Unit 2 Plan - Grade 7

Introducing Proportional Relationships with Tables

—

If the ratios between two corresponding quantities are always equivalent, the relationship between the quantities is called a proportional relationship.

This table shows different amounts of milk and chocolate syrup. The ingredients in each row, when mixed together, would make a different total amount of chocolate milk, but these mixtures would all taste the same.

Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to $4:1$ .

About the relationship between these quantities, we could say:

tablespoons of chocolate syrup	cups of milk
4	1
6	$1\frac{1}{2}$
8	2
$\frac{1}{2}$	$\frac{1}{8}$
12	3
1	$\frac{1}{4}$

The relationship between the amount of chocolate syrup and the amount of milk is proportional.
The table represents a proportional relationship between the amount of chocolate syrup and amount of milk.
The amount of milk is proportional to the amount of chocolate syrup.

We could multiply any value in the chocolate syrup column by $\frac14$ to get the value in the milk column. We might call $\frac14$ a unit rate, because $\frac14$ cup of milk is needed for 1 tablespoon of chocolate syrup. We also say that $\frac14$ is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup.

Green Paint (1 problem)

When you mix two colors of paint in equivalent ratios, the resulting color is always the same. Complete the table as you answer the questions.

How many cups of yellow paint should you mix with 1 cup of blue paint to make the same shade of green? Explain or show your reasoning.
Make up a new pair of numbers that would make the same shade of green. Explain how you know they would make the same shade of green.
What is the proportional relationship represented by this table?
What is the constant of proportionality? What does it represent?

cups of blue paint	cups of yellow paint
2	10
1

Show Solution

You need 5 cups of yellow paint for 1 cup of blue paint. You can see this by multiplying the first row by a factor of $\frac12$ . Alternatively, you have to multiply 2 by $\frac{10}{2} = 5$ to get 10. Multiplying 1 by 5 gives 5.
Any amounts equivalent to the ratio of 1 cup of blue paint to 5 cups of yellow paint. Sample response: 3 cups of blue paint mixed with 15 cups of yellow paint will also make the same shade of green. This can be obtained by multiplying the second row by a factor of 3 or choosing 3 for blue and then multiplying that by 5.
The relationship between the amount of blue paint and the amount of yellow paint is the proportional relationship represented by this table.
The constant of proportionality is 5. It represents the cups of yellow paint needed for 1 cup of blue paint.

Lesson 3

More about Constant of Proportionality

—

When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled.

The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.

We can multiply any number in the first column by $\frac23$ to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is $\frac23$ . This means that the bug’s pace is $\frac23$ seconds per centimeter.

Table with 2 columns and 4 rows of data. distance traveled (cm) and elapsed time (sec).

This table represents the same situation, except the columns are switched.

We can multiply any number in the first column by $\frac32$ to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is $\frac32$ . This means that the bug’s speed is $\frac32$ centimeters per second.

Table with 2 columns and 4 rows of data. elapsed time (sec) and distance traveled (cm).

Notice that $\frac32$ is the reciprocal of $\frac23$ . When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column by to get the values in the second.

Fish Tank (1 problem)

Mai is filling her fish tank. Water flows into the tank at a constant rate. Complete the table as you answer the questions.

How many gallons of water will be in the fish tank after 3 minutes? Explain your reasoning.
How long will it take to fill the tank with 40 gallons of water? Explain your reasoning.
What is the constant of proportionality?

time (minutes)	water (gallons)
0.5	0.8
1
3
	40

Show Solution

4.8. If the first row is doubled (scale by 2), there are 1.6 gallons after 1 minute. If the second row is tripled (scale by 3), there are 4.8 gallons after 3 minutes. Or the first row could be scaled by 6 to get 4.8 gallons after 3 minutes.
25 minutes. One way to find a scale factor to use is to divide 40 by 0.8. $\frac{40}{0.8} = 50$ and $50 \cdot 0.5 = 25$ .
1.6 (or equivalent). You can observe the amount of water that corresponds with 1 minute, or you can divide any value in the right column with its corresponding value in the left column.

Section A Check

Section A Checkpoint

Lesson 4

Proportional Relationships and Equations

—

In this lesson, we wrote equations to represent proportional relationships described in words and shown in tables.

This table shows the amount of red paint and blue paint needed to make a certain shade of purple paint, called Venusian Sunset.

Note that “parts” can be any unit for volume. If we mix 3 cups of red with 12 cups of blue, you will get the same shade as if we mix 3 teaspoons of red with 12 teaspoons of blue.

red paint (parts)	blue paint (parts)
3	12
1	4
7	28
$\frac14$	1
$r$	$4 r$

The last row in the table shows that if we know the amount of red paint, $r$ , we can always multiply it by 4 to find the amount of blue paint needed to make Venusian Sunset. If $b$ is the amount of blue paint, we can say this more succinctly with the equation $b=4 r$ . So, the amount of blue paint is proportional to the amount of red paint, and the constant of proportionality is 4.

We can also look at this relationship the other way around.

If we know the amount of blue paint, $b$ , we can always multiply it by $\frac14$ to find the amount of red paint, $r$ , needed to make Venusian Sunset. So, the equation $r=\frac14 b$ also represents the relationship. The amount of red paint is proportional to the amount of blue paint, and the constant of proportionality $\frac14$ .

blue paint (parts)	red paint (parts)
12	3
4	1
28	7
1	$\frac14$
$b$	$\frac14 b$

In general, when $y$ is proportional to $x$ , we can always multiply $x$ by the same number $k$ —the constant of proportionality—to get $y$ . We can write this much more succinctly with the equation $y=k x$ .

It’s Snowing in Syracuse (1 problem)

Snow is falling steadily in Syracuse, New York. After 2 hours, 4 inches of snow has fallen.

If it continues to snow at the same rate, how many inches of snow would you expect after 6.5 hours? If you get stuck, you can use the table to help.
Write an equation that gives the amount of snow that has fallen after $x$ hours at this rate.
How many inches of snow will fall in 24 hours if it continues to snow at this rate?

time (hours)	snow (inches)
	1
1
2	4
6.5
$x$

Show Solution

13 inches (Two inches fell in 1 hour, 6.5 is $1 \boldcdot (6.5)$ , and $2 \boldcdot (6.5) = 13$ .)
Sample response: $y=2x$ , where $x$ is the number of hours that have passed and $y$ is the inches of snow that has fallen.
48 inches ( $24 \boldcdot 2 = 48$ )

Lesson 6

Writing Equations to Represent Relationships

—

Remember that if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form $y = k x$ . Sometimes writing an equation is the easiest way to solve a problem.

For example, we know that Denali, the highest mountain peak in North America, is 20,310 feet above sea level. How many miles is that? There are 5,280 feet in 1 mile. This relationship can be represented by the equation

$\displaystyle f=5,280 m$

where $f$ represents a distance measured in feet and $m$ represents the same distance measured in miles. Since we know Denali is 20,310 feet above sea level, we can write

$\displaystyle 20,310=5,280 m$

Solving this equation for $m$ gives $m = \frac{20,310}{5,280}≈ 3.85$ , so we can say that Denali is approximately 3.85 miles above sea level.

More Recycling (1 problem)

Glass bottles can be recycled. At one recycling center, 1 ton of clear glass is worth $25. (1 ton = 2,000 pounds)

How many pounds of clear glass is worth $10?
How much money is 40 pounds of clear glass worth?
Write an equation to represent the relationship between the weight of clear glass and the value of the glass.

Show Solution

800 pounds, because $2,000 \div 25 = 80$ and $80 \boldcdot 10 = 800$
$0.50, because $40 = 80 \boldcdot 0.50$
Sample response: If $v$ represents the value, in dollars, of $p$ pounds of clear glass, then the equation could be either $p = 80v$ or $v = 0.0125p$ .

Section B Check

Section B Checkpoint

Lesson 7

Comparing Relationships with Tables

—

Here are the prices for some smoothies at two different smoothie shops:

Smoothie Shop A

smoothie size (fl oz)	price ($)	dollars per ounce
8	6	0.75
12	9	0.75
16	12	0.75
$s$	$0.75s$	0.75

Smoothie Shop B

smoothie size (fl oz)	price ($)	dollars per ounce
8	6	0.75
12	8	0.67
16	10	0.625
$s$	???	???

For Smoothie Shop A, smoothies cost $0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is $\displaystyle p=0.75 s$ where $s$ represents size in ounces and $p$ represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)

For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely not proportional.

In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation of the form $y = k x$ , then we are sure it is proportional.

Apples and Pizza (1 problem)

Based on the information in the table, is the cost of the apples proportional to the weight of apples?

pounds of apples cost of apples

2 $3.76

3 $5.64

4 $7.52

5 $9.40
Based on the information in the table, is the cost of the pizza proportional to the number of toppings?

number of toppings cost of pizza

2 $11.99

3 $13.49

4 $14.99

5 $16.49
Write an equation for the proportional relationship.

Show Solution

Yes, because the cost per pound of apples is the same in each row, 1.88 dollars per pound.
No, because the cost per topping is not the same in each row. (An equation is $C = 1.50T + 8.99$ but students do not need to provide an equation.)
$c = 1.88p$ , where $c$ represents the cost of the apples and $p$ represents the pounds of apples.

Lesson 9

Solving Problems about Proportional Relationships

—

Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.

When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
If an aardvark is eating termites at a constant rate, then there is a proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.

Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?

If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.

After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.

Folding Programs (1 problem)

Lin is folding programs for the school music concert. She wants to know how long it will take her to finish folding all the programs. What information would you need to know to write an equation that represents this relationship?

Show Solution

Sample responses:

Is Lin folding the programs at a constant rate?
How long does it take her to fold 1 program?
How many programs can she fold in 1 minute?
How many programs are there total?

Section C Check

Section C Checkpoint

Lesson 10

Introducing Graphs of Proportional Relationships

—

One way to represent a proportional relationship is with a graph. Here is a graph that represents different amounts that fit the situation, “Blueberries cost $6 per pound.”

Line graph. Weight in pounds. Cost in dollars. Horizontal axis, 0 to 5, by 1's. Vertical Axis, 0 to 40, by 10's.

Different points on the graph tell us, for example, that 2 pounds of blueberries cost $12, and 4.5 pounds of blueberries cost $27.

Sometimes it makes sense to connect the points with a line, and sometimes it doesn’t. For example, we could buy 4.5 pounds of blueberries or 1.875 pounds, or any other part of a whole pound. So all the points between the whole numbers make sense in the situation, and any point on the line is meaningful.

If the graph represented the cost for different numbers of sandwiches (instead of pounds of blueberries), it might not make sense to connect the points with a line, because it is often not possible to buy 4.5 sandwiches or 1.875 sandwiches. Even if only points make sense in the situation, though, sometimes we connect them with a line anyway to make the relationship easier to see.

Graphs that represent proportional relationships all have a few things in common:

Points that satisfy the relationship lie on a straight line.
The line that they lie on passes through the origin, $(0,0)$ .

Here are some graphs that do not represent proportional relationships:

Graph of a non-proportional relationship, x y plane, origin O.

These points do not lie on a line.

Line graph. Horizontal axis, 0 to 7, by 1's. Vertical Axis, 0 to 6, by 1's.

This is a line, but it doesn’t go through the origin.

Which Are Not Proportional (1 problem)

Which graphs cannot represent a proportional relationship? Select all that apply. Explain how you know.

A coordinate graph, all on an xy planes, origin O. — A

Show Solution

B and C. Sample reasoning: Since graph B does not go through the origin, it cannot be a proportional relationship. Since the points in graph C cannot be connected by a single, straight line, it cannot be a proportional relationship.

Lesson 11

Interpreting Graphs of Proportional Relationships

—

For the relationship represented in this table, $y$ is proportional to $x$ . We can see in this table that $\frac54$ is the constant of proportionality because it’s the $y$ value when $x$ is 1.

The equation $y = \frac54 x$ also represents this relationship.

$x$	$y$
4	5
5	$\frac{25}{4}$
8	10
1	$\frac{5}{4}$

Here is the graph of this relationship.

If $y$ represents the distance in feet that a snail crawls in $x$ minutes, then the point $(4, 5)$ tells us that the snail can crawl 5 feet in 4 minutes.

If $y$ represents the cups of yogurt and $x$ represents the teaspoons of cinnamon in a recipe for fruit dip, then the point $(4, 5)$ tells us that you can mix 4 teaspoons of cinnamon with 5 cups of yogurt to make this fruit dip.

We can find the constant of proportionality by looking at the graph: $\frac54$ is the $y$ -coordinate of the point on the graph where the $x$ -coordinate is 1. This could mean the snail is traveling $\frac54$ feet per minute or that the recipe calls for $1\frac14$ cups of yogurt for every teaspoon of cinnamon.

In general, when $y$ is proportional to $x$ , the corresponding constant of proportionality is the $y$ -value when $x=1$ .

Filling a Bucket (1 problem)

Water runs from a hose into a bucket at a steady rate. The amount of water in the bucket for the time it is being filled is shown in the graph.

Graph of a line, x y plane, origin O. time (seconds), water (gallons).

The point $(12,5)$ is on the graph. What do the coordinates tell you about the water in the bucket?
How many gallons of water are in the bucket after 1 second? Label the point on the graph that shows this information.

Show Solution

After 12 seconds, there were 5 gallons of water in the bucket.
$\frac{5}{12}$ (or equivalent). The point $\left( 1,\frac{5}{12} \right)$ should be labeled.

Lesson 12

Using Graphs to Compare Relationships

—

Here is a graph that shows the price of blueberries at two different stores. Which store has a better price?

Two line graphs. Weight in pounds. Cost in dollars. Horizontal axis, 0 to 5, by 1's. Vertical Axis, 0 to 40, by 10's.

We can compare points that have the same $x$ value or the same $y$ value. For example, the points $(2, 12)$ and $(3, 12)$ tell us that at Store B you can get more pounds of blueberries for the same price.

The points $(3, 12)$ and $(3, 18)$ tell us that at Store A you have to pay more for the same quantity of blueberries. This means Store B has the better price.

We can also use the graphs to compare the constants of proportionality. The line representing Store B goes through the point $(1, 4)$ , so the constant of proportionality is 4. This tells us that at Store B the blueberries cost $4 per pound. This is cheaper than the $6 per pound unit price at Store A.

Revisiting the Amusement Park (1 problem)

Noah and Diego left the amusement park’s ticket booth at the same time. Each moved at a constant speed toward his favorite ride. After 8 seconds, Noah was 17 meters from the ticket booth, and Diego was 43 meters away from the ticket booth.

Which line represents the distance traveled by Noah, and which line represents the distance traveled by Diego? Label each line with one name.
The graph of two lines in the coordinate plane with the horizontal axis labeled "elapsed time in seconds" and the vertical axis labeled "distance traveled, in meters." One line begins at the origin and moves steeply upwards and to the right. The other line also begins at the origin and moves steadily upwards and to the right.
Explain how you decided which line represents which person’s travel.

Show Solution

The steeper line represents the distance traveled by Diego.
Sample reasoning: Diego had gone farther after 8 seconds. If you pick a time and look at which line represents a person who has gone farther, that is the steeper graph. So that must be Diego’s line.

Section D Check

Section D Checkpoint

Unit 2 Introducing Proportional Relationships — Unit Plan