Ninth Grade Academy 2013
Teacher’s Guide
Mathematics
Denver Public Schools
1
2
Table of Contents
Part 1: Teaching Notes
Investigating Line of Fit . . . . . . . . . . . . . . . . . . . .
5
Investigating Order of Operations . . . . . . . . . . . . . . . .
9
Investigating Probability . . . . . . . . . . . . . . . . . . . . 11
Investigating Integers. . . . . . . . . . . . . . . . . . . . . . 15
Investigating Statistics. . . . . . . . . . . . . . . . . . . . . . 19
Investigating Laws of Exponents . . . . . . . . . . . . . . . . . 21
Investigating Proportions
. . . . . . . . . . . . . . . . . . . 27
Part 2: Student Pages
Investigating Line of Fit . . . . . . . . . . . . . . . . . . . .5-12
Investigating Order of Operations . . . . . . . . . . . . . . . 13-18
Investigating Probability . . . . . . . . . . . . . . . . . . . 19-26
Investigating Integers. . . . . . . . . . . . . . . . . . . . . 27-32
Investigating Statistics. . . . . . . . . . . . . . . . . . . . . 33-38
Investigating Laws of Exponents . . . . . . . . . . . . . . . . 39-48
Investigating Proportions
. . . . . . . . . . . . . . . . . . 49-52
Note: Questions labeled “Now Try This” are additional assignments which may be used if you
have extra time or for students who need an extra challenge.
3
4
Investigating Line of Fit
Student Pages 5-12
Objectives:
 Perform an experiment to collect data and create a scatter plot.
 Write an equation to model a set of data and make predictions based on the graph or
equation.
 Use technology to model data.
Materials: Graphing calculator (with AACLEAN installed), stopwatch/timer/clock with second
hand, rulers.
Lesson Outline: (Two days)
Day 1: Collect data. Create a scatter plot on paper, draw the line of fit and make predictions.
Share.
Day 2: Use a graphing calculator to create a scatter plot and line of fit. Use the line to make
predictions. Share.
Now Try This: Additional practice entering data, finding line of fit on calculator, and making
predictions.
Teaching Notes:
Day 1:
 After students collect and graph data, discuss line of fit (a line which models data, follows
the general trend of the data). Then have them draw their line of fit and continue with
predictions.
 With the class, compare student graphs, paying special attention to the scaling of the
graphs and to the line of fit.
Day 2:
 Be sure that you have installed the program AACLEAN onto your calculators. This will make
using the programs and resetting the calculators much easier.
 Do not use ZoomFit to set the window. Setting up the window helps student connect the
pencil/paper graph with the graph on the calculator. It also helps them focus on appropriate
domain and range for their function.
 Do not have students use a regression on the calculator to find the line of fit. Using a
“guess and check” method will support the development of students’ understanding of the
relationship between the equation and graph.
 Discuss the effect of changing a and b in the equation 𝑦 = 𝑎 + 𝑏𝑥.
5
Calculator Notes: (Screen shots are samples.)
 To run AACLEAN, select PRGM, arrow down to AACLEAN and press ENTER twice. This will
clear screens and lists and reset calculator to normal settings.
 Enter the data from your table into your calculator:
Press STAT, then 1 to select Edit. You are now in the lists. Clean out any data in the lists
before entering new data. Press the up arrow until L1 is highlighted, press CLEAR, and then
press ENTER. Repeat this for any other list that needs to be cleared. Enter the chain length
into L1 and the time into L2. Press ENTER after you type in each data value. (Sample data
shown)
 Set Up the STAT PLOT:
Press 2ND, then Y= to get to the STAT PLOT menu. Press 1 to get to Plot1. Use the arrow
buttons to move the cursor so that On is highlighted, then press ENTER. Use the arrow
buttons to highlight the scatter plot, then press ENTER. Use the arrow buttons so that the
cursor is next to Xlist. Xlist should be L1. To type in L1 press 2ND, then 1. Ylist should be L2.
Press 2ND, then 2.

Set up the WINDOW: Setting up the window helps student connect the pencil/paper
graph with the graph on the calculator. It also helps them focus on appropriate domain and
range for their function.
Press WINDOW. The following may help you in entering values.
Xmin = a number less than the minimum value for x.
Xmax = a number greater than the maximum value for x.
Xscl = the distance between tick marks on the x-axis.
Ymin = a number less than the minimum value for y.
Ymax = a number greater than the maximum for y.
Yscl = the distance between tick marks on the y-axis.
Xres = 1
View the graph:
Press GRAPH.
6
Answer Guide:
Human Chain: Wrist Experiment
As a class, form a circle with each person gently holding the wrist of the person to his or her right. One person is the beginning of
the chain and another person is the end of the chain. The experiment begins when the beginning person says “go” and carefully
but quickly squeezes the next person’s wrist, this next person squeezes a wrist, and so on. After the last person’s wrist is squeezed,
he or she should say “stop.” The amount of time from when the word “go” is spoken until the word “stop” is spoken is recorded.
Run the experiment ten times, using a different number of people in the chain each time. Record the data for the experiment in
the table below.
Sample data
Chain Length (# of
Time (seconds)
people)
20
7.52
10
3.40
15
4.41
17
4.97
5
1.55
40
10.77
30
8.86
60
16.05
22
7.72
44
12.52
Answers will vary for problems depending on class data.
5.
Changing b in the equation changes the slope of the line. A steeper line has a greater value for b.
Changing a changes the y-intercept which moves the line up and down.
9.
Explain why the data points do not form a straight line.
The points do not form a straight line because not every person has the same response time and the same people were not
used in all of the chains.
10. Compare your equation to the equations of other students in the class. Did every student get the same equation? Why or
Why not? Each student has a slightly different equation because there is more than one possible line of fit.
Now Try This
11. What is the y-intercept of your graph?
What does this point mean in terms of the wrist experiment problem? Answers will
vary. The y-intercept represents the number of seconds to say “Start” and “Stop”.
12. What is the slope of your line?
What does this point mean in terms of the wrist experiment problem? Answers will vary.
The slope of the line represents the average time it takes a person to squeeze the wrist of the next person.
13. Suppose you did this experiment again but each person squeezed the wrist of the person next to him/her and then gave
them a high five. What change do you think this would have on the data? How would that change the graph? How would
it affect the equation? Giving a high five also would increase the amount of time it takes for each person. This would
change the slope of the line and make the line steeper and would increase the “b” value of the equation. It would not
change the amount of time needed to say “Start” and “Stop”, so would not change the “a” or the y-intercept.
Now Try This
1. What is the average increase in the number of students that are applying early for the time period shown in the
USA TODAY Infograph? about 6197 students
3. Record your equation here: answers will vary. y = 41,664 + 6000x
Year
# of students
2004-2005
2005-2006
2006-2007
83,664
89,664
95,664
7
8
Investigating Order of Operations
Student Pages 13-18
Objectives:
 Practice using order of operations to evaluate expressions.
 Use a graphing calculator to evaluate expressions.
Materials: Graphing calculator
Lesson Outline: (One day)
Discuss order of operations. Students work on the activities 1-5 and discussion. Activity 6 is a
class competition.
Now Try This: The Graduation Year Puzzle can be used throughout the academy for students
with extra time.
Teaching Notes:
 Be sure that students identify common misconceptions about order of operations.
o Multiplication and division are done Left to Right—multiplication is not always done
first.
o Addition and subtraction are done Left to Right—addition is not always done first.

Be sure students identify common errors when using the calculator to evaluate integer
expressions.
o Implied grouping in fractions—parentheses may be required
Calculator Notes: Be aware that graphing calculators follow order of operations. Grouping
symbols should be used when appropriate.
Answer Guide:
Activities
x 2 key that allows you to find the square of a number.
5
the caret key, ^. For example, to evaluate 3 , press 3 ^ 5, and then press ENTER.
Many calculators have an
1.
To use exponents other than 2, you can use
Simplify each expression, showing each step in order:
 (4  2) 3 

4  8  32
(8) 3  512
4  23 =

2. Simplify
(2  5  4) 3
42
using paper and pencil. Then check your answer with a calculator.
665.5

9
2  5  4 3  4 2 equivalent to (2  5  4 3 )  4 2?
3. Is
Explain your answer.
Yes. For each expression, you would evaluate the exponents first, multiplication second, and addition last.

4.
Simplify each expression with pencil and paper. Check your answers with a calculator.

a. 3 2
29

f.
9
3
5
b. 3 (2
39
4+5
2−1
3
 5)
c. (3 2)
36
g. 4 + 5 ÷ 2 −

5.5
1

h.
1
2
8+6 ÷2
11
d. 3 2
12
2
e. 2
64

(32)

What did you learn about entering the fraction into your calculator? You have to use parentheses around the numerator and
denominator when they contain operations.
5.
Use a calculator to simplify each expression. Round your answers to the nearest hundredth.
a.
(2.1  5.6  4 3 )
64
0.28
6.

b.
[(2.1  5.6)  4 3 ]/64
c.
0.38
[(8.6  1.5) /2 3 ]
52
0.04
Answers will vary based on birth
year.
Now Try This

Some sample answers:
1: 4 − 1 − 2 + 0
2: 4 − 2 ∙ 1 + 0
3: 4 − 2 + 1 + 0
4: 41 + 2 ∙ 0
5: 4 + 1 + 2 ∙ 0
6: 20-14
10
Investigating Probability
Student Pages 19-26
Objectives:
 Simulate experimental probabilities on a calculator.
 Compare experimental and theoretical probabilities.
 Briefly introduce the difference between a combination and a permutation.
Materials: Graphing calculator, two dice per pair of students
Lesson Outline: (Two day lesson)
Day 1:
Discuss theoretical and experimental probabilities; make sure students know how to use the
calculators to simulate an experiment. Students work on Activities 1 and 2 in pairs. Discuss the
results of the first two activities and how students’ predictions compared with the results of the
experiments.
Day 2:
Students work on activities 3 and 4 in pairs. Allow students to share the experiments they
designed in Activity 3. Discuss how their predictions compared with the experimental and
theoretical probabilities.
Teaching Notes:
 When finding experimental probabilities, students may combine the data obtained from
rolling dice with data obtained from calculator simulation or may just look at simulation
data.
 Experiments in Part 4 are more complex. Be sure students understand each
experiment.
Calculator Notes:
All activities are done with the Probability Simulator. Press the APPS button, use the arrow
down key to :Prob Sim, hit ENTER, press any key to get into the Simulator.
Activity 1: From the Probability Simulator main menu select number 2 Roll Dice, press the
WINDOW key to roll the die. The +1 key will roll one additional die, +10 will roll 10 more dice,
+50 will roll 50 more dice. The calculator will keep track of how many times each number is
rolled. Pressing the side arrow key keys will show you the number of times each number was
rolled. Press the Y= key to escape.
Activity 2: From the Roll Dice main menu, press the ZOOM key to go to the settings. Change
the number of dice to two. Press the Graph key to OK the changes.
Activity 4: From the Probability Simulator main menu select number 6 Random Numbers. Press
the ZOOM key to go into the settings. Set the Numbers to 3, Range to 0 – 9, Repeat: Yes. Press
the GRAPH key to OK the changes. Press the WINDOW key to draw.
11
Answer Guide:
Probability Simulation Day 1
Activities
Part 1: 1a. Sample Answers
Die
Frequency
1
2
2
3
3
3
4
4
5
5
6
3
Total number of trials
20
3
3
4
5
5
10
6
4
Total number of trials
34
1b. Sample Answers
Die
Frequency
1
2
2
10
1c. Complete the table to find the experimental probability of each number.
Die
1
2
3
4
5
6
Experimental 2/34 10/34 3/34 5/34 10/34 4/34
Probability
.059 .294
.088 .147 .294
.118
Compare the experimental probabilities with the theoretical probabilities.
1d. The theoretical probability of getting any given number is 1/6 or .167.
1e. Answers may vary
Discussion questions:
Is it possible for the experimental probability to match the theoretical probability?
What do you think it would take for the experimental probability to match the theoretical probability?
Part 2: Now run a simulation for rolling a pair of dice at least 30 times. Escape back to the main Roll Dice menu. Go into the
settings and change the number of dice to two. OK the changes and roll the dice. Use the table to record how often each sum of
the two dice appeared.
2a. Predictions will vary
2b. Sample data
Sum
Frequency
2
0
3
1
4
1
5
3
6
2
7
3
8
4
9
3
10
2
11
0
12
1
Total number of trials
20
2c. Sample data
Sum
Frequency
2
1
3
2
4
1
5
3
6
8
7
6
8
10
9
6
10
3
11
0
12
1
Total number of trials
41
10
3/41
.073
11
0/41
0
12
1/41
.024
2d. Complete the table to find the experimental probability of each sum.
Sample data based on 2c
Sum
Experimental
Probability
2
1/41
.024
3
2/41
.049
4
1/41
.024
5
3/41
.073
6
8/41
.195
7
6/41
.146
8
10/41
.243
9
6/41
.146
12
2nd Die
1st Die
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Sum
2
3
4
5
6
7
8
9
10
11
12
Theoretical Probability
1/36
.028
2/36
.056
3/36
.083
4/36
.111
5/36
.139
6/36
.167
5/36
.139
4/36
.111
3/36
.083
2/36
.056
1/36
.028
Part 3
Answers will vary.
Part 4
Answers will vary
Students will score more points on the game where order doesn’t matter.
Now Try This
Actual order of frequency in the English language:
E T A O [N I ] S R H L D C U [P F] M W Y [B G]V K [Q X ][J Z]
(Those bracketed have the same frequency of occurrences.)
13
Investigating Integers
Student Pages 27-32
Objectives:
 Learn how to perform operations on signed numbers.
 Learn about absolute value.
 Practice order of operations.
Materials: TI-84 graphing calculator
Lesson Outline: (Two day lesson)
Day 1:
Activities 1-3: Overview, Observations, and Activity: What Is My Sign?
Day 2:
Activities 3: Integer Smash, numbers 4 and 5.
Teaching Notes:
 If students have trouble with integer operations, you may want to work with a chip
model. Students used red chips for negative numbers and black for positive in middle
school.
 What Is My Sign will keep going until the student quits. It might be a good idea to give
students a time limit for playing this activity (possibly 5 minutes).
 Be sure that students understand the difference between the subtract key and the
negative key on the calculator.
 Emphasize why (-3)2 is not the same as -32.
 Discuss the meaning of absolute value before starting that section.
Computer Notes:
Overview: To access the Overview press APPS then select ALG1PRT1. Press any key to
continue. Press 1 to select Number Sense, 1 to select Integers, 1 to select Overview. Use the
right arrow to scroll through the lesson and the left arrow to go back. Students will be
prompted to use their worksheet when they reach Observation 1 and Observation 2.
Activities:
What Is My Sign?
Press ENTER to select this activity. An expression will appear on the screen. If the expression is
positive, press the up arrow. If the expression is negative, press the down arrow. If the
expression equals zero, do nothing. Once you have selected your answer, press the left arrow
to see if your answer is correct.
When you are finished playing, press Y= to quit.
Integer Smash
Press the down arrow to highlight the activity, press Enter to select Integer Smash. Use the up
or down arrow to highlight a level of difficulty then press ENTER. Press WINDOW, ZOOM,
14
TRACE, or GRAPH to select an operation. An expression will appear on the screen. Type in the
missing number, then press ENTER.
When you are finished playing, press Y= to quit.
To exit out of the application press 2ND, QUIT, Y=.
To find absolute value, press MATH, press the right arrow, 1 to select abs(.
Answer Guide
Number Sense: Integers
Activities
Observation 1
How far apart are the bird and the fish? Should your answer
be positive or negative? Use complete sentences to explain.
The bird and the fish are 125 ft apart. The distance is positive.
Observation 2
What is the sign of the answer to the problem -3 * -11?
33
In your own words, state the rules for multiplying and dividing
signed numbers.
The product of a positive and negative number is negative.
The product of two numbers with the same sign is positive.
Rules for division are the same as the rules for multiplication.
What is My Sign?
The game will keep going until the student quits. It might be a good idea to give students a time limit for playing this activity
(possibly 5 minutes)
Integer Smash
Make sure you are in the Activities. To exit out of the application press 2ND, QUIT, Y=.
15
4. Calculate
5  4
-1

Calculate
5  3 2
-11

Calculate
5  (3 2)
=-11
Calculate
(5  3)  2
=4
 Notice that the graphing calculator uses Order of Operations.
For each of the last three expressions, describe the order in
which the operations are calculated (which operation was performed first, second, etc.).
 5  3 2 Multiplication first, followed by subtraction.
5  (3 2) Again, multiplication first, then subtraction.

(5  3)  2 This time subtraction is first because it is inside the parenthesis.

5.

a.
-5

e.
-6
3  2
1 6
b.
-2

f.
8
8  6
(4)(2)

c. 4  6
-10
d.
5
63
9
h.
g.
7

i. (−3)2
9

j. −32
-9

k. (−1)2

1
2  (3)
45
5
-9
l. (−1)3

-1
12
16
6.
Calculate the following problems by hand, then check your answers using the graphing calculator. Make sure you show all
of your steps.
a. -4 + -12 * -10
b. 4* -8 – -10 * 2
116
-12
c.
30
6
5
0
d. 12  28
e. | -4 * -8 – -10 * 2|
40
52
f.

-| -3 – 14 – -10|
-7

Now Try This
Answers will vary
17
Investigating Statistics
Student Pages 33-38
Objectives:
 Calculate mean and median for a data set.
 Use a graphing calculator to calculate mean and median.
 Conduct a survey.
Materials: Graphing calculator
Lesson Outline: (One day lesson)
Discuss mean and median with students.
Students work on activities 1-5 with discussions.
Students conduct a survey (may be done in groups).
Teacher Notes:
 Review mean, median and mode
 Discuss the meaning of the answers in relation to the problem context
 Stress x is the symbol statisticians use for mean
Calculator Notes:
First,enter the data into the lists. Press Stat then 1 to select edit. You are now in the lists. If
there is data in the lists, it needs to be cleared out before entering new data. Press the up
arrow until L1 is highlighted, press CLEAR, and then press ENTER.
If there is no L1 (or if a student accidentally deletes L1), press STAT, then 5 to select
SetUpEditor. Press ENTER and all of the lists should be restored back to their default names.
Type in the weekly allowances into L1. Press ENTER after you type in each data value. Once
you have finished typing in the data, press 2ND then MODE to return to the home screen.
Now, find the median and mean. Press STAT then the right arrow so that CALC is highlighted.
Press ENTER to select 1-Var Stats.
Sample:
Press 2ND then 1 to type in the name of the list where the data is stored. Press ENTER and the
calculator will display a list of statistics for the data.
18
The first value, x , is the mean. Use the down arrow until you see Med.
Answer Guide
Statistics: Mean & Median

Activities
20
10
5
5
10
15
5
10
5
10
5
5
5
5
5
1. 120/15=$8; I found the sum of the data values and then divided that by the number of data values.
2. 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 15, 20; $5; I put the data in order and found the middle number of the data set.
3. 8 and 5. Yes, they match what was calculated by hand.
4.
a.
6.4, 5.6, 7.3, 1.2, 5.7, 8.9; mean =5.85; median = 6.05
b.
-23, -13, -16, -21, -15, -34, -22; mean = -20.57; median = -21
c.
123, 423, 190, 289, 99, 178, 156, 217, 217; mean = 210.22; median = 190
d.
8.4, 2.2, 7.3, 5.3, 6.7, 4.3, 5.1, 1.3, 1.1, 3.2, 2.2, 2.9, 1.4, 68; mean = 8.53; median = 3.75
a.
b.
Answers may vary. Median because it’s closer to more of the data values.
mean = 4.16 median = 3.75
The median is the same but the mean is changed significantly.
The median is less likely to be affected by extremely high or low data values.
5.
c.
6. Answers will vary
Now Try This
1. Answers may vary
2. B
3. D, March to June
19
Investigating Laws of Exponents
Student Pages 39--48
Objective:
 Use graphing calculators to investigate the properties of exponents.
Materials: Graphing calculator
Lesson Outline: (One day lesson)
Have students work in pairs on the activity. Discuss and/or have students present their results
and the rules.
Teaching Notes:
 Be sure to allow time and discussion so students can make sense of exponent rules.
 If doing scientific notation section, introduce calculator scientific notation (e.g., 4.5x1012
would be written as 4.5E12 on the calculator.)
Calculator Notes: See student activity sheet.
Answer Guide
Investigating Laws of Exponents
In this activity, you will use the graphing calculator to investigate simplifying expressions with exponents.
Part 1
1.
2.
First make a prediction about how each of these could be simplified.
a.
4 4   4
d.
2 2   2
3
2
8
5
4
12
b.
3 3   3
e.
12.2 12.2   12.2
5
3
4
8
c.
5
5 5   5
2
4
6
9
Use a calculator to check your predictions. Correct any predictions that were wrong.
For example, is
4 4   4
3
2
6
or
4 4   4
3
2
5
? Use the home screen to test both possibilities. To get
43 on the
calculator, type in 4 on the home screen, then press the MATH button, and then choose option 3: 3
We can see from testing in the home screen that
4 4   4
3
2
5
since they are both equal to
1024.
20
3a.
4 4 
3
2
b.
 4  4  4 4  4 
d.
3 3 
5
3
 44444
 (3  3  3  3  3)(3  3  3)
 33333333
 45
 38
2 2 
8
4
e.
4a.
2
x x   x
5
5
2
4
12.2 12.2 
4
5
12.29
x x   x
3
5 5 
56
212
d.
c.
5
10
b.
e.
x x   x
5
2
7
c.
x x   x
6
3
9
x x   x
3
8
11
21
7.
am  an  amn
Sample answer: When multiplying expression that have the same base, the base stays the same and you add the
exponents.
8. 23 ∙ 24 = 128
47= 16384
Part 2
1. Make a prediction about how each of these could be simplified.
a.
4 
 46
b.
3 
d.
2 
 212
e.
12.2 
3 2
6 2
3 3
 39
2 4
c.
5 
2 4
 58
 12.28
2.
Use a calculator the way you did in Part 1 to check your predictions. Correct any predictions that were wrong.
3.
Expand each of the problems in number 1 to verify that what you’ve discovered from the calculator is correct.
a.
4 
3 2
b.
3 
e.
12.2 
3 3
c.
5 
2 4
 (43 )( 43 )
 444444
 46
d.
4.
5.
6.
2 
6 2
2 4
Based on what you’ve done so far, make a prediction about how the following expressions would be simplified.
a.
x 
 x6
b.
x 
d.
x 
 x 25
e.
x   x
3 2
5 5
5 3
3 8
 x15
c.
x 
6 3
 x18
24
Use the Y= menu and the table the same way you did in Part 1, #4 to check your predictions from above. Correct any
predictions that were wrong.
(a m )n  a mn
Sample answer: When you raise a power to another power, you multiply the exponents.
22
Part 3
1.
3.
5.
Make a prediction about how each of these could be simplified.
a.
5 /5   5
d.
2 / 2   2
9
8
3
6
4
b.
4
3 /3   3
5
3
e.
2
c.
12.2 /12.2   12.2
7
5
5 /5   5
6
2
4
2
Make a prediction about how each of these could be simplified.
a.
x /x   x
d.
x /x   x
5
8
3
5
2
b.
x /x   x
4
3
e.
x /x   x
3
7
5
3
2
c.
x /x   x
10
4
6
(a m ) /( a n )  a mn
Describe how to find the value of this expression.
Sample answer: When dividing expressions with the same base, you subtract the exponents.
Now Try This
1.
2.
Make a prediction about how each of these could be simplified.
a.
(3  23 ) 2  32  26
b.
(3  2)5  35  25
c.
32 (23 ) 2  32  26
d.
3(2  3) 2  33  22
e.
(35  2)(23  2)  35  25
f.
(3  2)6  36  26
( a  b) m  a m  b m
Sample answer: When there is an exponent outside parentheses, everything inside the parentheses is raised to that
power.
23
Now Try This
Problem 1 – Writing Scientific Notation in Expanded Form
Scientific notation is a way of writing very large and very small numbers. Numbers in scientific notation include two
parts, a number greater than or equal to 1 and less than 10, and a power of 10. Examples include:
5.6106 = 5,600,000
and
2.310-8 = 0.000000023
Write each of the following numbers in expanded form. Check your answer with a calculator.
1.
–3.77108 = –377,000,000
2. 1.202105=120,200
3.
4.22410-6 =0.000004224
4.
–5.2410-12= –0.00000000000524
Problem 2 – Writing Numbers in Scientific Notation
To write numbers in scientific notation, place a decimal point so there is one non-zero digit to the left of the
decimal. Count the number of decimal places the decimal point moved—this is the exponent for the 10. The
exponent will be positive or negative depending on the direction of the move.
156,000,000 = 1.56108
5.
and
0.0000045 = 4.510-6
The following are salaries for the five top paid players of the Cincinnati Bengals. Write each salary in scientific
notation.
6.
Player
Salary
Scientific Notation
Carson Palmer
$13,980,000
1.398107
Stacy Andrews
$7,455,000
7.455106
Chad Johnson
$6,415,370
6.41537106
Antwan Odom
$5,700,000
5.7106
Levi Jones
$5,266,666
5.266666106
The 2007 median American household income was $5.0233104. How does this compare to the salaries above?
$50,233 Much lower than players salaries
24
7.
Imagine you could fold a piece of paper 0.004 inches thick 50 times. About
how many inches thick would the resulting paper be after the 50th fold?
To solve, type 0.004 on the home screen and press enter. Then, type * 2
enter. Each time enter is pressed, the previous value is doubled.
4.503599627x1012
8.
What is the answer in expanded form? 4,503,599,627,000
9.
What is the percent chance of flipping a coin 40 times and having it come up
heads each time?
9.09494702E-13 = 0.000000000000909494702
25
Investigating Proportions
Student Pages 49-52
Objectives:
 Review fractions.
 Use proportional reasoning and problem solving skills to increase a family-size recipe to a
reunion -sized recipe.
Materials: Graphing calculators
Lesson Outline: (One day lesson)
Present the problem from the Investigating Proportions activity page. Have students work in pairs or
groups on the problem. Have two or three groups present their results to the class. After the
presentations, have students discuss the various approaches used. Discuss the similarities and
differences between the groups’ solutions. Have students verify that the calculations were done
correctly and evaluate how the calculator was used to help solve the problem.
Teaching Notes:

Encourage students to use fractions as they determine the ingredient amounts and use this as
an opportunity to review fraction operations.
Calculator Notes:
To convert a decimal to a fraction in the calculator you need to already have the decimal on the home
screen. Press the MATH button, then hit ENTER on 1:►Frac, and then hit ENTER again.
Answer Guide
The Problem
A Family Reunion is coming up. You are in charge of making your favorite Parmesan Macaroni Casserole for the reunion. The
recipe you have makes enough for 6 people, but you need to make enough to feed 20 people.
How many people could you feed if you multiply each ingredient by 3? 18
How many people could you feed if you multiply each ingredient by 4? 24
Should you multiply each ingredient by 3 or 4? Why do you think so? Answers will vary.
Ingredients for 6 Servings
12 ounces cream cheese
3
teaspoon garlic salt
4
Ingredient amounts × 4
48 ounces
3 teaspoons
26
1
1
cups milk
2
3
cup Parmesan cheese, shredded
4
6 cups
3 cups
6 ounces elbow macaroni
24 ounces
1
pounds ham, diced
4
5 pounds
1
3
cup celery, sliced
4
3 cups
1
cup green pepper
3
4/3 or 1 1/3 cups
In order to feed everyone without any leftovers, I would need to multiply each ingredient by 20/6 since 6 times 20/6 equals 20.
Ingredients for 6 Servings
12 ounces cream cheese
3
teaspoon garlic salt
4
1
1
cups milk
2
3
cup Parmesan cheese, shredded
4
6 ounces elbow macaroni, cooked, drained and
rinsed
1
1
pounds ham, diced
4
Ingredient amounts × __20/6___
(fraction)
40 ounces
5/2 or 2 ½ teaspoons
5 cups
5/2 or 2 ½ cups
20 ounces
25/6 or 4 1/6 pounds
3
cup celery, sliced
4
5/2 or 2 ½ cups
1
cup green pepper, finely chopped
3
10/9 or 1 1/9 cups
How do these amounts compare to the amounts you multiplied by 4? Answers will vary.
27
Now Try This
You are in charge of the food for the awards banquet for your school and are going to make the Parmesan Macaroni Casserole.
There will be 150 people at the banquet. How much of each ingredient will you need?
Ingredients for 6 Servings
12 ounces cream cheese
3
teaspoon garlic salt
4
1
1
cups milk
2
3
cup Parmesan cheese, shredded
4
6 ounces elbow macaroni, cooked, drained and
rinsed
1
1
pounds ham, diced
4
Ingredient amounts × __25___
300 ounces or 18 ¾ lbs
75/4 tsp or 6 ¼ Tbs
75/2 cups or 9 3/8
75/4 cups or 4 11/16
150 ounces or 9 3/8 pounds
125/4 pounds or 31 ¼ pounds
3
cup celery, sliced
4
75/4 cups or 4 11/16 quarts
1
cup green pepper, finely chopped
3
25/3 cups or 8 1/3 cups
The school cafeteria has large rectangular pans to cook the casserole. Each pan holds 3 gallons. About how many pans will you
need to cook enough food for 150 people? About 5 pans.
Now Try This
1. Diagram 1:
The height of the smaller copy = ½ of 6 inches = 3 inches; Width = 2 inches
Uses proportional reasoning correctly: For example: Height/width = 6/4 = 3/width or Size of photo/Size of copy = 6/3 = 4/width
Diagram 2:
The width of the smaller copy = ½ of 6 inches = 3 inches; Height = 4 ½ inches
Uses proportional reasoning correctly: For example: Height/width = 6/4 = height/3
2.
Diagram 1: 6 inches wide, 6 inches high
Diagram 2: 8.5 inches wide, 6 inches high
28