Question

problems 1 - 2: when deven makes chocolate milk, he mixes 2 cups of milk with 3 tablespoons of chocolate syrup. this table shows how to make batches of different sizes.

1. is there a proportional relationship between cups of milk and tablespoons of chocolate syrup?

show or explain your thinking

1. what is a constant of proportionality for this relationship?

problems 3 - 4: when you mix two colors of paint in equivalent ratios, the color you get is always the same.

1. complete the table so that each row makes the same color purple
2. what is a constant of proportionality for this relationship?

problems 5 - 7: here is some information about the side lengths of two scaled copies, triangle a and triangle b.

1. complete the table to determine the missing side lengths of each triangle.
2. what is a constant of proportionality in this relationship?
3. what does that constant of proportionality tell you about the triangles?

Explanation:

Step1: Check proportionality (Q1)

Proportional relationships have equal $\frac{\text{syrup}}{\text{milk}}$ ratios.
Given base ratio: $\frac{3}{5}$.
Check table values:

• $\frac{12}{20}=\frac{3}{5}$, $\frac{\frac{3}{2}}{1}=\frac{3}{2}\div1=\frac{3}{2}$? No, correction: Wait, correct table pairs (milk, syrup): (5,3), (2, $\frac{6}{5}$? No, given table: (2, $\frac{6}{5}$) no, visible table: (0,0), (1, $\frac{3}{5}$), (10,6).

$\frac{0}{0}$ (undefined, but 0 pairs work), $\frac{\frac{3}{2}}{1}$? No, correct given: Deven uses 5 milk : 3 syrup. So $\frac{\text{syrup}}{\text{milk}}=\frac{3}{5}$.
Check (10,6): $\frac{6}{10}=\frac{3}{5}$, (1, $\frac{3}{5}$): $\frac{\frac{3}{5}}{1}=\frac{3}{5}$, (0,0) fits proportionality. All ratios equal.

Step2: Find constant (Q2)

Constant is $\frac{\text{syrup}}{\text{milk}}$.
$\frac{3}{5}$ or $0.6$

Step3: Complete paint table (Q3)

Base ratio $\frac{\text{red}}{\text{blue}}=\frac{6}{2}=3$.
For blue=1: red $=1\times3=3$.

Step4: Paint constant (Q4)

Constant is $\frac{\text{red}}{\text{blue}}=3$ (or $\frac{\text{blue}}{\text{red}}=\frac{1}{3}$)

Step5: Complete triangle table (Q5)

Base ratio $\frac{\text{Triangle B}}{\text{Triangle A}}=\frac{6}{1}=6$.
For A=$\frac{1}{2}$: B=$\frac{1}{2}\times6=3$.
For B=8: A=$8\div6=\frac{4}{3}$.

Step6: Triangle constant (Q6)

Constant is $\frac{\text{Triangle B}}{\text{Triangle A}}=6$.

Step7: Interpret triangle constant (Q7)

The constant means Triangle B's sides are 6 times Triangle A's sides.

Answer:

1. Yes, it is proportional. All $\frac{\text{tablespoons of syrup}}{\text{cups of milk}}$ ratios equal $\frac{3}{5}$, and (0,0) is included, so the relationship is proportional.
2. $\frac{3}{5}$ (or 0.6)
3. Blue=1, Red=3
4. 3 (or $\frac{1}{3}$ if using $\frac{\text{blue}}{\text{red}}$)
5. For Triangle A=$\frac{1}{2}$, Triangle B=3; for Triangle B=8, Triangle A=$\frac{4}{3}$
6. 6
7. The constant of proportionality tells us that the side lengths of Triangle B are 6 times the side lengths of Triangle A, meaning Triangle B is a scaled copy of Triangle A with a scale factor of 6.