Lesson 3

In this warm-up, students are asked to determine the number of dots in an image and explain how they arrived at that answer. The goal is to prompt students to visualize and articulate different ways in which they can decompose the dots, using what they know about arrays, symmetry, and multiplication to arrive at the total number of dots. To encourage students to refer to the image in their explanation but not count every dot, this image is flashed for a few seconds and then hidden. It is flashed once more for students to check their thinking. Ask students how they saw the dots instead of how they found the number of dots, so they focus on the structure of the dots in the image.

As students share how they saw the dots, ask how the expressions they used to describe the arrangements and grouping of the dots in the two problems are similar. This prompts students to make connections between the properties of multiplication.

Tell students you will show them an image made up of dots for 3 seconds. Their job is to find how many dots are in the image and explain how they saw them. Display the image for all to see for 3 seconds and then hide it. Do this twice. Give students 1 minute of quiet think time between each flash of the image. Encourage students who have one way of seeing the dots to think of another way while they wait.

Ask students to share how many dots they saw and how they saw them. Record and display student explanations for all to see. (Consider re-displaying the image for reference while students are explaining what they saw.) To involve more students in the conversation, consider asking:

• “Who can restate the way ___ saw the dots in different words?”
• “Did anyone see the dots the same way but would explain it differently?”
• “Does anyone want to add an observation to the way ____ saw the dots?”
• “Who saw the dots differently?”
• “Do you agree or disagree? Why?”

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 they represent.

Ask students if they have ever cooked something by following a recipe. If so, ask them what they made and what some of the ingredients were.

Ask: “How might we use ratios to describe the ingredients in your recipe?” (The ratios could associate the quantities of each ingredient being used.)

Explain to students that in this task, they will think about the ratios of ingredients for a tuna casserole and how to adjust them for making different numbers of batches.

Students who are not yet fluent in fraction multiplication from grade 5 may have difficulty understanding how to find half or one-fifth of the recipe ingredient amounts. Likewise, they may have difficulty identifying one-third of a batch. Suggest that they draw a picture of \(\frac12\) of 10, remind them that finding \(\frac12\) of a number is the same as dividing it by 2, or remind them that \(\frac12\) of a number means \(\frac12\) times that number.

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 relate to the number of batches in each case. Draw out 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-fifth of a batch, students may speak in terms of dividing by 2 and dividing by 5. Point out that “dividing by 2” has the same outcome as “multiplying by one-half,” and “dividing by 5” has the same outcome as “multiplying by one-fifth.” (Students multiplied a whole number by a fraction in grade 5.) Later, we will want to state our general definition of equivalent ratios as simply as possible: as multiplying both \(a\) and \(b\) in the ratio \(a:b\) by the same number (not “multiplying or dividing”).

In this activity, students identify what equivalent ratios have in common (a ratio equivalent to \(a:b\) can be generated by multiplying both \(a\) and \(b\) by the same number) and generate equivalent ratios (MP8). It is at this point in the unit where students will explicitly define the term equivalent ratios (MP6).

Arrange students in groups of 3–4. Provide each group with tools for creating a visual display.

Summarize what we know so far about equivalent ratios. When we double or triple a color recipe, the ratios of the amount of ingredients in the mixtures are equivalent to those in the original recipe. For example, \(24:9\) and \(8:3\) are equivalent ratios, because we can think of \(24:9\) as a mixture that contains three batches of purple water where a single batch is \(8:3\).

When we make multiple batches of a food recipe, we say the ratios of the amounts of the ingredients are equivalent to the ratios in a single batch. For example, \(3:6\), \(1:2\), and \(9:18\) are equivalent ratios because they correspond to the amount of the ingredients in different numbers of batches of tuna noodle casserole, and they all taste the same.

In this activity, we’ll write a definition for equivalent ratios.

When students pause after question 5, have a whole class discussion about the first five questions. Then assign each group a different ratio to use as their example for their visual display. Some possibilities:

Students may incorporate recipes, specific examples, or batch thinking into their definitions. These are important ways of thinking about equivalent ratios, but challenge them to come up with a definition that only talks about the numbers involved and not what the numbers represent.

If groups struggle to get started thinking generally about a definition, give them a head start with: “A ratio is equivalent to \(a:b\) when . . .”

If students include “or divide” in their definition, remind them that, for example, dividing by 5 gives the same result as multiplying by one-fifth. Therefore, we can just use “multiply” in our definition.

Each group will share their visual display as they explain their definitions. Highlight phrases or explanations that are similar in each display. 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 \(a:b\) when both \(a\) and \(b\) are multiplied by the same number.