Teaching for Conceptual Understanding: Ratios and Proportional Relationships, Facilitator Handbook

Teaching for Conceptual Understanding: Ratios and Proportional Relationships

Professional Development

Facilitator Handbook

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Pearson School Achievement Services

Teaching for Conceptual Understanding: Ratios and Proportional Relationships

Facilitator Handbook

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Published by Pearson School Achievement Services, a division of Pearson, Inc.

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Printed in the United States of America.

ISBN 115504

Facilitator Agenda

Teaching for Conceptual Understanding: Ratios and Proportional Relationships

Section / Time / Agenda Items
Introduction / 20 minutes / Slides 1–4
• Welcome and Introduction
• Agenda
• Outcomes
• Content Warm-up
Activity: Intuitive Problem Solving
1: The Progression of Ratios and Proportional Relationships / 40 minutes / Slides 5–11
• Section 1 Big Questions
• Exploring the Standards
Activity: Stoplight Highlighting
Activity: Stoplight Bicycle Chain
• The Importance of Ratios and Proportional Relationships
• Revisit Section 1 Big Questions
• Reflection: The Journey Begins

2: Building the Foundation for Ratios and Proportional Relationships / 40minutes / Slides 12–21
• Section 2 Big Questions
• Defining Ratio
Activity: 3:1 Scenarios
• Vocabulary and Symbols in 6.RP.1
Activity: Sorting Representations
Activity: Word Wall Artifacts
Break / 15 minutes
2: Building the Foundation for Ratios and Proportional Relationships / 80minutes / Slides 22–35
• Additive versus Multiplicative Reasoning
Activity: Is it additive or multiplicative?
• Models and Strategies for Problem Solving in 6.RP.3
• Tables of Equivalent Ratios
Activity: Chocolaty and Nutty Tables of Equivalent Ratios
• Double Number Line Diagrams
Activity: Double Number Line Diagram Think-Pair-Share
• Tape Diagrams
Activity: Tape Diagram Think-Pair-Share
• Solving Problems Using 6.RP.3 Models and Strategies
Activity: 6.RP.3 Models and Strategies in Action
• Percent as a Rate per Hundred in 6.RP.3c
Activity: Using Double Number Lines for Percent Problems
• Grade 6 Progression Reflection
Activity: Grade 6 Progression Reflection
Lunch / 30 minutes
2: Building the Foundation for Ratios and Proportional Relationships (continued) / 90minutes / Slides 36–48
• Ratios with Rational Numbers in 7.RP.1
Activity: Super Sour Candies
• Defining Proportional Relationships
• Determining Proportional Relationships in 7.RP.2a
Activity: Examples and Nonexamples
• Representing Proportional Relationships in 7.RP.2b,c,d
Activity: Graphing Proportional Relationships
• Conceptual Understanding of Multiple Representations
Activity: Making Connections
• Multistep Problems
Activity: Different Context, Same Models and Strategies
• Revisit Section 2 Big Questions
• Reflection: Will Do

Break / 15 minutes
3: Making Connections within Mathematics / 35minutes / Slides 49–54
• Section 3 Big Questions
• Making Connections within Mathematics
Activity: Connections to Geometry Progression
Activity: CCSSM Connections Search
• Revisit Section 3 Big Questions

4: Planning Lessons with Ratios and Proportional Relationships / 50minutes / Slides 55–59
• Section 4 Big Questions
• Focusing on Instruction
Activity: Classroom Resources Brainstorm
• Planning So That Ratios and Proportions Make Sense
Activity: Four-Step Planning Process
• Revisit Section 4 Big Questions

Reflection and Closing / 5 minutes / Slides 60–66
• Outcomes
• Final Reflection
• Evaluation
• References
• Pearson Legal Statement

Total / 6 hours

Preparation and Background

Workshop Information

BigQuestions

• How are you responsible for teaching new content at yourgrade level?
• How does what you teach impact student understanding at the next grade level?
• What types of problems will yourstudents need to solve in order to solve problems with ratios and proportions at a conceptual level?
• How do the instructional strategies that youchoose to use help your students develop a conceptual understanding of ratios and proportions?
• How can yourstudents’ understanding of ratios and proportions impact the teaching and learning of standards in other content areas?
• How will you help your students make connections between ratios and proportions and topics that they will study when you address other standards?
• What key elements must you be mindful of when you planlessons that involve ratios and proportional relationships?
• How will yourlesson-planning process change to incorporate these key elements?

Assessments of Participants’ Learning during the Workshop

• Section Reflections
• Planning: Instructional Outcome Planning
• BigQuestions Review

Assessment Back in the School/Classroom

• Implement the instructional strategies identified as necessary for the conceptual understanding of ratios and proportional relationships.

Outcomes

• Articulate the learning progressions necessary for students to conceptually understand ratios and proportional relationships.
• Identify strategies for helping students build their mathematical understanding of ratios and proportional relationships.
• Use a planning template to build lessons that strategically support the conceptual development of ratios and proportional relationships.
• Identify strategies that support simultaneous development of conceptual understanding and problem-solving skills with the intentional use of purposeful student struggle, flexible grouping, and ongoing assessments.
• Articulate common misconceptions as opportunities for students’ conceptual understanding of ratios and proportional relationships.

Facilitator Goals

• Assist participants in understanding the progression of ratios and proportional relationship concepts as outlined in the CCSSM.
• Deepen participants’ conceptual understanding of ratios and proportional relationships.
• Guide participants to develop the instructional strategies needed for teaching the conceptual understanding of ratios and proportional relationships. The strategies include the following:
• Identifying important vocabulary and symbols
• Having students create examples and nonexamples
• Posting models
• Posting examples of solutions
• Usingmultiple representations
• Identifying important points

Session 2: Building the Foundation for Ratios and Proportional Relationships

(Slides 12–48)

Time: 210minutes

Big Questions

• What types of problems will yourstudents need to solve in order to solve problems with ratios and proportions at a conceptual level?
• How do the instructional strategies that you choose help your students develop a conceptual understanding of ratios and proportions?

Learning Objectives

• Participants will be able to articulate how they can teach ratios and proportions concepts through the problems that students solve.
• Participants will be able to articulate how the instructional strategies they choose assist students in developing a conceptual understanding of ratios and proportions.

Materials

• Chart paper
• Sticky tack
• Markers
• CCSSM document (CCSSM.pdf)
• Index cards
• Participant Workbook, pages 9–30

Topic / Presentation Points / Presentation Preview
Solving Problems Using 6.RP.3 Models and Strategies /
• Display Slide 29.
Activity: 6.RP.3 Models and Strategies in Action
NOTE: There are several different strategies that students can use to solve problems that involve ratios. When students first start using these strategies, their approaches may be very in-depth and lengthy. As they become more familiar with the relationships and comparisons among equivalent ratios, their strategies will become more abbreviated and efficient (The Common Core Standards Writing Team 2011, 6). The following problems will allow participants to use different strategies for solving problems involving ratios. Possible solutions and discussion points have been provided below for each problem.
• Direct participants to work in pairs to solve one of the problems from the problem setSolving Problems Using 6.RP.3 Models and Strategies A–D. Assign each pair a model to use—ratio tables and graphs, tape diagrams, or doublenumber lines.
NOTE: Make sure that participants know what the models are, especially the tape diagrams and double number lines. You can use the examples below as a reference for the model, but be careful to avoid giving away the solution.
• As participants complete their solutions using their assigned models, ask them to transfer their solutions to chart paper. Each pair will then present their solutions to either the large group or to smaller groups of combined tables depending on the size of the group and time available.
• After the participant presentations, discuss the following:
• How the models helped them to solve the problems
• The ratios involved
• Possible student misconceptions
• How the models help students struggle with the important mathematics to build conceptual understanding
/
PW: Pages18–19
• Display Slide 30.
Problem A
Ricky Racer can complete 3/4of a race in 24 minutes. If he continues at that same rate, how long would it take him to complete the whole race? If he wanted to cut his time by 25%, how long would it take him to complete 1/2of the race? (diagram or double number line)

The time to complete the whole race is 32 minutes.

Cutting the time by 25% means taking away one 1/4of the original time and spreading that time out over the whole race. In this case, it would be 24 minutes to complete the whole race. Completing half of the race at this time would take 12 minutes.
Double Number Line

The time to complete the whole race is 32 minutes.

Cutting the time by 25% means taking away one 1/4of the original time and spreading that time out over the whole race. In this case, it would be 24 minutes to complete the whole race. Completing half of the race at this time would take 12 minutes.
Go through the following discussion points (The Common Core Standards Writing Team 2011, 4):
• Diagrams can be a useful visual aid for solving problems. They allow students to “see” the relative size of the quantities they are working with and how they relate to one another.
• Students can use double number lines to represent equivalent ratios.
• Students can best use double number lines when the quantities being compared have different units.
• Double number line diagrams can help students see that there are an infinite number of pairs in the same ratio, including those with rational numbers entries.
/
PW: Page 18
• Display Slide 31.
Problem B
If two bags of candy cost \$5.00, how much will 12 bags of candy cost? How much will 3 bags of candy cost? What about 6 bags of candy? (table and graph or equation)
Table and Graph

The table and graph show that 12 bags of candy would cost \$30.00. Because one bag of candy costs \$2.50, multiply\$2.50 by 12 to get the total cost for 12 bags of candy. Find the cost for 3 and 6 bags of candy using the same method. Three bags of candy would cost \$7.50, and six bags of candy would cost \$15.00.
Equation
Participants can glean the equation for this problem from the table and graph due to the pattern displayed—multiply the number of bags purchased by \$2.50. Use n for the number of bags of candy and c for the cost. The equation is:
n(\$2.50) = cor c = n(\$2.50)
To create the equation without the use of the table and graph, still use n for the number of bags of candy purchased and c for the final cost. Two bags of candy cost \$5.00, so one bag of candy will cost 1/2of \$5.00. This is the amount thatis needed to multiply the number of bags of candy by to get the total cost, represented by the following equation:
n(1/2 ×\$5.00) = c or c = n(1/2 ×\$5.00)
Go through the following discussion points (The Common CoreStandards Writing Team 2011, 4–5):
• Students can graph a collection of equivalent ratios in the coordinate plane.
• The graph represents a proportional relationship.
• The graph of a collection of equivalent ratios lies on a line through the origin, and students can see the pattern of increases in the table in the graph as coordinated horizontal and vertical increases.
• The unit rate appears in the equation,in the graph as the slope of the line, and in the coordinate pair with the first coordinate 1.
/
PW: Page 18
• Display Slide32.
Problem C
At the farmers’ market, Jones sells tomatoes at 5 pounds for \$7 whereas Smith sells his tomatoes at 4 pounds for \$6. Which farmer sells his tomatoes at a cheaper price? (ratio table)
Ratio Table

• When each farmer sells the same number of pounds (20 pounds), the ratio tables that compare the two costs show that Farmer Jones’ tomatoes are actually \$2 cheaper than Farmer Smith’s tomatoes.
/
PW: Page 19
• Display Slide 33.
Problem D
To create her special paint color for the school project, Ms. Smith combines cups of red, blue, and white paint in a ratio of 2:2:4. How much of each paint will need to be mixed to create 48 gallons of paint? (tape diagram)
Tape Diagram
There are 16 cups in 1 gallon, so double the paint mixture for one gallon.

If 16 cups equals 1 gallon, then you can multiply 16 by 48 to find that 48 gallons equals total 768 cups. From the tape diagram,you can see that 1/2of the mixture will be white paint;1/4will be red; and 1/4will be blue. You can divide 768 by 2 to find that you will need 384 cups of white paint. You can then divide 384 by 2 to find that you will need 192 cups of red paint and 192 cups of blue paint.
Go through the following discussion points (The Common Core Standards Writing Team 2011, 4, 7):
• Tape diagrams represent collections of equivalent ratios and visually depict the relative size of the quantities.
• Students can be use tape diagrams when the two quantities have the same units.
• Students can use tape diagrams to solve problems and also to highlight the multiplicative relationship between the quantities.
• Measurement conversion provides other opportunities for students to use relationships given by unit rates. For example, recognizing that 16 cups equals 1 gallon as a rate can help students connect the concepts and methods developed for other contexts with measurement conversion.
• As you wrap up the discussion of the strategies used for solving each problem, do the following:
• Post Standards 6.RP.3 a, b, c, d and 6.EE.9 along with the appropriate posted solution strategy.
• Add examples of models and solutions to the ratios and proportions wall, and bring participants’ attention back to the wall. Point out the growth of the wall and the wealth of knowledge that it continues to build for students as a reference for continued use.
/
PW: Page 19
Percent as a Rate per Hundred in 6.RP.3c /
• Display Slide 34.
• Direct participants to page 42 of the CCSSM document (CCSSM.pdf) to review Standard 6.RP.3c.
Activity: Using Double Number Lines for Percent Problems
• Direct participants to the Participant Workbook to read through the directions and the word problems on percent(America’s Choice 2002, 15).
• Refer back to double number line model solutions that participants have completed so far, and then ask the following question:
• Why do you think the CCSSM require students to utilize double number lines in Grade 6?
NOTE: You are looking for participants to state that double number lines build conceptual understanding of ratios and proportions so that the equations and algorithms employed to solve problems will make sense beyond the rote memorization of things like “cross multiply.”
• Direct participants to work through the problem set for a few minutes independently and then direct ask themto pair up to compare solutions and complete any remaining problems.
• Direct participants to spend most of their time on Problems 1 and 2, and if they finish early then move on to Problems 3 and 4.
• Circulate throughout the room, and make note of participants’ solution methods.
• Conclude this activity by having select participants present and discuss their solutions methods. Ask participants to explain the connections between the multiple representations utilized.
/
PW: Page 20
Grade 6 Progression Reflection /
• Display Slide 35.
Activity: Grade 6 Progression Reflection
• Direct participants to engage in a think-pair-share about one of the following prompts:
• For Grade 6 teachers, how will you prepare to teach the CCSSM Grade 6 Ratio and Proportional Relationships concepts?
• For Grade 7 teachers, what will you expect your incoming Grade 7 students to know about ratios and proportions?
• Direct participants to the Participant Workbook to take notes on this discussion.
/
PW: Page 21
Ratios with Rational Numbers in 7.RP.1 /
• Display Slide36.
• Direct participants to page 48 of the CCSSM document (CCSSM.pdf) to review Standard 7.RP.1.
• Explain to participants that in Grade 7, students extend their reasoning about ratios and proportional relationships in several ways(The Common Core Standards Writing Team 2011, 8):
• They use ratios in cases that involve pairs of rational number entries, and they compute associated unit rates.
• They identify these unit rates in representations of proportional relationships.
• They work with equations in two variables to represent and analyze proportional relationships.
• They also solve multi-step ratio and percent problems, such as problems involving percent increase and decrease.
• At this grade, students will also work with ratios specified by rational numbers.
• Students continue to use ratio tables, extending this use to finding unit rates.
NOTE: The next four activities will take participants through problem situations that students will face in Grade 7 as teachers align instruction to the CCSSM. In the first of these four activities, you will ask participants to use ratios to solve problems with pairs of rational numbers.Possible solution strategies are provided below. /