Attachment A

GLE: NO.1.A.9

Math A :Line Up Cards

Prepare 3x5 cards with a variety of numbers, both rational and irrational.

For example,

2.5

0

-1

Give each student a card and ask the class to arrange themselves around the room in

numerical order. (Since the Math A exam allows the use of a scientific or graphing calculator, you may wish to

allow students to use their calculators when determining their place in the number line.)

This very quick activity will help students obtain a feeling of how numbers are

related to one another. It is particularly valuable in allowing students to see how

rational and irrational numbers compare.

51

52

There are many additional activities which also utilize these cards:

Ask all of the students holding "irrational" numbers to move to one side of

the room, and all students holding "rational" numbers to move to the opposite

side of the room.

Ask 5 students to stand in the front of the room in numerical order. You, the

teacher, holding a card, position yourself in the number line. Ask the class if

you are standing in the correct numerical location. This could also be a small

quiz or extra credit activity.

Ask 2 students to stand in the front of the room. Ask the class to determine

the size of the interval between the two numerical values.

Shuffle the cards and ask the students to place themselves in numerical order

a second time.

When the students are lined up in numerical order, ask all of the students to

step forward whose number:

-- is a multiple of 2.

-- is an irrational number.

-- is an odd number.

-- can be expressed as a fraction.

-- etc.

There are many possibilities -

let your imagination guide you.

Attachment A

GLE: NO.1.A.9

53

Attachment B

GLE: NO.2.B.9

Hidden Irrationals

On the dot paper below, the horizontal and vertical distance from one dot to the other is 1 unit.

Draw line segments with the following lengths and label with their lengths.

1. square root 2 2. square root 5 3. square root 8

4. square root 18 5. square root 20 6. square root 26

7. square root 32 8. square root 34 9. square root 40

10. Choose 3 other lengths of irrational numbers that you can draw by connecting dots. Draw and label.

11. Give 3 other irrational-number lengths that you can not draw connecting dot to dot.

12. Explain what has to be true for you to be able to draw the number lengths in this way.

13. Are all the diagonals irrational? Convince me with an argument or a counter-example.

54

Attachment B

GLE: NO.2.B.9

Input/Output Machine

An input/output machine can be used with almost any subject area. A mathematical equation, chapter

definition, or review question is written on one side of an index card. The answer to the problem/question

is written on the other side of the card. Students read the card, decide what they think the answer is, and

place the card in the slot, question side up. When it comes out of the machine, the answer side is

displayed. Students have fun making the cards for this machine, which serves as a reinforcement strategy

for them!

Materials

empty detergent box

contact paper

poster board

scissors or box cutter

items to decorate

glue gun

Cut a strip of poster board approximately 10 inches long and 3 inches wide. Cover this strip in contact

paper. Cut two 3 inch slits in the front of the box. Glue the poster board inside the box so that it slopes

down in a half circle and attach to the other slit. This forms a slide for the card to pass through. Now

cover your box in contact paper and decorate as desired.

Extension: Make a bundle of approximately 5 cards which illustrate the same mathematical relationship.

After the student has sent all cards through the input/output machine he/she should guess the rule.

Example:

Input

3

5

6

1

8

Output

11

15

17

7

21

Rule: Output = twice the input plus 5. (O = 2I + 5)

55

Attachment B

GLE: NO.2.B.9

Scavenger Hunt

Find:

1) a mixed number.

2) a number which is a perfect square.

3) a used car for less than $2000.

4) a fraction less than 1/2.

5) a formula.

6) a unit of measure.

7) a negative number.

8) a number which is a factor of 24.

9) a number which is a multiple of 7.

10) the final score for a sporting event.

11) an approximate number.

12) the amount of interest you earn on a CD.

13) a number which is a prime factor of 60.

14) two numbers whose sum is more than 100 but less than 200.

15) a pattern of three or more numbers.

16) a whole number used to indicate order.

17) a percent.

18) a person’s name on a page where the page number has a one in the ten’s place and a four in the one’s

place.

19) a number whose square root is irrational.

20) a number which names something and is not used as a quantity.

21) an equation.

22) a variable.

23) a metric system measurement.

24) a ratio.

I Have, Who Has?

Copy cards onto colored paper, laminate, cut apart, and decorate with stickers if desired. Large Group

Activity: Pass out all cards, one or two per student. Start by reading any card. The person with the answer

to the question on that card reads his or her card. Continue in this manner until you get back to the starting

card.

Small Group Activity: Start with any card. Place the cards on the table with the second card answering

the first card, the third card answering the second card, etc. The last card is answered by the first card. See

which group can finish first.

56

Attachment B

GLE: NO.2.B.9

I have 63 Who has that

divided by 9?

I have 7. Who has that

plus 3?

I have 18. Who has 12

more than that?

I have 30. Who has the

quotient of that and 6?

I have 10. Who has that

Who has that minus 7?

I have 3. Who had that

multiplied by 6?

I have 5. Who has the

product of that and 8?

I have 40. Who has 4 less

than that?

I have 36. Who has that

divided by 9?

I have 4. Who has the

sum of 9 and that

number?

I have 13. Who has one

less than twice that

number?

I have 25. Who has that

number times 3?

I have 75. Who has that

decreased by 3?

I have 72. Who has that

divided by 9?

I have 8. Who has that

increased by 9?

I have 17. Who has twice

that?

I have 34. Who has one

less than that?

I have 33. Who has that

divided by 3?

I have 11. Who has 9

more than that?

I have 20. Who has 3

times that?

I have 60. Who has the

difference of that divided

by 5?

I have 45. Who has that

divided by 5?

I have 9. Who has that

multiplied by 9?

I have 81. Who has that

plus 9?

I have 90. Who has that

decreased by 20?

I have 70. Who has that

minus 7?

57

Attachment C

GLE: NO.1.C.9

The Tale of Exponents

Emma exponent was so upset. She was labeled as a negative. Her thoughts were always

filled by the chance of being positive. Finally, the day arrived. As she was hanging out in the

denominator, her friend Matt Mathematician told her that if she moves to the numerator, she

will become positive. "Can this be?", thought Emma. It was definitely worth a try, so she put

on her walking shoes and made the trip to the top. When she arrived in the numerator, her

dream became reality. Emma Exponent was now positive.

Matt Mathematician was Emma's hero. Just as he was for Eric Exponent, who lived as a

negative in the numerator and slid down to become positive. Not to mention Zoe Zero who

was told of her magical powers to turn all of her bases into 1. Matt Mathematician made

many exponents' dreams come true.

Exponential Equations Graphing Activity

In our study of Algebra thus far, we have only studied equations of lines. The equation of a line

comes in three forms. Name these forms below and show their equation forms.

1.

2.

3.

Today we are going to take a short look at another type of equation, the exponential equation. The

graphing calculator is going to be used to learn about the different characteristics of these equations.

Using the name exponential equation as a clue, what do you think these equations must include?

The first exponential equation that we are going to graph is :

Y=2 to the x power.

Use the graphing calculator to draw the graph on the coordinate plane to the right (see Appendix A)

Now graph the following

Y= -2 to the x power

How does the negative sign change the graph of the function?

Moving on, how about:

Y = 1/2 to the x power

What is the relation between the numbers 2 and 1/2 ?

How does changing the reciprocal change the graph of the original function?

58

Attachment D

GLE: NO.1.C.9

At your part-time job you earn $5.15 per hour. Explain how you can figure out how much money you have earned. Write a variable expression appropriate to answer the question of how much you have earned. Be sure to identify a variable. Create a table of values of the money earned when the hours worked are 5, 10, 15, and 20.

Rubric(s)

Rubric: How Much Did I Make?

Trait: How Much Did I Make?

Performance Type:

Level 1: 4 Level 2: 3 Level 3: 2 Level 4: 1

A response at this level analyzes the full range of the problem correctly. It represents all of the information appropriately, and applies mathematical concepts to solve the problem correctly. This response thoroughly explains the processes used to solve the problem in the context of the problem. This response must have a correct answer.

A response at this level analyzes the problem correctly. It represents most of the information appropriately and applies mathematical concepts, which are essentially complete and correct, to solve the problem. It may contain minor flaws. This response explains the process(es) in the context of the problem.

A response at this level analyzes most of the problem correctly. It represents some of the information appropriately and applies mathematical concepts, which are essentially complete and correct, to solve the problem. It may contain major flaws or be incomplete. This response shows little or no attempt to explain the process(es) in the context of the problem.

A response at this level shows some attempt to solve the problem but analyzes the problem incorrectly. It represents little or no information appropriately and makes some attempt to apply mathematical concepts to solve the problem. This response makes little or no attempt to explain the process(es) in the context of the problem. It may have a correct answer with no supporting information or may have inappropriate mathematical concepts.

A response at this level shows no evidence of mathematical thinking or no response is given.

59

Attachment D

GLE: NO.1.C.9

A classmate you are working with wrote the following: 24 - 4 * 2 = 40. This is not true. Explain what your classmate did wrong. Rewrite this so that the left side is equal to 40. Justify your answer.

Rubric(s)

Rubric: What's Wrong Here?

Trait: What's Wrong Here?

Performance Type:

Level 1: 4 Level 2: 3 Level 3: 2 Level 4: 1

A response at this level analyzes the full range of the problem correctly. It represents all of the information appropriately and applies mathematical concepts to solve the problem. It may contain minor flaws. This response explains the process(es) and justifies clearly the conclusions in the context of the problem. This response must have a correct answer.

A response at this level analyzes the problem correctly. It represents most of the information appropriately and applies mathematical concepts, which are essentially complete and correct, to solve the problem. It may contain minor flaws. This response explains the process(es) in the context of the problem and shows some justification of the conclusion.

A response at this level analyzes most of the problem correctly. It represents some of the information appropriately and applies mathematical concepts to solve the problem. It may contain major flaws or be incomplete. This response shows little or no attempt to explain the process(es) or justify the conclusion.

A response at this level shows some attempt to solve this problem but analyzes the problem incorrectly. It represents little or no information appropriately and makes some attempt to apply mathematical concepts to solve the problem. flaws. This response makes little or no attempt to explain the process(es) used or to justify conclusions.

A response at this level shows no evidence of mathematical thinking or no response is given.

60

Attachment E

GLE: NO.2.D.9

© 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org

Power Up Game Materials:

• Battery cards (six cards per team)

• Die

• Transparent Spinner

Prior to class, copy the spinner onto a transparent sheet so that it can be projected on the overhead

projector. To make a spinner, unbend a paper clip, lay it on the spinner, and put a pencil point inside

the clip.

Also, copy the battery card sheets back-to-back on heavy paper or cardstock. When copying, be

sure that the cards align so that the positive value on the front matches the negative value on the

back. Cut out the cards. (Alternatively, you can copy the cards one-sided only. This makes the game

significantly more difficult.)

During class, go over the rules with students.

Object:

Earn points by lining up your battery cards end-to-end so that the sum equals the target number. The

first team to earn 10 points wins the game.

Playing the Game:

• Divide the class into teams of 2 to 4 students.

• Shuffle the cards, and give each team six cards. On one side of each card, the voltage is

negative; on the other side, the voltage is positive.

• Place the transparent spinner on the overhead projector and spin. The number on which the

pointer lands is the target number (the voltage sum).

• Your teacher will then roll the die. The number on the die tells you the number of battery

cards you must use to obtain the target number. If a 1 is rolled, the teacher will roll the die

again.

• Your team will be given 20 seconds to create a lineup that equals the target number and that

uses the correct number of cards.

o If you create a correct lineup in the allotted time, your team earns 2 points.

o If your team can not create a lineup in the allotted time, or if you create a lineup with

an incorrect total, your team receives 0 points.

o If your team creates a lineup with the correct total but the wrong number of cards,

you will receive an additional 10 seconds to correct the mistake. If you fix the lineup

correctly, your team will receive 1 point.

• Once during the game, at any time before the target number is determined and the die is

rolled, your team may request a new set of six battery cards. Your team must give the old set

to the teacher. © 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org © 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org © 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org

61

Attachment E

GLE: NO.2.D.9

62

Attachment E

GLE: NO.2.D.9

63

Attachment E

GLE: NO.2.D.9

64

Attachment F

GLE: NO.3.E.9

Golden Rectangle

A Golden Rectangle is a rectangle in which the ratio of the length to the width is

approximately 1.618 : 1.

The Golden Rectangle is described as one of the most pleasing shapes to the

human eye.

For over 2,000 years, people have used "golden rectangles" in art and architecture.

The Parthenon and the head of Mona Lisa were designed using the concept of the

"golden rectangle."

The very old mathematics movie "Donald Duck in Mathmagic

Land" does a wonderful job of illustrating the Golden Rectangle.

This old 16mm film has been updated toVHS videotape and can be

purchased for as little as $5.00 at most family chain stores such as

Walmart and KMart. All mathematics teachers should have their

own copy of "Donald" - it's a tradition!!! : )

After watching "Donald" (or discussing the Golden Rectangle), have students

complete the following exercise:

Item Length Width

Ratio of

length to

width

Golden

Rectangle?

3 x 5 card

8.5 x 11 sheet of paper

desk top

textbook

teacher's desk

ID card

Anything else of

interest....

65

Attachment G

GLE: NO.3.E.9

Summary: Prior to this task, the class will discuss, read, or watch pieces of Gulliver's Travels. They will then choose a real world object that could help Gulliver and enlarge or shrink it to an appropriate size using the concepts of proportion.

Resources: Gulliver's Travels (movie, book, or discussion notes)

Student Directions: Now that you have learned a bit about Gulliver's Travels (and troubles). You are assigned to rewrite a small piece of his story and supply him with a new large or small tool to help him. Choose a scene in which Gulliver needs some additional help, rewrite the scene and using the concepts of similarity create a larger or smaller tool for him. Include in your work the original measurements of the actual object and the solutions to the proportions you solve to scale it up or down. (note: the class should decide as a whole what scale factors to use)

Rubric(s)

Rubric: Two Scales Scale

Trait: Two Scales Scale

Performance Type:

Level 1: A

-Turns in a well-written scene which logically incorporates the new 'tool.' 20 points -Accurately measures and solves proportions to scale the tool. 40 points -Creates the new tool using the proportions. 40 points

Learning Activities: 1. The Perfect Person Activity: After a lecture on the concept of ratio, the teacher

introduces the Golden Ratio. The students each measure their height and distance from

their belly button to the floor and compare this ratio to the Golden Ratio.

2. Proportional portions lesson: After practicing solving proportions, the students bring

in recipes that they have "scaled" to feed the whole class. They share their academic

and culinary results. Included is a discussion of the importance of similarity in the real

world (models of objects too large to study otherwise, etc...)

3. Teach a Friend: Communication activity. Students write a letter to a friend who

missed class and needs to have an explanation of proportion.

4. Worksheet: After the previous lessons and activities, the students practice solving for

missing parts of similar figures and proving figures similar.

5. Discovery Activity: The students discover various proportion concepts using a ruler

and lined paper. (See lesson in Mathematics Teacher magazine. Vol 87 No 4. April

1994)

6. A Tale of Two Scales Activity: The students enlarge or shrink a real world object

using similarity to help Gulliver.

7. Test: The test will include skill-level and understanding of application questions.

66

Name ___________________________ Attachment H

GLE: AR.1.B.9-12

1. Given the following pattern, use words or an expression to describe the pattern. 1, 9, 25, 49, 81, …

________________________________________________________________________________

________________________________________________________________________________

The first three stages of a pattern are shown below. . 1 2 3

Stage Number 1 2 3 4 5 6 7

Area of Stage 1 3 5

2. Find the area of the next four rectangles in the pattern. 3. What is the area of the 20th stage? ______________________

4. Develop a formula for the area of the nth stage._________________________________

Write an equation beside each of the charts below to represent the pattern.

5.

x 1 2 3 4 5

y 1 -1 -3 -5

-7

6.

x 1 2 3 4 5

y 4.5 5 5.5 6

6.5

67

Name ___________________________ Attachment H

GLE: AR.1.B.9-12

Figure Number 1 2 3 4

Term 1 2 3 4

Perimeter 8 10 12 14

Area 3 5 7 9

7. Use the pattern generated above to find the perimeter and area of the 10th figure. _____________________________

8. Generalize the pattern to find the nth term for perimeter. __________________________

9. Generalize the pattern to find the nth term for the area. ___________________________

10. Use the formula/generalization created in question two to find the perimeter of the 105th figure. ________________________________ 11. Use the formula/generalization created in question three to find the area of the 105th figure. __________________________________

68

Attachment H

GLE: AR.1.B.9-12

Answer Key

1. This pattern takes each odd number, starting with 1, and squares it. The expression would be (2n-1)2. In recursive notation, the pattern is (the square root of Now + 2)2, with the first Now being one. 2.. The areas for the next four terms will be: 7, 9, 11, and 13 (for the 4th through 7th terms).

3. The 20th stage would have an area of 39.

4. The nth stage will be Area = 2∙n – 1 = 2n – 1.

5. y = -2x + 3 6. y = 1/2x + 4

7. The 10th figure’s perimeter is 26 and its area is 21.

8. One example of an explicit function for the perimeter would be P = 2∙figure number + 6.

For students who use recursive functions one example of perimeter would be

21 nn PP

with the first Pn-1 being 8.

9. One explicit example for area would be A = 2∙figure number + 1. A recursive example

for this problem with area could be 21 nn AA

.

10. The 105th figure would have a perimeter of 216. 11. The 105th figure would have an area of 211.

69

Name ___________________________ Attachment I

GLE: AR.1.C.9-12

A B C D E

Which of the graphs above could be used to describe the following situation?

Justify your selection including the identity of both the independent and dependent variables.

1. The grade Juan will receive on his math final will depend on the amount of time he studies.

_________________________________________________________________________________________

________________________________________________________________________________________

2. The population of the western salamander compared to the amount of pollution in its environment.

_________________________________________________________________________________________

_________________________________________________________________________________________

Julie is asked to graph the following two functions and was surprised to discover that the two equations generated

the same line.

3. Do you agree with Julie? _________________

4. Compare each function and describe why they are different representations of the same line. The description

should include discussion about slope and intercepts and be more than that they give the same values.

152

yx

52

5 xy

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

70

Attachment I

GLE: AR.1.C.9-12

Answer Key

1. There is a hope that with additional time studying Juan’s test score would improve. So one possible

answer for this problem would be to identify time he studies for the final as the independent variable

(what he controls and located on the x axis) and the grade he receives as the dependant variable (the

result and located on the y axis). Using this rationale, students could either answer D or E. Both sets

of students would need to defend what happens as their graph moves left to right. In both cases, as

time increases so does the score. In D they would need to explain how the score could continue to

rise. In E they would need to explain why the score would level off. It is possible that others might

be selected and the student will need to defend their answer with strong justification.

2. One possible answer would be to make the independent variable be the pollution in the environment

and the dependent variable to be the population of the western salamander. Students could select

graph A or B and would need supporting explanations.

3. Students should agree with Julie that the two equations are different representations of the same line.

4. . Explanations will vary but should include discussions about slope (where it can be seen or found in

the equation), and the intercepts (both x and y intercepts and where they can be seen or found in the

equation). In comparing each function they either know or discover that the first one shows the x and y

intercepts (and is generally called the intercept form of a linear equation). The second equation is more

common and is the slope-intercept form. Notice in both forms the slope appears. In the first equation the

rise and run are there and the “negative” value for the 5 should let the student know that it must be the

opposite as the y value must be moved to the other side to see typical rise over run form of the slope.

While the intercept form is not as common, it does have some interesting similarities to some of the conic

functions students will study in future mathematics courses.

Another possible solution would show the table of values for the two lines. The student should be able to

see that when the same values are used for the independent variable, students should find the same values

will resulting in each table for the dependent. The students could then find the slope and y-intercept from

the found table.

Example: x - y = 1 y = 5/2 x - 5

2 5

x y x y

0 -5 0 -5

10 20 10 20

20 45 20 45

From the table, it is apparent the two equations represent the same line. Notice the y intercept in both

cases is -5. Students can determine the slope by noticing that every time the y changes by 25 the x

changes by 10 in each case. This makes the slope 25/10 or 5/2 in each case.

71

Attachment J

GLE: AR.1.E.9

Hiking

The rate, time, distance relationship is explored. For example, different distances can be traveled in the

same time with different rates, or at the same rate in different times.

1. Students are given the "Two Hikers" problem, where one hiker starts at the top of a mountain trail and

the other starts at the bottom. They hike at different constant rates, which are given. Students must

represent each hiker with a linear graph on the same set of axes, and determine where and when they meet

on the trail. It may be necessary to review graphing lines using slope and intercepts at this point.

2. What will happen if the rate of hiking of one hiker is higher or lower?

3. Additional hiker problems are given, such as one hiker going up one day starting at a given time,

staying overnight then returning the next day starting at a different time of day but hiking at the same rate.

When was the hiker at the same spot on the trail both days? What if it were raining on one of the days?

4. Performance task "Train Schedule"

72

Attachment K

GLE: AR.1.E.9

Performance Activity

Train Schedule Student Directions: You may work with one partner on this project. Together you must turn in a complete proposal with each partner's contributions clearly labeled. You may choose to turn the project in before the due date. If you do, I will tell you if anything is missing or incorrect and give it back to you to complete before the due date. The following project is worth 100 points: As an engineer, you are to design a linear 4km single-track train route from the parking lot to the interior of MathWorld. Cone Mountain, which can be located anywhere from .5 to 1.5 km of the MathWorld stop, has not yet been built. At that location, the track will enter a tunnel through the mountain and split into two parallel tracks before joining back together as a single track. This will allow trains to run simultaneously in both directions. Your design must determine the location of the mountain, the constant speed of the train in each direction, and the train departure schedule so that trains traveling in opposite directions pass in the mountain. You must include a scale drawing or model, a labeled graph, a system of equations, and its solution. You must also do a self-assessment, describing how well you feel you did on each part and giving yourself a score. Your proposal will be presented to the chief engineer (Your Teacher.)

Context of Use: This is an end of unit performance assessment that can be assigned earlier in the unit.

Rubric(s)

Rubric: Train Schedule

Summary: Weighted Performance Checklist (100 points possible): 1. The location of the mountain, speed of each train, and the starting times are coordinated with work shown (15 points possible) 2. The speed of the train in each direction is meaningfully described (5 points possible) 3. A schedule for 3 round trips is presented (5 points possible) 4. The location of the mountain is meaningfully described (5 points possible) 5. A scale drawing or model is turned in (10 points possible) 6. A labeled graph of the system is turned in (15 points possible) 7. A system of linear equations with a solutions is provided (40 points possible) 8. A self-assessment is turned in (5 points possible)

Trait: Train Schedule

Performance Type: Written.

*You may add additional questions to change the parameter and have the student explain their answer.

73

Name___________________________ Attachment L

GLE: AR.1.E.9

An equation, y = 3x + 5, is graphed. A second equation is then given which changes the coefficient 3 to the number 2, thus giving the equation y = 2x + 5. 1. Describe the effect of the change on the x and y-intercepts. ______________________________

_________________________________________________________________________________

2. Describe the effect of the change on the slope of the line. _______________________________

________________________________________________________________________________

3. Describe the effect of the change on the graph. _______________________________________

________________________________________________________________________________

4. When velocity is constant, distance traveled, d, is given by the formula d = vt, where v equals velocity

and t equals time. What is the effect on velocity if twice the distance is traveled in half the time?

__________________________________________________________________________________

Explain your answer.

__________________________________________________________________________________

__________________________________________________________________________________

5. Describe the effect on the appearance of the graph of the function 2x – 4y = 12 if the y-scale is twice

the x-scale (i.e., the horizontal axis values increase by 1 and the vertical values increase by 2).

_________________________________________________________________________________

Explain your answer. ______________________________________________________________

________________________________________________________________________________

74

Attachment L

GLE: AR.1.E.9

Scoring Guide

Student answers will vary but should have discussion similar to the following:

1. The y-intercept is not effected by the change as in both cases it is (0,5).

On the other hand, the x-intercept changes because in the first equation it will be -5/3 while in the

second one it becomes -5/2.

2. The slope of the line in the first equation is 3 which means it goes up from left to right at a faster

rate than the second which has a slope of 2.

3. The graph in the second equation becomes less steep than the first. Both are linear functions.

4. . Exemplary response – Velocity would need to be quadrupled (4 times faster);

2d = v(2

1t); 4(

t

d) = v; or a real number validation.

2 points – a correct answer and explanation

1 point – a correct answer or correct explanation

0 points - other

5. Answers will vary. Students should be able to describe that when a graph’s scale is affected it will

change the appearance of the graph. In this case it will make the slope seem less steep (lower) than .5,

which will be the slope given the equation above.

This should make sense as for each change in x of 1 you would only need to go what would appear to

be half as far on the change of y as the y scale is going by 2 (instead of 1). This should also help point out

the importance of labeling the scale so one can make inferences from the graph that will match the

equation and situation.

75

Attachment M

GLE: AR.2.A.9

A Variation on Bingo

Here's an idea for a variation on the Math bingo game

Give each student a blank bingo card like the one below, and a list of

possible answers.

Ask the students to place the answers anywhere on the card. (There

are 25 answers listed below. If you wish to give a "free space", remove

one of the answers/questions.) Students will X-cross the appropriate

answer as each question is read. The questions can be read in any

order. Bingo can be obtained vertically, horizontally, or diagonally (or

with any other creative arrangement you wish to use.)

Below is a sample game appropriate for this unit of study, with a

teacher answer sheet, and a sheet for students. Try it for a change of

pace.

76

Student Answer Sheet (to be placed on the Bingo card)

1. 5+x 8. x-5 15. 2-3x 22. x-6

2. 6(x+1 9. 3x-2 16. 9-x 23. 9-2x

3. 2x-5 10. 6x+1 17. 2(x+5) 24. 3/x-2

4. 6x-1 11. 6-x 18. x/6 25. 3(x+2)

5. x+6 12. 5x-2 19. x+1

6. 5-2x 13. 9+x 20. 2x+5

7. 3/(x-2) 14. 5-x. 21. x-1

Attachment M

GLE: AR.2.A.9

77

Teacher Question Sheet with Answers

1. 5 more than a number 5+x

2. 6 times, a number increased by 1 6(x+1)

3. twice a number, diminished by 5 2x-5

4. 1 less than 6 times a number 6x-1

5. the sum of 6 and a number x+6

6. 5 decreased by twice a number 5-2x

7. the quotient of 3, and a number

decreased by 2 3/(x-2)

8. 5 less than a number x-5

9. 2 less than the product of 3 and a

number 3x-2

10. 6 times a number, increased by 1 6x+1

11. the difference of 6 and a number 6-x

12. 2 less than 5 times a number 5x-2

13. the sum of 9 and a number 9+x

14. 5 decreased by a number 5-x

15. 2 diminished by 3 times a number 2-3x

16. the difference of 9 and a number 9-x

17. the product of 2, and a number

increased by 5 2(x+5)

18. the quotient of a number and 6 x/6

19. a number increased by 1 x+1

20. 2 times a number, increased by 5 2x+5

21. 1 less than a number x-1

22. 6 less than a number x-6

23. 9 decreased by 2 times a number 9-2x

24. the quotient of 3 and a number,

diminished by 2 3/x-2

25. the quotient of 3, and 2 more than a

number 3(x+2)

Copyright ©1999-2006 Oswego City School District Regents Exam Prep Center

Attachment M

GLE: AR.2.A.9

78

Attachment N

GLE: AR.2.A.9

Student Prompt:

PART I

You have opened a savings account that earns simple interest. Your account began with $1,000, earns 6%

interest annually, and you deposit $50 each month. Create a chart to determine the amount of money you

would have in your account after 4 years. Write a symbolic algebraic expression to represent the amount

of money you would have after “n” years.

Part 2

Change the amount of money that you began in your account, the interest rate and the amount you deposit

each month. Create a chart to show how much money you would have in your account after 4 years.

Beginning account balance: ______________________

Annual interest rate: _______________

Monthly deposit: _____________________

79

Attachment N

GLE: AR.2.A.9

Answer Sheet

Part 1

Possible symbolic Algebraic Expressions:

[(a x 12) + c] x d = x

Year Amount in Account

0 $1,000 (Optional on chart)

1 $1,696

2 $2,332

3 $2,968

4 $3,604

Part 2

4 Points – Shows a clear understanding of a recursive relationship; chart is correct

3 Points – Shows an understanding of a recursive relationship; 3 of 4 years are correct.

2 Points - Shows some confusion of a recursive relationship; 2 of 4 years are correct.

1 Point - Shows confusion of a recursive relationship; 1 of 4 years is correct.

0 Points – Shows no understanding of a recursive relationship; chart is incorrect.

80

Name ___________________________ Attachment O

GLE: AR.2.B.9-10

When two angles are added and their sum is equal to 180°, the angles are said to be supplementary

angles.

Given: One of two supplementary angles is 4o more than one-third of the measure of the other angle.

1. Write an equation that represents the given information. __________________________________

2. Find the measure of each of the angles, showing all the work necessary.

Given the sequence below, two of your classmates identified equations describing the sequence in

different ways.

-4, 3, 10, 17, 24 …

Grant’s equation: y = 7x – 4

Denise’s equation: Next = Now + 7 with the first Now being –4

3. Explain how you can determine that both equations are correct. In your explanation, include why one

student is multiplying by 7 and the other is adding 7. __________________________________

_________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

Sam was working on simplifying the problem: 42

323 )()(

ba

aba

.

After completing the problem, Sam’s answer was27

1

ba .

4. Determine whether Sam’s answer is correct and justify your decision. ______________ _________________________________________________________________________ __________________________________________________________________________

81

Name ___________________________ Attachment O

GLE: AR.2.B.9-10

5. Simplify the following. Be sure to show the factoring of the numerator and denominator.

n2

+ 5n + 6

n2 – 8n – 33 =

Determine which of the following are equivalent to 26?

A. 2 · 25 B. 23 · 23 C. (22)3

D. (23)2 E. 23 · 22 D. 218 ¸ 23

6. Select the correct letter for each of the equivalent multiplications. __________________________

7. Use the properties of exponents to explain why A - F above are or are not equivalent to 26.

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

8. Solve |3x – 4| < 7 for "x" showing your work to complete the problem.

9. Graph the solution

82

Attachment O

GLE: AR.2.B.9-10

Answer Key

1. Let x = the first angle

Let 1/3 x + 4 be the second angle

18043

1

xx

2. x + (1/3 x + 4) = 180

1804

3

4x

4/3x + 4 – 4 = 180 - 4

4

3176

3

4

4

3 x

x = 132 132x and its supplement is 48

3. Answers may vary.

One example of a comparison would be the following: “Both of these students have equations that correctly describe the

sequence given. Grant’s equation is an explicit form and Denise’s equation is a recursive form. Both describe the way the

numbers are changing in the pattern. In Denise’s equation the rate of change is an added value because that is what is

happening for each 1 unit increase in the input value the output will have an additional 7. The start value is the same as the y

value in the y-intercept (-4). Grant’s equation is multiplying by 7 because in the explicit form you may go right to the nth term

that you are seeking, so you must be able to calculate the number of jumps that you have taken which is how many times you

have “added” 7 as multiplication can be considered as a repeated addition.”

Sam has made some mistakes in his calculations. Negative exponents have the result of reciprocating the

number (e.g.,

2

2

1c

c

). The correct answer for this problem would be

11

2

a

b

2.42

323 )()(

ba

aba

= a-6

b-2

a-3

/ a2b

-4

= a-9

b-2

/ a2b

-4

= b4

/ (b2a

9a

2)

= (b4) / (a

11b

2)

= b2

/ a11

83

Attachment O

GLE: AR.2.B.9-10

Answer Key 5. n

2 + 5n + 6

n2- 8n - 33

( n + 2 ) ( n + 3 )

( n + 3 ) ( n - 11 )

n + 2

n - 11

6. A, B, C, D are correct.

7. A and B use the fact that when you multiply powers with like bases you add the exponents. C and D are

powers raised to a power. In this case you multiply the exponents making each equal to 26.

In E, you would need to add the exponents, which will not result in the same solution. In F, you would need to

subtract the exponents, which would result in a larger answer than desired.

A. 21 · 25 = 21+5 B. 23 · 23 = 23+3

= 26 = 2

6

C. (22)3 = 22x3 D. (23)2 = 22x3

=26 = 2

6

E. 23 · 22 = 23+2 F. 218 ÷ 23 = 218-3

= 25 = 2

15

8. 3x – 4 < 7 and 3x – 4 > -7

+ 4 +4 +4 + +4

3x < 11 3x > -3

3x < 11 3x > -3

3 3 3 3

x < 11/3 and x > -1

-1 < x < 11/3

-1

0

4

84

Name __________________________ Attachment P

GLE: AR.2.D.9

If a linear function has a slope of –2/3 and an y-intercept of –4,

1. Write the equation of the function. _____________________________________________

2. Interpret the meaning of the y-intercept if y is a cost. ___________________________________

__________________________________________________________________________________

Two of your classmates are discussing these equations:

y = 2x + 7 and g = x – 3

Jon claims that y will always be greater than g because you multiply by 2 and add 7 rather than subtract

Jon supports his claim with this table and graph:

3. Do you agree or disagree with Jon? ____________Show your work or describe how you got your

answer.

4. Solve the system of equations showing all work. 3x + 4y = 10

2x – y = 6

X y g

0 7 -3

1 9 -2

2 11 -1

3 13 0

4 15 1

Equation numbers

-5

0

5

10

15

20

0 1 2 3 4 5

85

Attachment P

GLE: AR.2.D.9

Answer Key

1. . 4

3

2

xy

would be one equation that could match this problem.

2. In this situation the y-intercept is (0,-4). If this is a cost then one interpretation would be that the

cost for no items would mean there were other expenses not associated with the item described with the x

value. This could be fixed cost such as insurance or rent.

3. Exemplary response – Disagree with John because any value less than –10 makes g greater than y; or

student extends chart or draws graph to show points at which g is greater than y.

2 points – a correct answer and explanation

1 point – a correct answer or correct explanation

0 points - other

4. x = 34/11 and y = 2/11

Multiplying the second equation by 4 8x – 4y = 24

First equation 3x + 4y = 10

Add the equations 11x = 34 so x = 34/11

Multiply the first equation by 2 6x + 8y = 20

Multiply the second equation by –3 -6x + 3y = -18

Add the equations 11y = 2 so y = 2/11

86

Name __________________________ Attachment Q

GLE: AR.4.A.9

If a linear function has a slope of –2/3 and an y-intercept of –4,

1. Write the equation of the function. ____________________________________________

2. Interpret the meaning of the y-intercept if y is a cost. _______________________________

________________________________________________________________________________

3. Chris drove from Blue Springs (a city 20 miles from the Kansas state line) to Marshall Junction (a

location 57 miles from the Kansas state line) in 32 minutes at a constant speed. The speed limit is 70 mph.

Did Chris break the speed limit? _____________

Support your answer. _____________________________________________________________

_______________________________________________________________________________

_______________________________________________________________________________

4. The graph below shows the value of an automobile over five years. Use the graph to write a linear

equation describing the value as a function of year.

__________________________________________________________________

Automobile Value

8000

10000

12000

14000

16000

1 2 3 4 5 6

Year

Valu

e

87

Attachment Q

GLE: AR.4.A.9

Answer Key

1. 1. 4

3

2

xy

would be one equation that could match this problem.

2. In this situation the y-intercept is (0,-4). If this is a cost then one interpretation would be that the

cost for no items would mean there were other expenses not associated with the item described with the x

value. This could be fixed cost such as insurance or rent.

3. Exemplary response – No, Chris did not break the speed limit, because

32

37

= 60

x

; cross multiplying gives 32x = 2,220; x = 69.375 mph; or 37 32 = 1.16 miles per minute,

multiplied by 60 = 69.6 mph; or another valid process.

4. x = ownership of automobile in years y = value of tractor

y = -$1,250x +$16,250

This answer is found by looking at the slope found by using two points from the graph. It appears from

Year One the value is $15,000 and then in Year 5 the value is $10,000. Using the slope formula,

Slope = $10,000 - $15,000

5 -1

= -$5,000

4

= -$1,250

This makes the equation y = -$1,250x + b. If the graph is moved to Year 0

it would change by $1,250 making the y-intercept or original cost $16,250.

This results in the equation, y = -$1,250x + $16,250.

88

Attachment R

GLE: GSR.1.B.9

Creating Congruent

Triangles

Under what conditions are

triangles congruent?

Materials: rulers and protractors

(students may work alone or in groups)

1. Draw a triangle of any size and shape and label it ABC.

2. Using the ruler and protractor, measure and record the angles and sides of

the triangle. Remember that there are 180 degrees in the sum of the angles

of a triangle.

3. Draw triangle DOG where DO = AB, angle D = angle A,

and angle O = angle B.

4. Measure and record the remaining angles of triangle DOG.

5. Are triangles ABC and DOG congruent?

6. Is it possible to draw triangle DOG so that it is NOT congruent to triangle

ABC?

7. Draw triangle KIT where KI = AB, angle K = angle A, and

angle T = angle C.

8. Measure and record the remaining angles of triangle KIT.

9. Are triangles ABC and KIT congruent?

10. Is it possible to draw triangle KIT so that it is NOT congruent to triangle

ABC?

11. Discuss your finding from this activity.

89

Name __________________________________ Attachment S

GLE: GSR.1.B.9

Performance Event

A fast food restaurant recently gave toys away in its children’s meals. The toys came

inside a cardboard pyramid. Unfolded, the pyramid had this design:

3

2.5 inches

A. Calculate the area of wasted (scrap) material if each pyramid is stamped out of

a square piece of cardboard that is 7 inches on each side. Provide the work that shows

how you arrived at your answer and write your answer on the line.

Area: ____________ sq. inches

90

Name __________________________________ Attachment S

GLE: GSR.1.B.9

B. Can the amount of scrap material be reduced? If so, write a letter to the

president of the fast food restaurant company explaining how it can save money on

the production of these cardboard pyramids by determining the amount of scrap

material saved. Include drawings if you wish. If the amount of scrap material cannot be

reduced, explain why.

Write your letter to the president of the fast food restaurant here.

91

Attachment S

GLE: GSR.1.B.9

Answer Key

Exemplary response –

Part A – the piece of cardboard 7 inches on a side has 49 square inches; using the Pythagorean

Theorem, the box is about 21 square inches, so 28 square inches is wasted.

Part B – a piece of cardboard about 5 inches on a side (4.95 inches) could be used to make the

box, resulting in a savings of 24 square inches of cardboard for each box. Many valid processes could produce these results.

Many valid letters could be written to the CEO, but it should contain the above info and point out the

roughly 50% savings in materials used to make the toy boxes by using the smaller piece of cardboard.

Scoring Guide:

4 points: The student’s response fully addresses the performance event by:

demonstrating knowledge of mathematical principles/concepts needed to complete the event, such as

accurately computing the areas of the toy boxes

communicating all process components that lead to an appropriate and systematic solution, such as

illustrating the savings to the company by using the smaller piece of cardboard to make the toy

boxes

having only minor flaws with no effect on the reasonableness of the solution

3 points: The student’s response substantially addresses the performance event by:

demonstrating knowledge of mathematical principles/concepts needed to complete the event, such as a

generally accurate computation of areas of toy boxes

communicating most process components that lead to an appropriate and systematic solution

having minor flaws with minimal effect on the reasonableness of the solution

2 points: The student’s response partially addresses the performance event by:

demonstrating a limited knowledge of mathematical principles/concepts needed to complete the event,

such as computations of areas of toy boxes with errors

communicating some process components

having flaws or extraneous information

1 point – The student’s response minimally addresses the performance event by:

demonstrating a limited knowledge of mathematical principles/concepts needed to complete the event,

such as an inaccurate calculations

communicating few or no process components

having flaws or extraneous information that indicates a lack of understanding or confusion

0 points – Other; such as merely copying prompt information.

92

Student _________________________________ Attachment T

GLE: GSR.1.B.9

Building codes for Smallville state that no building may be more than 35 feet tall. A carpenter

constructing the building in the figure would like to build the roof so that the vertical rise of its two

inclines will be 10 inches for every foot of horizontal distance as they extend toward the peak of the

house.

20 feet

30 feet

Given the other dimensions shown, will the carpenter be able to build the roof the way he wants to and

still meet the code? Show work that supports your conclusion.

93

Name _______________________ Attachment U

GLE: GSR.2.A.9

A(1,5) B(4,5)

1. Given the vertices of parallelogram ABCD in this standard (x, y) coordinate plane,

what is the area of ΔABC in square units?

A. 10

B. 12

C. 15

D. 16

2. Find the straight-line distance from the store to the school. Round your answer to the

nearest hundredth of a mile. Provide the work that shows how you arrived at your

answer and write your answer on the line.

Each block grid = 0.75 mile ________________ miles

D(-1,-3) C(2,-3)

94

Attachment U

GLE: GSR.2.A.9

Answer Key

1. B

2. Distance = 7.21 miles or3 13

2; on the grid, the school is 4 over and 6 down; using the Pythagorean

Theorem, 42 + 6

2 = x

2; x =

3 13

2 or approximately 5.41; or another valid process.

95

Attachment V

GLE: GSR.3.A.9

Household Decoration

Topics: Reflections, Translations, Rotations, and Dilations

Many of the geometric figures that we see daily in our real world can change in size, in shape and can move from side to side or forward and backward. These changes are descriptions of transformational geometry that have applications found in the areas from science and architecture to music and history. The student Goal is to inform the customers of a shop about the beauty of geometric designs that are used and have been used through-out our world of shapes in motion. The student Role is to be an engineer of geometric designs that will be "hot-selling" decorative creations. The Audience is composed of fellow students and the teacher who wants to buy unusual items that appeal to the eye to display in their homes. The Situation for each student is to research, propose, design and create geometric designs that illustrate transformational geometry. The Product/Performance and Purpose is to design a proposal, a plan and a project. The goal is to convince the shop's proprietor of the plan (orally) and to inform the customers about the appeal of the designed works that may be displayed throughout their homes. The project should include 3 representations of transformational geometry with a written explanation of these types of transformational geometry illustrated in the design and why they were chosen. The project should be one that can be used in everyday life in customers' homes. Customers may ask questions about the design of the project. Also, a self-assessment will be performed at the time of the presentation.

96

Attachment V

GLE: GSR.3.A.9

Scoring Guide

3 points 2 points 1 point

Transformations

Includes 3 different

types of

transformations

Includes 2 different

types of

transformations

Includes 1 type of

transformation

Student knowledge

Clear understanding ,

able to identify the

transformations orally

Some understanding of

transformations

Very little

understanding of

transformations

Written

Explanation

Clear written

explanation of design

and the transformations

used

Written explanation of

design and the

transformations used

Explanation is unclear

or not provided

97

Attachment W

GLE: GSR.3.B.9

Student Directions: On a recent test students were asked how many hours they studied. The following list shows the results with X representing the number hours spent studying and Y representing the number of errors (wrong answers). (1,5) (2,4) (2,3) (3,1) (1,7) Graph this data. Be sure to label all axes as well as correctly scale the axes. Identify the function rule. Create graphs that show the translation and reflection of the function. Explain what the data indicates about the relationship between hours studied and performance on tests.

Rubric(s)

Rubric: What's the trend scoring

Trait: What's the trend scoring Performance Type:

4 3 2 1 0

Response shows a complete and thorough understanding of the mathematical concepts. All graphs correct and complete.

There may be minor errors in scaling that do not effect conclusions. Function is identified correctly Explanation of data is clear and concise

Response shows a complete understanding of concepts. Graphing is correct, some minor errors. Function is identified Explanation of data is generally correct

Response shows some understanding of concepts. Two of the three graphs are correct. Function has minor flaw Explanation of data has flaws.

Some effort was made that demonstrates some minimum of mathematically correct representation On of the three graphs is correct. Function has major flaws. Explanation of data has major flaws

Response shows no understanding of mathematical thought or no response was given.

98

Attachment X

GLE: DP.1.A.9

Learning Activities:

1. Students will make a collage of examples of statistics from news print, programs and/or

internet

a. Students will share their examples and discuss these examples to illustrate possible bias.

2. Introduce lesson on mean, median, mode, spread, and ways to represent data

3. In class project on data collection and graphing

4. Class activity on correct sampling techniques

a. Students devise criteria for reliable sample questions.

b. Students develop a set of survey questions based on criteria developed in class.

5. Peer group assignment on completing frequency table, calculating statistics, interpreting

results using calculators and software programs.

6. Performance Task: Working in groups, students create two surveys, one showing

quantitative data and one showing qualitative data. They will then collect and analyze the

data results. Students will make an oral presentation to the class using Powerpoint or charts

discussing their results

7. Students will do a self-reflection on their project

8. Students will evaluate the other group projects.

9. Throughout the units, students will be assessed using quizzes, tests, journal entries, and

other tools as needed.

99

Attachment Y

GLE: DP.1.C.9

Scoring Guide

4 Excellent

3 Good

2 Competent

1 Needs Improvement

Question formulated was relevant and appropriate Data was well organized Graph chosen and drawn presented the data in an easy to understand manner Graph(s) included clear statistical representations (measures of central tendency)

Question formulated was appropriate Data was organized in a logical manner Graph chosen and drawn presented the data in a fair manner Graph(s) included statistical representations

Question formulated resulted in poor generation of data Data was somewhat organized Graph chosen and drawn was hard to understand Graph(s) do not clearly represent statistical data

Question formulated was not able to generate enough data to create a study Data was not organized Graph chosen and drawn did not correspond to data Graph(s) did not include statistical representations or the statistics where not correct

100

Attachment Z

GLE: DP.2.A.9

Sample Problems on

Mean, Median and Mode

Situation A

There are three different basketball teams and each has played five games. You have each team's score

from each of its games.

Game 1 Game 2 Game 3 Game 4 Game 5

Jaguars 67 87 54 99 78

Wolves 85 90 44 80 46

Lions 32 101 65 88 55

1. Suppose you want to join one of the three basketball teams. You want to join the one that is doing the

best so far. If you rank each team by their mean scores, which team would you join?

2. Instead of using mean scores, you use the median score of each team to make your decision. Which

team do you join?

3. Pretend you are the coach of the Lions and you were being interviewed about your team for the local

newspaper. Would it be better for you to report your mean score or your median score?

Situation B

You and your friends are comparing the number of times you have been to the movies in the past year.

The following table illustrates how many times each person went to the movie theatre in each month.

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

John 1 3 2 5 2 3 1 4 2 3 2 1

Mary 1 2 1 1 1 3 3 2 2 4 1 2

Brian 1 3 2 2 1 4 5 3 2 2 1 3

Kelly 2 2 1 1 3 2 4 1 3 2 3 2

1. By comparing modes, which person went to the movies the least per month?

2. By comparing medians, which person went to the movies the most per month?

3. Rank the friends in order of most movies seen to least movies seen by comparing their means.

4. Which month, by comparing the means of movies seen in each month, is the most popular movie-

watching month?

5. By comparing medians, which month is the least popular month?

6. What is the mean of the medians for each month (the arithmetic average of the medians of the number

of movies seen in each month)?

Attachment W

101

Attachment Z

GLE: DP.2.A.9

Sample Problems on

Mean, Median and Mode

Answers

Situation A:

Answer 1: Jaguars (The mean score is 77)

Answer 2: Wolves (The median score is 80)

Answer 3: The mean score (The mean score is 68.2 and the median score is 65)

Situation B:

Answer 1: Mary (Her mode is 1)

Answer 2: They all went the same amount (The medians are all 2)

Answer 3: 1. John and Brian (Their mean is 2.4167), 2. Kelly (Her mean is 2.167), 3. Mary (Her mean is

1.9167)

Answer 4: July (The mean for July is 3.25)

Answer 5: January (The median for January is 1)

Answer 6: 2.0833

102

Attachment AA

GLE: DP.2.B.9

Graph Sketcher

Have paper, pencil and a calculator handy. For each exploration, build a chart like:

Choose the list of x values (-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 might be a good list) and then calculate the

corresponding y values. Use this chart to fill in the points to be plotted in simpleplot. To do the

explorations below, be sure to calculate many points (10 or more).

Exploration 1: Up and Down Choose any function, for example, x**2. Now graph, on the same picture,

x**2+1, x**2 +2, x**2+3, x**2-5, and so on (take the same function and add or subtract different

numbers). What happens to the graphs? Why? Try it with several different functions. What is different?

What is the same?

Exploration 2: Left and Right Choose any function, for example, x**2. Using different numbers,

substitute (x+number) or (x-number) for x in the function formula. Graph several such functions on the

same picture, for example, x**2, (x-1)**2, (x+4)**2. What happens? Why? Try it with several different

functions. What is different? What is the same?

Exploration 3: Stretch-1 Choose any function, for example, x**2. On the same picture, graph the

original function and the function multiplied by different numbers, for example, 3*(x**2), -5*(x**2),

.7*(x**2), and so on. What do you see? Why does it happen? Try it with several different functions. What

is different? What is the same?

Exploration 4: Stretch-2 Choose any function, for example, x**2. Using different numbers, substitute

(x*number) for x in the function formula. Graph several such functions on the same picture, for example,

(x*3)**2, (x*5)**2, (x*.1)**2 What do you see? Why does it happen? Try it with several different

functions. What is different? What is the same?

Exploration 5: Combine! Choose a function, for example, x**2, and do all or some of the things from

the above to it, for example, (2*x+3)**2-5. First graph only the original function and try to predict how

the modified function will look like. Then graph both the original and the modified function on the same

picture, and see if you predicted the shape correctly. Try to do modifications in a different order (e.g.,

2*x+3)**2-5 and (2*(x+3))**2-5). Does it make a difference? Why? Now do it for several different

functions.

103

Attachment BB

GLE: DP.2.C.9

Line of Best Fit

When data is displayed with a scatter plot, it is often useful to attempt to represent that data

with the equation of a straight line for purposes

of predicting values that may not be displayed on the plot.

Such a straight line is called the "line of best fit."

It may also be called a "trend" line.

A line of best fit is a straight line that best represents the

data on a scatter plot.

This line may pass through some of the points, none of the

points, or all of the points.

Materials for examining line of best fit: graph paper and a strand of spaghetti

Is there a relationship between the fat grams

and the total calories in fast food?

Sandwich Total Fat

(g)

Total

Calories

Hamburger 9 260

Cheeseburger 13 320

Quarter Pounder 21 420

Quarter Pounder with Cheese 30 530

Big Mac 31 560

Arch Sandwich Special 31 550

Arch Special with Bacon 34 590

Crispy Chicken 25 500

Fish Fillet 28 560

Grilled Chicken 20 440

Grilled Chicken Light 5 300

104

Attachment BB

GLE: DP.2.C.9

Can we predict the number of total calories

based upon the total fat grams?

Let's find out!

1. Prepare a scatter plot of the data.

2. Using a strand of spaghetti, position the

spaghetti so that the plotted points are as close to

the strand as possible.

Our assistant, Bibs, helps position

the strand of spaghetti.

3. Find two points that you think will be on the

"best-fit" line.

4. We are choosing the points (9, 260) and (30,

530). You may choose different points.

5. Calculate the slope of the line through your

two points.

rounded to three decimal places.

6. Write the equation of the line.

7. This equation can now be used to predict

information that was not plotted in the scatter plot.

Question: Predict the total calories based upon

22 grams of fat.

ANS: 427.141 calories

Choose two points that you think will form the

line of best fit.

Predicting: - If you are looking for values that fall

within the plotted values, you are

interpolating. - If you are looking for values that fall

outside the plotted values, you are

extrapolating. Be careful when

extrapolating. The further away from the

plotted values you go, the less reliable is

your prediction.

105

Attachment BB

GLE: DP.2.C.9

So who has the REAL "line-of-best-fit"?

In step 4 above, we chose two points to form our line-of-best-fit. It is possible, however, that

someone else will choose a different set of points, and their equation will be slightly

different. Your answer will be considered CORRECT, as long as your calculations are correct

for the two points that you chose.

We can answer the question "Who has the REAL line-of-best-fit?", with the assistance of a

graphing calculator.

106

Attachment CC

GLE: DP.3.A.9

Student Directions: *** Budget Cuts ***

Due to budget cuts, your local Board of Education has announced that they are being forced to cut most

extracurricular activities, including clubs and sports. They say they are making this decision because they believe

that, according to their findings, the average student only participates in 0.3 activities, and only 23% of the

student body participates in extracurricular activities.

You are upset about this decision, and disagree with the Board's findings. However, the Board will only listen to

sound data that conflicts with its findings.

Your task is to take a survey in your school to find out:

1) Is the average number of extracurricular activities that the average student participates in greater that the 0.3

that the Board of Education claims?

2) What is the average number of extracurricular activities the average student participates in (construct a

confidence interval)?

3) Is the proportion of the student body that participates in extracurricular activities greater that the 23% that the

Board of Education claims?

4) What is the proportion of students who participate in extracurricular activities (construct a confidence

interval)?

Write a report to the Board of Education explaining your findings and giving your recommendation for action

based on them. Be sure to explain how you got your sample, and why it is representative of the student body at

your school.

107

Attachment CC

GLE: DP.3.A.9

Rubric(s)

Rubric: Rubric - Budget Cuts

Trait: Rubric - Budget Cuts

Performance Type:

4 Points 3 Points 2 Points 1 Point 0 Points

The response

indicates

application of a

reasonable

strategy that

leads to a

correct solution

in the context

of the problem.

The

representations

are correct. The

explanation

and/or

justification are

logically sound,

clearly

presented, fully

developed,

support the

solution, and

do not contain

significant

mathematical

errors. The

response

demonstrates a

complete

understanding

and analysis of

the problem.

The response

indicates

application of a

reasonable

strategy that

may or may not

lead to a

correct

solution. The

representations

are essentially

correct. The

explanation

and/or

justification are

generally well

developed,

feasible, and

support the

solution. The

response

demonstrates a

clear

understanding

and analysis of

the problem.

The response

indicates an

incomplete

application of

a reasonable

strategy that

may or may

not lead to a

correct

solution. The

representations

are

fundamentally

correct. The

explanation

and/or

justification

support the

solution but

may not be

well

developed, is

plausible,

and/or may be

incomplete.

The response

demonstrates a

conceptual

understanding

and analysis of

the problem.

The response

indicates little or

no application of a

reasonable

strategy. It may or

may not have the

correct answer.

The

representations

are partially

correct. The

explanation and/or

justification reveal

serious flaws in

reasoning. The

explanation and/or

justification may

be incomplete or

missing. The

response

demonstrates a

minimal

understanding and

analysis of the

problem.

The response is

completely incorrect or

irrelevant. There may be

no response or the

response may state "I

don't know".