Download College Focus Student Workbook

Mathematics Course for High
School Seniors
2016 – 2017
STUDENT HANDBOOK
Version 2.0
Table of Contents
Student Handbook – Version 2.0
Lesson
Page Number
Lesson
Page Number
PR #1
S1
Alg. 2.5
S189
PR 1.1
S5
Alg. 3.1
S206
PR 2.1
S16
Alg. 3.2
S215
PR 2.2
S21
Alg. 3.3
S220
PR 2.3
S24
Alg. 3.4
S228
PR 3.1
S31
Alg. 3.5
S234
PR 3.2
S36
Alg. 3.6
S241
PR 3.3
S38
Alg. 3.7
S256
PR 3.4
S40
Alg. 3.8
S271
PR 3.5
S52
Alg. 3.9
S295
PR 3.6
S54
Alg. 4.0
S305
PR 3.7
S57
Alg. 4.1
S307
PR 3.8
S77
Alg. 4.2
S312
PR 4.1
S78
Alg. 4.3
S317
PR 4.2
S84
Alg. 4.4
S329
PR 4.3
S85
Alg. 4.5
S345
Alg. 1.1
S127
Alg. 4.6
S358
Alg. 1.2
S135
Alg. 4.7
S378
Alg. 2.0
S147
Alg. 4.8
S393
Alg. 2.1
S152
Alg. 4.9
S404
Alg. 2.2
S160
MTE Lesson 1
S556
Alg. 2.3
S163
MTE Lesson 2
S575
Alg. 2.4
S171
MTE Lesson 3
S593
PROPORTIONAL
REASONING
Non-Routine
first LESSON
Sums of Consecutive Numbers
Instructions


Individually, or in groups of two or three, work through the following math activity.
As you work, think about the strategies you use to solve the problem.
Math Activity
7 + 8 = 15
2+3+4=9
4 + 5 + 6 + 7 = 22
The expressions above are examples of the sums of consecutive numbers. The number 15 is shown as the
sum of two consecutive numbers. The number 9 is shown as the sum of three consecutive numbers. The
number 22 is shown as the sum of four consecutive numbers. In this activity, you will explore how to
make different numbers with sums of consecutive numbers.
1.
(Handouts for exploring consecutive sums)
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S1
Non-Routine
first LESSON
Problems to follow Table handouts
2. What patterns do you see in the chart? (You might look for numbers that can be made by a particular
length of sum, or how many different-length sums can make particular numbers, or which numbers can’t
be made by consecutive sums. Use your imagination!)
3. What predictions about sums of consecutive numbers can you make from these patterns? Write a rule
if you can. How general is your rule?
4. If you haven’t already, look just at the numbers that can be made by a sum of three consecutive
numbers. What patterns do you see in those numbers? How can you predict whether a number can be
made by a sum of three consecutive numbers? Which consecutive numbers will add up to these
numbers?
5. Now do the same thing for sums of five consecutive numbers. Can you predict which numbers can be
made by such sums and what consecutive numbers will make them?
6. Make a prediction about sums of other odd numbers of consecutive numbers. Test your prediction for
an example or two. Why does your prediction work?
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S2
Non-Routine
first LESSON
7. Does this rule work for sums of an even number of consecutive numbers? Why or why not? If not,
how might you modify the rule to make it work?
8. Now use your rule(s) to predict whether each of the following numbers can be made with two
consecutive numbers, three consecutive numbers, four consecutive numbers, etc. Explain why you made
the predictions you did. Then check them to see if they work.
a. 45
b. 57
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c. 62
d. 75
e. 80
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S3
PR
LESSON 1.1
Warm-Up: ONE WHAT?
A. The “unit” can mean different things. The principal said that there were exactly 4 full classes on the third
floor and no other students. How many students are on the third floor?
To answer this we need to know “What is a full class.” If I tell you that a full class is 31 students, what
can you determine?
B. When mom cuts and serves us oranges, each person gets half an orange. She has 5 oranges so, how
many people can she serve?
C. 35 grapes were enough to serve 5 kids at snack time. What was a single serving?
D. Let’s stick with the grapes for a minute. Below are enough grapes for one and a half snacks:
What is the number of grapes in one serving? What would be the total number of grapes needed to
have enough for 5 servings?
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S5
PR
LESSON 1.1
Activity 1
1. Suppose that below are enough raisins for of one serving of raisins. We want to figure out what six full
servings would require.
a. If the picture of raisins above represents
b. If you know
of a serving, can you figure out what
is?
of a raisin serving size, can you determine a whole serving size?
c. Now that you know a whole serving size, how many raisins do you need for 6 servings? 10 servings? 52
servings?
2. Locate
on the number line.
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S6
PR
3. Locate
LESSON 1.1
on the number line.
4. Locate
on the number line.
5. Locate
on the number line.
6. Locate on the number line.
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S7
PR
7. Suppose the picture below represents . We want to figure out how to represent . Will
LESSON 1.1
be fewer
or more circles?
a. If the picture above represents , then can you figure out
?
b. If you know , can you figure out 1 (a whole)?
c. If you know 1, can you represent
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?
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S8
PR
LESSON 1.1
Activity 2
8. Suppose the picture below represents .
a. If the picture above represents
, what would be 1? (Is there a question you ask yourself to figure
this out?)
b. How would you represent
?
9. Analyze your friend’s response to the problem below. Is it correct?
BJ and Reece bought a deck of baseball cards. BJ took 3/8 of the deck and Reece took 5/8 of the deck. BJ
took home 15 cards. How many cards were there in the deck?
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S9
PR
10. Imagine a single unit is represented as follows:
region
LESSON 1.1
. What is represented by the shaded
?
11. If the unit is
, what would the shaded region
represent?
12. In each case draw a picture of one unit. One method is to find a useful unit fraction to help you.
a.
b.
c.
d.
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S10
PR
13. Ruth’s diet allows her to eat
ordered
LESSON 1.1
pound of turkey or chicken breast, fresh fruit, and fresh vegetables. She
pound of turkey breast at the delicatessen. The sales person sliced 3 uniform slices, weighed
them, and said, “this is
of a pound.” What part of the order could Ruth eat and stay on her diet?
14. Answer the questions below:
Describe how you can see each quantity
named.
a. fourths
b. eighths
c. thirty-seconds
d. sixteenths
e. sixty-fourths
a.
b.
c.
d.
e.
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S11
PR
LESSON 1.1
Exit Ticket
Sixteen liters of water fill my fish tank to
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of its capacity. How many liters does it take to fill the tank?
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S12
PR
LESSON 2.1
Activity 1
1. Two quantities are given in each example. Using arrows and words, show how the quantities change
together or write NR when the quantities are not related (Lamon).
Example: People working on a project, time to complete the project
Think:
With fewer people working on the project, it will take more time to complete the project so
people ↓, time ↑. With more people working, the project can be completed in a shorter time, so people ↑,
time ↓.
a.
Unit cost, total cost
b.
Size of a gas tank, cost to fill your tank
c.
Distance in feet, distance in meters
d.
Diameter of your tires, gas consumption on a given trip
e.
Side length of a square, area of a square
f.
Percentage of discount, discount in dollars
g.
Price paid for an item, sales tax
h.
Population of a town, time in years
i.
Number of hours you drive, average speed over your trip
j.
Number of office colleagues who buy a lottery ticket, share of the winnings
k.
Number of work days in a month, number of vacation days in a month
l.
Price of a candy bar, number of candy bars you can afford to buy
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PR
LESSON 2.1
2. Use arrow notation to show the direction of change (↑, ↓, or =) for two related quantities. Then fill in
the blank. If this is impossible, tell why (Lamon).
a. 8 people clean a house in 2 hours; 2 people clean that house in ___ hours.
b. It costs $30 to play 15 games; 3 games will cost ___.
c. It takes 4 hours to play 1 game of golf; 3 games will take ___ hours.
d. John is 10 years old and his mother is 3 times as old as John; Ben is 12 years old, and his mother is
___ years old.
e. I pay 6% sales tax when I purchase 1 item; I will pay ___ % tax when I purchase 5 items.
f. Three people eat their dinner in 30 minutes; it will take 9 people ___ minutes.
g. 2 people deliver all the papers on a certain route in 30 minutes; 6 people do the route in ___
minutes.
3. Carter and Rico like to ride their bikes on the trails in town. Today, they both started riding at the
beginning of the trail; each rode continuously at a constant speed, making no stops, to the end of the trail.
Rico took longer than Carter to reach the end of the path.


Which boy was biking faster? How do you know?
What are the assumptions? What are the variables?
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PR
LESSON 2.1
4. Two pitchers of juice are on the table. Pitcher B contains weaker juice than pitcher A. Add one
teaspoon of instant juice crystals to pitcher A and one cup of water to pitcher B.

Which pitcher will contain the stronger juice? Why?
5. Jenson drives 100km in 2 hours and has 60km to go. Do you think Jenson will drive the other 60km in
more or less than 2 hours? Justify your answer.
6. Greg and Ross hammered a line of nails into different boards from one end to the other. Ross
hammered more nails than Greg. Ross’ board was shorter than Greg’s. On which board are the nails
closer together? Why?
(Source: Non‐Numeric Problems that Encourage Proportional Thinking:
http://www.edugains.ca/resources/LearningMaterials/ContinuumConnection/BigIdeasQuestioning_ProportionalReasoning.pdf )
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PR
LESSON 2.2
Activity 1: String Bean and Slim Task1
Jo has two snakes, String Bean and Slim. Right now, String Bean is 3 feet long and Slim is 10 feet long. Jo
knows that two years from now, String Bean will be about 8 feet long, while Slim’s length will be about
15 feet.
Over the next two year, will both snakes grow the same amount? Explain and justify your response.
University of Pittsburgh. (2010). Retrieved from http://schools.nyc.gov/NR/rdonlyres/A9F735CB-47E4-40F8-884FEA54D0AB5705/0/NYCDOEG6MathRatios_Final.pdf
1
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S21
PR
LESSON 2.3
Activity 1
In these cards there are descriptions of fizzy orange mixtures. Your task is to put the cards in
order of strength, from the least orangey (top) to most orangey (bottom). If you think more than
one card describes the same fizzy orange mixture, group them together.
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S26
PR
LESSON 3.1
Activity 1: Walk-a-thon
1.
The Walk-a-thon
Juliana participated in a walk-a-thon to raise money for cancer research. She recorded the total
distance she walked at several different points in time, but a few of the entries got smudged and can
no longer be read. The times and distances that can still be read are listed in the table below.
Time in hours
Miles walked
1
2
6
12
5
a. Assume Juliana walked at a constant speed. Complete the table.
b. How fast was Juliana walking in miles per hour? How long did it take Juliana to walk one mile?
c. Next year Juliana is planning to walk for seven hours. If she walks at the same speed next year,
how many miles will she walk?
2.
Gianna’s pay
Use a table to record steps in your work.
Gianna is paid $90 for 5 hours of work.
a. At this rate, how much would Gianna make for 8 hours of work?
b. At this rate, how long would Gianna have to work to make $60?
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PR
LESSON 3.1
Activity 2: Ice cream by the pound
Ahmed’s Ice Cream Shop sells ice cream by weight. They charge $2.50 per quarter pound.
1.
Suppose your serving weighs one-third of a pound. How much will it cost?
2.
You plan on buying ice cream for you and your date. You estimate buying half of a pound. How much
will it cost?
3.
You learn that your date loves ice cream and orders half a pound all to themselves. You order your
usual one-third of a pound. How much will it cost? How much total ice cream did you buy?
4.
Your teacher is collecting student deposits for the Thanksgiving ice cream party. She estimates that
she will need about 1/3 of a pound per student and there are 30 students. She’s trying to decide
between Häagen-Dazs, Ben & Jerry’s, and Ahmed’s Ice Cream. What should she do to determine the
costs for each choice? (You’ll need some additional research or estimates for this. Let us know if you
have any ice cream pricing expertise.) State the cost of a serving, the cost per pound, and the cost for a
class order.
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5.
LESSON 3.1
On Fridays Ahmed advertises buy one pound and get the second pound for 50% off.
a. Recalculate the cost of Ahmed’s ice cream for the class party. State the cost of a serving, the cost
per pound, and the cost for a class order.
b.
6.
With the discount the class president argues that they should order half a pound per person.
Determine the new class cost.
Gretchen the city health czar learned about the ice cream party and has reminded the principal of the
quarter pound ice cream limit per student. (The per student order can only be increased when the
event is linked to an exercise event.) Determine the new costs.
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LESSON 3.1
Exit Ticket
A sale is advertising 3 bottles of Shampoo for $9. The store only has 8 bottles remaining. If the store keeps
the sale price for all shampoo, how much should all 8 bottles cost? Create a table to show finding your
answer.
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LESSON 3.2
Activity 1: Worms Task
Worms
Use number lines to answer the following question:
 Andy Worm travels 6cm every 4 minutes.
 Betty Worm travels 15cm every 10 minutes.
Are Andy and Betty traveling at the same rate? If not, who is traveling “faster”? Explain how you know!
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PR
LESSON 3.2
Exit Ticket
Salty Slug is arguing with Sweet Snail claiming slugs can travel faster than snails. Salty’s argument is as
follows:
Slugs move at a faster rate because they can travel 100cm. in 20 hours, and snails can travel only 24cm. in 4
hours.
 Use a number line to illustrate if you agree or disagree with Salty Slug and justify your answer using a
table of values.
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LESSON 3.3
Activity 1
1.
You found the following recipe for Pad Thai to serve 4 people. You’re only cooking for 2 people and
you don’t want any left overs. Determine how much you will need of each ingredient.
●
●
●
●
●
●
2.
2 to 3 Tbsp. tamarind paste, to taste
6 Tbsp. fish sauce
2 Tbsp. soy sauce
1 to 2 tsp. chili sauce, OR 2/3 to 1 ½ tsp. cayenne pepper, to taste
6-8 Tbsp. palm sugar OR brown sugar
18 ounces of rice noodle
After cooking a scrumptious Pad Thai, you found that 10 of your friends want to try it. Determine how
much of each ingredient you will need to make enough for 10.
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LESSON 3.3
Practice Set 1
Solve these problems using any method you like.
1. This ratio table shows the number of cans of paint needed to cover various lengths of fence
Cans of
Paint
Feet of
Fence
2
4
6
8
10
12
30
60
90
120
150
180
a. Find the number of cans needed to paint 75ft. of fence.
b. Find the number of cans needed to paint 140ft. of fence.
2. The Holmes family has two adults and three children. The family uses an average of 1,400 gallons of
water per week. Last month they had two houseguests who stayed for a week. Assuming that the water
usage is proportional to the number of people in the house, about how much water did the household
use during the guests’ visit?
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LESSON 3.4
Activity 1
Mixture A
Mixture B
Mixture C
A. Consider mixtures A, B, and C. Would they generate paint of the same color? If not, rank them from
lightest to darkest.
B. Create a mixture that would have the exact same color as Mixture B. State the ratio of blue to white for
mixture B and the ratio of blue to white for your mixture. What is true of these two ratios?
C. Jerry wants to create a mixture with nine pails that is the exact same color as mixture A. How many
white and how many blue pails will he need?
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LESSON 3.4
Activity 2
1.
In the junior softball league, you keep records on players’ batting averages. To motivate young players,
they create a fresh batting average for every game. Below is a table of six players’ hits and the number
of times they were at bat.
Player
Mike
Julian
Suraj
Lekha
Antonio
Maya
Hits
4
10
8
10
6
9
At Bat
11
15
12
18
12
16
First section: No calculator 
a. Who seems better Mike or Julian? Why?
b. Who seems better Mike or Suraj? Why?
c. Who seems better Suraj or Julian?
d. Try to rank these players from lowest to highest batting average without using a calculator.
e. Jon has missed a few practices because of a new job he started. He is pleading with the coach to
let him play in the next game even though he missed practice. He says, “Come on coach, let Mike
pitch me five balls and I’ll show you that I’m better than Suraj.” How many would Jon have to hit to
show that he is better than Suraj?
f. If Mike pitched five and Jon hit two, could it be true that Jon is better than Suraj?
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LESSON 3.4
g. Owen is Suraj’s twin brother and he plays on a different team. The two of them are identical in
everything. They even maintain the same batting average! What are different possible “hits” and
“at bats” for Owen?
(Please grab a calculator!) In each case, determine whether the pricing is proportional and explain why or
why not.
1.
The Green Grocery sells tomato sauce in 6oz. cans for $0.49 and in 15oz. cans for $0.69.
2.
The Town Market sells its store brand sugar in a 4 pound bag for $2.99, a 5 pound bag for $3.75, and a
10-pound bag for $11.74.
3.
At the farmers' market, strawberries were priced at $2.98 for a pound, and $1.49 for 8 ounces.
4.
I can buy 25 fabric softener sheets for $1.69, 50 sheets for $2.99, or 100 sheets for $5.98.
5.
A certain breakfast buffet charges $0.89 for each year of a child's age, up to the age of 12.
6.
Stop N Shop sells 12-can soft drink boxes for $3.29 each or a bundle of 3 boxes for $10.19.
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LESSON 3.4
Activity 3: Do the facts agree?1
1.
“In 2012, 123 African-Americans were shot dead by police. There are currently more than 43 million
blacks living in the U.S.A.,” O’Reilly said on his program. “Same year, 326 whites were killed by police
bullets. Those are the latest stats available.”
a. What would the population of whites have to be for the two rates to be proportional?
b. What is the US population of whites?
2.
“Nicholas Kristof wrote this week (12/1/2014) that young black men are 21 times more likely to be shot
and killed by police than young white men.” His source, ProPublica, stated, "The 1,217 deadly police
shootings from 2010 to 2012 captured in the federal data show that blacks, age 15 to 19, were killed at
a rate of 31.17 per million, while just 1.47 per million white males in that age range died at the hands
of police."
a. How does Kristof come to his conclusion?
b. James, a student in class, graphed the following two points: (31.2, 1000000), and (1.5, 1000000).
He then showed how he could draw a line between the two points and claimed that the rates were
the same. Is he correct? Why or why not?
1
http://www.cnn.com/2014/12/02/politics/kristoff-oreilly-police-shooting-numbers-fact-check/
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LESSON 3.4
3.
St. Louis County officers shot four times more African Americans than whites, even though African
Americans make up only 25 percent of St. Louis County’s population. (http://www.kmov.com/specialcoverage-001/Police-records-show-far-more-blacks-than-whites-shot-by-local-officers280371252.html)
On the second bar represent the fraction or percent of African Americans shot among all individuals
shot.
4.
In 2011 there were 11,944 law enforcement agencies that employed 535,651 officers.
a. Law enforcement agencies reported that 54,774 officers were assaulted while performing their
duties in 2011. What proportion of officers were assaulted?
b. Of all officers who were assaulted in 2011:
 33.3 percent were responding to disturbance calls (family quarrels, bar fights, etc.).
 12.6 percent of the officers assaulted were handling or transporting prisoners.
 14.7 percent of the officers assaulted were attempting other arrests.
How many police officers were assaulted responding to disturbance calls?
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5.
LESSON 3.4
a. Why is $26,000 per year proportional to $500 per week? What are the equivalent fractions? What
is done to determine the weekly rate?
b. In 2011 the average annual pension for police serving 35 years or more was $97,793. Determine
the rate of pension per month, per week and per day.
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LESSON 3.4
Problem Set 2
1.
Consider these triangles.
a. Are the side lengths of Triangle ABC proportional to those of Triangle XYZ? Explain how you know.
b. Are the triangles similar? Explain.
2.
For each rectangle, find the ratio of the length to the width. Is the relationship between the lengths
and widths of these rectangles proportional? Explain
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3.
4.
LESSON 3.4
On the cards below, the top numbers are proportional to the bottom numbers. Find another card that
belongs in this set. Explain how you know your card belongs.
1
1.5
2
2.5
2
3
4
5
Chris created this tile pattern:
a. Describe how the pattern of white and colored tiles changes from one stage to the next
b. For the stages shown, is the number of white tiles proportional to the number of purple tiles?
Explain
c. Starting with Stage 1 above, draw the next two stages for a tile pattern in which the number of
white tiles is proportional to the number of purple tiles.
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LESSON 3.4
Exit Ticket
“Mixing Drinks (source: map.mathshell.org)
When Sam and his friends get together, Sam makes a fizzy orange drink by mixing orange juice with soda.
On Friday, Sam makes 7 liters of fizzy orange by mixing 3 liters of orange juice with 4 liters of soda. On
Saturday, Sam makes 9 liters of fizzy orange by mixing 4 liters of orange juice with 5 liters of soda.
1.
Does the fizzy orange on Saturday taste the same as Friday’s fizzy orange, or different?
If you think it tastes the same, explain how you can tell.
If you think it tastes different, does it taste more or less orangey? Explain how you know.
2. On Sunday, Sam wants to make 5 liters of fizzy orange that tastes slightly less orangey than Friday’s
and Saturday’s fizzy orange drink. For every liter of orange, how many liters of soda should be added to
the mixture? Explain your reasoning.
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LESSON 3.5
Problem Set 1
1.
Abby is inviting friends over for brunch. Best West Bagels charges $4.50 for three bagels and Connie’s
Bagel House charges $6.75 for half a dozen bagels. At which store will Abby get more for her money?
Explain.
2.
Jon is bringing muffins to the brunch. The East Village Bakery advertises “Two muffins for $2.99.”
Mollie’s Muffins charges $15 per dozen. Where will Jon get a better deal? Explain.
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LESSON 3.5
Problem Set 2
1.
Yudelca has a long distance phone plan that charges higher rates for calls outside of the continental
US. On her bill she notices that a 17-minute call to Mexico cost $7.14. A 25-minute call to Toronto cost
$8.00. How would you compare these rates? Explain.
2.
The West Corner Deli charges $3.52 for four mangoes while Mario’s Supermarket charges $5.52 for six
mangoes. Which store offers the better price for mangoes? Explain your answer.
3.
Go back and solve problems 1 and 2 by an alternative method. Compare the method you used initially
with your alternative method. (Why might one method be better for a problem? Would it always be
the better method? Would your preference be different without a calculator?).
4.
Julio and Marco are arguing over whether or not the price for a dozen eggs is a unit rate. What do you
think and why?
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LESSON 3.6
Activity 1
Jefferson High School has 600 students, and Memorial High School has 450 students. For each of these
problems, use this percent diagram to estimate an answer.
100%
600
450
75%
400
300
50%
200
150
25%
0
1.
0%
Jefferson
Memorial
0
A survey of the two schools finds that 300 Jefferson students watch more than 1 hour of television
every night, while 270 Memorial students watch more than 1 hour per night.
a. What percentage of Jefferson students watch more than 1 hour of TV each night?
b. What percentage of Memorial students watch more than 1 hour of TV each night?
c. Comparing percentages, do more Jefferson students or more Memorial students watch
more than 1 hour of TV each night?
2. Jefferson has 275 girls and Memorial has 350 girls. Which school has a greater percentage of girls?
Explain.
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3.
4.
LESSON 3.6
At each school, 75% of the students play a musical instrument.
a.
About how many students at Jefferson play a musical instrument?
b.
About how many students at Memorial play a musical instrument?
Here are two statements about Jefferson and Memorial schools:
A. Jefferson has 275 girls and Memorial has 250 girls, so there are more girls at Jefferson.
B. Jefferson is almost 46% girls and Memorial is almost 56% girls, so there are more girls at Memorial.
 Which statement uses a common scale to make a comparison?
 Which statement do you think is a better answer to the question, “Which school has more
girls?” Explain your thinking.
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LESSON 3.6
Exit Ticket
Michaela was making $8 per hour and just received a 50 cent raise. Annie was making $960 per month and
received a raise of $40 per month. They are trying to decide who got the better deal. What can you
determine and why?
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LESSON 3.7
Warm-Up
One month Rob spent $8.02 on his phone. The next month he spent $6.00. To work out the average amount
Rob spends over the two months, you could press the calculator keys:
(8.02+6)÷2=
1.
Tom usually earns $40.85 per hour.
He has just heard that he has had a 6% pay raise.
He wants to work out his new pay on his calculator.
It does not have a percent button.
Which keys must he press on his calculator?
Write down the keys in the correct order. (You do not have to do the calculation)
2.
Maria sees a dress in a sale. The dress is normally priced at $56.99
The ticket says that there is 45% off
She wants to use her calculator to work out how much the dress will cost.
It does not have a percent button.
Which key smust she press on her calculator?
Write down the keys in the correct order. (You do not hav eto do the calculation)
3.
Last Year, the price of an item was $350. This year it is $450.
Lena wants to know what the percentage change is.
Write down the calculation she will need to do to get the correct answer.
(You do not hav eto do the calculation)
4.
In a sale, the prices in a shope were all decreased by 20%
After the sale they were all increased by 25%
What was the overall effect on the shop prices?
Explain how you know.
Increasingand Decreasing Quantities by a Percent, © 2012 MARS. Shell Center. University of Nottingham
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LESSON 3.7
Warm-Up 2
Barry’s Bargain Basement Sale
Everything’s a Bargain!
All items are marked with an arrival date.
1 month old, 10% off the original price!
2 months old, 20% off the original price!
3 months old, 30% off the original price!
4 months or older, 50% off the original price!
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Steve’s Super Savings Store
We won’t be undersold!
All items are marked with an arrival date.
If it’s been here 1 month, take 10% off!
If it’s been here 2 months, take another 20% off
the already reduced price!
If it’s been here 3 months or longer, take
another 40% off the already reduced price!
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LESSON 3.7
Problem Set 1
1)
Rebecca went to Barry’s Bargain Basement to buy a pair of jeans. The jeans had been in the store for 1
month. The original price was $40.
a) What was the sale price?
b)
2)
What percentage of the original price was the sale price?
Julie was a smart shopper. The pair of shoes she wanted had an original price of $50 at both Barry’s
and Steve’s. The shoes had been in both stores 2 months.
a) How much would Julie pay for the shoes at Barry’s? At Steve’s?
b)
Explain why these sales prices are different.
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3)
4)
LESSON 3.7
Doug, a frequent shopper at Barry’s, has his eye on a hat with an original price of $20.
a) The hat has been in the store for 1 month. How much will Doug save?
b)
Doug is considering waiting another month to get a better price. How much will he save after 2
months?
c)
Is the amount Doug would save with the 10% discount double the amount he would save with
the 20% discount?
d)
How can you easily determine how much Doug would save after 3 months?
Joy found a sweatshirt at Steve’s with an original price of $30.
a) If the sweatshirt has been in the store long enough to earn the 10% discount, how much will Joy
save?
b)
If the sweatshirt has been there long enough to earn the 20% discount, how much will she save
from the original price?
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c)
LESSON 3.7
Is the dollar amount of the second discount twice the dollar amount of the first discount? Explain
why or why not.
d)
Joy calculated the final price this way: “First I take 10% off, and then I take 20% off, so all together I’m
taking 30% off. That’s $9, so the final price is $21.” Is Joy correct? Explain.
e)
If Joy buys the sweatshirt after it’s been in the store 2 months, what is the total percent discount from
the original price? Show how you found your answer.
5)
An item originally marked at $100 is in both stores for more than 3 months. Where would you get the
better buy? Explain.
6)
Is the ad for Steve’s—which claims they won’t be undersold—true when Steve’s prices are compared
with Barry’s prices? Explain.
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7)
LESSON 3.7
Suppose you reduce an 8 inch by 10 inch picture by 90% and then increase the copy by 90%.
a)
Will you get the same size picture you started with? Explain.
b)
What is the percent change from the original size to the final size
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LESSON 3.7
Problem Set 2
In a recent election by a seventh grade class with 90 students, Jesse received 36 votes and Kiran received 54
votes.
*These problems are taken From Impact Mathematics, Course 2
1)
Tell whether each of the following statements presents this information accurately. For each accurate
statement, explain or give a calculation to show why it is accurate.
a)
The ratio of the number of students who voted for Jesse to the number who voted for Kiran is
2:3.
b)
40% of the voters preferred Jesse.
c)
Kiran received 20% more of the total vote than Jesse received.
d)
50% more people voted for Kiran than for Jesse.
e)
18 more people voted for Kiran than for Jesse.
f)
Kiran received 3/5 of the votes.
g)
The number of people who voted for Kiran is 1/5 times the number who voted for Jesse.
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LESSON 3.7
2)
Which of the accurate statements in Problem 1 seems to give the best impression of the class’s
preference for Kiran? Why?
3)
Which of the accurate statements in Problem 1 seems to minimize the class’s preference for Kiran?
Explain.
4)
Do some of the statements seem more informative than others?
5)
Some ratios compare a part of some group to the whole group, like the number of students who voted
for Jesse, 36, compared to the total number of voters, 90. Such ratios are called part-to-whole ratios.
Other ratios compare a part of a group to another part of the same or a different group, like the number of
students who voted for Jesse, 36, compared to the number who voted for Kiran, 54. These kinds of ratios are
called part-to-part ratios.
a)
Which statements in Problem 1 make part-to-whole comparisons?
b)
Which statements in Problem 1 make part-to-part comparisons?
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LESSON 3.7
Exit Ticket
In a sale, the prices in a shop were all decreased by 20%. After the sale they were all increased by 25%. What
was the overall effect on the shop prices? Explain how you know.
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NRP
LESSON: PR 3.8
Activity 1: Summer Jobs1
The headline reads, “Students Protest Lack of Summer Jobs.”
The students at Gridville High, like many students across the country, are worried about summer jobs. The
editor of Coordinate Opinions, the school newspaper, began writing an editorial on the deterioration of
summer job opportunities, especially tor girls. When word of the editorial was leaked to the mayor, he
demanded to be given air time on the local news broadcast. His response to the proposed editorial:
“The editor of Coordinate Opinions should be removed from her job! Her allegations that the summer job
situation for students has deteriorated is totally unfounded. More students have summer jobs today than
when I was in high school. And as far as girls not having adequate opportunities in Gridville, more girls will
be employed this summer than boys.”
The school principal, in an attempt to appease the mayor, removed the school editor from her post.
However, in a last-ditch effort to get the story out, the editor made copies of two data displays that she had
planned to use in her article. The data displays in Figure I circulated throughout Gridville High.
Analyze the data to make the case for the dismissed editor. Feel free to use tables, ratios, fractions,
decimals, percents, or any other data displays that will strengthen you argument.
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LESSON 4.1
Warm- up1
Think and Discuss
Many percent problems can be solved by setting up a proportion.
For example, suppose you’re making a shade of light blue paint with 80% blue paint and the remainder
white paint. If you want a total of 7 gallons of paint, how many gallons of blue are needed?
You could set up the following proportion:
It can also be helpful to consider a percent diagram as below. What would be the proportion that would
correspond with the percent diagram below?
1
Activities in this lesson are adapted from IMPACT Mathematics Course 2, McGraw-Hill.
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LESSON 4.1
Activity 1
1.
Consider this question: What percent of 150 is 6?
a.
b.
c.
2.
Explain a method to figure this out.
Draw a percent diagram to represent the question.
Express the question as a proportion and solve it.
Now consider this question: What is 185% of 20?
a.
b.
Draw a percent diagram to represent the question.
Express the question as a proportion and solve it.
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LESSON 4.1
3.
A gardener planted 13% of his tulip bulbs in the border of his garden and the rest in the garden bed. In
spring, every one of the bulbs grew into a tulip. He counted 45 tulips in the border. How many bulbs
had he planted all together? Show how you found your answer.
4.
Of 75 flights leaving from Hartsfield Airport in Atlanta, 33 went to the West Coast. What percentage of
the 75 flights went to the West Coast? Show how you find your answer.
5.
The largest land animal, the African bush elephant, may weigh as much as 8 tons. However, that is only
about 3.9% of the weight of the largest animal of all, the blue whale. How heavy can a blue whale be?
Show how you found your answer.
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LESSON 4.1
6.
A In 1995, Americans spent about $403 billion on recreation, including books, toys, videos, sports, and
amusement parks. They spent 186% of that amount on housing. How much money did Americans
spend on housing in 1995?
7.
In 1950, the estimated world population was 2.6 billion. In 2015 it was 7.3 billion.
a.
Describe the population in 2015 as a percentage of the 1950 population.
b. Describe the 1950 population as a percentage of the 2015 population.
c.
In 2015, about 324 million people lived in the United States. What percentage of the world's
population lived in the United States in 2015?
8.
Complete the following.
a.
Write a proportion that expresses this statement: 32% of 25 is b. Solve your proportion to find b.
b. Write a proportion that expresses this statement: n% of a is b. Explain why this proportion is
useful for solving percent problems.
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LESSON 4.1
Activity 2
Zoe and Luis both had a resting pulse rate of 84 beats per minute. After jogging for a short period, they both
had a pulse rate of 105 beats per minute. Consider the following statements from the two friends about
how they each calculated the percent increase.
ZOE said:
105 is what percent of 84? I can solve this proportion:
So, n is 125. That means my active pulse rate is 125% of my resting pulse rate. That’s a 25% increase.
LUIS said:
The difference between the two numbers is 105 – 84 = 21. This difference is what percentage of my resting
pulse rate?
So n is 25. That’s 25% and I Know it’s an increase because my active pulse rate was greater
from Impact Math Course 2
Questions:
1.
Check to see if the two methods above give the same result.
2.
Create a similar problem with slightly different numbers.
3.
Will these two methods always give you the same answer? Explain how you know.
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LESSON 4.1
Exit Ticket
1.
Increase 100 by 20%. Decrease your new result by 20%.
•
•
•
2.
Is the result the original number, 100?
Try this again with a different starting number.
Explain the result.
Use algebra to show what happens. Let A be you starting amount. Increase A by r percent. Now
decrease your new result by r percent. By how much does your new result differ from the starting
amount?
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LESSON 4.2
Activity 2
Consider the following.
Kerry said that the Japanese Bullet Train takes about 6 minutes to travel 22.2km. Jerry said that at this rate,
he could travel around the world at the equator in less than 8 days. Kerry disagrees – she thinks it will take
longer.
[The diameter of Earth is approximately 12,740km.]
•
Who is correct? Justify your response.
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LESSON 4.3
Mini-Lesson Starter Task
When the choir calendar sale began, Ms. Gomez bought the first 6 calendars. From his purchase alone,
the choir raised $15.00. By the end of the first day of the sale, they had raised $67.50. How many
calendars had been sold by the end of the first day?
Take a few moments to solve this by a method of your choice.
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LESSON 4.3
Mini-Lesson (Part 2): Anna, Bob, Carla, and Darnell
Here are three different methods for solving the problem
Anna: I found out how much they would raise if they sold 1 calendar. Then I divided 67.50 by that
amount.
Unit rate = $15/6 = $2.50 per calendar
$67.50/$2.50 = 27
Bob: They raised $15 for 6 calendars. I made a table and multiplied up to get 67.50 to something.
Dollars raised
Num. of Calendars
15
6
30
12
60
24
7.50
3
67.50
27
Carla: I wrote an equation. I knew the amount raised would be proportional to the number of calendars.
I let n be the number of calendars and I wrote
and then I solved this equation to find the
answer.
Then Carla showed that her method was really simpler if she wrote her proportion in the following way.
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LESSON 4.3
After seeing Carla’s work Darnell said he preferred to solve those with the arrow method:
= 27
Before calling any of these methods better than the others, it is important to try the methods out on a few
problems.
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LESSON 4.3
Problem Set 1
Recall the methods Anna (unit rate), Bob (table), and Carla/Darnell (A couple of algebraic methods) used.
Solve the following problems using any method you like. Once you have completed the 8 problems, select
4 problems to solve by an alternative method.
1.
Pierre went to the bank and exchanged 50 Canadian dollars for 32.28 U.S. dollars. Later that day
Monica went to the same bank to exchange 72 Canadian dollars for U.S. dollars. How many U.S.
dollars did she receive? (Assume the exchange rate was the same throughout the day.)
a. Set up two proportions you could use to solve this proportion.
b. Solve one of your proportions. How much money in U.S. dollars did Monica receive?
2.
This ratio table shows the number of cans of paint needed for painting various lengths of fence.
Cans of
Paint
Feet of
Fenc
e
2
4
6
8
10
12
30
60
90
120
150
180
a. Find the number of cans needed to paint 75ft of fence.
b. Find the number of cans needed to paint 140ft of fence.
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LESSON 4.3
3.
The Holmes family has two adults and three children. The family uses an average of 1,400 gallons of
water per week. Last month they had two houseguests who stayed for a week. Assuming that the
water usage is proportional to the number of people in the house, about how much water did the
household use during the guests' visit?
4.
Paul is using a trail map to plan a hiking trip. The scale indicates that ½ inch on the map represents
1½ miles. Paul chooses a trail that is about 5½ inches long on the map. How long is the actual trail?
5.
Georgeanne loves mixed nuts. The local farmer's market sells a mix for $2.79 per pound. How many
pounds can she buy for $10.85?
6.
Jack is making his famous waffles for Sunday brunch. He knows that 5 waffles usually feed 2 people.
If he wants to serve 11 people, how many waffles should he make?
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LESSON 4.3
7.
The week before Thanksgiving, turkeys go on sale for $1.29 per pound at the EatMore grocery store.
How much will a 14-lb turkey cost?
8.
At Camp Poison Oak there are 2 counselors for every 15 campers. The camp director expects 75
campers next summer. How many counselors will they need?
Write-up on problems 1-8:
For your first solution method did you use the same strategy for solving all the? If so, explain it. If
not, choose one of the strategies you used and explain it.
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LESSON 4.3
Problem Set 2
Solve each of the problems below with any method you choose. Show all of your work.
Compare your solutions with your group and discuss your preferences for different methods for each
problem
1.
Priya uses seven tubes of toothpaste every ten months. How many tubes would she use in 3 years?
In 10 years? How long would it take her to use 100 tubes?
2.
Rajan often cleans his teachers’ mini-wipe boards. He can clean 15 boards in 8 minutes. At this rate,
how long would it take him to clean 90 mini wipe boards? Now reverse the problem: How many
mini wipe boards could Rajan clean in one hour?
3.
The Dance Club is having a tamale sale! The school has 1600 students, but the club members are not
sure how many tamales to make. One day during lunch, the club asked random students if they
would buy a tamale for a buck. They found that 15 out of 80 students surveyed said they would
definitely buy a tamale. How many tamales should the Dance Club expect to sell?
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LESSON 4.3
4.
A typical small bag of colored candies has about 135 candies in it, 27 of which are blue. At this rate,
how many blue candies would you expect in a pile of 1000 colored candies?
5.
Ten graphing calculators cost $1049.50. How much would 100 cost? 1000? 500?
6.
Tickets to 50 home baseball games would cost $1137.50. How much would it cost to get tickets for
all 81 home games? How many games could you go to for $728?
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7.
LESSON 4.3
Terrence weighs 160 pounds and is on a diet to gain two pounds a week so that he can make the
football team. Carlos weighs 208 pounds and is on a diet to lose three pounds a week so that he can
be on the wrestling team in a lower weight class.
If Terrence and Carlos stick to these goals and keep dieting, when will they weigh the same, and how
much will they weigh at that time? Clearly explain all of your method.
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LESSON 4.3
Problem Set 3
Solving Proportional Equations. Solve for x. Check your solutions.
1.
2.
4.
3.
5.
7.
6.
8.
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10.
11.
12.
13.
LESSON 4.3
14.
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LESSON 4.3
Exit Ticket
Solve the following proportional equation and provide justifications for your steps.
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ALGEBRA
ALGEBRA
LESSON 1.1
Activity 1: Amazon Island
On Amazon Island, there are many different merchants and people selling things. The people of Amazon
Island often found themselves seeking out best prices and adding up purchases to find totals.
In order to help them do their calculations, they created two binary operations: “minimize” and “join.” They
used the symbol ⊕ for minimize and ⊞ for join.
5⊕8=5
5 ⊞ 8 = 13
You take the minimum of 5 and 8 to get 5
Generally: a ⊕ b = min{a,b}
You join 5 and 8 to get 13 (it’s really just like adding but they call it joining);
Generally: a ⊞ b = a+b
In this lesson we will explore some properties, equations, and solutions by investigating how they might
apply on the Amazon Island.
A solution to an algebraic equation is the set of things put in place of the variable that make it true.
1.
Solve the following (i.e. find everything that would make it true.) There may be one solution, more
than one solution, or no solutions.
a.10 ⊕ 20 = x
b. a ⊕ 6 = 3
c. 4 ⊕ (5 ⊕ 6) = p
d. b ⊕ 5 = 5
e. 6 ⊕
=x
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LESSON 1.1
f. b ⊕ 8 = 12
g. -7 ⊕ c = -7
2.
Create three of your own simple one-variable equations using ⊕.
• Find a solution for your equation.
• Can you find additional solutions for your equation?
• How many solutions are there for the equation?
• Are any conjectures emerging?
3.
In traditional mathematics, we say that addition is commutative because switching the order of two
numbers we’re adding doesn’t change the answer. We say that it is associative because when adding
three numbers, it doesn’t matter whether we add the first two first or the last two first.
• Do you think ⊕ is commutative?
• Do you think ⊕ is associative?
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ALGEBRA
4.
LESSON 1.1
Does the following work for ALL real numbers a, b, and c?
a ⊕ (b⊕c)=(a⊕b) ⊕c
5.
Does the following work for all real numbers x and y?
x⊕y = y⊕x
6. In traditional mathematics, zero and one can be special in addition and multiplication.
• Why would we call zero special in addition?
• Is there a similar number for the Amazon operation of minimize, ⊕?
In addition we call -5 the additive inverse or opposite of 5. Is there something similar for ⊕?
7. A teacher without a plan:
Teacher: Today we’re going to investigate the operation of ⊞ on from Amazon Island.
Student: I don’t need to do this.
Teacher: Why not?
Student: Because it’s just addition!
Teacher: Shoot!
Is ⊞ just addition?
Is it commutative?
Is it associative?
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LESSON 1.1
Activity 2
Does the island of Amazon need an order of operations? An expression might be calculated in different
ways if a common order of operations does not exist.
1.
Create a few expressions such as 3⊞4⊕5 and explore whether you think an order of operations will
make a difference in the result
2.
Consider the expression 5⊕6⊞ 7. Determine the result if …
a) the joining operation is done first
b) the minimize operation is done first
c) calculations are done left to right
d) calculations are done right to left
e) the square-with-a-plus operation is done first
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LESSON 1.1
Exit Ticket
After a lot of debate, the Amazon Math Council has determined that joining will precede minimizing unless
parentheses are used to make minimizing the priority. Explain what this will mean by using an example that
you create.
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LESSON 1.2
Activity 1: A Planet without Order
The planet Krypton is a place without order. They refuse to agree to adopt a single order of
operations. The Western Lefties think that order should just come from left to right. On an Eastern
continent, they want to go from right to left. Those on an island without much ink and pencil lead
want to reason out a logical choice that everyone on the planet follows. Others see no reason to
create rules for the community to follow.
1. If there is no order for operations, find at least two different values that are possible for each of the
given expressions:
a. 10 – 3 2
b. 48
23
c. 40 – 20
2 + 15 x 1.5
d. a+b3 when a=2 and b = 5
e. ab3 when a=2 and b = 5
f. 1+2c+3c2 when c=
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LESSON 1.2
2. Two Kryptonites were struggling in emailing each other to discuss a problem. The Islander felt like
1+2c+3c2 should equal 321 when c = 10. The Western Leftie just calculated from left to right and came
up with 108,900. Explain each of the answers they came to.
3. When Western Leftie figured out the problem, he just sent a new email using a few extra parentheses
for the Islander:
{[((1+2) x c) +3] c}2
What does the expression above lead to when c=10?
When Islander wanted to get Leftie to see the answer as 321, she had
to write, (1+(2 × c))+(3 × (c^2))
The Islander hated all of the parentheses and considered leaving the planet.
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LESSON 1.2
Problem Set 1
Luckily on the planet earth we have come to agreements on an order of operations that enables all of us to
communicate with only a reasonable use of parentheses. Some would say that the precedence of
multiplication over addition is probably a consequence of natural language.
5 apples and 2 bananas is something like 5×a+2×b or 5a+2b where the five next to a implies
multiplication.
This may have to do with the fact that multiplication is sometimes expressed by adjectives in the natural
language, while summation is expressed by a conjunction.
The critical result is that people around the world can communication mathematical expressions without
excessive use of parentheses. Multiplication and division precede addition and subtraction. And exponents
apply to what is immediately next to them. Solve the following using the correct order of operations:
1. 40 —
+ 15 x 15
2. 10 — 3 x 2
3. 46 — 24 ÷ (3 — 1)
4. 48 ÷ 23
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LESSON 1.2
5. (1 + 9)3 - 102 x 8
6. 100 ÷ 52 + 4
7. 32 x 22 — 6 x (9 — 5)
8. 90 - 5 x 32
9. 18 ÷ (100 — 99)9 + 23
10. 92 — 14 x 3
11. (2 +3)3 — (11 — 4)2 x 2
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LESSON 1.2
Activity 2: More exploring on earth
1a. Write an expression to represent the total area of this diagram:
1b. Which compound area diagram below represents the expression: 5 + 3 x 2?
1c. Create two of your own multiple choice problems like the problem above. Explain the answer.
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LESSON 1.2
2a. Explain the difference in meaning between 10 x 4 + 5 and 10 x (4 + 5).
Use words or diagrams to help your explanation.
2b. Explain the difference in meaning between 82 + 22 and (8 + 2)2.
Use words or diagrams to help your explanation.
2c.
Explain the difference in meaning between
and 8 + 6 ÷ 2.
Use words or diagrams to help your explanation.
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LESSON 1.2
Activity 31: Matching Cards
1. Take turns at matching pairs of cards that you think belong together.
2. Each time you do this, explain your thinking clearly and carefully. Your partner should either explain that
reasoning again in his/her own words or challenge the reasons you gave.
3. If you think there is no suitable card that matches, write one of your own on a blank card.
4. Once agreed, stick the matched cards onto the poster paper writing any relevant calculations and
explanations next to the cards.
You both need to be able to agree on and to be able to explain the placement of every card.
1
Adopted from http://map.mathshell.org/materials/download.php?fileid=1358
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LESSON 1.2
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LESSON 1.2
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LESSON 1.2
Exit Ticket
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LESSON 2.0
Activity 1: Lunchroom Tables
Leonard Nimoy High School (LNHS) is purchasing new trapezoidal tables for their lunch room from
TablesRUs. The principal, Ms. Counter needs to figure out how many to order. Five students can sit around
one of these tables as shown below.
She can also put two tables together and seat eight students.
She can also put more than two tables together.
1.
Consider the cases below:
a. What happens if you put 3 tables in a line? How many can sit around the table row?
b. What if you put 8 tables in a line? How many can sit around the tables?
c. How many students can sit around 20 tables?
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LESSON 2.0
d. How many students can sit around 100 tables?
e. How many students can sit around t tables?
2.
At LNHS they usually have about 450 students at fourth period lunch and 300 students during fifth
period lunch. Sometimes these tables are taken to the gym for different events. How many tables
would Ms. Counter need for lunch?
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LESSON 2.0
Activity 2: Tables, continued
3.
The janitor reminded Ms. Counter that the cafeteria wouldn’t hold all of the tables in one line. He
estimated that the length of the cafeteria was long enough to set up about 20 tables. With this in
mind, determine the fewest number of tables needed and the expected set-up to seat 450 students.
4.
Mr. Splotch the art teacher is borrowed 15 tables to use for a special joint paper poster assignment.
He’ll be setting them up in three rows. How many students will he be able to seat around them?
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LESSON 2.0
5.
A competing table store, TablesForMore, is selling a new trapezoidal table that seats 7 students around
one table – one on each end, three on the long parallel side and two on the short parallel side. These
tables cost $80 each whereas the TableRUs tables cost $60. Would it be worth it for the school to buy
these tables instead? Show your exploration of the question, work, and findings.
6.
The principal often borrows the trapezoid tables for meetings and sometimes rents them out for
community events. For a small meeting, about 6 people she arranges them as below:
For larger community events they use a set up like the one below or even longer:
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LESSON 2.0
a.How many sit around two tables in the first figure? How many sit around 6 tables in the second figure?
b. Consider a longer table arrangement of this kind. How many would sit around 10 tables? 18 tables?
50 tables? n tables?
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LESSON 2.1
Activity 1
1.
Kayla has exactly 28 cents in her pockets.
What can you say for sure about the coins in her pocket?
2.
Consider the cases below:
a. If Kayla has exactly three nickels, how many coins total does she have?
b. What is the greatest number of coins Kayla can have?
c. If Kayla has at least one nickel, what is the greatest number of pennies she could have?
d. If Kayla has more nickels than pennies, how many coins does she have, in total?
e. Can Kayla have an odd number of coins in her pocket?
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LESSON 2.1
Activity 1 (alternate)
1.
Kayla has exactly 28 cents in her pockets. What can you say for sure about the coins in her pocket?
2.
If Kayla has three nickels, how many coins total does she have?
3.
What is the greatest number of coins Kayla can have?
4.
If Kayla has at least one nickel, what is the greatest number of pennies she could have?
5.
If Kayla has more nickels than pennies, how many coins does she have, in total?
6.
Can Kayla have an odd number of coins in her pocket?
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LESSON 2.1
Activity 2
1. Rommel brought $15 to a school fundraiser and spent it all on hot dogs and game tickets. Hot
dogs were $3 each and tickets were $0.25 each.
a. What can you say about his purchases?
Number of hot dogs ($3
each)
4
Number of tickets
($0.25 each)
8
b. If Rommel bought 23 items total, how many
game tickets did he have?
c. If Rommel bought more hot dogs than game
tickets, how many tickets did he have?
d. How many game tickets does Rommel have
to give up for every additional hot dog he
eats?
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LESSON 2.1
Problem Set 1
Write your questions here:
Q1:
Q2:
Q3:
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LESSON 2.1
Answer the following questions by using the next page:
Q4:
Q5:
Q6: If you only have 3-cent coins and 5-cent coins, what total amounts can you make with just these
two kinds of coins?
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LESSON 2.1
Activity 3
1.
Pose three questions about the given information.
2.
Malika joined Staywell Gym for one month and chose Plan A. Try out some numbers and then find
Malika’s total cost for the month if she makes v visits to the gym.
3.
A toad is 25 feet north of a bullfrog and both of them are jumping north. Every time the toad jumps 1
foot, the bullfrog jumps 3 feet. Write a good question to turn this into a word problem. Then, answer
it.
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LESSON 2.1
Activity 4
1a.
Raj has three times as many one-dollar bills as he does five-dollar bills. He has a total of $32. How
many of each bill does he have?
1b. If Raj has three times as many one-dollar bills as he does five-dollar bills, could he have $42? $40?
Describe the possible amounts of money Raj could have under this condition.
2.
Jacob has exactly 84 cents in his pocket. He only has dimes and pennies. Use some guess and check to
find the number of pennies (p) Jacob has if he has d dimes.
Try some numbers first:
If Jacob has 3 dimes …
If Jacob has 7 dimes …
If he has 4 dimes …
If he has d dimes
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LESSON 2.1
Problem Set 2
Carla’s going to a county fair, and she’s trying to figure out how much money to bring.
She knows she’s going to want to buy exactly 4 food tickets, and she’ll spend the rest of her money playing
games. Food tickets cost $3 each, and games cost $2 per turn.
Write an equation for the total amount of money (m) Carla must bring if she plays g games.
Try some numbers first.
If Carla plays 8 games…
If she plays ___ games…
If she plays ___ games…
If she plays g games…
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LESSON 2.2
Activity 1
1.
There are about 240 million egg-laying hens in the United States.
A laying hen produces roughly 250 to 300 eggs per year.
On July 4, 2015, the population of the United States was estimated to be about 321 million people.
Write three questions that make sense to ask about this situation.
2.
Jay has two more sisters than brothers. How many more daughters than sons do Jay’s parents have?
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LESSON 2.2
Activity 2
3.
In his wallet, Luis only carries one-dollar bills, five-dollar bills, and ten-dollar bills. The total amount in
his wallet right now is $43.
a) What is the largest number of bills he could have?
b) The smallest?
4. Suppose you also know that Luis has four times as many one-dollar bills as ten-dollar bills, and that his
total is still $43.
How many five-dollar bills does he have?
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LESSON 2.2
Problem Set 1
1.
Maria sees a sign for the school’s book fair.
She brought $11 to the fair. She bought nothing but books or tote bags. What can you say for sure
about this situation?
2.
At the same book fair, Maria also sees the following sign.
Write three questions that make sense to ask about this situation.
3.
Jing wants to design and sell cell phone covers. It will cost her $80 a day to rent a kiosk in the mall. The
materials to make one cell phone cover cost about 60 cents.
Write 3 questions that make sense to ask about this
situation. What additional information might Jing
need in order to answer those questions?
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LESSON 2.3
Activity 11
1. Pick a number.
• Add 3
• Multiply it by 2
• Add 4
• Divide it by 2
• Subtract your original number.
• Is your result five?
Briefly explain any ideas you have that might explain the result.
2. Now try this one: Pick a number.
•
•
•
•
•
•
Subtract 1.
Multiply it by 5
Add 4.
Multiply by 2
Add 2.
What is your final number?
3. Start with your ending number from problem 2
•
Walk through the steps of problem two in reverse. So for example the last step was “Add 2” so I will
take my final number and subtract two.
(What is the reverse operation of multiplication?)
• Did you get to your starting number?
1
Some questions taken from COMAP’s Mathematics Modeling Our World and Harold Jacob’s Elementary Algebra.
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LESSON 2.3
Activity 22
1.
Below is a different way of thinking about the problem. Take a couple of minutes to figure out what
the picture is representing. With a partner, take turns explaining the steps:
The number thought of:

Add seven:

Multiply by two:

Subtract four:

Divide by two:

Subtract the number first thought of:
2.
Ian, Maria, and Jacob are working through a similar problem. The bag represents their starting number.
Directions
Draw it
Ian
Mari
Start with a number
4
0
Add 6
10
Multiply by 3
a
Jacob
24
Subtract 12
Divide by 3
Subtract your original
number
a. Explain to a partner how to interpret the table.
b. Complete the table.
c. When Jacob multiplied by 3 he got 24. What was Jacob’s number before he multiplied by 3?
2
Problems adapted from EDC’s Transition to Algebra and Harold Jacob’s Elementary Algebra.
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LESSON 2.3
Problem Set 1
1.
Complete the table.
Directions
Draw it
Kayl
Start with a number
Multiply by 2
a
Raj
Ben
7
18
Add 8
68
Divide by 2
Subtract your original
number
2.
If
+ 1 = 16, what is
?
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3.
If
+ 4 = 16, what is
?
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4.
LESSON 2.3
Complete the table below.
Directions
Start with a number
Draw it
Jing
Kelly
Ian
58
Multiply by 5
15
Add 8
23
43
Subtract your original
number
20
36
12
10
5.
If
6.
If
+ 2 = 34, then
+ 5 = 50, then
= ____?
= ____?
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LESSON 2.3
Problem Set 2
1.
Complete the table.
Directions
Start with a number
Draw it
Add 4
Imani
2
Rob
6
14
Multiply by 3
Eva
42
Subtract your original
number
26
16
Subtract your original
number
13
6
2.
If Malika also got 13 when the picture was
3.
If
= 24, what is
4.
If
+ 5 = 23, what is
, what was her original number (
)?_____
?
?
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5.
Complete the table.
Directions
Start with a number
Draw it
Mali
2
Ash
3
Add 5
Luis
16
28
32
30
35
Divide by 5
75
15
Subtract your original
number
4
6.
If Brandon writes 10 when the picture is
7.
Fill in the table
1
8.
LESSON 2.3
7
6
13
10
3
, what is his number (
14
)?
94
.5
76
6
5
If you were doing math magic with someone in problem 7, what instruction would you give after “start
with a number?”
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LESSON 2.3
Activity 3
1.
Below are directions for another number trick
a.
Complete the table
b. Will the “trick” here work for any number?
c.
Create diagrams for the trick using some symbol for the unknown and circles for the ones.
d. Do your diagrams prove that the trick will work for any number?
2.
The pictures below illustrate the steps of some math magic. Describe what is happening in each step in
words.
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
STEP 6


    
 
 

3.
Below is the beginning of a number trick. Work with a partner to add steps to it that will get a person
to the number 2 from any starting number. Pictures may help.
STEP 1: Start with a number
STEP 2: Triple it
STEP 3: Add Twelve
…
4.
Create a set of steps for a number trick that ends with the number from the start.
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LESSON 2.3
Exit Ticket
Create a “trick” problem and provide a visual model to explain the solution.
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LESSON 2.4
Activity 1
1.
If
=
, then
= _____
2. If
= 16, then
(HINT: If you don’t know what to do, try a guess and see if it works!)
3.
If   + 8 = 30, then  = _____
5.
What number can replace
your answer.
so that the statement “
6a.
What number can replace
in the statement “
6b. What number can replace
in the statement “
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4.
= _____
If   + 1 =  + 5, then  = _____
=
=
” will be true? Briefly explain
?”
=
?”
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7a.
LESSON 2.4
This mobile always balances, why?
7b. This mobile balances when the square is replaced by a certain number. What number is that?
7c.
This mobile never balances. Why is that?
7d. Does this mobile balance always, sometimes, or never? If it does, when is that?
7e. Draw a mobile that always balances:
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7f.
Create a mobile that never balances:
7g.
Draw a mobile that only balances if the square equals 3.
LESSON 2.4
7h. How many moons are in one triangle?
8.
If
= 4, would the statement “
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=
” be correct? Why or why not?
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LESSON 2.4
Problem Set 1
1.
Complete the following questions:
a. If
=
, then
=
b. If
c. If   = 12, then  =
=
, then
=
d. If    = 21, then  =
e. Kylie starts with a number. If she doubles it, the answer is 24. What was her original number?
g. If  - 4 = 5, then  =
f. If  + 3 = 40, then  =
h. What number can replace
2.
so that the statement “
=
” will be true?
The mobile in each picture is balanced. Find the weight of each shape.
A.
B.
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C.
D.
E.
F.
G.
H.
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I.
J.
K.
L.
M.
N.
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O.
P.
Q.
R.
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LESSON 2.4
Activity 2
1.
Complete the table:
3
5
9
10
75
22
0
2.5
3000
2a.
These two mobiles mean the same thing. Solve them both
3a.
These two mobiles also mean the same thing. Finish the second one and solve both.
Translate the picture into algebraic notation:
4. ◊ ◊ + 5 = ◊ + 12
5.
Turn the equation into a mobile:
11. 3c+ 1 = c + 9
12. What is the value of □ in question 10?
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2b
□ □ □ □ + 3 = □ □ + 13
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LESSON 2.4
Practice Set 2
1.
Fill in the missing parts of the table to see some math magic
Directions
Draw it
Start with a number
Describe your
drawing
A bag
Algebraic
Expression
b
Add 3
A bag and 3
b+3
Multiply by 3
3b + 9
Subtract 3
Divide by 3
Subtract your
original
number
2.
Fill in the missing parts of the table to see some math magic
Directions
Draw it
Start with a number
Describe your
drawing
A bag
Add 3
Algebraic
Expression
b+3
Multiply by 2
Add 4
Divide by 2
Subtract your
original
number
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3.
LESSON 2.4
Start with a number
Directions
Draw it
Start with a number
Describe your
drawing
A bag
Algebraic
Expression
Multiply by 2
2 bags
2b
Add 2
Multiply by 2
Divide by 4
Subtract your
original
number
4.
5.
Fill in the table to figure out the pattern. If “n” can be replaced with any number, describe the pattern
using n.
4
7
.5
16
28
2
3
0
20
50
10
5
Fill in the table to figure out the pattern
6
4
10
5
7
48
32
80
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100
4n
3
0
64
8
n
n
72
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Translate these equations into algebraic notation.
6. ◊ ◊ ◊ ◊ – 2 = ◊ ◊ ◊ + 3
7.
 – 5 =  + 6
8.
 + 5 = 20
3d – 2 = d + 6
9.
LESSON 2.4
10. b + 5 = 20
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ALGEBRA
LESSON 2.4
Activity 3
1.
Do these two math magic tricks always give the same result?
TRICK 1
Start with a number
Add 6
Multiply by 2
Subtract 4
Divide by 2
TRICK 2
Start with a number
Multiply by 2
Add 6
Subtract 4
Divide by 2
Do they give the same result? _____ Why or why not?
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2.
LESSON 2.4
Using these magic steps to create a sequence that always results in the number you started with. Test
your magic on four numbers
ADD 10
DIVIDE BY 2
Description
MULTIPLY BY 2
Test 1
Test 2
Test 3
SUBTRACT 5
Test 4
Algebraic
Expression
Start with a
number
n
n
3.
Use the table to figure out the pattern
3
11
5
101
9
33
15
303
8
32
14
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1
10
30
29
n
0
98
20
3n – 1
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ALGEBRA
4.
LESSON 2.4
Akini is describing her work on this problem but part of her answer is missing. Fill in the missing
information
What number can replace
so that the statement “
=
” is true?
“First, I rewrote the problem with an algebraic expression like this: 3b + 5 = 2b + ____
Then, I crossed out ___ bags from each side so that one side had none
So, I simplified the expression by writing: b + 5 = _____
Next I subtracted ________
Finally, I could see that b, the bag, always has to be ____ for the statement to be true.”
5.
Which of these three moves will always keep the mobile balanced?
A. Move all the diamonds from the left side to the right side.
B. Remove a moon from the left side, and a diamond from the right side
C. Remove 2 moons from both sides.
Solve these equations:
6.
If
+2=
7.
If 2b + 2 = b + 7, then b = _____?
8.
If 2b = 22, then b = _____?
9.
If
+4=
+ 7, then
+ 7, then
= _____?
= _____?
10. If 2b + 4 = b + 7
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ALGEBRA
LESSON 2.4
Practice Set 3
1. Fill in the table and find the pattern.
2
4
8
12
2.
24
48
6
7
18
0
56
30
n
6
If you were doing the same math magic in the first problem with someone else, what would you tell
them to do after “start with a number?”
3.
4.
6.
+5=
+2=
+ 8, then
=
+ 12
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5.
7.
2b + 5 = b + 8, then b =
3b + 2 = b + 12, then b =
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8.
9.
Fill in the missing parts of the table
3
11
5
101
6
14
8
12
28
16
1
104
-3
25
3
0
LESSON 2.4
10
n
7
2n + 6
Fill in the missing parts
Directions
Draw it
Start with a number
Algebraic
Expression
Value
4b
Add 4
4b + 4
16
4
Subtract 1
10. Fill in the missing parts
Directions
Start with a number
Draw it
Algebraic
Expression
b
Multiply by 2
Value
28
2b + 14
26
b+7
Subtract your original
number
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11.
+ 8 = 37, then
13. If ◊
=
12. If
◊ ◊ = 24, then ◊ =
14.
+ 23 =
+ 13, then
– 5 = 4, then
LESSON 2.4
=
=
15.
16. Fill in the missing parts
3
11
5
101
7
15
9
105
21
45
27
315
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46
1
-1
10
3
n
4
18
3n + 12
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ALGEBRA
LESSON 2.4
17. Represent this equation using a picture:
3b + 8 = 4b + 6
18. Translate this equation into an algebraic expression:
 – 8 =  + 4
19. Mali starts with a number. If she multiplies her number by 7, the answer is 42. What was her original
number?
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ALGEBRA
LESSON 2.5
Activity 1: Who Am I? Puzzles
Algebraic Habits of Mind: Puzzling & Persevering1
To solve “Who Am I?” puzzles you have to play with the digits both separately and together to find the
mystery number. When solving puzzles, always start with what you know. These clues will eventually
help you narrow the answer, but there can be many ways to get there. Try anything that looks hopeful.
0
1
4
9
16
1
Transition to Algebra, EDC, 2014
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ALGEBRA
LESSON 2.5
Activity 2
1.
8.
For each pair of numbers figure out the distance between the two. Use the number line to help.
a.
5 and 9
b.
-1 ½ and -2
c.
3 ½ and 4
d.
-2 and 8
e.
7.5 and 9
f.
7 ¼ and 8
g.
6 and 7 ¼
h.
9 ¾ and 10
i.
-1 and 1
j.
2 ½ and 8
k.
-4 and 11
l.
5 and 5
What is the distance between -3 and 4 ¼ ?
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9.
LESSON 2.5
What is the distance between 5 1/3 and 10?
10. Answer True or False for each statement a-d. If the statement is false give a specific example that
proves the statement is false. Since the variable can be any number, all you need is one example to
show the statement is false.
a. 5r is always positive (false) EX. If r is -2, 5r =
b. -m is always negative (false)
c. If p is a positive number, then p + 10 is always positive (true)
d. If k is a positive number, then k – 8 is always negative (false)
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ALGEBRA
LESSON 2.5
Problem Set 1
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G.
LESSON 2.5
Identify six pairs of numbers that are 5 units apart.
(For example -100 and -105 are 5 units apart; I could also say -3 and 2 are five units apart)
H.
Identify two pairs of numbers that are 3.5 units apart.
Reflection: Do you feel like you have improved your skills in solving the “Who Am I?” problems? What are
your strategies?
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ALGEBRA
LESSON 2.5
Algebra 2.5: Activity 32
Try these:
2
Source: Lesson adapted from TTA from EDC.
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ALGEBRA
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LESSON 2.5
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ALGEBRA
LESSON 2.5
Algebra 2.5: Activity 43
In each problem, multiply the middle number by itself and the two outside numbers together. Try to work
without a calculator.
1.
2.
3.
4.
5.
3
6.
7.
8.
Source: Lesson adapted from TTA from EDC.
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ALGEBRA
LESSON 2.5
Some patterns are easy to describe with words. For example “start with 1, add 2, then add 2 again and
again.” Other patterns can be much harder to describe with words. Algebra provides a language that makes
it easier to describe mathematical patterns.
9. How can you complete the empty boxes?
10. Set up problems to explore comparing a number squared with the product of the number that is two (2)
greater than the center number with the number that is two (2) less than the center number. Below is an
example:
a. 1 x 5 = 5 and 3 x 3 = 9: They have a difference of ___.
b. 6 x 10 = 60 and 8 x 8= 64: They have a difference of ___.
c. Is there a conjecture you might pose?
d. What would explain why these results occur?
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ALGEBRA
LESSON 2.5
Algebra 2.5: Problem Set 24
Solve the following:
4
Source: Lesson adapted from TTA from EDC.
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ALGEBRA
LESSON 2.5
Complete the problems below:
F.
G
H.
I.
Find the products of the numbers below without a calculator. Use your work from the number lines to
help.
J. 999 x 1001=
K. 7 x 9=
L. 49 x 51=
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ALGEBRA
LESSON 2.5
Solve the MysteryGrid puzzles and Mobile problem:
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LESSON 3.1
Warm-up Activity
Task: Dae gives directions from the baseball field to her house:
•
•
•
•
•
•
You should be on Clark right now, heading south.
Turn right on Addison
After about 4 blocks, turn left on Western.
At the next street, turn right on Belmont.
When you see a police station on the corner, turn right.
My housed is the fourth building on the left.
Create a map and directions to return to the baseball field from Dae’s house.
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ALGEBRA
LESSON 3.1
Activity 1
Another way to represent equations is through arrow diagrams. Consider the equation 2x + 1 = 17. It can be
represented with the following arrow diagram:
1. Explain why this arrow diagram is equivalent to the given equation.
To solve this equation using the arrow diagram, you need to work in reverse beginning with the 17:
a. What is the opposite of adding 1?
b. What is the opposite of multiplying by 2? Is anything else considered the opposite of multiplying
by 2?
c. What would be the bucket and circles picture corresponding to this?
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2.
a.
Consider the following:
Write the expression 3(a + 2) – 1 as a sequence of steps describing the operation.
b.
Write the steps to reverse the operations.
3.
a.
What about a problem such as the following: 5(3x+2)-1 = 69
Create an arrow diagram for the equation.
b.
Solve the equation by working through the arrow diagram in reverse.
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LESSON 3.1
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ALGEBRA
4.
LESSON 3.1
Which step can you use to undo the operation in the equation Error! Bookmark not defined.
? Select A, B, C, or D.
I. Multiply by
A. I only
5.
II. Multiply by
B. III only
C. II or III
III. Divide by
IV. Divide by
D. I or IV
Jasmine asks you to choose a number, add 10 and then find the remainder when you divide by 7.
When you tell her your ending number, can Jasmine always find your starting number? Explain.
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ALGEBRA
LESSON 3.1
Problem Set 1
Solve the following equations in at least 3 ways: a) using arrow diagrams, b) using buckets and circles, c)
using only equations, d) staring and thinking about it.
1. 36 = 2x-10
2. 3(x-6)+2=38
3.
Consider the following equation: c = 5(p + 2) – 3
Create an arrow diagram for it.
Write an equation solving for p. Let your initial arrow diagram begin with p.
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4.
Solve each of the following equations:
a.
3(a + 2) – 1 = 17
b.
3(a + 2) – 1 = 8
c.
d.
3(a + 2) – 1 = 0
3(a + 2) – 1 = 19
LESSON 3.1
5.
Problem 3 involved two variables, p and c. The four equations in problem 4 involved only one variable
in each. What kinds of differences does this create?
6.
Andrew explains, “I choose a number. I add 1 and multiply by -4. Then I add 2. My ending number is
22.” What is Andrew’s starting number?
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LESSON 3.1
7. Create arrow diagrams for the following equations and then solve them:
a.
b. 3(a — 1) — 5 = 34
8. Explain how working backwards with the arrows can simplify solving an equation.
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LESSON 3.1
Problem Set 2
1.
For each of the following, create an arrow diagram to represent the given equation. Then solve the
equation using the arrow backward method:
a. –5x + 13 = 93
b. 13 – 5x = 93
c. –7x + 12 = –45
d. 12 – 7x = –45
e. –3x – 17 = 4
f. –17x – 3x = 4
Hint for b. What if you rewrite it as 13 + -5x = 93. What was the change made? Would that always work?
Does it look familiar?
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2.
LESSON 3.1
Create an arrow diagram to represent the given equation. Then solve the equation using the arrow
backward method:
a. 10(x + 3) = 73
b. 10x + 3 = 73
c. 73 = 10(x + 3)
d. 73 = 10x + 3
e.
f.
g.
h. 73(3x) = 10
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LESSON 3.3
Activity 1
Use the area model to solve each problem
1.
7 x 13 = _____ + _____ = ______
2.
15 * 24 = ______
Use the area models to help fill in the blanks in the problems below.
3.
4.
15 * 44 = _____
25 * 14 = _____
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5.
LESSON 3.3
6.
26 * 51 = ______
541 * 62 = _______
7. Which of the following expressions corresponds to the diagram? Circle all that apply
A. (3 + 3) (5 + 7)
B. 15 + 21 + 15 + 21
C. (3 + 5) (3 + 7)
8. What do these three expressions have in common?
i. (6 + 3) (4 + 5)
ii. (7 + 2)2
iii. 9 * (4 + 5)
A They all have the same final answer
B They are just different ways of writing 9 * 9
C They have nothing in common
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Use the area models to complete the following.
9.
LESSON 3.3
10.
2 * (30 + 4) = ______
11.
2 * (3x + 4) = _____
12.
4 * (20 + 6) = _____
4 * (2b + 6) = _____
The rectangles below are made of smaller rectangles. Fill in the missing parts as shown in the example.
Then find the base, height, and area of the large rectangle. Note: The drawings are not to scale.
Example:
Area = 7 (x + 2) (base x height) = 7x+14 (total)
13.
Area = 6 * (2x + 5) (base x height) = _____________ (total)
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LESSON 3.3
Fill in the missing pieces in these multiplication problems.
14.
15.
-6 (b + 5) = ________________
16.
-5 (3b + 5) = __________________
17.
7 (y – 2) = __________________
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7 (4k + 2) = __________________
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LESSON 3.3
Fill in the missing pieces in these multiplication problems.
18.
(b – 5)(3c – 4) = _____________________
19.
(2b – 5)(3c – 4) = _____________________
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LESSON 3.3
Problem Set 1
Fill these in completely.
1.
2.
32 * 101 = _____ + _____ + _____ + _____
= __________________
= _______
3.
4.
10 * 63= __________________
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5 * (30 + -1) = __________________
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5.
LESSON 3.3
6.
10 (y + 3) = __________________
2 (4x – 3) = __________________
8.
7.
h (k – 7) = __________________
h (7 – k) = __________________
9.
10.
(y – 3)(x + 6) = __________________
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(2a + 5)(b – 3) = __________________
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LESSON 3.3
11. Use an area model (on paper or in your head) to solve the problems below:
a. 2 x 43
b. 31 x 52
12. What number am I?
• I am a multiple of 6
• I am a two digit number with t as my tens digit and u as my ones digit.
• t=2u
• I am between 52 and 82 but I am not a perfect square.
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LESSON 3.4
Activity 1: Rewriting expressions1
One way to visualize distribution for a problem is by writing it out. For example, 5(x+1) can be written out as
a repeated addition problem and then rearranged.
5(x + 1) = (x + 1) + (x + 1) + (x + 1) + (x + 1) + (x + 1)
=x+x+x+x+x+1+1+1+1+1
= 5x + 5
1. Use the method shown above to prove that:
a.2(x + 6) = 2x + 12
b. 4(x + y) = 4x + 4y
c. 3(x2 + 2) = 3x2 + 6
1
Sources: Jacob Algebra I; TTA
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LESSON 3.4
Another way to show the result is through an area model. 5(x+1) could be the area of a rectangle with sides
5 and (x+1).
2a. Express the area of the given rectangle. (We express a rectangle’s area as length x width.)
i.
ii.
iii.
iv.
2b. Use the distributive property (or its reverse) to express the areas above in alternative ways.
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LESSON 3.4
Problem Set 1
1.
Draw a rectangle for each of the following and rewrite the area of the rectangle in an expression
without parentheses.
a) 3(w + 5)
b) 2(x + y)
c) 4(x2 + 2)
2.
Use the method from the first example in Activity 1 to justify a version of the following without
parentheses.
a) 3(w - 5)
b) 2(x - y)
c) 4(x2 + - 2)
d) 3(x - 20)
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e) 3(-x - 20)
LESSON 3.4
f) 3(x + - 20)
g) -3(x - 20)
You may be familiar with the results above as coming from the distributive property. By visualizing it with
the area model, your work above is another way to see why it is true.
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LESSON 3.4
Activity 2
We know that 2(x + y) = 2x + 2y. Is it also true that (x + y)2 = x2 + y2? To answer this it is helpful to consider
the following cases.
1.
3.
Evaluate (x + y)2
2. Now evaluate x2 + y2
a)
if x is 2 and y is 0
a)
if x is 2 and y is 0
b)
if x is 0 and y is 6
b)
if x is 0 and y is 6
c)
if x is 3 and y is 4
c)
if x is 3 and y is 4
d)
if x is 9 and y is 1
d)
if x is 9 and y is 1.
What can you conclude about (x + y)2 and x2 + y2?
Might checking only two cases mislead you?
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4.
LESSON 3.4
Find the number of times you need to repeat an operation to get to the ending number. Keep track of
your trials and thinking. If you cannot get to the ending number, explain why.
Starting Number
a. 7
b. 1
c. 15
d. 0
e. 12
f. 4
g. 7
Repeated Operation
Add 4
Multiply by 2
Subtract 5
Multiply by 8
Divide by 2
Multiply by -1
Multiply by 7
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Ending Number
91
2,048
-100
0
3/32
4
more than 1000
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LESSON 3.6
Activity 1
1.
Each expression below is the left side of an equation. Write an expression for the right side so that x =
3 is a solution of the equation.
Hints: Think of the problem in a couple of ways. What do you want your last step to be?
Would you want to use arrows and working backwards? Could there be more than one solution?
Would simplifying what is given first help?
a. 2(x + 1)
b. 3x — 1
c. 7x + 5 + 2x — 1
d. Make up your own expression
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2.
LESSON 3.6
Re-do problems a thru c from the first problem by writing an expression for the right side so that x =
10.
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LESSON 3.6
3. Solve each equation:
a.
5d — 2 = 2d + 10
b. 5d — 2 = 3d + 10
c.
5d — 2 = 4d + 10
d. 5d — 2 = 5d + 10
e.
5d — 2 = 6d + 10
f. 5d — 2 = 7d + 10
g.
5d — 2 = 8d + 10
h.
Which of the above equations has a different number of solutions?
i.
Describe any other pattern you see in the solutions from a-g.
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4.
Solve each equation:
a.
4c + 2 = 4c — 1
b. 4c + 2 = 4c
c.
4c + 2 = 4c + 1
d. 4c + 2 = 4c + 2
e.
Which of the above equations has a different number of solutions?
f.
Describe any other pattern you see in the solutions
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LESSON 3.6
Activity 2
Here are two basic moves in solving equations:
Assumption: The Basic Moves of Solving Equations
1. If you start with an equation and add the same number to each side, you do not change the solutions
of that equation.
In symbols, for any three numbers a, b, and c
a = b if, and only if, a + c = b + c
2. If you start with an equation and multiply each side of the equation by the same nonzero number, you
do not change the solutions of that equation.
In symbols, for any three numbers a, b, and c, where c =/= 0
a = b, if an only if, ac = bc
1.
Provide the steps for solving the equation below using the “basic moves” described above:
73x – 15 = 48x + 99
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LESSON 3.6
2.
Describe the one basic move you can use to transform the equation 3t + 13 = 5t + 6 into each given
equation below:
a.
3t + 7 = 5t
b. 2t + 13 = 4t + 6
c.
3t + 113 = 5t + 106
d. 13 = 2t + 6
e.
- 2t + 13 = 6
f. 15t + 65 = 25t + 30
3.
Choose three equations from the six above in number 2 and solve them. How do you explain the
results?
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LESSON 3.6
4.
Solve the equation below first using the arrow method and then using the basic moves method.
4(x – 7) + 13 = 27
5.
Determine whether the given solution is correct or whether it involves a mistake. If there is a mistake,
explain the mistake.
b.
c.
a.
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LESSON 3.6
Problem Set 21
1
1.
Describe the one basic move you need to make in order to change the equation 2x + 17 = 36 into each
equation below:
a.
2x + 13 = 32
b. 2x — 12 = 7
c.
2x = 19
d. 6x + 51 = 108
e.
x + 8.5 = 18
f. 5x + 17 = 3x + 36
g.
2x — 19 = 0
h. x + 17 = 36 — x
i.
What can you predict will be true if you solve the eight equations above?
Source: (CME , page 155)
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2.
LESSON 3.6
Here is one of Maya the Magnificent’s number tricks:
Choose any number.
Multiply by 3.
Add 5.
Multiply by 4.
Add 16.
Divide by 12.
Subtract the starting number.
Maya says, “I know what your final answer is!”
a.
Let your starting number equal x. Record the result and simplify the expression after each step.
b.
Does Maya know for sure what your final answer is? Explain.
3a.
For how many numbers n is (5n + 12) equal to (5n + 13)? Explain.
3b. Solve the equation 5x + 12 = 5x + 13. What are your results?
c.
What does the resulting equation tell you about the solutions to the starting equation?
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4.
a.
LESSON 3.6
STANDARDIZED TEST PREP: Solve the equation 17 — (5 — p) = 2(5p — 16). What is the value of p?
b. 4
c.
d. 5
5. Solve the equation 5(x+1) = 5x+5. What does your result indicate?
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LESSON 3.6
Activity 3
1. Solve each equation.
a.
5x = 2x + 21
b. 5(x — 90) = 2(x — 90) + 21
c.
5(x — 100) = 2(x — 100) + 21
d. 5(x — 8) = 2(x — 8) + 21
e.
5(x — 40) = 2(x — 40) + 21
f. 5(x + 100) = 2(x + 100) + 21
g.
What is a pattern in your calculations and the result?
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LESSON 3.6
Activity 4: Multi-step to Two-step
Multi-step to Two-step
A multi-step process can be thought of as a two-step process in disguise. Below is an example of how a
multi-step process can be expressed as a simple two-step process using the distributive property:
Consider the expression:
Distribute the 3 by multiplying:
Simplify multiplications
Combine 24 – 12 to get 12
3(2x + 8) – 12
3(2x) + 3(8) – 12
6x + 24 – 12
6x + 12
So we have shown that the multistep process 3(2x + 8) – 12 is really equivalent to 6x + 12. Use this in
working on the problems below.
1. Simply each expression:
a.
x + 2x + 3x + 4x + 5x — 12x
d.
x + 2x + 3x + 4x + 5x — 15x
b.
x + 2x + 3x + 4x + 5x — 13x
e.
x + 2x + 3x + 4x + 5x — 16x
c.
x + 2x + 3x + 4x + 5x — 14x
f.
x + 2x + 3x + 4x + 5x — 17x
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LESSON 3.6
2.
a.
Use the distributive property to convert the multistep processes below into two-step processes.
y = 3(5x + 4) — 6
b. y = 2(2x + 12) + 15
c.
y = 5(x + 3) — 9
d. y = 8(10x — 3) — 12
3. Simplify the following into two-step expressions.
a.
4(x + 2) + 11
b. x + 2(5 + 2x)
c.
9(2x — 5) — 3
d. 5(x — 1) + 8(x + 1)
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e. 7(x + 1) + (7x + 7)
LESSON 3.6
f. 7(x + 1) + (—1)(7x + 7)
4. Simplify the following:
a.
2(x + 4) + 7
b. 13 + 3(1 + 2x)
c.
3(2x — 5) — 8
d. 4(x + 3) + 7(x + 3)
e.
6(3 — 2x) — 3(x + 1)
f. 4(x — 7) — 2(2 — 3x)
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LESSON 3.6
Exit Ticket
Solve the given equation using the two “basic” moves where appropriate. Explain your steps in solving
the equation.
-3(x + 2) + 6 = x - 40
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LESSON 3.7
Activity 11: If this mobile balances…
If the mobile in the first column balances, does the mobile in the second column also balance? How do you
know?
This mobile DOES BALANCE.
Will this one balance?
How do you know
A
B
C
D
1
Transition to Algebra, Educational Development Center, 2014.
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LESSON 3.7
E
F
G
H
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LESSON 3.7
Activity 2
In each problem, the first mobile is balanced. Figure out whether or not the second mobile must
balance and why.
1
This mobile DOES BALANCE.
Will this one balance?
How do you know
2
3
4
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LESSON 3.7
Translate each mobile into an algebraic expression. Use c for the circle, s for the square, and m for the
moon.
5.
6.
7.
8.
Which of the following moves would keep the mobile balanced?
a. Add a triangle to both sides
b. Add 5 droplets to both sides
c. Move all the triangles to the right side
d. Switch the droplet and the circle
e. Add a circle to the right side
f. Remove one triangle from both sides
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LESSON 3.7
14. Look over your answers. What are moves allowed on a mobile? How do they compare to what is
allowed with an equation? What are moves you can’t make?
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LESSON 3.7
Problem Set 1
1.
Which of the following moves would keep the mobile balanced? Circle all that apply
c.
2.
a.
Add 3 circles to the right side
b. Add 2 rhombuses to both sides
Move the shapes so that all 5 circles are on the left side
and both squares are on the right side
d. Subtract a circle from both sides
e. Cross out both squares
f.
Subtract 3 circles from the left side
Which of the following moves would keep the mobile balanced? Circle all that apply
c.
3.
a.
Add 4 drops to both sides
b. Remove a square from both sides
Move all the buckets to the left side and the droplet to the
right side
d. Add a bucket to both sides
e. Switch the rhombus and the circle
f.
Switch the rhombus and the droplet
Which of the following equations must also be balanced? Circle all that apply
a.
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c+s+s+c=m+m+c
b. 2s + c = 2m
c.
6c + 6s = 6m + 3c
d. 2m + c = c + 2s + c
e. 3s + 2c = 2m + c + s
f.
3c = 2m + 2s
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LESSON 3.7
4.
5.
6.
7.
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LESSON 3.7
Activity 3
What is your first step?
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LESSON 3.7
Activity 4
1-4. Solve both the mobile and the equation and check that you get the same answer:
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5.
LESSON 3.7
Which of the following would work as a first step to solving this equation (use s for square).
a.
b.
c.
d.
e.
Subtract s from both sides
Subtract 9 from both sides
Add 12 and 27 so the right side becomes 39 + s
Move all the squares to the left and all the numbers to the
right to get 5s = 48
Combine all the squares on the left side to get 4s + 9 = 12 +
27 + s
6.
Read the box below and complete the questions that follow.
Algebraic Habits of Mind: Using Tools Strategically
• Mobiles are useful tools for visualizing balance. They help make sense of algebra since solving
algebra problems always requires the equation to stay balanced.
• Mobiles are not always useful, however. For instance, how would you represent ½ or -5 on a
mobile?
• You can’t draw a mobile for every equation. But every equation balances. And anything you do to
an equation has to keep the balance.
a.
Create a problem with a negative number and show how you might solve it.
b.
Solve the following: Donald uses a clicker to help him figure out costs. Every time a student walks into
the gym, he clicks. Every student will receive a coupon for a burger costing $4.59, and Donald buys
these at face value. If there were 28 students that came in, how much will Donald need for all for all of
the coupons?
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LESSON 3.7
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LESSON 3.7
Problem Set 2
1.
2.
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3.
LESSON 3.7
a. Which of the following would work as a first step to solving this equation:
10 + 3d + 2 = 6 + 5d
i.
ii.
iii.
iv.
Add 10 + 2 = 12 on the left
Subtract 3d from both sides
Subtract 6 from both sides
Subtract 2 from both sides
b. Solve the equation for d.
Directions
Start with a number
4. Think of a number
Draw it
Algebraic
Expression
Ian
Maria
Jacob
Multiply by 3
Add 5
3b + 5
Multiply by 2
-2
Subtract 8
5.
29
20
Complete the number line below by filling in the missing spaces. The number line is drawn to scale.
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6.
LESSON 3.7
Use an area model to multiply these expressions.
a.
b.
c.
d.
7. Solve the given Mystery Grid. Explain why you can be certain of your answers to various boxes.
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LESSON 3.7
Exit Ticket
Which of the following would work as a first step to solving this equation:
5c + 4 = 2c + 18 + c
a.
b.
c.
d.
e.
f.
On the right side, turn 2c + c into 3c
Subtract 2 from both sides
Subtract 3c from both sides
Combine all the c’s and all the numbers to get 8c = 22
Subtract c from the right and add it to the left side to get 6c + 4 = 2c + 18
Subtract c from the left side and move it to the right to get the same number of c’s on both sides.
4c + 4 = 4c + 18
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LESSON 3.8
Activity 1
1.
Create expressions for each of the following:
a. 10 is less than x
b. 10 less than x
c. x less than 10
d. x is less than 10
Let x = 50. How can you think about and evaluate a-d above? Which statements become
impossible?
Fill in the missing information.
2.
Instructions
Start with a number
Multiply by 4
Result
c
3c
(3c – 6)
4(3c – 6)
4(3c – 6) + 13
3.
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Instructions
Result
Start with a number
m
LESSON 3.8
Add 2
4.
Multiply by 7
7(
Divide by 2
7(
Subtract 8 from the result
7(
Instructions
Result
Start
g
2
2
)
)
g-3
5.
Divide by 4
g–3
4
Instructions
Result
Start with a number
v
Multiply by 3
Subtract the result
from 8
Multiply that by 2
(use parentheses)
Add 11
*Test out the language to see the difference between “Subtract from 8” and “Subtract 8”
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6.
7.
LESSON 3.8
Answer the problems below.
Instructions
Result
Start
n
Add 11
8.
Instructions
Result
Start with a number
z
Subtract 3
Multiply by 4
Add 10
Divide by 5
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LESSON 3.8
Problem Set 1
Fill in the missing information.
1.
Instructions
Result
r
r+3
Divide 23 by the result
* Test the language to see the difference between “Divide 23 by the result” and “Divide by 23”
2.
Instructions
Result
j
Divide 30 by the result
Subtract 24
Multiply by 6
Add 72
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3.
LESSON 3.8
Work with a partner to place the following numbers on the number-line:
1.5, 1.05, 0.5, 0.05, , , , , ,
4.
5.
Write an algebraic expression corresponding with the area model below.
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LESSON 3.8
6.
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LESSON 3.8
Activity 2
Blank Start with a Number Table. (Instructions from teacher)
Instructions
Result
Start with a number
n
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LESSON 3.8
Activity 3
Arrange the steps below to create the correct sequence that will produce the final result (in the last
row). The steps for both problems 1 and 2 are listed below.
1.
Add 12
Divide by 5
Multiply by 7
Instructions
Result
Start with a number
w
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2.
Add 12
Divide by 5
Multiply by 7
Instructions
Result
Start with a number
d
LESSON 3.8
Subtract 3
The steps for problems 3 and 4 are listed below.
Divide by 7
Subtract 31 Add 4
Multiply by 5
3.
Instructions
Result
Start with a number
m
4.
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Instructions
Start with a number
LESSON 3.8
Result
f
For 5-7 Write out the steps needed to arrive at the algebraic expression
5.
7.
6.
__________________
___________________
___________________
__________________
___________________
___________________
__________________
___________________
___________________
__________________
___________________
___________________
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LESSON 3.8
Match each algebraic expression on the left with the set of directions on the right. (Hint: Use B twice)
A.
Start with a number
8.
Multiply by 5
Divide by 3
Add 14
9.
B.
Start with a number
Multiply by 5
Add 14
Divide by 3
C.
Start with a number
Add 14
Multiply by 5
Divide by 3
D.
None of the above
E.
Start with a number
Subtract 2
Multiply by -7
Add 11
Divide by 4
F.
Start with a number
Subtract 2
Multiply by -7
Divide by 4
Add 11
G.
Start with a number
Multiply by -7
Subtract 2
Divide by 4
Add 11
10.
11.
12.
13.
14.
15.
16.
H. None of the above
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LESSON 3.8
Problem Set 2
Write an algebraic expression to match the given sequence of steps.
1.
Start with a number.
Multiply by 4.
Add 8.
Divide by 3.
2. Start with a number.
Add 5.
Divide by 12.
Subtract 18.
Multiply by -6
3-6. Complete the blanks on each of the following:
3.
4.
5.
6.
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7.
8.
(4y – 8)(7 – y) =
9.
LESSON 3.8
(x + 4)(x + 5) =
Complete the puzzle by using each numbers 1 through 5 only onetime in each column and row. Your
clues are in each section bordered by a thick line: The numbers in those sections will be combined using
the given arithmetic sign (+, -, x, ÷) in order to equal the resulting number that is provided. For example,
in the upper-left corner, the clue is “40, x” and you have 3 squares (the corner square shares a row with
a second square and a column with the third square). What combinations of three numbers would fit
this scenario? How about 4 x 5 x 1? Or 2 x 2 x 5, which would work only if the 5 is in the corner square so
that the two 2s are in separate rows and columns.
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LESSON 3.8
Activity 4
Sort each of the following expressions by the indicated final instruction. For example, in expression A,
the final instruction is “Multiply by 4.”
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LESSON 3.8
Activity 4: Alternative Problems
7(8m – 6)
(m + 3)/9 – 7
(7m + 6)/8
7(m/3 + 8)
7(m + 3) + 8
6(4m + 1) – 7
(m – 7)/3 + 8
(7(m – 2) + 6) / 8
2 – 8m
8 + 6/(m – 2)
(2 – (m + 8))/6 – 7
7(8 + (2m + 6)/3)
2 – 7(m + 9)
6 / (7(m + 8) – 7)
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LESSON 3.8
Activity 5
Can you figure out the last step? For problems 1 and 2 below, state what the last instruction had to be. Write
the last instruction for each Start with a Number problem
1.
2.
Instructions
Result
Start with a number
d
Instructions
Result
Start with a number
z
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LESSON 3.8
For 3-6 write the last instruction for the Start with a Number problems and use the words “the result”
to refer to the rest of the expression. Remember to think about the language for subtraction and division
(“from/by the result” and “the result from/by …”)
Example:
___Divide 15 by the result____
3.
9 – 4k
__________________________
4.
3(7r – 5) – 4
__________________________
5.
3 + 6(y – 7)
__________________________
6.
7.
__________________________
Make up an expression with the variable m and with at least 3 steps and with the final instruction being
“Add 25 to the result.”
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LESSON 3.8
Activity 6
1-18 Match the expression on the left with the description on the right
1.
10 - h
a. 10 less than h
2.
h - 10
b. 10 more than h
3.
h + 10
c. h is less than 10
4.
h < 10
d. h less than 10
5.
10 < h
e. 10 is less than h
6.
6 – r2
a. Subtract 6 from r and then square the result
7.
(r – 6)2
b. Square r, then subtract 6
8.
r2 – 6
c. Square r, then subtract the result from 6
9.
(r – 6)2
d. Subtract r from 6, then square the result
10.
2s – 8
a. Subtract 6 from r and then square the result
11.
s–8
b. Square r, then subtract 6
12.
8 – 2s
c. Square r, then subtract the result from 6
13.
8–s
d. Subtract r from 6, then square the result
14.
s – 16
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15.
25/(c + 4)
a. Divide 25 by c, then add 4 to the result
16.
(25/c) + 4
b. Divide c by 25, then add 4 to the result
17.
(c/25) + 4
c. Add 4 to c, then divide by 25
18.
(c + 4)/25
d. Add 4 to c, then divide 25 by the result
LESSON 3.8
Write the expression for the following statements:
19. Four less than twice some number n.
20. Four minus twice some number n.
21. Subtract 3 from half some number n.
22. One less than the square of some number.
23. Sujiter’s age is twice that of his sister. If Sujiter is 16, how old is his sister?
24. Prasanna’s age is twice that of his brother. If his brother is 16, then how old is Prasanna?
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LESSON 3.8
Problem Set 3
Write the last instruction for each Start with a Number problem. Use the words “the result” to refer to the
rest of the expression.
1.
____________________________
2.
____________________________
3.
____________________________
4.
____________________________
5.
7(n + 1)
____________________________
6.
____________________________
7.
____________________________
8.
____________________________
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LESSON 3.8
Match the algebraic expression on the left with the description on the right
9. Add 40 to x, then divide the result by 16
A.
10. Divide x by 16, then add 40 to the result
B.
11. Divide 16 by x, then add 40 to the result
C.
12. Add 40 to x, then divide 16 by the result
D.
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LESSON 3.8
Write an algebraic expression to match each sentence, using n for the unknown number
13.
Think of a number n, divide it by 7, then add 2
14.
Think of a number n, multiply it by -6, then add 11 to the result
15.
Think of a number n, subtract 5 from it, then multiply the result by -2
16.
17. Use the balances below to figure out a-c:
a.
b.
c.
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LESSON 3.8
18. Complete the puzzle by using each numbers 1 through 4 only onetime in each column and row. Your
clues are in each section bordered by a thick line: The numbers in those sections will be combined using
the given arithmetic sign (+, -, x, ÷) in order to equal the resulting number that is provided.
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LESSON 3.9
Activity 1
1.
17 –
4.
15
⨁
= 2, What is
?
= 3, What is ⨁?
2.
5.
Solve 7-14 and show your process.
7. 15 – (k – 2) = 10
k - 2 = _____
k = _____
9.
3 + 4(y – 3) = 23
17 – 3x = 2
15
𝑏𝑏+2
3.
6.
=3
8.
54
𝑦𝑦+2
17 – (b + 8) = 2
15
5𝑦𝑦
=3
=9
y+2 = _____
y= _____
10. 27/(2n – 11) = 3
4(y - 3) = _____
y - 3 = _____
y = _____
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11. Think of a number, subtract 7, multiply by 5,
add 3. Tanuja followed these instructions
and got 28 as her final result. What number
did she think of?
13.
3𝑘𝑘+6
4
=9
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LESSON 3.9
24
12. 16 − 𝑎𝑎−5 = 10
14. Think of a number, double it, add six,
multiply the result by 3. Kavita followed
these instructions and got 72 as her final
result. What number did she start with?
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LESSON 3.9
15-16. Work with a partner to solve the following:
17. Now create your own equation to solve using the chunking or blob strategy and then show your
solution.
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LESSON 3.9
Problem Set 1
Solve each equation. Show your process.
A. 14 – 5x = 4
C.
30
𝑎𝑎+4
=5
B.
6 + (m – 3) = 20
D.
𝑝𝑝+4
= 12
100
=4
E
24 – (y + 8) = 12
F.
G.
6(10 – k) + 7 = 13
H.
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5
3𝑏𝑏+7
8
7−ℎ=3
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I.
1 + 4(3c + 2) = 81
K.
M.
J.
LESSON 3.9
10
7 − 8−6𝑦𝑦 = 2
L.
Draw an area model and solve the equation below:
( ___ + ___ ) ( ___ + ___ ) = x2+11x+30
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N.
LESSON 3.9
Write an algebraic expression to match each sentence, using u for the unknown number.
Subtract a number u from 12
12 minus a number u
12 less than a number u
_______________________
________________________
_______________________
O.
Draw a number line to answer P and Q.
P. I am exactly halfway between 16 and 34. Who am I?
Q. I am exactly halfway between -0.03 and 0.07
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LESSON 3.9
Activity 2
Solve each of the following and show your process.
1.
(c + 2)2 = 36
2.
(y – 1)2 = 81
4.
(3x + 4)2 = 100
c + 2 = _____ OR c +2 = _____
c = _____ OR c = _____
3.
(10 – n)2 = 64
10 - n = ____ OR 10 - n = ____
n = ____ OR n = ____
5.
3(h + 2)2 = 75
(h + 2)2 = ____
h + 2 = ____ OR h + 2 = ____
6.
Think of a number, subtract 3, square the
result. If you end up with 9, what are the
only two numbers you could have started
with?
h = ____ OR h = ____
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LESSON 3.9
Problem Set 2
Solve and show your process.
A. (a + 8)2 = 100
B.
(w – 7)2 = 64
C.
(11 – b)2 = 36
D.
(2n – 3)2 = 49
E.
(h – 12)2 + 7 = 88
F.
32 – (m + 1)2 = 7
G.
9(8 – x)2 = 36
H.
(𝑏𝑏+4)2
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3
= 27
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J.
L.
17 – (c + 7)2 = 16
Directions
Start with a number
K.
LESSON 3.9
4(9 – k)2 – 11 = 25
Result
Ian
Maria
Jacob
6
9
Add 6
Multiply by 3
30
Subtract 12
M.
N.
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O.
LESSON 3.9
P.
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LESSON 4.0
Algebra Lesson 4.0
Activity 1, Part 1
Sitting around the pond in central park, my friend and I found ourselves surrounded by squirrels and
ducks. My friend asked me to count the heads while he counted the feet. He counted 70 feet and I
counted 19 heads. Then some kids came by and scared all the ducks and squirrels away. How
many ducks and how many squirrels had there been?
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LESSON 4.0
Activity 1, Part 2
Sharing solutions
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LESSON 4.1
Algebra Lesson 4.1
Mini-Lesson
Tamika bought 4 pairs of jeans and 2 sweaters on sale for $116.
Pose a few questions from the statement given above.
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LESSON 4.1
Activity 1, Part 1
In Mr. Tomatoehead’s garden, there are three times as many broccoli plants as there are corn plants.
Individually generate three questions based on the statement above.
Rotate questions and wait for teacher instructions.
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LESSON 4.1
Activity 1, Part 2
In Mr. Tomatoehead’s garden, there are three times as many broccoli plants as there are corn plants.
Who set up a table? What did your table look like?
What is a rule for the table’s pattern?
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LESSON 4.1
Activity 2
In Mr. Tomatoehead’s garden, there are three times as many broccoli plants as there are corn plants.
His garden has 108 plants altogether from the broccoli and corn.
Pose 1 or 2 questions and work on answering them.
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LESSON 4.1
Exit Ticket
Tamika bought 4 pairs of jeans and 2 sweaters on sale for $116. Rosa paid $107 for 3 of the
same types of jeans and 4 of the same sweaters. How much did a pair of jeans cost? How much did
a sweater cost?
Show your work in determining your solutions.
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LESSON 4.2
Algebra Lesson 4.2
Warm-Up Activity
Carl drove his Prius 55 miles per hour from noon to 2pm:
How many hours was he driving?
How far did he go?
If Carl had kept driving until 5pm, how far would he travel?
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LESSON 4.2
Activity 1
Michael is a faster runner than Tahira. They decide to have a race. To be fair, Michael gives Tahira
a head start.
• Tahira starts at noon and runs three miles per hour.
• Michael runs four miles per hour; he starts one hour later.
Will Michael ever catch up to Tahira? If so, when? If not, why not?
Directions 1
1.
2.
3.
Work on the problem for 2-3 minutes on your own.
After your independent time have some idea of what you are doing, or have a question
Talk with a partner:
First ask any questions you have. Make the questions as specific as possible.
If there are no questions—and you have some idea of a direction to take—explain your method
to your partner.
4.
5.
6.
1
Work alone for another 2 minutes.
Come back and share with your group.
Decide how you are going to proceed as a group.
Edited from New Design High School Lesson Study
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LESSON 4.2
Poster Feedback Directions
With your group look at the poster and try to understand the thinking of the group.
Look at each part of the poster and make sure you understand what the group did and why.
Wherever you do not understand, or wherever you think the group is being unclear, write down a
question or comment on a post-it and put it on the poster.
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LESSON 4.2
Activity 2
Consider the original problem with Michael and Tahira:
Michael is a faster runner than Tahira. They decide to have a race. To be fair, Michael gives Tahira
a head start.
Tahira starts at noon and runs three miles per hour.
Michael runs four miles per hour; he starts one hour later.
Will Michael ever catch up to Tahira? If so, when? If not, why not?
Questions:
• How can we change the question so that Michael never catches up to Tahira?
• How can we change the problem so that the distance between Michael and Tahira stays the
same?
• How can we change the problem so that Michael catches up to Tahira more quickly?
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LESSON 4.2
Problem Set 1
Solve each of the following problems in at least two different ways.
1.
Scott takes off from New Design High School on his bicycle at 9 in the morning. He bikes at 6
miles per hour. Laurie takes off two hours later, biking at 10 miles per hour.
Will Laurie ever catch up to Scott?
If so, when?
2.
A truck traveling at a constant rate of 45 miles per hour leaves Albany. One hour later a car
traveling at a constant rate of 60 miles per hour also leaves Albany traveling in the same
direction on the same highway. How long will it take for the car to catch up to the truck, if both
vehicles continue in the same direction on the highway?
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LESSON 4.3
Algebra Lesson 4.3
Activity 11
When you are pricked with a pin, there is a short time delay before you say, “Ouch!” This reaction
time varies linearly with the distance between your brain and the place you are pricked.
Dr. Hollers pricks Leslie Morley’s finger and toe, and measures reaction times of 15.2 and 22.9
milliseconds, respectively. Leslie’s finger is 100 cm from the brain, and her toe is 170 cm from the
brain.
a. Pose a few questions.
b. Work with a partner on answering your questions and your partner’s questions.
c. Create a table with Reaction time and Distance from brain. Which measure should be the
“input” variable? (The output variable would be a function of the input variable.)
d. Sketch a graph of Reaction time versus Distance from brain.
For E-G, keep track of all of your work as you solve each problem and think about using
proportional reasoning.
e.
f.
g.
h.
i.
j.
How long would it take Leslie to say “Ouch!” if pricked in the neck, 10 cm from the brain?
How long would it take Leslie to say “Ouch!” if pricked in the shoulder, 15 cm from the brain?
How long would it take Leslie to say “Ouch!” if pricked in the knee, 110 cm from the brain?
Express the pattern of the table as a rule.
With a partner work on interpreting your rule. What do the numbers in your rule indicate?
How fast do impulses travel in centimeters per second? (You may need to do some
𝟏𝟏
of a second.)
conversions. A millisecond is
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
Source: Edited from Foerster, Paul A. (1994). Algebra I: Expressions, Equations, and Applications.
Addison-Wesley, New York.
1
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LESSON 4.3
Activity 22
What does a calorie mean? How far would you have to run to burn off a Big Mac? Take a guess 
• Consider LeBron James: How long do you think LeBron James would have to play basketball
to burn off all the calories in a Big Mac?
• The number of calories you burn depends on two things: the exercise you perform and your
weight.
• Sitting for one minute, for example, burns 0.009 calories for every pound of body weight
(see the table below).
• A Big Mac has about 550 calories.
• LeBron James weighs about 250 pounds.
Calories Burned in One Minute of Exercise
Sitting 0.009 cal/lb
Walking 0.019 cal/lb
Bowling 0.023 cal/lb
Biking, slow 0.029 cal/lb
Golf 0.033 cal/lb
Soft-Baseball 0.038 cal/lb
Weight Training 0.039
cal/lb
Biking, fast 0.045 cal/lb
1.
2
Ice Skating 0.053 cal/lb
Tennis 0.061 cal/lb
Basketball 0.063 cal/lb
Jogging 0.063 cal/lb
Swimming 0.064 cal/lb
Soccer 0.076 cal/lb
Jump Rope 0.083 cal/lb
Pose three questions to help yourself think about this problem.
Source: Edited from http://pblu.org/projects/new-tritional-info
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LESSON 4.3
2.
Work with a partner to answer each other’s questions.
3.
Work with a partner to figure out how long it would take for LeBron James to burn off the Big
Mac playing basketball
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LESSON 4.3
Problem Set 13
1.
Use the information below to determine how long Justin Timberlake would have to walk to burn
off a Big Mac.
Calories Burned in
One Minute of
Exercise
Walking 0.019 cal/lb
• Justin Timberlake weighs about 160 pounds.
• A Big Mac has about 550 calories.
2.
Determine how long it would take for the following people to walk off a Big Mac.
• LeBron James, 250 pounds _________________
• Selena Gomez, 125 pounds _________________
• A 10 year old weighing 75 pounds _________________
3.
3
Calories per minute: Complete the table below to find out how many calories each person burns
in one minute of exercise.
Problems and images from Mathalicious: http://www.mathalicious.com/
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LESSON 4.3
Your friend, Anita, is designing a website to promote Jumping Rope. She wants to include a
chart of calories burned per 10 minutes of exercise versus weight so that users can enter their
weight and see the calories burned.
a. Consider a few different weights and determine the calories burned per ten minutes. A
table may be helpful.
b. Find a general rule for calories burned in ten minutes given weight. [Use C(w) ]
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LESSON 4.3
Activity 3
Different types of exercise burn different numbers of calories.
1. Choose one of the celebrities below and three types of exercise. For each activity, how long
would it take to burn off each of the McDonald’s items below? Keep track of and show all of
your work.
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LESSON 4.3
2. In 2012, McDonald’s began posting calorie information in all of its restaurants in the United
States. Do you think it makes a difference in people’s purchasing?
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LESSON 4.3
Activity 4
Shoe Size Problem. On the planet Grog, a person’s shoe size varies linearly with the length of his or
her foot. The smallest adult shoe is a size 5 and fits a 9-inch long foot. A size 11 shoe fits an 11inch long foot.
a. Write the particular equation expressing shoe size in terms of foot length.
b. If Gizmo’s foot is one foot long, what size shoe does he need?
c. Imagine your own shoe was made on the planet Grog. Figure out the length of your foot in
inches based on the size you now wear.
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LESSON 4.3
d. Bob Lanier, who once played basketball for the Detroit Pistons, wears a Size 22 shoe. If his
shoe was made on Grog, how long is his foot?
e. Plot the graph of adult shoe size versus foot length.
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LESSON 4.3
f. Gadget’s foot grew two inches over the past year. By how much will her shoe size change?
Activity 5: Skill Review
The following lists some approximations of √126.
To the nearest tenth
To the nearest hundredth
To the nearest thousandth
11.2
11.22
11.225
1. Square each of the numbers below. Use your calculator if you like. Write your answers in
the space beside. Which decimal values are smaller and which are larger than √126?
(11.2)2 =
(11.22)2 =
(11.225)2 =
2. Find √200 rounded to two decimal places with your calculator. Record your answer in the
space below.
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LESSON 4.3
3. Find √2 rounded to two decimal places with your calculator. Record your answer in the
space below.
4. Using the fact that √200 = √2 ⋅ √100, simplify √200, keeping it in its radical form. (No
decimals).
5. Describe how you could determine what √200 is, rounded to two decimal places, if you knew
only (a) but not part (b).
Problem Set 2
1. What number should replace the blank space in each of the following equations to make it
true?
a. √4 ⋅ 13 = ____√13
b. √5 ⋅ 81 = ____√5
c. √9 ⋅ 49 = ____
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LESSON 4.3
d. √64 ⋅ 64 = ____
e. √3 ⋅ 3 ⋅ 7 = ____√7
f. √15 ⋅ 11 ⋅ 15 = 15√____
g. √16 ⋅ 5 ⋅ 100 = ____√5
h. √2 ⋅ 2 ⋅ 2 ⋅ 23 = 2√_____
2. What expression should replace the blank space in each of the following equations to make it
true? Assume that 𝑥𝑥 > 0.
a. √36𝑥𝑥 = ____√𝑥𝑥
b. √10𝑥𝑥 2 = ____√10
c. √5𝑥𝑥 3 ⋅ 5𝑥𝑥 3 = ____
d. √𝑥𝑥 6 = _____
e. √𝑥𝑥 11 = ____√𝑥𝑥
f. √9𝑥𝑥 16 = ____
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LESSON 4.4
Algebra Lesson 4.41
Activity 12
1.
Measure and record your height and arm span in centimeters: height: ____ arm span: _____
2.
Mark your arm span on the number line below.
3.
On the graph below Maria place a point to represent her arm span which is 166 cm and her
height which is 160 cm. Place another point on the graph to represent your arm span and
height.
Materials in this lesson are taken from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra materials, Unit
6.
2 Source: Edited from TTA Unit 6 page 8.
1
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LESSON 4.4
4.
Imagine a person whose point on the graph above is lower and to the right of your point.
What could you say about their arm span and height?
5.
Imagine an alien planet where everyone is exactly the same height, but the aliens have all
different arm spans. What might a graph of arm spans and heights look like on that planet?
Describe what their graph would look like or sketch one.
6.
If you were to graph the measures from 3 of your friends, would you expect points to lie on a
line? Why or why not?
7.
Lazy Larry fusses to get out of his seat in class. He knew that he is 5 feet 9 inches tall and he
knew that there are about 2.5 centimeters in an inch. How tall is he in centimeters? How could
he figure it out?
8.
Yudelca tells Larry that there are actually about 2.54 centimeters in an inch and that he needs
to go check his work with a real measure. Will he find that his result from 6 is less or more than
what it should be?
Just a note: Does it matter whether we label the graphs used as “Height versus Arm Span” or
the switch, “Arm Span versus Height”? The convention on earth is that we place the oftendependent and vertical axis measure first. To communicate effectively, we continue the
convention.
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LESSON 4.4
Activity 2
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5.
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LESSON 4.4
Ricardo, Sam, Tyrell, Ursula and Vinny measure their heights and forearm spans. The results
are displayed in the graph at the bottom.
a. Who is the shortest person? What is that person's height?
b. Who has the longest forearm? How long is that person's forearm?
c. Put the five students in order from tallest to shortest. If any students are the same height,
show that in your answer.
d. Put the five students in order from longest to shortest forearm. If any students are have
the same length forearm, show that in your answer.
e. Which students have a forearm length that appears to be greater than their height based
on the graph? Make up two more students, "W" and "X", whose forearms appear longer
than their height based on the graph.
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LESSON 4.4
Problem Set 1
Where Am I?
Use the hints to figure out the location of each point. Fill your answers in, then plot and label your
solutions on the graph below.
1.
2.
3.
I am point G.
My horizontal position is 2.
My vertical position is 3.
Where am I?
( , )
I am point H.
My x-coordinate is -4.
My y-coordinate is 3.
Where am I?
( , )
I am point I.
My x-coordinate is 5.
My vertical position is equal to my
horizontal
position (y = x).
Where am I?
( , )
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4.
5.
6.
I am point J.
My horizontal position is -2.
My vertical is -5.
Where am I?
( , )
I am point K.
My horizontal position is 1.
My y-coordinate is triple my horizontal
position (y = 3x).
Where am I?
( , )
I am point L.
My x-coordinate is 5.
My y-coordinate is one half the value of my
xcoordinate (y = ½x)
Where am I?
( , )
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LESSON 4.4
I am point M.
My x-coordinate is 3.
My y-coordinate is the opposite of my xcoordinate
(y = -x).
Where am I?
( , )
The table below shows real data from a 'hybrid' algebra class (half the class was online, half in a
classroom). The study recorded how many hours students spent studying online during the whole
course, and their grade at the end of it.
1.
Plot the data.
2.
In general, does the graph show a relationship between amount of time studying online and
grade in the course?
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LESSON 4.4
3.
Based on this graph, what amount of time would you spend online if you wanted to score higher
than 80% in the course?
4.
A discussion of the value of the graph:
Do you agree with the Michael and Jay? Why or why not?
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LESSON 4.4
Imagine that your favorite television show has just been made valuable to download online at $4
per episode.
5.
Use the table to list sample scenarios. Wherever the number of episodes downloaded is
missing, fill in a number, and then determine the number of dollars spent. Then, plot each
sample scenario on the graph.
6.
How many episodes would you have bought if you spent $240?
7.
Would the point (81, 324) be on the graph if the grid were large enough to plot it? Why or
why not?
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8.
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LESSON 4.4
Would the point (101, 408) be on the graph if the grid were large enough to plot it? Why or
why not?
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LESSON 4.4
Problem Set 2
Part 1: Use the hints to figure out the location of each point. Fill your answers in, then plot and label
your solutions on the graph below.
1.
2.
3.
4.
I am point A.
My horizontal position is -1.
My vertical position is 2.
Where am I?
( , )
I am point B.
My x-coordinate is 2.
My y-coordinate is -4.
Where am I?
( , )
I am point C.
My horizontal position is -3.
My vertical position is -2.
Where am I?
( , )
I am point D.
My x-coordinate is 3.
My y-coordinate is double my x-coordinate.
Where am I?
( , )
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5.
6.
7.
8.
I am point E.
I have the same y-coordinate as (4,-4) and
the same x-coordinate as (3,5).
Where am I?
( , )
I am point F.
My x-coordinate is -2.
My y-coordinate is triple my x-coordinate (y
= 3x).
Where am I?
( , )
I am point G.
I am halfway between (2,2) and (6,6).
Where am I?
( , )
I am point H.
I am halfway between (-1,0) and (-1,6)
Where am I?
( , )
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LESSON 4.4
Part 2: Use the number lines to answer each of the questions.
9.
Finish labeling this number line.
10. What is the distance between -10 and 41?
11. How far apart are -17 and 40?
12. How far apart are -55 and -24?
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LESSON 4.4
13. What is the distance between 97 and 509?
14. What is the distance between -4 and 405?
15. How far apart are 2 and -4?
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LESSON 4.4
Activity 3: Skill Review
Radical expressions, like variable expressions, can only be simplified if they have like terms. For
example:
5𝑥𝑥 + 11𝑥𝑥 = 16𝑥𝑥
Both of the above equations follow the same algebra rules.
5√2 + 11√2 = 16√2
Just like with variables, when an expression doesn’t have like terms, you cannot simplify it. For
example:
7𝑥𝑥 + 13𝑦𝑦
7√2 + 13√5
Both of the above equations cannot put together into one term to be simplified.
Many of the same rules that work with variables also work with radicals. One big difference with
radicals though, is that sometimes the number inside the square root can be simplified, which can then
let you simplify the expression. For example:
6√28 − 3√7
The 28 in this problem can be factored so that one of its factors is a perfect square. In other words:
28 = 4 ⋅ 7
If you replace the 28 in the problem with these two factors, then you can write the expression out as:
6√4 ⋅ 7 − 3√7
Since √4 = 2, then you can also rewrite the expression as:
6 ⋅ 2√7 − 3√7
And now you can multiply 6 times 2 to get 12.
12√7 − 3√7
This is a radical expression that now has the same radicals being subtracted! So it can be combined
together and simplify to:
9√7
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LESSON 4.4
Problem Set 3
1. If possible simplify each of the sums and differences listed below.
a. 6√2 + 5√2
b. 6√2 − 5√2
c. 6√2 + √2
d. √6 + √7
e. √7 − √6
f. 6√2 − √2
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2. Simplify each of the sums and differences below by simplifying the radical terms first.
a. √12 + 3√3
b. √20 + √5
c. 8√2 − √8
d. √11 + √99
e. √54 − √24
f. √24 − √54
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LESSON 4.4
3. Write a multiplication problem equivalent to each of the following addition problems.
a. √5 + √5 + √5
b. √3 + √3 + √3 + √3 + √3
c. √𝑥𝑥 + √𝑥𝑥 + √𝑥𝑥 + √𝑥𝑥 + √𝑥𝑥 + √𝑥𝑥 + √𝑥𝑥
d. √11𝑥𝑥 + √11𝑥𝑥 + √11𝑥𝑥 + √11𝑥𝑥
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LESSON 4.5
Algebra Lesson 4.5
Mini- Lesson1
Sketched graph
A sketched graph can show the relationship between two quantities without specifying actual
numerical quantity precisely.
Handout
1
Edited from EDC’s Transition to Algebra
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LESSON 4.5
Handout
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LESSON 4.5
Activity 1
Cut out card sets A and B. Then match the scenarios from the handouts by grouping them together.
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LESSON 4.5
Journey Home
Samantha bikes home along a straight road from her friend’s house, a distance of 8 miles. The
graph shows her journey. 2
1.
Describe what may have happened. You should include details like how fast Samantha rides
her bike.
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
2.
Are all sections of the graph realistic? Fully explain your answer.
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Mars Shell Center, Interpreting Distance-Time Graphs,
http://map.mathshell.org/download.php?fileid=1680
2
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LESSON 4.5
__________________________________________________________________________________________
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LESSON 4.5
Problem Set 1
1.
Circle the graph that best matches the given situation.
“When Jenny planted her orange tree, it was two feet tall. After planting, the tree continued
to grow every year at a constant rate.”
2.
Sketch your own graph to match each of the given situations below.
a. Jane was driving her car on the highway. She slowed down to exit the highway, and
then sped back up again.
b. A scuba diver jumps into the water and starts to descend. Once he reaches a certain
depth, he begins to ascend back to the surface.
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LESSON 4.5
Choose the situation that best matches the graph. Then, label the vertical axis to match
whichever situation you select.
a. Jon turned on the water to fill up the bathtub.
Once the bathtub was full, he turned off the water.
b. Aisha's sunflower didn't grow much in the cool
springtime weather, but grew very rapidly when the
warm summer weather arrived.
c. The flu virus began with just a few patients. Soon
patients began spreading the virus to friends and family
members until almost the entire town had the flu.
4.
Use the graph below to answer the questions.
a. Draw a graph to represent your commute home from school each day. Be sure to
include any stops you make on your way home.
b. Compare your graph with Samantha's graph from Journey Home.
c. Imagine that you are traveling home from a friend’s house instead of from school. How
might your graph look different?
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LESSON 4.5
Activity 2
Olivia and Cynthia debate how to work with Integers
Olivia
Hey, Cynthia. How did you solve problem #3? It says:
Cynthia
−7 − 3(−4 − 9)
Oh, I think I distributed the −3 first. So after I did that I wrote out this:
And then this:
Olivia
Cynthia
−7 − 3(−4) − 3(−9)
−7 + 12 + 27
How did you get that −3(−9) in the first part?
Mr. Diller told us that when we distribute, we have to multiply by the outside number
that’s right next to the parentheses by every number that shows up on the inside. In
this problem, the outside part is the −3. So that’s where the −3 comes from and the
Olivia
Cynthia
Olivia
−9 was from the inside.
Oh I see it now. I did it differently though.
Well when I finished mine up, I got the answer of 32. How did you do it?
I just followed PEMDAS and made the inside easier first. So first I wrote the problem
down:
−7 − 3(−4 − 9)
Then I combined the −4 and the −9 on the inside to get this:
Cynthia
Olivia
−7 − 3(−13)
How did you get −13? I thought two negatives made a positive.
That’s only when you multiply. When it’s addition or subtraction, I think of the number
line. We start at −4 and then go to the left 9 spots. That would land me even
deeper into the negative numbers. When I plugged just the inside part into my
calculator, so just the −4 − 9 and it said it was −13, it made sense to me that way.
Cynthia
Oh okay. That makes sense to me.
Olivia
Yeah, so then I multiplied the −3 by the −13 and got this:
−7 + 39
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LESSON 4.5
Cynthia
And that equals 32 also! Which is the same as my answer.
Olivia
I think then it doesn’t matter how we do it, as long as we get the same final answer.
Use the conversation above to recreate the work that both Olivia and Cynthia did on their
worksheets.
Olivia
Cynthia
−7 − 3(−4 − 9)
−7 − 3(−4 − 9)
Problem Set 2
Use Olivia and Cynthia’s discussion and your own knowledge to simplify the following expressions.
1.
−5 − 4
2.
−9(−5)
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3.
−7 + −8
4.
−14(−2)
5.
−8 ÷ −4
6.
−9 + 3
7.
5(−7)
8.
5 − (3 − 18)
9.
−(−25 + 6)
10.
2(3 − 9) − 3(2 − 9)
11.
−4 + 5(−3)
12.
7(2 − 11) + 2(7 − 11)
13.
−7(11)(13) − 13(11)(7)
14.
7 − (2 − 12)
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15.
−4−9
17.
−(1 − 61 − 13)
−5
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16.
LESSON 4.5
(2 − 5)(2)3
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LESSON 4.6
Algebra Lesson 4.61
Mini-Lesson
My equation: __________________
1 Some materials in this lesson are adapted and edited from: Education Development Center, Inc. (2011 & 2013.) Transition to
Algebra, Unit 6.
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LESSON 4.6
Activity 12
1.
Circle the points that make the hint true. If the points fit, plot them on the grid below. There
are multiple solutions, so be sure to check each point.
Hint: y= x + -3
*Don’t forget the y-coordinate always equals x+ -3 in this problem.
2.
Find five more solution points for this equation. You might not be able to plot them all on the
graph.
Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra,
Unit 6.
2
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LESSON 4.6
Connecting your solution points may help you notice that there are other solution points in
between the ones you found and plotted. List two such points, and check to see if they are
solutions.
( __, __ )
( __, __ )
Circle the points that make each equation true and plot them if possible. Then, find five of your own
solutions, plot them if possible, and connect all of the solutions points.
4.
y= 2x + 2
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LESSON 4.6
y= x2 + 2
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LESSON 4.6
Problem Set 13
Circle the points that make each equation true and plot them if possible. Then, find five of your own
solutions, plot them if possible, and connect all of the solution points.
1.
y = -x + 2
2.
y = 2x2 - 4
3 Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra, Unit 6.
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I am point Y.
◊ y=x+-5
◊ x=-3
3.
LESSON 4.6
Activity 24
1.
I am point X.
◊ My y-coordinate
satisfies y=x+-5.
◊ My x-coordinate is 4.
2.
What is my coordinate
address?
What is my coordinate
address?
I am point Z.
◊ y=x+-5
◊ x is 12.
What is my coordinate
address?
4.
A, B, and C are all solution points for the equation y=x+-5 because each point gives a pair of
numbers that make the equation true. What would be a point that is not a solution for y=x+5?
5.
Which of the following points are also solution points for y = x + -5? Circle the solution points
and cross out the non-solution points.
(45, 40)
(3, -2)
(1, -4)
(5 , )
(½, -4½)
(-15, -10)
(-6, 1)
(10, 5)
(2, -3)
(5, 0)
Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra,
Unit 6.
4
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LESSON 4.6
6.
Plot all the solution point from problems 1-5 that fit on the grid:
7.
There are infinitely many more solution points for y=x+-5. Find six more, record them in the
table and plot them too if they fit on the graph above.
(x,y)
(0, -5)
8.
From simply looking at the graph, these points look like they might be solution points to the
equation in number 6. Are they?
a. (1.2, -3.7)
9.
b. (-0.5, -5.5)
The graph of y = x+-5 is a line connecting all the solution points. Explain why it makes sense to
connect the points when graphing the relationship.
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LESSON 4.6
(x,y) pairs that make x+y=6 true are solution
points for that equation.
a. Find a solution if x=1. (1, __ )
b. Find a solution point if x = -3.
c. Find a solution point if y=0.
d. Find a solution point if x=100.
e. Find three more solution points.
f. You know have seven solution points for
this equation. Some of them don’t fit on
the graph paper shown here. Plot the
ones that do and use them to draw the
graph of x+y=6
11. y= -2x + 3
a. Find a solution point if x = 0.
b. Circle the solution points; cross out nonsolution points.
(4, -5)
(10, -17)
(2 ,0)
(-1, 5)
(-2, 7)
(-10, 17)
c. Use the solution points to draw the graph
of y=-2x+3. Check more solution points,
if necessary.
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LESSON 4.6
12. The graph of y = x2 - 5 is not a straight line.
a. Find a solution point if x = 0.
b. Circle the solution points; cross out non-solution points.
13.
(-2,-1)
(3,4)
(4,9)
(1,-4)
(2,-1)
(5,-20)
(2,0)
(-1,-4)
(-6,31)
(-5,20)
(-3,4)
(10,95)
Solve for x
a. 2x + 5 = 7
b. 2x + 5 = 1
c. 2x + 5 =-3
(hint: If blob + 5 = 7,
What is blob?)
d. These equations are all specific cases of
2x+5 = y. By solving the equations
above, you have found 3 solutions points.
What are they?
(___, 7) (___, 1) ( ___, ___ )
e. Use these points (and more if you want) to
graph 2x+5 = y.
f. Use the graph to find what seems to be the
solution point if y = -1.
g. You’ve found a possible solution point from
the graph. Solve the equation 2x+5=-1 to
know for sure.
h. What point can help you solve 2x+5=5?
i. Find x: 2x+5=5
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LESSON 4.6
14.
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LESSON 4.6
Problem Set 25
Use the hints to determine the coordinate point in each question.
1. I am point Q.
2. I am point S.
My x-coordinate is 0.
My y-coordinate is 0.
I lie on y = x – 4.
I lie on y = x – 4.
You can find me at (
,
)
You can find me at (
3. I am point R.
My x-coordinate is 0.
I lie on y = x – 4.
You can find me at (
,
)
,
)
4. I am point T.
I lie on y = x – 4.
You can find me at (
,
)
*How many answers are there to number
4?*
5. Find pairs of points, (x,y), that make the equation y = -x + 2 true. These are the solution points.
a. Find a solution point where x = 0.
b. Find a solution point where x = -3.
c. Find a solution point where y = 0.
d. Find a solution point where y = -10.
e. Find three more solution points.
f. Plot the solutions points that fit on the graph.
Connect them to draw the graph of
y = -x + 2.
Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra,
Unit 6.
5
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LESSON 4.6
Activity 36
Fill in the tables with the solution points for each equation. Then, cut out each card and match up
each of the cards from Set A with its corresponding card from Set B.
Card Set A
1.
y=x+3
x
-3
0
2.
1
x
y
4.
y=3
-3
0
1
4
-2
0
y
5.
6.
x = -2
x
y = x2 - 4
x
-2
0
7.
y=
x
1
y = -3x - 1
x
y
y
0
y
3.
x
y = 2x - 2
-2
0
2
−1
2
-2
y = -x2 + 4
x2 + 2
y
2
y
8.
4
0
x
-1
0
2
y
Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra,
Unit 6.
6
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LESSON 4.6
Card Set B
A.
B.
C.
D.
E.
F.
G.
H.
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LESSON 4.6
Activity 47
Based on the following graphs, write a scenario that describes the duration of a trip and the speed
of travel and/or how much distance was covered. The first graph is done for you as an example.
1.
Graph I shows the distance traveled on a trip from home to the beach and back. What can you
say about the trip from the graph? Make up a scenario to correspond with the graph.
2.
Graph II shows a slightly different trip. What does Graph II show compared to Graph I?
7
Edited from EDC's Transition to Algebra
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3.
How does the trip that Graph III describes differ from the other trips?
4.
Say what might have been doing on during the trip that Graph IV represents.
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LESSON 4.6
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LESSON 4.6
Problem Set 38
Solve for x.
1. x2 – 9 = -8
4.
5.
6.
2. x2 – 9 = -5
3. x2 – 9 = 0
These equations are all special cases of x2 – 9 = y. By solving the equations above, you
should have found six solution points to the equation x2 – 9 = y. What are they?
(
, -8 )
(
, -5 )
(
,
)
(
, -8 )
(
, -5 )
(
,
)
Solve x2 – 9 = -9. Why do you get only one solution point?
Use these points (and more if necessary) to
graph x2 – 9 = y.
8 Materials adapted from: Education Development Center, Inc. (2011 & 2013.) Transition to Algebra, Unit 6.
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LESSON 4.6
7.
Explain why it makes sense from this problem that a graph of an equation with x2 must curve
back on itself.
8.
x2 – 9 = -10 has no solution. That means there is no solution point for x2 - 4 = y that has a yvalue of -10. How does this fact show up in the graph?
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LESSON 4.6
Lesson Formative Assessment
Use solution points to draw the graph of y = x2 - 5. Check more solution points if necessary.
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LESSON 4.7
Algebra Lesson 4.71
Activity 12
1.
Look at each of the lines below. As you look from left to right along each line, does it go up,
down or neither? The answer to this question tells us if the slope of each line. When a line goes
up from left to right, it has a positive slope, and when a line goes down from left to right, it has
a negative slope. The slope is zero if the line is horizontal, meaning it shows no increase or
decrease from left to right. Determine if the slopes of the lines below are positive, negative, or
zero.
2.
Now put lines A-G in order of steepness, from steepest in the negative direction to steepest in
the positive direction.
Steepest Negative
3.
Steepest Positive
To make precise comparisons, we can measure steepness of a line using slope. You already
know how to find the sign of the slope of a line. To measure the slope, find two points on the
line and use the ratio of the y-distance to the x-distance of the points.
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
2 Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
1
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LESSON 4.7
Use the slope ratio to find the slope of each line. Don’t forget the sign of each slope!
A
B
C
D
E
x-distance: ______
distance: ______
x-distance: ______
x-distance: ______ x-distance: ______ x-
y-distance: ______
distance: ______
y-distance: ______
y-distance: ______ y-distance: ______ y-
slope: ______
______
slope: ______
slope: ______
slope: ______
slope:
For problems 4-6, imagine a line with two points, P and Q marked on it.
4.
Imagine the line has a very steep positive slope. Which distance between P and Q is greater,
the x-distance or the y-distance? How do you know?
5.
What is true about the x-distance and y-distance between P and Q when the slope is zero?
How do you know?
6.
What is true about the x-distance and y-distance between P and Q when the slope is one?
How do you know?
7.
Imagine 4 new lines, each with one of the four following slopes:
𝟏𝟏 𝟏𝟏 𝟏𝟏 𝟏𝟏
, , ,
𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟔𝟔
Which line is the steepest? Explain how you know, using x-distances and y-distances.
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8.
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LESSON 4.7
Imagine a that goes through the points (-3,2) and (7,6). Determine the x-distance, y-distance,
and slope, including the sign of the slope.
x-distance: ______
9.
ALGEBRA
y-distance: ______
slope: ______
Imagine a that goes through the points (5,7) and (2,-6). Determine the x-distance, y-distance,
and slope, including the sign of the slope.
x-distance: ______
y-distance: ______
slope: ______
10. Imagine a that goes through the points (9,26) and (39,86). Determine the x-distance, ydistance, and slope, including the sign of the slope.
x-distance: ______
y-distance: ______
slope: ______
11. Imagine a that goes through the points (-50,50) and (-100,250). Determine the x-distance, ydistance, and slope, including the sign of the slope.
x-distance: ______
y-distance: ______
slope: ______
12. Next, imagine a line with a slope of zero. Name three points on the line. What do they have
in common?
(
,
)
(
,
)
(
,
)
13. Tahir and Bella are discussing the line that passes through points B and T. Tahir says, “I think
this line has a negative slope. If you start at point T and move to point B, the line goes down.”
Bella responds, “That’s not how you read a graph. I think the slope of this line is positive.”
Who is correct? What is the correct way to read the graph?
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LESSON 4.7
Practice Set 13
Determine the answers to the questions based on the information given.
1.
2.
3.
x-distance: ______
x-distance: ______
x-distance: ______
y-distance: ______
y-distance: ______
y-distance: ______
slope: ______
slope: ______
slope: ______
4.
The points (0, 0) and (-3, 6) lie on the same line. What is the sign of the slope of the line?
5.
The points (0, 0) and (-401,-296) lie on the same line. What is the sign of the slope of the line?
6.
The points (11, 4) and (2, 11) lie on the same line. What is the sign of the slope of the line?
7.
The points (-5,-100) and (4,-50) lie on the same line. What is the sign of the slope of the line?
8.
The points (-5, 50) and (-15,250) lie on the same line. Find the:
sign of the slope: ______
x-distance: ______
y-distance: ______
slope: ______
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
3
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Supported by the NYS Education Department
S381
NAME
9.
DATE
ALGEBRA
LESSON 4.7
The points (10, 80) and (-2,88) lie on the same line. Find the:
sign of the slope: ______
x-distance: ______
y-distance: ______
slope: ______
10. The points (-64, 22) and (-54, 722) lie on the same line. Find the:
sign of the slope: ______
x-distance: ______
y-distance: ______
slope: ______
11.
Find the missing quantities in the mobile.
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S382
NAME
DATE
ALGEBRA
LESSON 4.7
12. Complete the MysteryGrid
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S383
NAME
DATE
ALGEBRA
LESSON 4.7
Activity 24
Determine the answers to the questions based on the information given.
1.
2.
3. Which line is steeper?
a)
b)
c)
d)
x-distance: ______
x-distance: ______
y-distance: ______
y-distance: ______
slope: ______
slope: ______
𝟒𝟒
This line has a slope of
5.
Draw two lines on the grid that have the same slope as number 4.
𝟏𝟏𝟏𝟏
. Write 5 other names for
𝟒𝟒
4.
𝟏𝟏𝟏𝟏
Line 1
Line 2
Both lines are equally steep
Cannot be determined from
the given information
.
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
4
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S384
NAME
6.
DATE
ALGEBRA
LESSON 4.7
Draw two lines on the grid that have the same slope as the given line.
Numbers 7-9 below look like they have the same slope, but they don’t! Find each slope.
*Hint: Notice the scale of the axes *
7.
8.
slope: ______
© 2016 CUNY Collaborative Programs
9.
slope: ______
slope: ______
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S385
NAME
DATE
ALGEBRA
LESSON 4.7
Practice Set 25
1.
𝟑𝟑
Draw a line through point A with a slope of .
𝟓𝟓
𝟓𝟓
Draw a line through point A with a slope of .
𝟑𝟑
𝟔𝟔
𝟓𝟓
Which is greater, or ?
𝟓𝟓
𝟔𝟔
Write two more names for the slope of the steeper line.
2.
𝟒𝟒
Draw a line through point B with a slope of .
𝟑𝟑
𝟐𝟐
Draw a line through point B with a slope of .
𝟒𝟒
𝟐𝟐
Which is greater, or ?
𝟑𝟑
𝟏𝟏
𝟏𝟏
Write two more names for the slope of the steeper line.
3.
This line has a slope of
−𝟔𝟔
𝟗𝟗
.
Write five other names for
4.
−𝟔𝟔
𝟗𝟗
.
Draw two more lines on the graph that have the same slope.
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
5
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S386
NAME
DATE
ALGEBRA
LESSON 4.7
Activity 36
1.
How many lines can you draw through the point (5, 3) that have a slope of
line you can that has these properties.
−2
3
? Draw every
You should have found out that there is only one line like that, which is useful to know and
understand. It means that you know exactly where the line is, and it also means you can determine if
a given point is on the line or off the line.
2.
By drawing that one line, you are connection all of the solution points for a line that runs
−2
through (5,3) and has a slope of . Since this line extends infinitely (forever) in both directions,
3
there are an infinite (unlimited) number of points on the line. List six points on the line here.
3.
Are points L, M, and N collinear (on the same line)? How do you know?
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
6
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S387
NAME
DATE
ALGEBRA
LESSON 4.7
4.
Imagine a line between points F and G that has a slope of 3. Now, imagine a line between
points J and K that also has a slope of 3. Does that mean that points F, G, J and K are all
collinear?
5.
Imagine a line between points A and B that has a slope of -4. Now, imagine a line between
points B and C that also has a slope -4. Does that mean that points A, B and C are collinear?
6.
Imagine a line between points P and Q that has a slope of 2. Does that mean that points P, Q
and R are all collinear?
7.
Imagine a line between points L and M that has a slope of
1
points M and O that also has a slope of
collinear?
−12
18
−4
6
. Now imagine a line between
. Does that mean that points L, M and O are all
8.
Imagine three points: X (5,920), Y (35,720) and Z (65,520). Do X, Y and Z all lie on the same
line? How do you know?
9.
Imagine three points: S (-20,-400), U (20,100) and N (60,350). Do S, U and N all lie on the
same line? How do you know?
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S388
NAME
DATE
ALGEBRA
LESSON 4.7
Activity 47
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
7
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Supported by the NYS Education Department
S389
NAME
DATE
ALGEBRA
LESSON 4.7
Practice Set 38
3
1.
Draw a line with a slope of that runs through the point (6, 2). Name six points on the line.
2.
Draw a line with a slope of
5
−2
3
that runs through the point (8,-1). Name six points on the line.
3. Imagine three points are plotted on a graph: F (20, 60), G (30,160) and H (60,510). Are F, G
and H collinear? How do you know?
4. Imagine three points are plotted on a graph: C (10, 7), A (20, 1) and T (25,-2). Are C, A and
T collinear? How do you know?
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
8
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S390
NAME
5.
DATE
ALGEBRA
LESSON 4.7
The graph shows Tamra and Saul’s distance from work over time.
a. Are they traveling to work or from work?
b. Who started traveling first?
c. Who is traveling faster?
d. How many miles per hour does Saul travel?
e. How many miles per hour does Tamra travel?
6.
Five friends experience their growth spurts at the same time. One friend grows 2 centimeters,
one grows 4 centimeters, one grows 6 centimeters, one grows 8 centimeters, and one grows 10
centimeters. Determine how many centimeters Camilo grew based on the clues below.
• Annie grew more than Beverly
• Douglass grew 4 more centimeters than Elijah, but fewer than Beverly
• The difference in growth between Camilo and Elijah is 2 centimeters
7.
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S391
NAME
DATE
ALGEBRA
LESSON 4.7
8.
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S392
NAME
DATE
ALGEBRA
LESSON 4.8
Algebra Lesson 4.81
Mini-Lesson
Question: On the graph below, draw all of the lines that go through (2,3) with a slope of ½ .
(Remember that ½ has many other fractions equivalent to it.)
(Provide graph paper)
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
1
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S393
NAME
DATE
ALGEBRA
LESSON 4.8
Activity 12
4
A line goes through the point (9,8) and has a slope of . Marcel and Jahnia
5
discuss the following questions in math class.
Is (-1,0) on the line?
Marcel: The x-distance is 10.
The y-distance is 8.
The slope is positive.
8
The slope is .
10
Yes, (-1,0) is on the line.
Is (5,13) on the line?
Jahnia: The slope is negative.
The y-distance is 5.
The x-distance is 4.
−4
The slope is .
5
No, (5,13) is not on the line.
Use the work above to answer the following:
1.
Jahnia says, “Marcel, how did you get your x-distance of 10?” How might Marcel respond?
(Use a rough sketch to help yourself think about it.)
2.
Marcel asks Jahnia, “How did you know your slope is negative? Both of the distances are
positive.” How might Jahnia respond?
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
2
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S394
NAME
3.
DATE
ALGEBRA
LESSON 4.8
Consider these methods with the following dialogue to think about how can you test the point (x,
y) to see if it is on this line.
Marcel: So, next we have to figure out if the point (x,y) is on the line. How can we do that?
Jahnia: That problem looks similar to the ones we just completed. The only difference is we
are using the letters x and y instead of a pair of numbers. Let’s try to complete it the same
way as the other problems. What would be the x-distance?
Marcel: Before we had to subtract to get the distance. So let’s subtract here. The x-distance
would be the difference between x and 9, so x-9 is the x-distance.
Jahnia: Ok, that sounds good. So that means the y-distance would be ___________.
Marcel: Now lets’ try and figure out if the slope is negative or positive. We can’t plot the
point to see where it is, because we don’t know where (x,y) is. Hmm…
Jahnia: We do know where (x,y) is! It’s on the line. So that means we know the slope has to
4
be . We don’t have to answer “yes” or “no,” we just have to figure out a pattern for the
5
points on the line.
Marcel: Ok, I think I understand. It’s like the points we had before were guesses, and we
were using the slope to check to see if the points fit a pattern. Now we are just using x and
y in the pattern instead of two numbers from a point.
Jahnia: Exactly! So if (x,y) is a point, then
𝑦𝑦−8
𝑥𝑥−9
4
= .
5
Marcel: So that equation should work for every (x,y) point that is on the line.
Jahnia: That means the equation for the line is just the y-distance over the x-distance set
equal to the slope.
4.
Follow-up: Could
y−8
x−9
4
= be the equation of the line?
5
© 2016 CUNY Collaborative Programs
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S395
NAME
DATE
ALGEBRA
LESSON 4.8
Activity 23
1.
Plot the points C (9,4) and D (1,2). Then determine the slope of the line.
2.
Is (4,3) a point on this line? How do you know?
3.
Is (17,6) a point on this line? How do you know?
4.
Write your own point. Then determine if it lies on the line.
5.
How would you check to see if (x,y) lies on the line?
6.
Circle the solution points that are on the line.
7.
Fill in the blanks so that each point lies on the line.
(50,____)
(-15,____)
(____,9)
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
3
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S396
NAME
8.
DATE
ALGEBRA
LESSON 4.8
An equation describes the pattern for a set of points. For parts A and B, use the pattern to
help you complete the questions.
a. Describe a pattern you see for how the points below are related.
(4,7)
(8,11)
(9,12)
(50,53)
(-6,-3)
(826,829)
Add 3 more points that follow the same pattern.
(____,____)
(____,____)
(____,____)
Write an equation for this line that includes every point.
b. Describe a pattern you see for how the points below are related.
(5,42)
(21,201)
(8,72)
(351,3502)
(-4,-48)
(-59,-598)
Add 3 more points that follow the same pattern.
(____,____)
(____,____)
(____,____)
Write an equation for this line that includes every point.
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S397
NAME
DATE
ALGEBRA
LESSON 4.8
Activity 34
Use the clues to determine the missing coordinate(s) for each point. You can use the graph below for
help.
1.
I am point A.
My x-coordinate is 12.
I am on the line 𝑥𝑥 + 𝑦𝑦 = 14.
Where am I?
(
2.
,
)
I am point B.
I am on the x-axis.
My y-coordinate is two less than my x-coordinate.
Where am I?
(
,
)
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
4
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S398
NAME
3.
DATE
ALGEBRA
LESSON 4.8
I am point C.
My x-coordinate is 11.
I am on line 𝑦𝑦 = 𝑥𝑥 − 13.
Where am I?
(
4.
,
)
I am point D.
I am on line 𝑥𝑥 + 𝑦𝑦 = 0.
My x-coordinate is 5.
Where am I?
(
5.
,
)
I am point E.
I am on line 𝑥𝑥 = −6𝑦𝑦.
My y-coordinate is -1.
Where am I?
(
6.
,
)
I am point F.
I am on line 𝑦𝑦 = 𝑥𝑥 − 2.
My x-coordinate is -3.
Where am I?
(
,
)
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S399
NAME
DATE
ALGEBRA
LESSON 4.8
Practice Set 15
1.
Plot the points (3,5) and (11,1). Then, draw the line that goes through both points.
2.
What is the slope of the line?
3.
Is the point (1,6) on the line? How do you know?
4.
Is (19,-3) on the line? How do you know?
5.
Write your own point. Then determine if it lies on the line.
6.
How would you check to see if the point (x,y) is on the line?
Materials in this lesson are adapted from: Education Development Center, Inc. (2012.) Transition to
Algebra, Unit 9.
5
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S400
NAME
DATE
ALGEBRA
LESSON 4.8
7.
Circle the solution points that are on the line. Use your equation and/or graph for help.
8.
Fill in the blanks so that each point lies on the line.
(11,___)
9.
(-7,___)
(___,-2)
Consider a line with a slope of 8 that goes through the point (40,-20).
a. Is the point (100,220) on the line? How do you know?
b. Is the point (0,-180) on the line? How do you know?
c. Write your own point. Then determine if it lies on the line.
10. Consider a line with the equation
11. Consider a line with the equation
© 2016 CUNY Collaborative Programs
𝑦𝑦+6
𝑥𝑥−3
𝑦𝑦−3
𝑥𝑥+9
4
= . Circle the solution points that are on the line.
=
9
5
. Circle the solution points that are on the line.
−4
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NAME
DATE
ALGEBRA
LESSON 4.8
12. Fill in the blanks so that each point lies on the line.
(8,___)
(-7,___)
(___,48)
13. For each of the following, adjust the equation to find an equivalent form in which y is isolated
and it is simplified, with parentheses removed. (See hint box.)
a)
𝑦𝑦+4
𝑥𝑥−3
=
2
7
b)
𝑦𝑦−3
𝑥𝑥+9
=
5
−4
c)
𝑦𝑦−2
𝑥𝑥+2
=
8
4
14. Determine the value of each shape.
© 2016 CUNY Collaborative Programs
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S402
NAME
15.
DATE
ALGEBRA
LESSON 4.8
Complete the Mystery Grid.
© 2016 CUNY Collaborative Programs
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S403
NAME
DATE
ALGEBRA
LESSON 4.9
Algebra Lesson 4.91
Activity 12
Complete the following exercises by using solution points and graphs.
1.
Draw each line separately. Use as much space and as many points as you need to draw an
accurate graph.
𝒚𝒚 = 𝟐𝟐𝟐𝟐 + 𝟑𝟑
𝒙𝒙 + 𝒚𝒚 = 𝟗𝟗
Some materials in this lesson are adapted from: Education Development Center, Inc. (2012.)
Transition to Algebra, Unit 9, Impact Mathematics Course 3, and MARS.
2 Some materials in this lesson are adapted from: Education Development Center, Inc. (2012.)
Transition to Algebra, Unit 9, Impact Mathematics Course 3, and MARS.
1
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NAME
DATE
ALGEBRA
LESSON 4.9
2.
Now draw both lines on the same graph.
3.
The lines 𝑦𝑦 = 2𝑥𝑥 + 3 and 𝑥𝑥 + 𝑦𝑦 = 9 are two different lines that involve different relationships
and solution points. But there is one solution point that is the same for both of these lines. This
special point is called the point of intersection. What is the solution point, or point of intersection?
4.
Determine the values for x and y that make both mobiles true.
© 2016 CUNY Collaborative Programs
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S405
NAME
5.
DATE
ALGEBRA
LESSON 4.9
Graph both linear relationships on the grid below.
𝑦𝑦 = 3𝑥𝑥 − 12
𝑦𝑦 = 𝑥𝑥 + 6
6.
Bradley looks at the lines in number 5 and says “Those lines don’t cross on the graph, so they
must never cross. There is no point of intersection.” Do you agree with Bradley? Why or why
not?
7.
You can’t see where the lines cross, but we can figure it out using algebra. Since both
relationships are equal to the y-value at this point, we can set both relationships equal to each
other and solve to find the x-value of the point of intersection.
𝟑𝟑𝟑𝟑 − 𝟏𝟏𝟏𝟏 = 𝒙𝒙 + 𝟔𝟔
8.
Now use the x-value of the intersection point to find the y-value of the intersection point. Then
write out the entire intersection point.
9.
Is there a way to know if these lines will ever cross again? How can we be sure that these lines
won’t cross again when 𝑥𝑥 = −126 or 𝑦𝑦 = 2,298?
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S406
NAME
10.
DATE
ALGEBRA
LESSON 4.9
a. Draw the relationship 𝑦𝑦 = 𝑥𝑥 2 on the graph below. Use at least 5 points.
1
b. Draw the relationship 𝑦𝑦 = (2𝑥𝑥 + 2) + 1 on the graph below. Use at least 5 points.
2
c. Draw the relationships from parts a and b on the graph below.
d. How many points do the two relationships have in common? What are the common
point(s)?
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S407
NAME
DATE
ALGEBRA
LESSON 4.9
Problem Set 13
1.
a. Draw the relations y=2x+1 and y=2x-1 using as many points for each as you need.
b. Will these lines intersect? Why or why not?
2.
a. Draw the relations y=2x+1 and y=.5x using as many points for each as you need.
b. Will these lines intersect? Why or why not?
3
Materials adapted from: Education Development Center, Inc. (2012.) Transition to Algebra, Unit 9
© 2016 CUNY Collaborative Programs
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S408
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 24
1.
2.
Write as many different equations as possible for the following mobiles:
3.
Write as many different equations as possible for the following mobiles:
4
Materials adapted from: Education Development Center, Inc. (2012.) Transition to Algebra, Unit 9
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S409
NAME
DATE
ALGEBRA
LESSON 4.9
Challenge Problem
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S410
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 35
1.
Solve the mobile below.
Let h stand for hearts and s stand for stars.
Use h and s to write at least 2 equations to represent the
mobile.
Share your equations with a classmate. Then write down
any new equations you learn through sharing with your
classmate.
2.
5
Solve the mobile. Then, write at least 2 equations to represent the mobile. Let a represent
arrow, d represent diamond, and s represent semicircle.
Materials adapted from: Education Development Center, Inc. (2012.) Transition to Algebra, Unit 9
© 2016 CUNY Collaborative Programs
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S411
NAME
DATE
ALGEBRA
LESSON 4.9
3.
Solve the mobile. Then, write at least 2 equations to represent the mobile. Let t represent
triangle, and q represent quadrilateral.
4.
Solve the mobile. Then, write at least 2 equations to represent the mobile. Let h represent
hexagon, and q represent quadrilateral.
5.
Use the two equations to draw one or two mobiles showing the relationship(s) between t
(triangle) and a (arrow).
4𝑡𝑡 + 𝑎𝑎 = 20
10𝑡𝑡 = 20
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S412
NAME
DATE
ALGEBRA
LESSON 4.9
6.
Use the two equations to draw one or two mobiles showing the relationship(s) between s
(semicircle) and q (quadrilateral).
𝑠𝑠 + 𝑞𝑞 = 22
4𝑠𝑠 + 6𝑞𝑞 = 48
7.
More Mystery Numbers!
You’ve found the following calculations from an ancient civilization. You know that the
calculation symbols, plus, times, and equal, are the just like our algebra. You’ve also figured
out that each symbol is a different digit. Examine the equations to determine the value of each
symbol.
8.
Use the clues to determine the missing number.
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S413
NAME
DATE
ALGEBRA
LESSON 4.9
Problem Set 26
1.
Solve the mobile below.
Let h stand for hearts and s stand for stars.
Use h and s to write at least 2 equations to represent the mobile.
2.
Solve the mobile. Then, write at least 2 equations to represent the mobile. Let t represent
triangle, and q represent quadrilateral.
6
Materials adapted from: Education Development Center, Inc. (2012.) Transition to Algebra, Unit 9
© 2016 CUNY Collaborative Programs
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S414
NAME
DATE
ALGEBRA
LESSON 4.9
3.
Solve the mobile. Then, write at least 2 equations to represent the mobile. Let d represent
diamond and s represent semicircle.
4.
You find this puzzle in an old algebra text book. Any letter can stand for any number, and
different letters can stand for the same number, unless you are otherwise told. No unknown
numbers are negative, but at least one is greater than ten. Can you solve for any of the
missing numbers?
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S415
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 47
1.
Are Dan and Emma correct?
If you think Dan is wrong, explain the mistake and explain what you think the equation means.
If you think Emma is wrong, explain the mistake and explain what you think the equation means.
2.
7
Figure out for yourself the number of pens and the number of notebooks sold in the store.
MARS: http://map.mathshell.org/download.php?fileid=1730
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S416
NAME
DATE
ALGEBRA
LESSON 4.9
Problem Set 38
Joe is mixing nuts and raisins to sell at a school fair. He buys nuts in 4-pound bags and raisins in 1pound bags:
x = number of bags of nuts he buys.
y = number of bags of raisins he buys.
The following equations are TRUE:
3x = y
4x + y = 70
1.
Explain in words the meaning of each equation.
2.
Find two pairs of values for x and y that satisfy the first equation.
3.
Find two pairs of values for x and y that satisfy the second equation.
4.
Find pairs of values for x and y that satisfy both equations simultaneously.
8
MARS: http://map.mathshell.org/download.php?fileid=1730
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S417
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 59
The owner of a hat business has fixed expenses of $3,000 each week. She has these expenses
regardless of how many hats she makes. The remaining expenses are variable costs, such as
materials and labor. These costs are proportional to the number of hats made. For example, in a
week in which 500 hats were made and sold, her variable costs were $7,500. In a week in which
1,000 hats were made, they were $15,000.
All the hats were sold to department stores at $20 each. Let n equal the number of hats sold and I
equal income in a week.
1.
Write an equation relating the income (I) to the number of hats sold (n). Income would be the
money received from the department stores for the hats sold to them.
2.
Consider the hatmaker’s costs:
a. Write an expression for the variable costs (VC) to produce n hats. That is, if the shop made
n hats in a week, what would the variable costs be? (Think about the clues given in the first
paragraph)
b. Write an equation relating the total cost per week (TC) and the number of hats made and
sold that week (n).
3.
What does profit mean? Write a relationship (equation) connecting profit (P), income (I), and
costs (TC).
4.
The difference between cost and income is the owner’s profit. If the hatmaker has no sales, she
will still have to pay the fixed costs, and therefore will have a large loss. If she has high sales,
she will cover her fixed costs and make a good profit.
a. In one particular week, the hatmaker made and sold 500 hats. Did she make a profit or
suffer a loss? How much? (She sold her hats for $20 each).
b. In another week, she made and sold 1,000 hats. Did she make a profit or suffer a loss?
How much?
5.
Somewhere between no sales and many sales is a number of sales called the breakeven point. This is the point at which costs and income are equal, so there is zero profit but also
zero loss.
a. On one set of axes you need to graph the equations for income (question 1) and costs (2b).
9
Impact Math Course 3. P261.
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S418
NAME
DATE
ALGEBRA
LESSON 4.9
I. What would be a reasonable scale for n on the graph?
II. What would be a reasonable scale for profit on the graph?
III. Label each graph with its equation. You will need to read amounts from your graph, so use an
appropriate scale and be as accurate as you can.
How can you use your graph to estimate the break-even point?
From your graph, estimate the number of hats needed to make and sell each week to break even.
Calculate the costs and income for that number of items to check your estimate. Improve your
estimate if necessary.
Use your graph to estimate the number of hats she needs to sell each week to make a profit of
$1,000 per week. Calculate the costs and income for that number of items to check your estimate.
Improve your estimate if necessary.
Use the equations you wrote for Problems 1 and 2 to write an equation that gives the value of n for
which costs and income are equal. Solve your equation to find the break-even point.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S419
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 6
A.
Basketball 10: Carinne scored 23 points in last night’s basketball game. Altogether, 10 of her
shots went in; some were 2-point shots and the others were 3-point shots. Let a stand for the
number of 2-point shots and b the number of 3-point shots.
a. Write an equation for the total number of Carinne’s shots that went in.
b. Write an equation for the total number of points Carinne scored.
c. How many 2-point shots did Carinne make? How many 3-point shots did she make?
B.
Economics 11: A manager of a rock group wants to estimate, based on past expenence, how
many tickets will be sold in advance of the next concert and how many tickets will be sold at
the door on the night of the concert. At a recent concert, the 1,000-seat hall was full. Tickets
bought in advance cost $30, tickets sold at the door cost $40, and total ticket sales were
$38,000.
a. What are the two different types of tickets?
b. Keep track of guesses you make by writing them down and then checking to see if they
work. Look for patterns in your work.
c. Estimate the number of advance sales and the number of door sales made that night.
d. Check that your estimates fit the conditions by substituting them into equations.
C.
Movie Economics 12: The Webber and Searle families went to the movies. Admission prices were
$9 for adults and $5 for children, and came to a total of $62 for the 10 people who went.
a. Write two equations to represent this situation.
b. Solve your system to determine how many adults and how many children went to the movies.
Impact 3. P 265
Impact 3. P 277
12 Impact 3. P279
10
11
© 2016 CUNY Collaborative Programs
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S420
NAME
DATE
ALGEBRA
LESSON 4.9
Activity 7
A "system" of equations is a set or collection of equations that you deal with together. In a system,
you seek a “solution” that makes all of the equations true.
Think about the problems you have been doing in this section:
• First you graphed two lines and found the point where they intersected.
• Then you worked on several algebra balance problems and other puzzles where you were
given multiple clues and had to find the “solution” or values for each variable.
• You also worked on some scenarios where you had to find solutions that fit multipe conditions.
All these problems have involved systems of equations.
1.
Consider these five equations 13:
i. 2x + 3 = 7
ii. x2+5x+6=0
iii. (x - 2)2 = 0
iv. 2x + y = 7
v. 3x - y=3
a. Equations i, ii, and iii each involve one variable. How many solutions does each equation
have? Find the solutions using any method you choose.
b. Equations iv and v each involve two variables. How many solutions does each equation
have? Can you list them? Explain why or why not.
c. Consider Equations iv and v. If you were to graph the solutions of 2x + y = 7 and 3x – y
= 3 together on the same graph, what would you expect to see? What would be
especially significant on your graph? Explain.
d. Draw graphs of 2x + y = 7 and 3x - y = 3. At what point or points do they meet?
13
Impact 3 p 257
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S421
NAME
DATE
ALGEBRA
LESSON 4.9
e. What values of x and y make both x + y = 7 and 3x — y = 3 true? That can mean, what
is the solution of this system of two equations? How do these values relate to your answer
to Part d?
It is possible to find solutions of systems of equations that are not linear!
2.
Use a graph to solve this system of equations. Rewrite the equations before graphing them, if
necessary. How many solutions can you find?
y – x - 1=0
y=x2 - 3x + 4
3.
The problem below is from Activity 3. Is this an example of a system of equations? Why or
why not? 14
14
Edited from EDC’s Transition to Algebra
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S422
NAME
4.
DATE
ALGEBRA
LESSON 4.9
Consider the equation x + 1 = x2 - 3x + 4 :
a. Follow these steps to solve this equation using a table:
• Enter two equations, y1 = x + I and y2 = x2 - 3x + 4, into a graphing calculator.
• Use features of your calculator to find the points of intersection as accurately as you can.
• Examine the table for values of x for which y1 and y2 are equal. Create new tables,
using smaller increments as necessary.
b. How do the solutions of this equation relate to what you found in Problem 2? Explain why
the solutions are related in this way.
c. Now use a table or graph to solve x2 - 4x + 3 = 0. How do the solutions of this equation
compare to your result for Part a? Explain.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S423
NAME
DATE
ALGEBRA
LESSON 4.9
Problem Set 4
Find solutions for the following systems of two equations:
1.
y = 2x - 8
y = -3x + 42
2.
3x -2y = 75
y = 150
3.
y=x2
y=9
5.
2x-3y = -9
-5x+3y=0
6.
x+y = 3
-2x +2y = 18
8.
3x -3y = -6
x + 2y = 10
9.
y = 2x - 90
y = -3x +210
Solve by graphing:
4.
y=x+1
y = 3x -5
Solve any way
7.
2x - y = -50
-3x + y = -50
10. After doing some research, you find that there are two companies in your town with jobs for the
summer. Both of them make T-shirts. Tops, Inc. pays its employees $1000 per month plus
$2.50 for every shirt they complete in the month. Vees, Inc. pays its employees $500 per
month plus $3.50 for every shirt they complete in a month.
a. Consider if you were a student looking for full-time summer work. Pose two questions you
would want to ask?
b. Jon knows he can usually work faster than all of his friends. Explain which company might
be a better choice for him.
c. Matt doesn't like stressful, fast-paced environments. Which company might be better for
him? Why?
d. Create one table for Tops and a second one for Vees. Record any questions you might
have as you are completing the tables.
e. Shaneece worked making shirts in another city last year. She knows that in 40 hours she
can usually make about 120 shirts. She thinks that she would probably be able to work
about 160 hours in a month. Where do you think she should apply?
f. Use your tables from d to create a graph of Monthly Income (vertical axis) versus Shirts
completed (horizontal axis) for both Tops and Vees.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S424
NAME
DATE
ALGEBRA
LESSON 4.9
g. At Tops, Inc., every time employees double their monthly production rate, they earn an
extra $500. How would this change the choices and strategies for someone seeking a
job?
Activity 8: Skill Review
Yulia and Angelique discuss how to work with Absolute Value. They are given the problem:
Simiplify: |−14 + 5|
Yulia
I remember last year, when I had Mr. Umbridge, learning about these. What are they
called again?
Angelique
Something with number, or value in it? Absolute number?
Yulia
No, it was Absolute value! That was it.
Angelique
Yeah! That was totally it. I had Mrs. Lockhart and she talked about how the Absolute
Value bars are kinda like parentheses.
Yulia
Like, you have to do whatever is inside of them first?
Angelique
Yeah, like it’s the P in PEMDAS. So |−14 + 5| becomes |−9|.
Yulia
Yeah, but I forget what else Mr. Umbridge said. I remember him drawing a picture
and talking about distance. What did your teacher last year tell you to do?
Angelique
I think Mrs. Lockhart talked about how after you get to just one number inside the
Absolute Value bars, then it’s always a positive number.
Yulia
That’s a little different from what Mr. Umbridge said. I think he talked about how
once you got just one number inside, the answer inside was how far away it was from
zero.
Angelique
−10 −9
Is that the same thing?
−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
Use the number line above to tell how far apart is −9 from 0 on the number line. Whose teacher
last year used the number line?
What is |−9| equal to?
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S425
NAME
DATE
ALGEBRA
LESSON 4.9
Problem Set 5
Perform the operations indicated.
1.
|−1| + |13|
2.
|−1| + |−13|
3.
|−9| − |−4|
4.
|−4| − |−9|
5.
|−12 + 9|
6.
|13 − 18|
7.
|−8 − 7|
8.
|5 − (−4)|
9.
|−8 + 4| − |5 − 7|
10.
|16 − 22| − |9 − 10|
12.
|8 − 9| − |4 − 14|
11. |5 − 12| − |12 − 5|
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S426
MORE TOOLS
FOR
EQUATIONS
NAME
DATE
MTE
LESSON 1
More Tools for Equations Lesson 1
Warm Up Activity
1A.
2.
1B.
Rewrite the following expressions changing subtraction to adding the opposite. If possible,
simplify the expression:
A. x – 2y
B. 10 – 2
© 2016 CUNY Collaborative Programs
C. 10 – -2
D. -5y – 5y
Supported by the NYS Education Department
S556
NAME
DATE
MTE
LESSON 1
Mini Lesson 11
Introduction: Because division is un-multiplication, the same area model can provide the information
for three different equations.
Below are the three different equations that would correspond to the model:
Multiply the two factors:
4(b+3) = 4b+12
Divide by a factor:
4𝑏𝑏+12
= 𝑏𝑏 + 3
4
Divide by the other factor:
4𝑏𝑏+12
=4
𝑏𝑏+3
1.
Consider the following:
(2y + 5)(y – 3)
We can use 2y+5 as one dimension of the rectangle and y+ -3 as the other:
2y
5
y
-3
The areas of the internal sections are filled in below:
1
2y
5
y
2y2
5y
-3
-6y
-15
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S557
NAME
DATE
MTE
LESSON 1
Now I can extract 3 equations (two division and one multiplication):
2𝑦𝑦 2 + −1y+ −15
2𝑦𝑦+5
2𝑦𝑦 2 + −1y+ −15
𝑦𝑦−3
= 𝑦𝑦 − 3
= 2𝑦𝑦 + 5
(2𝑦𝑦 + 5)(𝑦𝑦 − 3) = 2𝑦𝑦 2 + −1𝑦𝑦 + −15
Turn to a partner, and explain where these equations come from.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S558
NAME
DATE
MTE
LESSON 1
Activity 12
Name: _________________________________________
1.
Use each area model to write three equations: one using multiplication and two using division.
A.
B.
C.
4y
2
D.
y
-9
4y2
-36y
m
6
m
m2
6m
5
5m
30
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S559
NAME
2.
DATE
MTE
LESSON 1
Draw an area model for each expression and use it to answer the multiplication or division
problem.
A.
C.
4(3𝑡𝑡 2 + 5) =
3𝑔𝑔2 +6𝑔𝑔
𝑔𝑔+2
B.
20𝑏𝑏−15
5
=
=
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S560
NAME
3.
DATE
MTE
LESSON 1
For the problems below, the inside of the area model is given. Complete the outside of the
model and write two or three equations corresponding to the model.
A.
B.
C.
D.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S561
NAME
4.
DATE
MTE
LESSON 1
With some expressions, the area model doesn’t give any new information. For example, it is
not more helpful to fill in the outside of the model below because b2 and 5 have no common
factors besides 1.
For the problems below, if it makes sense to factor, fill out the outside of the area model.
Otherwise, cross it out.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S562
NAME
DATE
MTE
LESSON 1
Problem Set 13
Name: _________________________________________
1.
Use each area model to write three equations: one using multiplication and two using division.
A.
B.
C.
D.
E.
3
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S563
NAME
2.
3.
DATE
MTE
LESSON 1
Draw an area model, and use it to answer the multiplication or division problem:
A.
B.
C.
D.
For these problems, only the inside of the area model is filled in. Complete the outside of the
model and use your work to write at least one equation represented by your model
(multiplication or division).
A.
B.
C.
D.
© 2016 CUNY Collaborative Programs
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S564
NAME
DATE
E.
4.
MTE
LESSON 1
F.
Find the following:
A
B
C
D
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S565
NAME
DATE
MTE
LESSON 1
Think!
Here are four ways to set up the expression b2 + 8w + 12 inside an area model. Three of the ways
don’t help or don’t work when you try to fill out the outside. Cross out the three that don’t work, and
complete the one that does.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S566
NAME
DATE
MTE
LESSON 1
Mini-Lesson 24
Introduction: When we say, “What are factors of 15?” we think about 1, 3, 5, and 15. Turn to a
partner and discuss how factoring and division are related.
Here are four ways to set up the expression n2 + 10n + 21 inside an area model. Three of the
ways don’t help or don’t work. Cross out the three that don’t work, and complete the one that does:
1.
Write the algebraic expression in the area model in three ways: A) as a sum of terms, B) as
two factors, and C) as one division equation.
Discuss writing the expression with like terms combined for A above.
4
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S567
NAME
2.
DATE
MTE
LESSON 1
Write the algebraic expression in the area model in three ways: A) as a sum of terms, B) as
two factors, and C) as one division equation.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S568
NAME
DATE
MTE
LESSON 1
Activity 25
Name: _________________________________________
1.
Write the algebraic expression in the area model in three ways: A) as a sum of terms, B) as
two factors, and C) as one division equation.
A.
2.
B.
Fill in the missing information, and complete the equations:
A.
B.
C.
5
D.
Transition to Algebra
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S569
NAME
3.
DATE
MTE
LESSON 1
Challenge Problems!
Arrange the expression inside the area model in a way that lets you fill out the outside. Fill out the
outside, too.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S570
NAME
DATE
MTE
LESSON 1
Activity 36
Name: _________________________________________
1.
Draw an area model and use it to answer the multiplication or division problem.
A. (x + 2)(y + 6)
C.
2.
6
B.
D.
The two problems below differ in a small but relevant way. Figure out how they differ, and
then solve them.
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S571
NAME
3.
DATE
MTE
LESSON 1
Match the area models below to the expressions they describe. There will be two expressions
for each area model.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S572
NAME
DATE
MTE
LESSON 1
Problem Set7
Name: _________________________________________
1.
7
Solve these area model puzzles, and write algebraic equations to describe them:
A.
B.
C.
D.
E.
F.
Materials adapted from: Transition to Algebra, Unit 10, p 4-13.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S573
NAME
2.
DATE
G.
H.
I.
J.
MTE
LESSON 1
Match the area models to the expressions they describe. There are two expressions for each
area model:
Area models K & L
Area models M & N
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S574
NAME
DATE
MTE
LESSON 2
More Tools for Equations Lesson 2
Mini Lesson 11
There are at least 10 different ways you could fill out this area model. Pick 2 different ways and
complete the model.
1
Materials are adapted from: Transition to Algebra.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S575
NAME
DATE
MTE
LESSON 2
Activity 12
Name: _________________________________________
5.
List all the pairs of integers (positive or negative) whose product is 12:
6.
Which pair has a sum of 7? ________ of 8? _________ of 13? _________
of -7? _________
2
This lesson is available online at: http://www.heinemann.com/shared/onlineresources/E05750/TTA_U10.pdf
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S576
NAME
DATE
MTE
LESSON 2
15. List all the pairs of integers (positive or negative) whose product is 30:
16. Which pair has a sum of 11? ________ of -13? _________ of 31? _________
of -17? _________
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S577
NAME
© 2016 CUNY Collaborative Programs
DATE
MTE
LESSON 2
Supported by the NYS Education Department
S578
NAME
DATE
MTE
LESSON 2
Activity 2
Name: _________________________________________
1.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S579
NAME
DATE
MTE
LESSON 2
2.
3.
a. Draw how you would arrange 5 roads so there are 6 intersections in the village.
b. Sketch a village map in which 7 roads form 10 intersections.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S580
NAME
c.
DATE
MTE
LESSON 2
d.
e. Suppose the people said they wanted 12 intersections but didn’t care how many roads they
had. How many different possible arrangements could you make? Draw the maps. What is
the smallest number of roads that will still give them 12 intersections?
f. With exactly 10 roads, what is the smallest number of intersections there can be? What is the
greatest number of intersections?
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S581
NAME
DATE
MTE
LESSON 2
Problem Set 1
Name: ____________________________________
Factor the expressions below. You may need to draw your own area model to complete these.
1.
3.
5.
© 2016 CUNY Collaborative Programs
2.
4.
6.
Supported by the NYS Education Department
S582
NAME
DATE
7.
MTE
LESSON 2
8.
10.
9.
11.
© 2016 CUNY Collaborative Programs
12.
Supported by the NYS Education Department
S583
NAME
DATE
MTE
LESSON 2
Mini Lesson 23
3
A.
B.
C.
D.
E.
F.
Transition to Algebra
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S584
NAME
DATE
MTE
LESSON 2
Activity 3
Name: _________________________________________
When factoring, we need to find two numbers that have a particular sum (from adding them), and
also have a particular product (from multiplying them). Sometimes the factors can be guessed easily
but other times it is not obvious. It becomes useful to have a system to organize your search.
Draw your own area model for the following, and then factor:
Example:
2.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S585
NAME
DATE
MTE
LESSON 2
3.
4. Decide if the factors in each problem will involve addition, subtraction or both
a.
b.
c.
•
Explain how you knew the answer for the problem above!
5.
6.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S586
NAME
DATE
MTE
LESSON 2
Problem Set 2
Name: _________________________________________
1.
2.
3.
4.
5.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S587
NAME
DATE
MTE
LESSON 2
6.
7.
9.
8.
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S588
NAME
DATE
10.
11.
12.
13.
© 2016 CUNY Collaborative Programs
MTE
LESSON 2
Supported by the NYS Education Department
S589
NAME
DATE
MTE
LESSON 2
Activity 44
Factoring Game Directions
You need:
•
•
•
•
1.
2.
3.
4.
5.
6.
a partner
pens or pencils in two different colors
A Factoring Game game board (see above for sample)
Blank paper with a t-table to keep score.
Player A chooses a number or expression on the game board and circles it. This will be Partner
A’s score for that round.
Using a different color, Partner B circles all the proper factors of Player A’s number or
expression. The proper factors of a number are all the factors of that number except the
number itself. Partner B lists the factors as her score.
Player B then circles a new number or expression. Player A circles all the remaining factors of
that number or expression. Then, play continues in this manner.
The players take turns choosing numbers or expressions and circling factors.
The game is over when on consecutive plays both players have been unable to circle at least
one un-circled factor from their opponent’s choice.
The player with the larger sum of factors and products is the winner.
Optional Rules:
Scoring could be number of boxes circled
Flip a coin in the beginning. The winner gets to choose a value for the ending x. At the end players
calculate the “total” from all of their expressions summed together.
Roll the dice and that will be the value of x
To Do: Create more game boards with factoring variations. Build an App! Or a site.
See https://illuminations.nctm.org/Activity.aspx?id=4134
4
Adapted from: https://illuminations.nctm.org/Activity.aspx?id=4134
© 2016 CUNY Collaborative Programs
Supported by the NYS Education Department
S590
NAME
DATE
MTE
LESSON 2
1
2
3
4
x
-x
- 2x
2x
3x
- 3x
x-1
x+1
x-2
x+2
x-3
x+3
- 2x2
- x2 + 2x
2x2 + 2x
- x2 - 2x
x2 + x - 2
x2 + 3x + 2
x2 - 4
2x2 - 2x
2x2 - 4x
- x2 + 3x
x2 - 4x + 3
x2 - 2x - 3
x2 - 5x + 6
x2 - 3x + 2
x2 - 4x + 4
- x2 - 3x
x2 + 2x - 3
x2 + 4x + 3
x2 + x - 6
x2 - x - 2
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LESSON 2
Activity 5
Sasha felt like she could always find a way to do things on her graphing calculator. She considered
the problem x2 – 13x – 48= (x + u)(x + v). She knew that she needed uv = - 48 and u + v = -13.
• Create a grid with a u-axis and a v-axis and then graph uv= -48 and u + v= -13. (Hint: Put
u on the x-axis and v on the y-axis and create graphs.)
• How can you use the intersections of the graphs to factor the problem?
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LESSON 3
More Tools for Equations Lesson 3
Warm-Up Activity
1.
If the product of two numbers is negative, what can you say about the numbers?
2.
If the product of two numbers is positive, what can you say about the numbers?
3.
Find the product of the following:
0.000001 x 0.000001?
899,999,831.345 x 0
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LESSON 3
Mini Lesson 1
1.
If the product of two numbers is zero, what must be true about the numbers?
2.
If the product of two numbers is not zero, what can you say about the numbers?
3.
The sum of a and b is zero. What can you say about a and b?
4.
If the difference of two numbers is zero, what can you say about the numbers?
5.
If the product of three numbers is positive, what can you say about the numbers?
6.
If the product of three numbers is negative, what can you say about the numbers?
Think about your answers to these questions as you solve the equations below:
A. If 2y = 0, y = _____
B.
C.
If 2(v – 1) = 0, v = _____
If pq = 0, p = _____ or q = ______
D. If (t – 1)r = 0, t = _____ or r = ______
E.
F.
If x – 7 = 0, x = ______
If y(y – 8) = 0, y = ______ or ______
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LESSON 3
G. If (x – 7)(x + 7) = 0, x = ______ or _____
H. If (7 + t)(6 – t) = 0, t = _____ or _____
I.
If (z + 4)(z + 5) = 0, z = ____ or _____
J.
(𝑥𝑥 − 5)(𝑥𝑥 − 5) = 0, x = ____ or _____
The concept you have been thinking through and using is called the “Zero Product Property.”
Activity 11
Name: _____________________________________
1.
You found a notebook with “Decoder” written on its cover. Inside, you found the following
message:
Replace each letter with a number, if:
a x b x c = 0,
a x b = 12 and
b – a = 1.”
What is the value of each letter? a = _____
2.
b = _____ c = _____
What is the value of z in the following:
q*r*s*t*u*v*w*x*y*z=z
z * z = z and
z–z=z
1
Materials adapted from TTA…
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3.
DATE
y2 + 8y + 14 = 0
____ ____
____
____
____
____
)(
LESSON 3
4. y2 – 9y + 20 = 0
____ ____
(
MTE
)=0
y = _____ or _____
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(
)(
)=0
y = _____ or _____
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MTE
LESSON 3
Solve the equations below. You may need to factor first. Use an area model, table, or anything
else that will help you.
5.
If x2 + 8x – 9 = 0 , x = _____ or _____
6.
If x2 + 5x – 36 = 0, x = _____ or _____
7.
If x2 + 17x + 30 = 0, x = _____ or _____
8.
If x2 + 7x + 6 = 0, x = _____ or _____
Use factoring to solve the equations below. To use the zero product property, you may need to
rewrite the equation. Use an area model, table, or anything else that will help you.
9.
x2 + 5x + 11 = 5
10.
x2 + 9x + 9 = -9
11. Create an equation that has only one solution.
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LESSON 3
12. Create an equation that has exactly two solutions.
13.
(From Transition to Algebra, 2014)
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LESSON 3
Problem Set 1
Name: ____________________________________
Solve each of the following equations:
1.
3f = 0
2.
6(p + 2) = 0
3.
c2 = 0
4.
y(y + 3) = 0
5.
(y + 3)y = 0
6.
(s + 3)(s – 4) = 0
7.
(x – 6)(x + 6) = 0
8.
r2 – 4 = 0
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MTE
LESSON 3
Factor the following expressions. Use whatever method helps you, including area models or tables.
9.
x2 + 10x + 24
11. x2 – 7x + 6 = 0
10. x2 + 9x – 36
12. x2 + 5x – 6 = 0
Use factoring to solve the equations below. To use the zero product property, you may need
to rewrite the equation. Use an area model,
table, or anything else that will help you.
13. (x + 5)(x – 7) = 0
14. (x + 9)(x – 2) = 0
15. x2 + 5x – 2 = 4
16. x2 - 5x – 2 = 4
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MTE
17. x2 + 6x + 8 = 0
18. x2 + 7x + 10 = 0
19. x2 - 10x – 11 = 0
20. x2 + 4x – 60 = 0
21. x2 + 17x – 60 = 0
22. x2 + 10x + 16 = 0
LESSON 3
Challenge problems! Solve for x, y and z in the following:
A. (x – 4)(y + 2)(y – 7) = 0
(x – 4)(y – 7) = -9
x = _____
y = _____
© 2016 CUNY Collaborative Programs
B. (x + 2)(y + 3)(z – 2) = 0
(y + 3)(z – 2) = 0
(x + 2)(y + 3) = 0
(y + 3) = 4
x = _____
y = _____ z = _____
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NAME
25.
DATE
MTE
LESSON 3
(Source: Transition to Algebra, 2014)
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NAME
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MTE
LESSON 3
Activity 22
Consider the following problem without solving it:
(x + 3)(y – 2)(z – 2) = 0
(x + 3)(y – 2) = -7
(x + 3) = 7
Consider some of the strategies we used previously to find solutions for equations, such as replacing
parts of the expression with a “Blob,” as follows, so that each expression is its own puzzle:
(Blob1)(Blob2)(Blob3) = 0
(Blob1)(Blob2) = -7
(Blob1) = 7
• IF (Blob1) = 7, then x = _____ when x + 3 = 7.
• SO if (7)(Blob2) = -7, Blob2 = _____, and
if Blob2 = _____, then y – 2 = _____ and y = _____.
• FINALLY, if (Blob1)(Blob2)Blob3) = 0, then (-7)(Blob3) = 0, and Blob3 = ______,
so that
(z – 2) = 0 and z = _____.
Can you create your own problem when we give you the solution?
• Begin with the following: a = 2
b=5
c=0
• Create the clues that could lead to your solution.
• Now create your own problem. Select values for a, b, and c, then create your clues. Put your
clues on a card and trade them with a partner in order to solve each other’s puzzles.
2
The following is adapted from Transition To Algebra, 2014
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LESSON 3
4. Scott is claiming that his solution of x = 2 or x= 5 can have more than one possible equation.
Would the following equations work?
𝑥𝑥 2 − 7𝑥𝑥 + 10 = 0
2𝑥𝑥 2 − 14𝑥𝑥 + 20 = 0
𝑥𝑥 2 − 7𝑥𝑥 + 5 = −5
Activity 3: Skill Review
The price of a 3D movie ticket to see the new Star Wars movie is $16. A group of 4 friends, Mary,
Josie, Tricia, & Theresa decide on their walk to school that they are going to see the movie this
Saturday.
Price Per ticket in dollars
𝟏𝟏𝟏𝟏
Number of tickets purchased
𝟒𝟒
Total paid in dollars
𝟏𝟏𝟏𝟏 ⋅ 𝟒𝟒
But Shawn overhears them making their plans, and says he wants to go too and bring his best friend
Ricardo. The girls agree to bring them along.
Price Per ticket in dollars
16
𝟏𝟏𝟏𝟏
© 2016 CUNY Collaborative Programs
Number of tickets purchased
4
𝟔𝟔
Total paid in dollars
16 ⋅ 4
𝟏𝟏𝟏𝟏 ⋅ 𝟔𝟔
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MTE
LESSON 3
Later in the day at the lunch table, Marcus finds out about it and also says he was looking for some
people to go see the movie with.
Price Per ticket in dollars
16
Number of tickets purchased
4
Total paid in dollars
16 ⋅ 4
𝟏𝟏𝟏𝟏
𝟕𝟕
𝟏𝟏𝟏𝟏 ⋅ 𝟕𝟕
Price Per ticket in dollars
16
Number of tickets purchased
4
Total paid in dollars
16 ⋅ 4
16
7
16 ⋅ 7
16
6
16 ⋅ 6
Then Mary makes the suggestion that they should just invite the whole class. Before 6th period she
announces to everyone their plans, and to raise their hand if they want to join. They count that 25
total people want to go see the movie this Saturday.
16
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏
6
𝟐𝟐𝟐𝟐
𝒏𝒏
16 ⋅ 6
𝟏𝟏𝟏𝟏 ⋅ 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏 ⋅ 𝒏𝒏
The total amount for all the tickets depends on how many students are going to see the movie. The
movie ticket price does not change, it is always $16. What changes is the number of possible
students going to see the movie.
If we don’t know the total number of students who went, we don’t know how much money will be
spent
But we can represent it as something! If there are 𝑛𝑛 students, then we know that the total amount can
be represented as 16 ⋅ 𝑛𝑛.
Once we know how many students, then we can fill in for 𝑛𝑛.
The total amount paid if 10 students go is 16 ⋅ 10 dollars. So in this case we evaluate the
expression for 𝑛𝑛 = 10.
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LESSON 3
The total amount paid if 15 students go is 16 ⋅ 15 dollars. In this case we evaluate the expression
for 𝑛𝑛 = 15.
Evaluating Expressions are helpful ways to sometimes work with problems.
When the students go to see the movie, some may decide they want popcorn. Complete the table
below.
Price Per popcorn in dollars
Total paid in dollars
6
Number of popcorns
purchased
1
6
3
6 ⋅__
6
6
6
6
6
© 2016 CUNY Collaborative Programs
2
4
5
6
𝑝𝑝
6 ⋅__
6 ⋅__
6 ⋅__
6 ⋅__
6 ⋅ __
6 ⋅ __
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MTE
LESSON 3
Mini-Lesson
Example 1: Evaluate 3𝑥𝑥 2 for 𝑥𝑥 = 5
Rewrite 1: 3( )2
Then you can fill in the empty space with the value that is given.
Rewrite 1: 3(5)2
Then you can simplify the expression using the Order of Operations.
3(5)2 = 3 ∙ 25 = 75
Example 2: Evaluate −4𝑏𝑏 − 2𝑐𝑐 for 𝑏𝑏 = 2.5 and 𝑐𝑐 = 10
Rewrite 2: −4( ) − 2( )
Then you can fill in the empty space with the value that is given.
Rewrite 2: −4(2.5) − 2(10)
Then you can simplify the expression using the Order of Operations.
−4(2.5) − 2(10) = −10 − 20 = −30
Complete each of the examples and simplify them to one number.
Example 1: Evaluate 3𝑥𝑥 2 for 𝑥𝑥 = 5
Example 2: Evaluate −4𝑏𝑏 − 2𝑐𝑐 for 𝑏𝑏 = 2.5 and 𝑐𝑐 = 10
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MTE
LESSON 3
Problem Set 2
Evaluate the following expressions.
1. 2𝑥𝑥 − 5𝑦𝑦 for 𝑥𝑥 = 9 and 𝑦𝑦 = −7
2. 6𝑥𝑥 2 + 9𝑥𝑥 − 4 for 𝑥𝑥 = −2
3. 7𝑦𝑦 − 4𝑧𝑧 + 2𝑤𝑤 for 𝑦𝑦 = −1, 𝑧𝑧 = 2, and 𝑤𝑤 =
1
2
4. 3𝑎𝑎2 − 4𝑏𝑏2 for 𝑎𝑎 = 1 and 𝑏𝑏 = −1
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LESSON 3
5. 5𝑡𝑡 − 14𝑠𝑠 + 𝑠𝑠𝑠𝑠 for 𝑠𝑠 = 9 and 𝑡𝑡 = −4
6. 4𝑥𝑥 3 + 2𝑥𝑥 2 − 7𝑥𝑥 + 12 for 𝑥𝑥 = −3
7. 2𝑥𝑥 2 + 5𝑥𝑥𝑥𝑥 − 3𝑦𝑦 2 for 𝑥𝑥 = 3 and 𝑦𝑦 = −2
1
8. The formula for the area of a triangle is 𝑏𝑏ℎ where 𝑏𝑏 is the base and ℎ is the height. What
2
is the area of a triangle with the base of 5 and a height of 12?
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MTE
LESSON 3
Activity 5: Skill Review
There are times in algebra when using guess and check along with evaluating can help lead to an
answer. With multiple choice problems, you can use the answers to give you your guesses too!
Follow Deno and Eddie to see how they reached the first answer on their exams. Then answer the
questions that show up after.
A.
B.
C.
D.
1
−1
−5
5
#1: Solve for 𝑡𝑡 if 5𝑡𝑡 − 3(𝑡𝑡 + 6) = −8
Deno is thinking: It’s the test and uh oh! I can’t
remember how to do these! Okay, calm down, I can
do this! Okay, what do I remember?
I know I can guess. Let me try plugging answer A.
5(1) − 3(1 + 6) = −8
Now I know I can just follow PEMDAS. Let’s see,
parentheses first.
5(1) − 3(7) = −8
5 − 21 = −8
Okay, 5 − 21 on my calculator gives me −16 not
−8, so it’s not A.
Let me give B a shot. What if 𝑡𝑡 is −1? Ugh! I hate
negatives. Good thing I have a calculator for this
exam.
5(−1) − 3(−1 + 6) = −8
Working through that he gets:
−20 = −8
That’s not right either. I’m glad there is no time limit
for this exam. I can take my time. Let me try C.
5(−5) − 3(−5 + 6) = −8
Then he gets:
−28 = −8
Okay, that didn’t work either. This last choice better
work!
5(5) − 3(5 + 6) = −8
Finally he gets:
−8 = −8
Hey that worked! The answer must be D. Okay, let
me try the next one. Whew!
© 2016 CUNY Collaborative Programs
Eddie is thinking. Okay, yes! I know how to do
these. First let me write the question down.
5𝑡𝑡 − 3(𝑡𝑡 + 6) = −8
Great, okay, now I think my first step is to distribute.
5𝑡𝑡 − 3𝑡𝑡 − 18 = −8
Oh man, I’m glad I studied distributing yesterday.
Last week I made the mistake where I forgot to
multiply the negative part!
Next step, I gotta put the two 𝑡𝑡 things together. Five
𝑡𝑡s minus three 𝑡𝑡s is gonna be two 𝑡𝑡𝑡𝑡.
2𝑡𝑡 − 18 = −8
Now it looks normal! I know I just have to move the
18 over. So I’m gonna add 18 to both sides.
2𝑡𝑡 = 10
Then I just divide both by 2:
𝑡𝑡 = 5
Awesome! That’s there.
I really want to do well on this test though, so before
I pick it and move on, I’m gonna check my work.
If I plug in 5, I should get −8.
5(5) − 3(5 + 6) = −8
5(5) − 3(11) = −8
25 − 33 = −8
−8 = −8
Perfect! Alright, let me move on to the next one.
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MTE
LESSON 3
Answer the following questions about Deno and Eddie.
1. Who answered the two questions correctly?
2. What method did Deno use when he forgot how to solve the equations?
3. Does Eddie need to check his solution in a multiple choice test?
Problem Set 3
Solve the following multiple choice problems in whatever way you like. Use Deno or Eddie to help.
1. 4𝑥𝑥 − 5 − 13𝑥𝑥 = 13
13
A.
22
B. −2
C. 2
8
D. −
9
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LESSON 3
2. 13𝑦𝑦 + 2(𝑦𝑦 − 8) = −16
A. 15
8
B. −
C.
8
5
5
D. 0
3. 4𝑥𝑥 − (3 + 2𝑥𝑥) = 3𝑥𝑥 + 5
A. −8
B. 8
C. 5
D. −4
4. 12𝑧𝑧 + 2(2 − 5𝑧𝑧) = −8
A. 6
B. 2
C. −2
D. −6
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LESSON 3
5. −5𝑥𝑥 − 3(𝑥𝑥 + 4) = −2𝑥𝑥
A. 2
B. −2
2
C.
3
D. −
3
2
6. −4 − 2(𝑥𝑥 + 3) = 7
17
A. −
2
B. −4
C. 4
17
D.
2
7. 5 = −3𝑥𝑥 + 7(𝑥𝑥 + 3)
1
A.
2
B. 11
C. −4
D. −11
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LESSON 3
8. 4 + 2𝑥𝑥 (2 − 5) = −4
A. 0
B. 12
4
C. −
D.
4
3
3
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