter Plot 2

  • Is there a relationship?

Yes.

  • If there is a relationship, does it appear to be linear?

Yes.

  • If the relationship appears to be linear, is it a positive or negative linear relationship?

The slope is upward so the slope is negative.

3. Scatter Plot 3

  • Is there a relationship?

No.

  • If there is a relationship, does it appear to be linear?

Not applicable.

  • If the relationship appears to be linear, is it a positive or negative linear relationship?

Not applicable.

4. Scatter Plot 4

  • Is there a relationship?

Yes.

  • If there is a relationship, does it appear to be linear?

No. (The U-shape means it is quadratic!)

  • If the relationship appears to be linear, is it a positive or negative linear relationship?

Not applicable.

5. Scatter Plot 5

  • Is there a relationship?

Yes.

  • If there is a relationship, does it appear to be linear?

Yes.

  • If the relationship appears to be linear, is it a positive or negative linear relationship?

Negative.

In addition to looking for a general pattern in a scatter plot, you should also look for other interesting features that might help you understand the relationship between two variables. Two things to watch for are as follows:

CLUSTERS: Usually the points in a scatter plot form a single cloud of points, but sometimes the points may form two or more distinct clouds of points. These clouds are called clusters. Investigating these clusters may tell you something useful about the data.

OUTLIERS: An outlier is an unusual point in a scatter plot that does not seem to fit the general pattern or that is far away from the other points in the scatter plot.

The scatter plot below was constructed using data from a study of Rocky Mountain elk (“Estimating Elk Weight from Chest Girth,” Wildlife Society Bulletin, 1996). The variables studied were chest girth in centimeter (x) and weight in kilogram (y).

  • Do you notice any point in the scatter plot of elk weight versus chest girth that might be described as an outlier? If so, which one?

The point in the lower left hand corner of the plot corresponding to an elk with a chest girth of about 96 cm and a weight of about 100 kg could be described as an outlier. There are no other points in the scatter plot that are near this one. This point corresponds to an observation for an elk that is much smaller than the other elk in the data set, both in terms of chest girth and weight.

  • Do you notice any clusters in the scatter plot? If so, how would you distinguish between the clusters in terms of chest girth? Can you think of a reason these clusters might have occurred?

Other than the outlier, there appear to be three clusters of points. One cluster corresponds to elk with chest girths between about 105 cm and 115 cm. A second cluster includes elk with chest girths between about 120 cm and 145 cm. The third cluster includes elk with chest girths above 150 cm. It may be that age and sex play a role. Maybe the cluster with the smaller chest girths includes young elk. The two other clusters might correspond to females and males if there is a difference in size for the two sexes for Rocky Mountain elk. If we had data on age and sex, we could investigate this further.

Summary

  • A scatter plot might show a linear relationship, a nonlinear relationship, or no relationship.
  • A positive linear relationship is one that would be modeled using a line with a positive slope. A negative linear relationship is one that would be modeled by a line with a negative slope.
  • Outliers in a scatter plot are unusual points that do not seem to fit the general pattern in the plot or that are far away from the other points in the scatter plot.
  • Clusters occur when the points in the scatter plot appear to form two or more distinct clouds of points.

Some Linear Relationships Are Stronger than Others

Below are two scatter plots that show a linear relationship between two numerical variables x and y.

  • Is the linear relationship in scatter plot 3 positive or negative?

This scatter plot is not as clear-cut as the previous examples. In general, there are more points in the scatter plot that describe a positive relationship.

  • Is the linear relationship in Scatter Plot 4 positive or negative?

This scatter plot also indicates a positive relationship. For most of the points, x-values increase, the y-values also tend to increase.

It is also common to describe the strength of a linear relationship. We would say that the linear relationship in scatter plot 3 is weaker than the linear relationship in scatter plot 4.

A scatter plot that has all of the points on a line with a positive slope indicates the strongest possible positive linear relationship. 

A scatter plot with the strongest possible negative linear relationship would be one in which all of the points are on a line with a negative slope. This line has a negative slope.

Consider the three scatter plots below. Place them in order from the one that shows the strongest linear relationship to the one that shows the weakest linear relationship.

Scatter Plot 7 is the strongest and Scatter Plot 5 is the weakest.

The Correlation Coefficient

The correlation coefficient is a number between –1 and +1 that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter r.

Several scatter plots are shown below. The value of the correlation coefficient for the data displayed in each plot is also given.

r = 1
r = 0.71
r = 0.32
r = –0.10
r = –0.32
r = –0.63
r = –1.00

 

  • When is the value of the correlation coefficient positive?

The correlation coefficient is positive when as the x-values increase, the y-values also tend to increase.

  • When is the value of the correlation coefficient negative?

The correlation coefficient is negative when as the x-values increase, the y-values tend to decrease.

  • Is the linear relationship stronger when the correlation coefficient is closer to 0 or to 1?

As the points form a stronger negative or positive linear relationship, the correlation coefficient gets farther from 0. Students note that when all of the points are on a line with a positive slope, the correlation coefficient is + 1. The correlation coefficient is – 1 if all of the points are on a line with a negative slope.

You will not be asked to calculate the correlation coefficient. Technology will take care of that for you!

Previously, you learned to use two points compute the slope and use that information to come up with a line through those two points. You could use this method to come up with a line through two of the points of the scatter plot. But which two points would you use?

Fortunately, there are easier ways to come up with what is called the line of best fit!

Correlation vs. Causation

The following is from Engage NY Algebra I Module 2 Topic D Lesson 19 under a Creative Commons license.

Correlation Does Not Mean There is a Cause-and-Effect Relationship Between Variables

It is sometimes tempting to conclude that if there is a strong linear relationship between two variables that one variable is causing the value of the other variable to increase or decrease. But you should avoid making this mistake. When there is a strong linear relationship, it means that the two variables tend to vary together in a predictable way, which might be due to something other than a cause-and-effect relationship.

For example, the value of the correlation coefficient between sodium content and number of calories for the fast food items is r = 0.79, indicating a strong positive relationship. This means that the items with higher sodium content tend to have a higher number of calories. But the high number of calories is not caused by the high sodium content. In fact sodium does not have any calories. What may be happening is that food items with high sodium content also may be the items that are high in sugar or fat, and this is the reason for the higher number of calories in these items.

Similarly, there is a strong positive correlation between shoe size and reading ability in children. But it would be silly to think that having big feet causes children to read better. It just means that the two variables vary together in a predictable way. Can you think of a reason that might explain why children with larger feet also tend to score higher on reading tests?

It is very difficult to prove causation. Be very careful in making any assumptions about causation!

Here are some interesting examples taken from Wikipedia: en.wikipedia.org/wiki/Correlation_does_not_imply_causation

Example 1:

As ice cream sales increase, the rate of drowning deaths increases sharply.

Therefore, ice cream consumption causes drowning.

The aforementioned example fails to recognize the importance of time and temperature in relationship to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.

Example 2:

Young children who sleep with the light on are much more likely to develop myopia in later life.

Therefore, sleeping with the light on causes myopia.

This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature,[5] the study received much coverage at the time in the popular press.[6] However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children's bedroom.[7][8][9][10] In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.

Click the mathematician image or the link below to launch the video to help you better understand this material.

Scroll down to the Guided Practice section and work through the examples before submitting the assignment.

10.08 Scatter Plots, Correlation and Causation - Explanation Video Link (Math Level 1)

Optionally: use the link above to view the explanatory math video.

10.08 Scatter Plots, Correlation and Causation - Explanation Videos (Math Level 1)

See video


10.08 Scatter Plots, Correlation and Causation - Extra Link (Math Level 1)

I highly recommend that you click on the links above and work through the material.

Guided Practice
After watching the video try these problems. The worked solutions follow.

Example 1: 

The table shows the average prices of movie tickets and the number of movie admissions. Make a scatter plot of the data in the table. Graph movie admissions on the horizontal axis.

Year Number of Admissions (Millions) Average Ticket Price
1990 1189 $4.23
1992 1173 $4.15
1994 1292 $4.18
1996 1339 $4.42
1998 1481 $4.69

 

Answer

10.08 Scatter Plots, Correlation and Causation - Worksheet (Math Level 1)

teacher-scored 50 points possible 60 minutes

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


10.09 Data Analysis Project (Math Level 1)

Choose the best method to display and analyze statistical data.

Something to Ponder

Why is it important to your life to understand statistics?

Mathematics Vocabulary

Data analysis: process of inspecting, cleaning, transforming, and modeling data to surface useful information, suggest solutions and lead to better informed decisions

Learning these concepts

Click the mathematician image or the link below to launch the video to help you better understand this "mathematical language."

10.09 Data Analysis Project - Explanation Video Link (Math Level 1)

10.09 Data Analysis Project - Explanation Videos (Math Level 1)

See video


10.09 Data Analysis Project - Worksheet (Math Level 1)

teacher-scored 20 points possible 60 minutes

At the end of this module you have the opportunity to put all of your statistical knowledge towards a project.

You will be able to:

  • Compute means and standard deviation
  • Create frequency tables, histograms, box plots and scatter plots
  • Display and analyze data you collect

Activity for this lesson

  1. Print the worksheet. Work all the problems showing ALL your steps.
  2. Digitize (scan or take digital photo) and upload your worksheet activity.

Pacing: complete this by the end of Week 9 of your enrollment date for this class.


10.10 Writing Assignment (Math Level 1)

teacher-scored 30 points possible 45 minutes

 

Writing Assignment

As a mathematics teacher, I often hear the question, “When am I ever going to use this?” from students who fail to understand the practical worth of mathematical competency.

Write an essay (at least 3 paragraphs and at least 100 words) answering that question regarding specific topics presented this quarter.

If necessary, research possible occupations you are considering. If you can’t think of any possible way you will use this, research possible reasons for studying this type of math.

Rubric

Criteria Description Points
Introduction (one paragraph) Stage is set for the body of the essay 6
Sentence Structure Complete and correct sentences; sentence variation - simple, compound, complex. 5
Mechanics Proper punctuation, capitalization, grammar and spelling. 5
Organization Clear and logical order; smooth transitions among sentences, ideas, and paragraphs 8
Conclusion Nice summary statement(s) 6

 

 

Pacing: complete this by the end of Week 8 of your enrollment date for this class.