Unit 2 Introducing Ratios — Unit Plan

A ratio is an association between two or more quantities. There are many ways to describe a situation in terms of ratios. For example, look at this collection:

Here are some correct ways to describe the collection:

• The ratio of squares to circles is $6:3$.
• The ratio of circles to squares is 3 to 6.

Notice that the shapes can be arranged in equal groups, which allow us to describe the shapes using other numbers.

• There are 2 squares for every 1 circle.
• There is 1 circle for every 2 squares.

Ratios can be represented using diagrams. The diagrams do not need to include realistic details. For example, there are 2 pairs of scissors and 6 glue sticks in a bin. Instead of this:

We can draw something like this:

This diagram shows that the ratio of glue sticks to pairs of scissors is 6 to 2. We can also see that for every pair of scissors, there are 3 glue sticks.

A recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.”

To double this recipe, we would use 10 cups of cranberry juice with 4 cups of soda water. To triple this recipe, we would use 15 cups of cranberry juice with 6 cups of soda water.

This diagram shows a single batch of the recipe, a double batch, and a triple batch:

We say that the ratios $5 : 2$, $10 : 4$, and $15 : 6$ are equivalent. Even though the amounts of each ingredient within a single, double, or triple batch are not the same, they would make fizzy juice that tastes the same.

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.

$3:2$ is also equivalent to $9:6$, because both 9 and 6 are multiplied by the same number: $\frac13$.

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

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

You can use a double number line diagram to find many equivalent ratios.

For example, a recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” The ratio of cranberry juice to soda water is $5:2$. Multiplying both ingredients by the same number creates equivalent ratios.

This double number line shows that the ratio $20:8$ is equivalent to $5:2$. If you mix 20 cups of cranberry juice with 8 cups of soda water, it makes 4 times as much fizzy juice that tastes the same as the original recipe.

The unit price is the price of 1 thing—for example, the price of 1 ticket, 1 slice of pizza, or 1 kilogram of peaches.

If 4 movie tickets cost $28, then the unit price would be the cost per ticket. We can create a double number line to find the unit price.

This double number line shows that the cost for 1 ticket is $7. We can also find the unit price by dividing, $28 \div 4 = 7$, or by multiplying, $28 \boldcdot \frac14 = 7$.

When we talk about two things happening at the same rate, we mean that the ratios of the quantities in the two situations are equivalent. There is also something specific about the situation that is the same.

• If two ladybugs are moving at the same rate, then they are traveling at the same constant speed.
• If two bags of apples are selling for the same rate, then they have the same unit price.
• If we mix two kinds of juice at the same rate, then the mixtures have the same taste.
• If we mix two colors of paint at the same rate, then the mixtures have the same shade.

For example, do these two paint mixtures make the same shade of orange?

• Kiran mixes 9 teaspoons of red paint with 15 teaspoons of yellow paint.
• Tyler mixes 7 teaspoons of red paint with 10 teaspoons of yellow paint.

To know if the two ratios describe the same rate, we can write an equivalent ratio for one or both ratios so that one quantity has the same value. Then we can compare the values for the other quantity.

Here is a double number line that represents Kiran's paint mixture. The ratio $9:15$ is equivalent to the ratios $3:5$ and $6:10$.

For 10 teaspoons of yellow paint, Kiran would mix in 6 teaspoons of red paint. This is less red paint than Tyler mixes with 10 teaspoons of yellow paint. The ratios $6:10$ and $7:10$ are not equivalent, so these two paint mixtures would not be the same shade of orange.

A table is a way to organize information. Each horizontal set of entries is called a row, and each vertical set of entries is called a column. (The table shown has 2 columns and 5 rows.) A table can be used to represent a collection of equivalent ratios.

Here is a double number line diagram and a table that both represent the situation: “The price is $2 for every 3 mangos.”

Finding a row containing a “1” is often a good way to work with tables of equivalent ratios. For example, the price for 4 lbs of granola is $5. At that rate, what would be the price for 62 lbs of granola?

Here are tables showing two different approaches to solving this problem. Both of these approaches are correct. However, one approach is more efficient.

• Less efficient

  

• More efficient

  

Notice how the more efficient approach starts by finding the price for 1 lb of granola.

Remember that dividing by a whole number is the same as multiplying by a unit fraction. In this example, we can divide by 4 or multiply by $\frac14$ to find the unit price.

To solve problems about something happening at the same rate, we often need:

• Two pieces of information that allow us to write a ratio that describes the situation.

• A third piece of information that gives us one number of an equivalent ratio. Solving the problem often involves finding the other number in the equivalent ratio.

Suppose we are making a large batch of fizzy juice and the recipe says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” We know that the ratio of cranberry juice to soda water is $5:2$, and that we need 2.5 cups of cranberry juice per cup of soda water.

We still need to know something about the size of the large batch. If we use 16 cups of soda water, what number goes with 16 to make a ratio that is equivalent to $5:2$?