# Lesson 1

Representing Ratios with Diagrams

### Lesson Narrative

Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).

Students should see diagrams as a useful and efficient ways to represent ratios. There is not really a right or wrong way to draw a diagram; what is important is that it represents the mathematics and makes sense to the student, and the student can explain how the diagram is being used. However, a goal of this lesson is to help students draw useful diagrams efficiently.

For example, here is a diagram to show 6 cups of juice and 3 cups of soda water in a recipe.

When students are asked to draw diagrams, they often include unnecessary details such as making each cup look like an actual cup, which makes the diagrams inefficient to use for solving problems. Examples of very simple diagrams help guide students toward more abstract representations while still relying on visual or spatial cues to support reasoning.

Diagrams can also help students see associations between quantities in different ways. For example, we can see there are 2 cups of juice for 1 cup of soda water by grouping the items as shown below.

While students may say “for every 2 cups of juice there is 1 cup of soda,” note that for now, we will not suggest writing the association as $$2:1$$. Equivalent ratios will be carefully developed in upcoming lessons. Diagrams like the one above are referred to as “discrete diagrams” in these materials, but students do not need to know this term. In student-facing materials they are simply called “diagrams.”

The discrete diagrams in this lesson are meant to reflect the parallel structure of double number lines that students will learn later in the unit. But for now, students do not need to draw them this way as long as they can explain their diagrams and interpret discrete diagrams like the ones shown in the lesson.

Teacher Notes for IM 6–8 Accelerated
In the launch for the first activity for this lesson (A Collection of Snap Cubes, if you use it, otherwise Blue Paint and Art Paste), explain that we can talk about the quantities in different categories using something called ratios. Tell students: “A ratio is an association between two or more quantities.” We use a colon, or the word “to,” between two values we are associating. For example, the ratio of senators to representatives in US Congress is 100 to 435 which can also be written as 100 : 435.

### Learning Goals

Teacher Facing

• Coordinate discrete diagrams and multiple written sentences describing the same ratios.
• Draw and label discrete diagrams to represent situations involving ratios.
• Practice reading and writing sentences describing ratios, e.g., “The ratio of these to those is $a:b$. The ratio of these to those is $a$ to $b$. For every $a$ of these, there are $b$ of those.”

### Student Facing

Let’s use diagrams to represent ratios.

### Required Preparation

For the Card Sort: Spaghetti Sauce activity, make 1 copy of the blackline master for each group of 2 students, plus a few extras. The blackline master shows the correct matches. Keep the extra copies whole to serve as answer keys. Cut up the rest of the slips for students to use, and throw away the cut slips that say “The above diagram also matches this sentence.” It may be helpful to copy each group's slips on a different color of paper, so that misplaced slips can quickly be put back.

### Student Facing

• I can draw a diagram that represents a ratio and explain what the diagram means.
• I include labels when I draw a diagram representing a ratio, so that the meaning of the diagram is clear.

Building On