Lesson 1

Introducing Ratios and Ratio Language

1.1: What Kind and How Many? (5 minutes)

Warm-up

In this warm-up, students compare figures and sort them into categories.

Launch

Display the image for all to see. Give students 1 minute of quiet think time followed by 2 minutes of partner discussion.

Student Facing

A collection of shapes that are each made up of squares. Each shape is one of four colors: white, blue, green, and yellow. Please ask for additional help.

Think of different ways you could sort these figures. What categories could you use? How many groups would you have?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students struggle to create their own categories, prompt them to consider a specific attribute of the figures, such as the size, color, or shape.

Activity Synthesis

Record all the ways students answered the question for all to see. Ask a student to explain how they sorted the figures. Ask if anyone saw it a different way until all the different ways of seeing the shapes have been shared. Emphasize that the important thing is to describe the way they sorted them clearly enough that everyone agrees that it is a reasonable way to sort them. Tell students we will be looking at different ways of seeing the same set of objects in the next activity.

1.2: The Teacher’s Collection (10 minutes)

Activity

This activity introduces students to ratio language and notation through examples based on a collection of everyday objects. Students learn that a ratio is an association between quantities, and that this association can be expressed in multiple ways.

After discussing examples of ratio language and notation for one way of categorizing the objects in the collection, students write ratios to describe the quantities for another way of categorizing objects in the collection.

As students work, circulate and identify those who:

  • Create different categories from the given collection.
  • Create categories whose quantities can be rearranged into smaller groups (e.g. 6 A’s and 4 B’s can be expressed as “for every 3 A’s there are 2 B’s”).
  • Express the same ratio in opposite order or by using different words (e.g. “the ratio of A to B is 7 to 3,” and “for every 7 A’s there are 3 B’s”).

Have a collection of objects ready to display for the launch. Make sure there are different ways the collection can be sorted. For example, the dinosaurs below can be categorized by color (green, orange, and purple), by the number of legs they stand on (standing on 4 legs or on 2 legs), or by the features along their backs (crest, white stripe, or nothing).

A collection of dinosaur figurines. 

Familiar classroom objects such as binder clips or pattern blocks can also be used to form collections. This picture shows a collection of binder clips that could be categorized by size (small, medium, and large) or by color (black, green, and blue) .

A collection of binder clips. 

Launch

Display a collection of objects for all to see. Give students 2 minutes of quiet think time to come up with as many different categories for sorting the collection as they can think of. Record students’ categories for all to see. Sort the collection into one of the student-suggested categories and count the number of items in each. Record the number of objects in each category and display for all to see. For example:

category A: green category B: orange category C: purple
3 2 4

Explain that we can talk about the quantities in the 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.

Share the following examples (adapt them to suit your collection) and display them for all to see. Keep the examples visible for the duration of the lesson.

  • The ratio of purple to orange dinosaurs is 4 to 2.
  • The ratio of purple to orange dinosaurs is \(4:2\).
  • The ratio of orange to purple dinosaurs is 2 to 4.
  • The ratio of orange to purple dinosaurs is \(2:4\)

Explain that we can also associate two quantities using the phrase “for every \(a\) of these, there are \(b\) of those.” Add the following examples to the display.

  • For every 3 green dinosaurs there are 4 purple dinosaurs.
  • There are 4 purple dinosaurs for every 2 orange dinosaurs.

Finally, find two categories whose items can be rearranged into smaller groups, e.g. 4 purple dinosaurs to 2 orange dinosaurs. Point out that in some cases we can associate the same categories using different numbers. Share the following example and add it to the display.

  • For every 2 purple dinosaurs, there is 1 orange dinosaur.
A collection of dinosaur figurines. 

Have students write two or three sentences to describe ratios between the categories they suggested.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: ratio and category.
Supports accessibility for: Conceptual processing; Language

Student Facing

  1. Think of a way to sort your teacher’s collection into two or three categories. Count the items in each category, and record the information in the table.
    category name      
    category amount      

    Pause here so your teacher can review your work.

  2. Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio:

    • The ratio of one category to another category is ________ to ________.

    • The ratio of one category to another category is ________ : ________.

    • There are _______ of one category for every _______ of another category.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may write ratios with no descriptive words. \(8:2\) is a good start, but part of writing a ratio is stating what those numbers mean. Draw students’ attention to the sentence stems in the task statement; encourage them to use those words.

Activity Synthesis

Invite several students to share their categories and sentences. Display them for all to see, attending to correct ratio language. Be sure to include students who express the same categories in reverse order, in different words, or with a different set of numbers (which students will later call an equivalent ratio). Leave several sentences displayed for students to see and use as a reference while working on the next task.

1.3: The Student’s Collection (20 minutes)

Activity

In this activity, students write ratios to describe objects in their own collection. They create a display of objects and circulate to look at their classmates’ work. Students see that there are several ways to write ratios to describe the same situation.

Launch

Invite students to share what types of items are in their personal collections. If students did not bring in a collection, pair them with another student, or provide them with an extra collection that you have brought in for that purpose.  

Provide access to tools for creating a visual display. Tell students they will pause their work before creating a visual display to get their sentences approved.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students consider audience when preparing to display their work. Display the list of items that should be included on the display and ask students, “what kinds of details could you include on your display to help a reader understand the ratios you’ve used to describe the objects in your collection?” Record ideas and display for all to see. Examples of these types of details or annotations include: the order in which representations are organized on the display, attaching written notes or details to certain representations, using specific vocabulary or phrases, or using color or arrows to show connections between representations. If time allows, ask students to describe specific examples of additional details that other groups used that helped them to interpret and understand their displays.
Design Principle(s): Maximize meta-awareness; Optimize output

Student Facing

  1. Sort your collection into three categories. You can experiment with different ways of arranging these categories. Then, count the items in each category, and record the information in the table.

    category name      
    category amount      
  2. Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio:

    • The ratio of one category to another category is ________ to ________.
    • The ratio of one category to another category is ________ : ________.
    • There are _______ of one category for every _______ of another category.

    Pause here so your teacher can review your sentences.

  3. Make a visual display of your items that clearly shows one of your statements. Be prepared to share your display with the class.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Use two colors to shade the rectangle so there are 2 square units of one color for every 1 square unit of the other color.
  2. The rectangle you just colored has an area of 24 square units. Draw a different shape that does not have an area of 24 square units, but that can also be shaded with two colors in a \(2:1\) ratio. Shade your new shape using two colors.

A 4 by 6 grid of squares.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Watch for students simply writing a numerical ratio, such as \(3:7\), without any descriptive words. Draw their attention to the sentence stems in the task statement.

Activity Synthesis

Once students have had enough time to create their displays, circulate through each display and listen to how students describe their ratios.

As students present their displays, point out the various ways that students chose to showcase their work, including different ways to say the same ratio. Ask students who used two sets of numbers to describe the same categories (e.g., 8 to 2 and “4 for every 1”) to demonstrate the two ways of grouping the objects.

Lesson Synthesis

Lesson Synthesis

This lesson is all about how to use ratio language and notation to describe an association between two or more quantities. Wrap up the lesson by drawing a diagram for all to see of, say, 6 squares and 3 circles.

A discrete diagram of squares and circles such that the top row contains 6 squares and the bottom row contains 3 circles.

Say, “One way to write this ratio is, there are 6 squares for every 3 circles. What are some other ways to write this ratio?” Some correct options might be:

  • The ratio of squares to circles is \(6:3\).
  • The ratio of circles to squares is 3 to 6.
  • There are 2 squares for every 1 circle.

Display this diagram and the associated sentences the class comes up with somewhere in the classroom so students can refer back to the correct ratio and rate language during subsequent lessons.

Consider posing some more general questions, such as:

  • Explain what a ratio is in your own words.
  • What things must you pay attention to when writing a ratio?
  • What are some words and phrases that are used to write a ratio?

1.4: Cool-down - A Collection of Animals (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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:

A discrete diagram of squares and circles such that the top row contains 6 squares and the bottom row contains 3 circles.

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.

A discrete diagram of 6 squares and 3 circles organized into 3 equal groups of 2 squares and 1 circle each.
  • There are 2 squares for every 1 circle.
  • There is 1 circle for every 2 squares.