Author:
Angela Vanderbloom
Subject:
Mathematics
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Tags:
  • 6.RP.A.1 Ratios and Ratio Language
  • License:
    Creative Commons Attribution Non-Commercial
    Language:
    English

    Education Standards

    6.RP.A.1 Lesson 1

    Overview

    In this lesson, students use collections of objects to make sense of and use ratio language. Students see that there are several different ways to describe a situation using ratio language. For example, if we have 12 squares and 4 circles, we can say the ratio of squares to circles is 12:4 and the ratio of circles to squares is 4 to 12. We may also see a structure that prompts us to regroup them and say that there are 6 squares for every 2 circles, or 3 squares for every one circle (MP7).

    Expressing associations of quantities in a context—as students will be doing in this lesson—requires students to use ratio language with care (MP6). Making groups of physical objects that correspond with “for every” language is a concrete way for students to make sense of the problem (MP1).

    Warm Up: What Kind and How Many?

    In this warm-up, students compare figures and sort them into categories. The first two questions are very straightforward to allow all students access to the activity and prime them to think about other ways the figures can be sorted. Students create their own categories in the third question and explain their reasoning.

    Launch

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

    Student Response

    1. Four: red, blue, green, yellow
    2. Four: 2, 3, 4, 5 square units
    3. Answers vary. Sample responses:
      • Two: rectangles and non-rectangles.
      • Three: rectangles, two different squares glued together, and L-shapes.
      • Four: squares, rectangles, two different squares glued together, and L-shapes.
      • Seven: small, medium, and large rectangles, 2 by 2 squares, small L, big L, and a small and a big square glued together.

    Activity Synthesis

    After briefly inviting students to share responses to the first two questions, record all the ways students answered the third 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.

    Introduction to Ratios

    Lesson Guide

    In this lesson, students learn the formal definition of a ratio and then use it to solve problems. The start of the lesson introduces the concept of a ratio as a way of comparing numbers of objects using division, which is an alternative to comparing numbers using subtraction. Have students look at the picture of stars and triangles and then read the text.

    Ask:

    • The ratio of triangles to stars is 3:10. What is the ratio of stars to triangles? (Answer: 10:3)

    ELL: Keep in mind that some students may not feel comfortable reading aloud. Be prepared to support them in this task.

    Opening

    Look at the picture of stars and triangles and read the following information.

    • One way that you can compare the number of stars and the number of triangles is to say that there are 7 more stars than there are triangles. This comparison looks at the difference between two quantities; it uses the operation of subtraction.
    • Another way that you can compare the number of stars and the number of triangles is to say that for every 3 triangles there are 10 stars. You can say that the ratio of triangles to stars is 3 to 10 or 3:10. This comparison uses the operation of division.
    • The value of the ratio of triangles to stars is 3/10, or 0.3. 

    Determining and Writing Ratios

    Students will be given various situations in which they will write a ratio.

    Answers

    1.  5 to 9; 5 : 9; and 5 / 9

    2.  4 to 12; 4 : 12; and 4/12

    3.  5 to 9; 5 : 9; and 5 / 9

    4.  4 to 7; 4 : 7 , and 4 / 7

    5.  3 to 1; 3 : 1; and 3 / 1

    Work Time

    Write a ratio for each situation.

    1.  For every 9 diet sodas a burger shop sold, there were 5 regular sodas sold.  What is the ratio of regular sodas sold to diet sodas sold?

    2.  Paige sawed a 4 ft piece from a 12 ft board.  What is the ratio between the shorter piece and the longer piece?

    3.  In a neighborhood, there are 9 old homes for every 5 new homes.  Write the ratio of new homes to old homes.

    4.  In a bird cage, there are 4 male birds for every 7 female birds.   What is the ratio of male birds to female birds?

    5.  In the library, there are three fiction books for every nonfiction book.  Write the ratio of fiction to nonfiction books.

    Blue Paint and Art Paste

    In this activity, students draw connections between a diagram and the ratios it represents. Students work in pairs to discuss different ways to use ratio language to describe discrete diagrams. They first identify statements that would correctly describe a given diagram. Then, they create both a diagram and corresponding statements to represent a new situation involving ratio.

    As students work, monitor for different ways in which students draw and discuss diagrams of the paste recipe. Identify a few pairs who draw different diagrams and use ratio language differently to share later. A few things to anticipate:

    • Some students may draw very literal drawings of cups and pints. Encourage them to use simpler representations.
    • Students may choose to draw letters (X’s) or other symbols or marks instead of squares and rectangles.
    • Students may use equivalent ratios to describe a situation, even though these have not been explicitly taught (e.g., they may say the ratio of cups of flour to pints of water is 4:1 instead of 8:2). Though this is correct, be careful here. We have previously regrouped objects and might say, for example, that with a ratio 8:2, “for every 4 cups of flour there is 1 cup of water,” but we have not asserted that this ratio can be written as 4:1 yet. The idea of equivalent ratios is sophisticated and will be developed over the next several lessons.
    • Correct descriptions may include fractions (e.g., for every tablespoon of blue paint, there is 13 cup of white paint). Although students are not expected to work with fractions in this lesson, responses involving fractions are fine.

    Launch

    Arrange students in groups of 2. Provide them with the tools needed for creating a large visual display for the second part of the task. Ensure students understand they are supposed to select more than one statement for the first question. Consider having students take turns reading each statement and deciding whether they think it describes the situation or not.

    Support for English Language Learners

    Lighter Support. Ask students to generate numerical representations for each statement on the card.

    Heavier Support. Provide numerical representation for each statement on the cards. Demonstrate how to justify whether or not the statement is true or false: “It’s false because in this diagram, for every cup of water...”.

    Support for Students with Disabilities

    Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.

    Student Response

    1. The following statements describe the paint mixture:
      A. The ratio of cups of white paint to tablespoons of blue paint is 2:6.
      C. There is 1 cup of white paint for every 3 tablespoons of blue paint.
      D. There are 3 tablespoons of blue paint for every cup of white paint.
      F. For every 6 tablespoons of blue paint, there are 2 cups of white paint.
      G. The ratio of tablespoons of blue paint to cups of white paint is 6 to 2.

      The following statements do not describe the paint mixture:
      B. For every cup of white paint there are 2 tablespoons of blue paint.
      E. For each tablespoon of blue paint there are 3 cups of white paint.

    2.  Answers vary. Sample responses:

    • The ratio of cups of flour to pints of water is 8:2.
    • The ratio of pints of water to cups of flour is 2 to 8.
    • For each pint of water, there are 4 cups of flour.
    • For every 8 cups of flour, there are 2 pints of water.
    • For every 4 cups of flour, there is 1 pint of water.
    • There are 2 pints of water for every 8 cups of flour.

    Anticipated Misconceptions

    Some students may think all of the statements about the paint mixture are accurate descriptions. If so, suggest that there are two false statements. Have students discuss the statements again in determining which two are false.

    Activity Synthesis

    Select students to share their paste diagrams and sentences with the class. Sequence the diagrams from most elaborate to most simple. Connect the many ways in which the paste can be represented and described. Compare more detailed pictures with a discrete diagram; point out how the discrete diagram is a more efficient way of showing the paste recipe.

    Elena mixed 2 cups of white paint with 6 tablespoons of blue paint.

    Here is a diagram that represents this situation.

    "A discrete diagram of squares that represent the amount of paint. The top row is labeled "white paint, in cups" and contains 2 large squares. The bottom row is labeled "blue paint, in tablespoons" and contains 6 small squares."

     

    1.  Discuss the statements that follow, and circle all those that correctly describe this situation. Make sure that both you and your partner agree with each circled answer.

               a.  The ratio of cups of white paint to tablespoons of blue paint is 2:6.

               b.  For every cup of white paint, there are 2 tablespoons of blue paint.

               c.  There is 1 cup of white paint for every 3 tablespoons of blue paint.

               d.  There are 3 tablespoons of blue paint for every cup of white paint.

               e.  For each tablespoon of blue paint, there are 3 cups of white paint.

               f.  For every 6 tablespoons of blue paint, there are 2 cups of white paint.

               g.  The ratio of tablespoons of blue paint to cups of white paint is 6 to 2.

    2.  Jada mixed 8 cups of flour with 2 pints of water to make paste for an art project.

              a.  Draw a diagram that represents the situation.

              b.  Write at least two sentences describing the ratio of flour and water.

     

    Card Sort: Spaghetti Sauce

    Materials

    Writing and using ratio language requires attention to detail. This task further develops students’ ability to describe ratio situations precisely by attending carefully to the quantities, their units, and their order in the ratio.

    Students work in pairs to match ratios of sauce ingredients to discrete diagrams and to explain reasoning (MP3).

    Launch

    Arrange students in groups of 2. Place two copies of uncut blackline masters in envelopes to serve as answer keys.

    Demonstrate how to set up and play the matching game. Choose a student to be your partner. Discuss what all the symbols mean. Mix up the cards and place them face-up. Point out that the cards contain either diagrams or sentences. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, e.g., by explaining your mathematical thinking, asking clarifying questions, etc.

    Give each group cut-up cards for matching. Tell students to check their matches after they complete the activity using the answer keys.

    Support for English Language Learners

    Heavier Support. Ask students with the sentence cards to circle the two quantities being compared (e.g., ‘tablespoons of oil’ and ‘cups of tomato sauce’). Draw or project an annotated sentence (with connections to diagrams) to support students in interpreting the statements. 

    Support for Students with Disabilities

    Expressive Language: Eliminate Barriers. Provide sentence frames for students to explain their reasoning (i.e., ____________ and _____________ are a match because _____________.)

    Student Response

      • Diagram A matches with sentence 4.
      • Diagram B matches with sentences 2 and 8.
      • Diagram C matches with sentence 1.
      • Diagram D matches with sentence 5.
      • Diagram E matches with sentences 3 and 7.
      • Diagram F matches with sentence 6.
    1. No answer necessary.
      1. Diagram B matches with sentences 2 and 8.
      2. Diagram E matches with sentences 3 and 7.
    2. Answers vary. Sample responses:
      • For diagram A, the ratio of cups of tomato sauce to tablespoons of oil is 3:1.
      • For diagram D, the ratio of tablespoons of oil to cups of tomato sauce is 2 to 5.

    Anticipated Misconceptions

    If students disagree about a match, encourage them to figure out the correct answer through discussion and use of the answer key. Make sure that when students use the answer key, they discuss any errors rather than just make changes.

    Students may think the shapes in the diagram need to be drawn in the same order the ingredients appear in the description. This is not the case. You could turn a diagram card upside down and it would still represent the same situation. The diagram just shows ingredients that get mixed together in a pot. It is the case, however, that within the description, the order of the words in the sentence must correspond with the terms within the ratio.

    Your teacher will give you cards describing different recipes for spaghetti sauce. In the diagrams:

    • a circle represents a cup of tomato sauce
    • a square represents a tablespoon of oil
    • a triangle represents a teaspoon of oregano

    1.  Take turns with your partner to match a sentence with a diagram.

           a.  For each match that you find, explain to your partner how you know it’s a match.

           b.  For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

    2.  After you and your partner have agreed on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.

    3.  There were two diagrams that each matched with two different sentences. Which were they?

    • Diagram _______ matched with both sentences ______ and ______.
    • Diagram _______ matched with both sentences ______ and ______.

    4.  Select one of the other diagrams and invent another sentence that could describe the ratio shown in the diagram.

    Cool - Down: Paws, Ears, and Tails

    Student Response

    1. Answers vary. Sample response:

      1. The ratio of ears to paws to tails is 6:12:3.
      2. There are 4 paws for every tail.
      3. There are 4 paws for every 2 ears. This means that there are 2 paws for every ear.

    Anticipated Misconceptions

    In the second question, students may not realize that the order of the words in the sentence must correspond with the terms within the ratio. Ears : paws : tails must correspond with 6:12:3. In the fourth question, students may not write the sentence for every one ear. If this is the case, prompt them to draw a circle around each set of two paws and one ear to help them see this relationship.

    Lesson 1

    There are 3 cats in a room and no other creatures. Each cat has 2 ears, 4 paws, and 1 tail.

    "A diagram of 3 identical cats."

    1. Draw a diagram that shows an association between numbers of ears, paws, and tails in the room.
    2. Complete each statement:

      1. The ratio of ______________ to ______________ to ______________ is ______ : _______ : ______.
      2. There are ______ paws for every tail.
      3. There are ______ paws for every ear.