Author:
Angela Vanderbloom
Subject:
Mathematics
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Tags:
  • 6.RP.3a Equivalent Ratios
  • License:
    Creative Commons Attribution Non-Commercial
    Language:
    English

    Education Standards

    6.RP.3a Lesson 2

    Overview

    In this lesson, they work with equivalent ratios more abstractly, both in the context of recipes and in the context of abstract ratios of numbers. They understand and articulate that all ratios that are equivalent to a:b can be generated by multiplying both aand b by the same number (MP6).

    By connecting concrete quantitative experiences to abstract representations that are independent of a context, students develop their skills in reasoning abstractly and quantitatively (MP2). They continue to use diagrams, words, or a combination of both for their explanations. The goal in subsequent lessons is to develop a general definition of equivalent ratios.

    Tuna Casserole Part 1

    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.

    Launch

    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.

    Support for English Language Learners

    Lighter Support: MLR 8 (Discussion Supports). For the above terms, act out or use images that show these in the context of cooking.

    Heavier Support. Allow students to translate the terms 'recipe', 'ingredients', and 'batches' into their first language.

    Support for Students with Disabilities

    Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.

    Executive Functioning: Eliminate Barriers. Provide a task checklist which makes all the required components of the activity explicit.

    Opening

    Here is a recipe for tuna casserole.

    Ingredients

    • 3 cups cooked elbow-shaped pasta
    • 6 ounce can tuna, drained
    • 10 ounce can cream of chicken soup
    • 1 cup shredded cheddar cheese
    • 112 cups French fried onions

    Instructions

    Combine the pasta, tuna, soup, and half of the cheese. Transfer into a 9 inch by 18 inch baking dish. Put the remaining cheese on top. Bake 30 minutes at 350 degrees. During the last 5 minutes, add the French fried onions. Let sit for 10 minutes before serving.

     

    Tuna Casserole Part 2

    Student Response

    1. The ratio of the ounces of soup to cups of shredded cheese to cups of pasta is 10:1:3.
    2. The ratio of these ingredients for different numbers of batches are:
      1. 20 ounces, 2 cups, 6 cups
      2. 5 ounces, 12 cup, 112 cups
      3. 50 ounces, 5 cups, 15 cups
      4. 2 ounces, 15 cup, 35 cup
    3. The ratio of cups of pasta to ounces of tuna is 3:6.
    4.  
      1. 3 batches
      2. 6 batches
      3. 13 batch

    Anticipated Misconceptions

    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 12 of 10, remind them that finding 12 of a number is the same as dividing it by 2, or remind them that 12of a number means 12 times that number.

    Activity Synthesis

    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 samevalue.

    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”).

    Work TIme

    Ingredients

    • 3 cups cooked elbow-shaped pasta
    • 6 ounce can tuna, drained
    • 10 ounce can cream of chicken soup
    • 1 cup shredded cheddar cheese
    • 112 cups French fried onions
    1. What is the ratio of the ounces of soup to the cups of shredded cheese to the cups of pasta in one batch of casserole?
    2. How much of each of these 3 ingredients would be needed to make:

                 a. twice the amount of casserole?

                 b. half the amount of casserole?

                 c. five times the amount of casserole?

                 d. one-fifth the amount of casserole?

            3.  What is the ratio of cups of pasta to ounces of tuna in one batch of casserole?

            4.  How many batches of casserole would you make if you used the following amounts of ingredients?

                a. 9 cups of pasta and 18 ounces of tuna?

                b. 36 ounces of tuna and 18 cups of pasta?

                c. 1 cup of pasta and 2 ounces of tuna?

    What Are Equivalent Ratios?

    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).

    Launch

    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 8:3are 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.

    Support for Students with Disabilities

    Executive Functioning: Eliminate Barriers. Provide a task checklist which makes all the required components of the activity explicit.

    Support for English Language Learners

    Lighter Support: MLR 1 (Stronger and Clearer Each Time). Use this with successive pair shares to give students a structured opportunity to revise and refine their ideas and verbal output. Before writing, have pairs talk about what they will write and how to most clearly explain this important idea. Have each student meet with two other partners in a row to provide extra practice and idea sharing three times. Students should borrow ideas and language from each partner to strengthen the final product. They can return to the first partner and create the visual.

    Student Response

    1. 15:12 is not equivalent to 5:3 because 15 is 5⋅3 but 12 is 3⋅4.
    2. 30:18 is equivalent to 5:3 because 30 is 5⋅6 and 18 is 3⋅6.
    3. Answers vary and might include 15:9, 20:12, and 50:30.
    4. Answers vary and should include some version of “multiply both parts by the same number.”
    5. Answers vary. Sample response: A ratio is equivalent to a:b when both a and b are multiplied by the same number.

    Anticipated Misconceptions

    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.

     

    Work Time

    The ratios 5:3 and 10:6 are equivalent ratios.

    1.  Is the ratio 15:12 equivalent to these? Explain your reasoning.

    2.  Is the ratio 30:18 equivalent to these? Explain your reasoning.

    3.  Give two more examples of ratios that are equivalent to 5:3.

    4.  How do you know when ratios are equivalent and when they are notequivalent?

    5.  Write a definition of equivalent ratios.

    Cool Down: Why Are They Equivalent?

    Student Response

    1. Answers vary. Sample responses: 2:3, 16:24, 400:600.
    2. Answers vary. 2:3 is equivalent to 4:6 because both 4 and 6 are multiplied by 12. 16:24 is equivalent because both 4 and 6 are multiplied by 4. 400:600 is equivalent because both 4 and 6 are multiplied by 100.

     

    1. Write another ratio that is equivalent to the ratio 4:6.
    2. How do you know that your new ratio is equivalent to 4:6? Explain or show your reasoning.