Lesson 3Defining Equivalent Ratios

Learning Goal

Let’s investigate equivalent ratios some more.

Learning Targets

If I have a ratio, I can create a new ratio that is equivalent to it.
If I have two ratios, I can decide whether they are equivalent to each other.

Lesson Terms

equivalent ratios

Warm Up: Dots and Half Dots

Dot Pattern 1:

(Tactile is recommended for this image) An image of dots arranged in groups. There are 6 groups of 9 dots arranged in a 3 by 3 square pattern.

Dot Pattern 2:

(A tactile is recommended for this image.) An image of half-dots arranged in columns and rows. There are 6 columns of 7 half-dots.

Activity 1: Tuna Casserole

Here is a recipe for tuna casserole.

Ingredients

3 cups cooked elbow-shaped pasta
6 ounce can tuna, drained
10 ounce can cream of chicken soup
1 cup shredded cheddar cheese
$1 \frac{1}{2}$ cups French fried onions

Instructions

Combine the pasta, tuna, soup, and half of the cheese. Transfer into a 9-inch-by-18-inch baking dish. Put the remaining cheese on top. Bake 30 minutes at 350 degrees. During the last 5 minutes, add the French fried onions. Let sit for 10 minutes before serving.

Problem 1

What is the ratio of the ounces of soup to the cups of shredded cheese to the cups of pasta in one batch of casserole?

Problem 2

How much of each of these 3 ingredients would be needed to make:

twice the amount of casserole?
half the amount of casserole?
five times the amount of casserole?
one-fifth the amount of casserole?

Problem 3

What is the ratio of cups of pasta to ounces of tuna in one batch of casserole?

Problem 4

How many batches of casserole would you make if you used the following amounts of ingredients?

9 cups of pasta and 18 ounces of tuna?
36 ounces of tuna and 18 cups of pasta?
1 cup of pasta and 2 ounces of tuna?

Are you ready for more?

The recipe says to use a 9 inch by 18 inch baking dish. Determine the length and width of a baking dish with the same height that could hold:

Twice the amount of casserole
Half the amount of casserole
Five times the amount of casserole
One-fifth the amount of casserole

Activity 2: What Are Equivalent Ratios?

The ratios $5 : 3$ and $10 : 6$ are equivalent ratios.

Is the ratio $15 : 12$ equivalent to these? Explain your reasoning.
Is the ratio $30 : 18$ equivalent to these? Explain your reasoning.
Give two more examples of ratios that are equivalent to $5 : 3$ .
How do you know when ratios are equivalent and when they are not equivalent?
Write a definition of equivalent ratios.
Pause here so your teacher can review your work and assign you a ratio to use for your visual display.
Create a visual display that includes:
- the title “Equivalent Ratios”
- your best definition of equivalent ratios
- the ratio your teacher assigned to you
- at least two examples of ratios that are equivalent to your assigned ratio
- an explanation of how you know these examples are equivalent
- at least one example of a ratio that is not equivalent to your assigned ratio
- an explanation of how you know this example is not equivalent
Be prepared to share your display with the class.

Lesson Summary

All ratios that are equivalent to $a : b$ can be made by multiplying both $a$ and $b$ by the same number.

For example, the ratio $18 : 12$ is equivalent to $9 : 6$ because both 9 and 6 are multiplied by the same number: 2.

A diagram for a ratio. At the top, the expression 9:6 is indicated. Directly below it, the expression 18:12 is indicated. Two arrows are drawn, one starting at 9 and pointing to 18 and the other starting at 6 and pointing to 12. On the left of the first arrow, multiplied by 2 is labeled and to the right of the second arrow, multiplied by 2 is labeled.

$3 : 2$ is also equivalent to $9 : 6$ , because both 9 and 6 are multiplied by the same number: $\frac{1}{3}$ .

A diagram like the last except the sides are show multiplied by one-third and the bottom ratio is 3:2

Is $18 : 15$ equivalent to $9 : 6$ ?

No, because 18 is $9 \cdot 2$ , but 15 is not $6 \cdot 2$ .

A diagram like the last but the bottom ratio is 18:15 with the left side being multiplied by 2, but the word "Nope" written on the right