PRINT VERSION - Equivalent Ratios

This toolkit will address how to apply concepts with equivalent ratios to describe and compare the relationship between two quantities by utilizing a variety of real-world applications and strategies.

Goals and Standards

---

Colorado Academic Standards:

• 6.RP.A. Ratios & Proportional Relationships: Understand ratio concepts and use ratio reasoning to solve problems.
  • 6.RP.A.1 Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities
  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
    • 6.RP.A.3.a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Mathematical Practice Standards:

• MP2. Reason abstractly and quantitatively. 
• MP3. Construct viable arguments and critique the reasoning of others. 
• MP7. Look for and make use of structure.

Learning Goal:

• Students will be able to use various representations to create equivalent ratios that represent the same real-world situation to solve for unknown values

Next Section:

Progression of Skills (outline)

Before the lesson:

• Students should be able to use words and numbers to accurately describe ratios. For example, if a student can accurately complete the Ratios Check for Understanding with less than three errors, then they are ready to access the material in this toolkit.

During this toolkit lesson:

• Students can use various representations to create equivalent ratios that represent the same real-world situation.

After the lesson:

• Students will be able to determine if two quantities are in a proportional relationship and they will be able to represent a proportional relationship with a verbal description, table, graph and equation.

Lesson

This toolkit is divided into three sessions that will support students with reaching the learning goal.

Anticipate Misconceptions:

• Using the wrong operations to find equivalent ratios
  • Some students think that you find equivalent ratios by adding or subtracting instead of multiplying or dividing. Reinforce the context behind the problem. For example, if I have 4 dogs for every 2 cats, then I can add another 4 dogs as long as I add another 2 cats which would be the same as doubling the ratio. Students should not add 1 dog for every 1 cat as that is not an equivalent ratio.
• Always needing to simplify ratios
  • Students typically feel the need to always simplify ratios. Explain to the students that sometimes we need to find several equivalent ratios in order to find unknown values, but that they can use simplifying as a way to check their work.
  • This is sometimes called “reducing” which is a word we no longer use in math because it can cause further misconceptions.
• Understanding the scale of units of measure
  • Students often have a difficult time comparing units of measure. For example, if I mix 1 cup of water to 2 tablespoons of powdered drink mix, they might assume that there is less water even though a cup is bigger than a tablespoon. Consider using several measuring spoons to demonstrate to students the differences between each unit of measure. 

Things to Consider:

• Decoding Word Problems
• Students may struggle with decoding word problems. For this toolkit, students need to be able to take information from a word problem and organize it onto a diagram. If students are struggling with this aspect, consider using instructional routines three reads and/or numberless word problems.

Session 1: Exploring and Defining Equivalent Ratios

This session will take around 35-40 minutes to complete.

Directions:

1. Start with the Student Activity on Desmos. Refer to the “teacher moves” sections for guidance on implementing this activity. Please note that if your students do not have access to a computer, complete the activity together by projecting the Desmos activity and allowing students to take turns doing the activity. Give the students the opportunity to discuss if they agree or disagree with their peers while taking turns to do the activity.
   1. Slides 1-2: Introduce the activity to the students. Demonstrate how to use the sliders on slide 1 and then give the students 3 minutes to complete slide 1. Ask the students to discuss similarities and differences with a partner. Next, give the students 2 minutes to complete slide 2 and 1 minute to share their noticings and wonderings (“What do you notice? What do you wonder?”). While they are sharing, pay close attention to students who make the connection between multiplying or dividing one ratio to get the next. This is a much needed connection.
   2. Slide 3: Give the students 2-3 minutes to complete slide 3. Ask students to discuss the following questions: Did you find an equivalent ratio on your first try? If so, how did you create your equivalent ratio? If you did not get it on your first try, what did you do that led you to an incorrect response? How did you adjust your thinking?
   3. Slide 4: Give the students 3 minutes to complete slide 4. Instruct the students to check 3 different equivalent ratios based on the fruit they selected. Share their work with a partner.
   4. Slide 5-7: Give students 7 minutes to complete slides 5-7. While students are working, check student progress. If students are struggling to get started then support that student 1-on-1. Once students are finished, have them share their responses with a partner and revise their work as needed. As a class, discuss slides 5, 6 and 7 to ensure that students understand the correct math operation associated with equivalent ratios. At this point, make sure students understand that we do not add to find equivalent ratios.
   5. Slides 8-9: Give the students 3 minutes to complete slides 8 and 9. Once students have completed these slides, check responses for slide 9 to make sure all students can find equivalent ratios. Allow students the time to go through each answer choice and describe why or why not that answer choice is an equivalent ratio as a whole group discussion.
   6. Slide 10: Give the students 1 minute to rate their understanding of equivalent ratios on slide 10.
2. Next, give the students 5-7 minutes to complete the Student Note Catcher with Desmos. Please note that page 1 of this document is the student copy and page 2 is the answer key. This document will give the students the opportunity to collect and record their learning from the Desmos activity.
3. To conclude this session, give the students the Exit Ticket for Lesson #1.  Students should be able to demonstrate their understanding from this session, so please allow students the time they need to complete the exit ticket the best they can. This exit ticket will assess students' ability to find equivalent ratios. Use the Exit Ticket for Lesson #1 Answer Key to determine if your students are ready to move on to session 2 or if they need additional practice with equivalent ratios.

Session 2: Representing Equivalent Ratios on a Double Number Line

This session will take around 35-40 minutes to complete.

Directions: 

1. Start with the Double Number Lines Intro Activity. Use the Double Number Lines Intro Activity Answer Key to support student learning during this activity.
   1. Question #1: Read the situation aloud to the students. Allow the students 2-3 minutes to write down their noticings and wonderings. This should be intentionally independent work time. (Note: if students struggle with organizing their thoughts, you might want to suggest using a T-chart.) Once question 1 is done, allow students to share their thoughts with a partner. Point out specific vocabulary terms such as equivalent ratios and double number lines. 
   2. Questions #2-3: Allow students 2-3 minutes to answer questions 2 and 3. Ask that students use highlighters or colored markers to identify where in the double number line they could represent their answers. STOP here and ask the students the following questions: 
2. Are these ratios in the double number line equivalent ratios? How do you know? 
3. How might these ratios be represented with a drawing similar to the first representation?
   1. Question #4: Lastly, give the students 5 minutes to answer question 4 in complete sentences. Once question 4 is complete, group the students together based on who agrees and who disagrees. Note: if all students disagree, then ask them to share their explanations and compare responses. If all students agree, ask them to draw a diagram to represent the situation and guide them to identify the misconceptions and relate them to the previous equivalent ratios. Ask the students the following question: What math operation was used to get from one equivalent ratio to the other? Is this the same operation that Mark used? Give the students the opportunity to revise their work if needed. To close out this activity, demonstrate on the double number line the math operations that can be performed to get from one to the next.
4. Next, give the students the Double Number Lines Practice Activity. Use the Double Number Lines Practice Activity Answer Key to support student learning during this activity.
   1. Question #1: Read the situation aloud to the students. Ask students the following question: Can you think of another equivalent ratio that will give Elena the same shade of light blue? Let the students explore this idea before they complete question 1. Next, give the students 3 minutes to create the double number line diagram. Make sure you watch for students that do not label the double number line.
   2. Question #2: Give students the opportunity to compare their double number line with a partner. Allow the students to give verbal feedback and discuss any differences. While they are comparing their diagrams, pay close attention to their conversations. Watch for errors in the students’ work and support pairs as needed. Next, give students the chance to revise their diagram as needed. Give the students 3 minutes to compare and 2 minutes to revise.
   3. Question #3-4: Allow students 2-3 minutes to answer questions 3 and 4. Ask that students to use highlighters or colored markers to identify where in the double number line they could represent their answers. STOP here and ask the students the following question: How did you determine the number of batches? Where is this represented on the double number line?
   4. Question #5-6: Allow students 2-3 minutes to answer these questions. Challenge your students to try to find an equivalent ratio that is different from their peers. 
5. To conclude this session, give the students the Exit Ticket for Lesson #2. Students should be able to demonstrate their understanding from this session, so please allow students the time they need to complete the exit ticket the best they can. This exit ticket will assess students' ability to use double number lines to find unknown values in a real-world situation. Use the Exit Ticket for Lesson #2 Answer Key to determine if your students are ready to move on to session 3 or if they need additional practice with double number lines.

Session 3: Representing Equivalent Ratios on a Table

This session will take around 35-40 minutes to complete.

Directions:

1. Start with the Tables Practice Activity. Use the Tables Practice Activity Answer Key to support student learning during this activity.
   1. Question #1: Read the situation aloud to the students. Ask students the following question: Can you think of another equivalent ratio that will give you a double batch of Nihaizu? Next, give the students 2 minutes to complete the table. Once they complete the table, ask students to take turns explaining their thinking while you annotate the table to model demonstrating their thinking. Refer to the Tables Practice Activity Answer Key for an example of how to show thinking on a table.
   2. Question #2: Give students 5 minutes to complete question 2. Circulate the room and support students one on one if they are struggling to get started. Emphasize that all students need to show their thinking on the table. Refer to the Tables Practice Activity Answer Key for an example of how to show thinking on a table.
2. Next, give the students the Comparing Double Number Lines and Tables Activity. Use the Comparing Double Number Lines and Tables Activity Answer Key to support student learning during this activity.
3. Give the students 10 minutes to complete the activity. It is good to let the students engage in productive struggle by asking focusing questions when necessary (questions focused on the students’ thinking, not a particular strategy). Let the students use problem solving skills and use the strategies to find the unknown values. If students do not know how to get started, model how to half the ratio and then release them to continue working. If students need longer than 10 minutes, use your professional judgment to make that call.
4. Once the students have completed the activity, have students compare their double number lines and revise as needed. Next, have students compare their table and revise as needed.
5. Lastly, have an open discussion about question 3. Give the students an opportunity to decide on which strategy works best for them and why. Emphasize that students can use these strategies to solve problems with unknown values if they are struggling to reason about the real-world problem.
6. Lastly, allow the students to complete the Card Sort Activity. Use the Card Sort Activity Answer Key to support student learning during this activity.
7. For the card sort activity, be sure to cut out the cards and have the students match the verbal descriptions, tables and double number lines. There are a total of 6 groups that need matching. While students are working, support as needed. If students struggle to get started, suggest they start with two representations first, or model how to make one set by reasoning about looking at the equivalent ratios and units. 
8. As an extension activity, have students create their own group of cards that have a verbal description, table and double number line for a different real-world problem. 
9. To conclude this session, give the students the Exit Ticket for Lesson #3. Students should be able to demonstrate their understanding from this session, so please allow students the time they need to complete the exit ticket the best they can. This exit ticket will assess students' ability to use tables to find unknown values in a real-world situation and their ability to compare double number lines with tables. Use the Exit Ticket for Lesson #3 Answer Key to determine if your students are ready to move on to the post-assessment or if they need additional practice with tables.

• Recipes: Students can use equivalent ratios with recipes to make half, double, or triple batches based on how much of the recipe they need.
• Shopping: Students can use equivalent ratios while shopping to determine the cost of multiple items.
• Painting: Students can use equivalent ratios while mixing paint colors to ensure that they make the same shade of a color to complete a painting project.
• Reading Extension: Read the book titled, “Multiplying and Dividing at the Bake Sale” by Tony Hyland to see real-world examples of using equivalent ratios.

• Double number line: a double number line is two parallel lines that represent equivalent ratios by matching the numbers up vertically.
• Equivalent Ratios: equivalent ratios are two or more ratios that are multiples of each other.
• Ratio: a ratio is a comparison between two or more quantities.
• Table: a table is a diagram that organizes information using rows and columns