In this activity, students use a realistic food recipe to find equivalent ratios that represent different numbers of batches. Students use the original recipe to form ratios of ingredients that represent double, half, five times, and one-fifth of the recipe. Then they examine given ratios of ingredients and determine how many batches the amounts represent.

Display the recipe for all to see. Ask students to share and explain their responses. List their responses—along with the specified number of batches—for all to see.

Ask students to analyze the list and describe how the ratio of quantities relates to the number of batches in each case. Elicit the idea that each quantity within the recipe was multiplied by a number to obtain each batch size, and that each ingredient amount is multiplied by the same value.

In finding one-half and one-fourth of a batch, students may speak in terms of dividing by 2 and dividing by 4. Point out that “dividing by 2” has the same outcome as “multiplying by one-half,” and “dividing by 4” has the same outcome as “multiplying by one-fourth.” Later, we will want to state our general definition of "equivalent ratios" as simply as possible: as multiplying both and in the ratio by the same number (not “multiplying or dividing”).

In this activity, students identify what equivalent ratios have in common, generate equivalent ratios, and write a definition for "equivalent ratios." Through repeated reasoning (MP8), students recognize that a ratio equivalent to can be generated by multiplying both and by the same number. As they prepare and present their best definition of "equivalent ratios," examples of equivalent ratios, and an explanation of how they know that the examples are equivalent, students practice attending to precision (MP6). 

This activity uses the Compare and Connect math language routine to support mathematically precise language in discussion.

Invite each group to share their visual display and explain their definition. After all groups have presented their work, use Compare and Connect to help students compare, contrast, and connect the definitions of "equivalent ratios" and explanations about whether pairs of ratios are equivalent. Highlight phrases or explanations that are similar in each display. Ask questions such as:

• “What do these definitions have in common?”
• “What details or language helped to make certain explanations clearer to you?”
• “What additions or adjustments might clarify these definitions (or explanations) even more?”

Then make one class display that incorporates all valid definitions. This display should be kept posted in the classroom for the remaining lessons within this unit. It should look something like:

Equivalent Ratio: A ratio is equivalent to when both and are multiplied by the same number.