Unit 2 Introducing Proportional Relationships — Unit Plan

Recipes 1 and 2 will taste the same. Sample reasoning: Recipe 3 is different because it requires $1\frac13$ cups of pineapple juice for every 1 cup of orange juice. Recipes 1 and 2 both require $1\frac12$ cups of pineapple juice for every 1 cup of orange juice.

Double number line diagrams can be used to compare the recipes, for instance, by noting that for Recipes 1 and 2, you use 2 cups of orange juice for every 3 cups of pineapple juice, whereas with Recipe 3, you use $2\frac14$ cups of orange juice for 3 cups of pineapple juice.

1. 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.
2. 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. 
3. The relationship between the amount of blue paint and the amount of yellow paint is the proportional relationship represented by this table.
4. The constant of proportionality is 5. It represents the cups of yellow paint needed for 1 cup of blue paint.

1. 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.
2. 25 minutes. One way to find a scale factor to use is to divide 40 by 0.8. $\frac{40}{0.8} = 50$ and $50 \cdot 0.5 = 25$.
3. 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.

1. 13 inches (Two inches fell in 1 hour, 6.5 is $1 \boldcdot (6.5)$, and $2 \boldcdot (6.5) = 13$.)
2. Sample response: $y=2x$, where $x$ is the number of hours that have passed and $y$ is the inches of snow that has fallen. 
3. 48 inches ($24 \boldcdot 2 = 48$)

1. 50 and $\frac{1}{50}$; Sample response: They are reciprocals of each other.
2. $t = \frac{1}{50} d$

1. 800 pounds, because $2,000 \div 25 = 80$ and $80 \boldcdot 10 = 800$
2. $0.50, because $40 = 80 \boldcdot 0.50$
3. Sample response: If $v$ represents the value, in dollars, of $p$ pounds of clear glass, then the equation could be either $p = 80v$ or $v = 0.0125p$.

1. Yes, because the cost per pound of apples is the same in each row, 1.88 dollars per pound.
2. 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.)
3. $c = 1.88p$, where $c$ represents the cost of the apples and $p$ represents the pounds of apples.

1. When the tables are apart: $c = 6t$ (or $t = \frac16 c$).
   When the tables are together: $c = 4t + 2$ (or $t = \frac14 c - \frac12$).
2. This relationship is proportional. Sample reasonings:
   • It can be represented with an equation of the form $c = kt$ (or $t = kc$).
   • There are 6 chairs per table no matter how many tables.
3. This relationship is not proportional. Sample reasonings:
   • The number of chairs per table changes depending on how many tables there are.
   • The quotient of chairs and tables is not constant.
   • The relationship cannot be expressed with an equation of the form $c=kt$.

As shown in this table, the number of chairs per table is the same when the tables are apart, but it is not the same if the tables are pushed together.

With tables apart:

With tables end-to-end:

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?

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.

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

1. The steeper line represents the distance traveled by Diego.

   

2. 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.

1. $0.25, because $2.50 \div 10 = 0.25$.
2. 4 stickers, because $10 \div 2.5 = 4$.
3. $n=4c$ and $c=0.25n$ (or equivalent).
4. Students are only asked to create one of these graphs. It is not necessary that they plot and label any points, but it could be a helpful step in creating a reasonably accurate graph.

Sample response: In a 100-yard, three-legged race, distance in yards and time in minutes are proportional since each value of distance could be multiplied by $\frac{1}{40}$ to get the time. The constant of proportionality they used was $\frac{1}{40}$.