# SFUSD Mathematics Core Curriculum Development Project · 2020-02-17 · 3 SFUSD Mathematics Core...

### Transcript of SFUSD Mathematics Core Curriculum Development Project · 2020-02-17 · 3 SFUSD Mathematics Core...

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

8th Grade

8.4 Linear Relationships

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task CPM CCC3 Lesson 1.1.2, 1-9 through 1-11 HW: CPM CC3 Lesson 1.1.2

1 per pair CPM eBook

Poster paper Markers Rulers

9 Lesson Series 1 CPM CCC3 Lesson 1.2.1 (2 pages) HW: CPM CC3 Lesson 1.2.1 CPM CC3 Lesson 4.1.1 (2 pages) HW: CPM CC3 Lesson 4.1.1 CPM CC3 Lesson 4.1.2 (3 pages) Resource Page 4.1.2 CPM CC3 Lesson 4.1.3 (2 pages) HW: CPM CC3 Lesson 4.1.3 CPM CC3 Lesson 4.1.4 (2 pages) HW: CPM CC3 Lesson 4.1.4 CPM CC3 Lesson 4.1.5 (2 pages) HW: CPM CC3 Lesson 4.1.5 CPM CC3 Lesson 4.1.6 (2 pages) HW: CPM CC3 Lesson 4.1.6

1 per pair CPM eBook 1 per pair CPM eBook 1 per pair 1 per student 1 per pair CPM eBook 1 per pair CPM eBook 1 per pair CPM eBook 1 per pair CPM eBook

Resource Page 4.1.1A – see teacher notes for how to copy Poster paper Markers or colored pencils Graph paper Rulers

1 Apprentice Task CPM CCC3 Lesson 4.1.7 (2 pages) HW: CPM CCC3 Lesson 4.1.7

1 per pair CPM eBook

Class set of graphing calculators Rulers

3 Lesson Series 2 CPM CCC3 Lesson 7.2.1 (2 pages) HW: CPM CCC3 Lesson 7.2.1 CPM CCC3 Lesson 7.2.2 (4 pages) Resource Page 7.2.2 HW: CPM CCC3 Lesson 7.2.2 CPM CCC3 Lesson 7.2.3 (4 pages) Resource Page 7.2.3 HW: CPM CCC3 Lesson 7.2.3

1 per pair CPM eBook 1 per pair 1 per pair CPM eBook 1 per pair 1 per student CPM eBook

Poster Paper Markers or colored pencils Rulers

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

1 Expert Task Stacking Cups Handout (3 pages) 1 per pair Rulers 6-10 styrofoam cups per group of 4 Graph paper

3 Lesson Series 3 CPM CCC3 Lesson 7.2.4 (3 pages) HW: CPM CCC3 Lesson 7.2.4 CPM CCC3 Lesson 7.2.5 (3 pages) HW: CPM CCC3 Lesson 7.2.5 Ironman Triathlon Handout (2 pages)

1 per pair CPM eBook 1 per pair CPM eBook 1 per pair

Graph paper

1 Milestone Task Milestone Task (3 pages) 1 per student Rulers, calculators

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Unit Overview

Big Idea

Linear relationships can be displayed using multiple representations, and data can be analyzed and compared within these representations. The generic equation y=mx+b can be used to express all linear relationships.

Unit Objectives

● Students will be able to graph proportional relationships. ● Students will be able to interpret the unit rate as the slope of the graph. ● Students will be able to compare two different proportional relationships. ● Students will be able to compare linear relationships represented in different ways (graph, table, equation, diagram, etc.). ● Students will be able to use similar triangles to explain why the slope m is the same between any two distinct points on a line in a coordinate plane. ● Students will be able to derive the equation y = mx for a line that goes through the origin (proportional relationship). ● Students will be able to derive the equation y = mx + b for a line that intercepts the vertical axis at b.

Unit Description

Students will explore the connection between proportional relationships, lines, and linear equations. They will do this by first examining what exactly a proportion is, and understanding that a rate is in fact the slope of the graph. Students will examine two different proportional relationships that are represented in various ways (graph, table, equation, diagram, etc.) and analyze data and information from these representations. Students will analyze data to find specific information (e.g., distances, times, speeds, etc.). After having an understanding of proportional and linear relationships, students will use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. They will then use this information regarding the slope m to derive the equation y = mx for a line through the origin (a proportional relationship) and the equation y = mx + b for a line intercepting the vertical axis at b (linear relationship).

CCSS-M Content Standards

Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In sixth grade, students had experience with ratios and proportions from their work with tape diagrams and number line diagrams. In seventh grade, students began working with ratios and analyzing proportional relationships, and used them to solve real-world and mathematical problems. Additionally, they computed unit rate. They also began recognizing proportional relationships between two quantities and representing these relationships using equations. Lastly, they will have experience working with scale drawings.

Students will make connections between proportional relationships, lines, and linear equations by displaying multiple representations and analyzing and comparing these representations. They will do this first by examining and extending patterns from tiles. They will use acquired knowledge regarding the slope, m, to derive the equation y = mx for a line through the origin (a proportional relationship) and the equation y = mx + b for a line intercepting the vertical axis at b (linear relationship).

In future units, students will delve further into linear relationships by solving linear equations algebraically, analyzing solutions, and solving systems of equations (graphically/algebraically/with a table). Students will apply this knowledge to the concept of functions, understanding the rule that assigns one output to each input. Students will use their background in linear relationships to delve further into linear equations, linear inequalities, and systems of linear equations and inequalities in ninth grade mathematics. They will also use their understanding of a linear function as a basis for other relationships: quadratic, exponential, logarithmic, etc., that are studied in ninth grade and beyond.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Unit Design

All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 Day 9 Days 1 Day 3 Days 1 Day 3 Days 1 Day

Total Days: 19 days

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Entry Task How Can I Generalize a Pattern?

Apprentice Task What Are the Connections?

Expert Task Stacking Cups

Milestone Task Summative Assessment

CCSS-M Standards

8.EE.6 8.EE.5 8.EE.5, 8.EE.6 8.EE.5, 8.EE.6

Brief Description of Task

Students extend tile patterns and learn how to generalize the geometric description of the pattern.

Students take a pattern and represent it as a table and a graph, and then explore the connections between tables, graphs and rules.

Students represent the height of stacked cups with a table, graph, and equation, and recognize connection of rate of change using data on graph.

Students will show what they have learned in the unit by working on a series of tasks that address proportional and linear relationships, including determining the slope from graphs and tables, arguing if a relationship is proportional, describing the meaning of slope in a specific situation, deriving equations from representations in the form of y = mx and y = mx + b, and analyzing various relationships to solve for x or y.

Source CPM CCC3 1.1.2 CPM CCC3 4.1.7 SFUSD Teacher-Created, Adapted from http://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/ and http://fawnnguyen.com/2013/02/05/20130204.aspx

Adapted from NYC Department of Education, Grade 8 Expressions and Equations

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

8.EE.5, 8.EE.6

8.EE.5, 8.EE.6 8.EE.5, 8.EE.6

Brief Description of Lessons

Students investigate proportional relationships in graphs and tables. Students apply various strategies to solve proportion problems. Students discover connections between multiple representations of a pattern: a graph, a table, a geometric presentation, and an equation. Students also look at different ways to represent the connections.

Students compare ratios and rates using different representations, including numbers, tables, and graphs. Students measure the steepness of a line on a graph.

Students have identified proportional relationships in both tables and graphs. In this lesson series, they look at the connection between the slope and the graph of a proportional relationship.

Sources

CPM CCC3 1.2.1 and 4.1.1 through 4.1.6 CPM CCC3 7.2.1 to 7.2.3

CPM CCC3 7.2.4 and 7.2.5 Mathalicious: Domino Effect CMP 3 Function Junction: The Families of Functions Problem 1.4 Extension: Engage NY Grade 8 “Picture Frame” Extension: Illuminations http://illuminations.nctm.org/Lesson.aspx?id=947

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Entry Task

How Can I Generalize a Pattern?

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will learn how to extend a tile pattern. ● Students will learn how to generalize the geometric description of the

pattern. CCSS-M Standards Addressed: 8.EE.6 Potential Misconceptions

● Students might think that an input can have more than one output.

Launch: Students will begin studying tile patterns and using them to make predictions. Then they will draw and predict the geometric description of the pattern from each tile pattern. They will create and predict the next figures. During: Students will continue to examine, predict and justify the new tile patterns. They will derive a solution for each geometric pattern. While students work, monitor students’ progresses and support students. Closure/Extension: Students prepare a team poster to explain and justify clearly the description of the pattern and the prediction.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

How Can I Generalize a Pattern?

How will students do this?

Focus Standards for Mathematical Practice: 4. Model with mathematics. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Students will identify patterns and figure number. Structures for Student Learning: Academic Language Support:

Vocabulary: pattern, input, output, x and y, generalize, rule, function, equation, slope, y = mx +b, rise/run, Sentence frames: I notice that when __________, the ___________(increases/decreases). The rule for this problem is _____________________.

Differentiation Strategies: Provide each student with a table/chart to write down and draw the geometric tile patterns and prediction. For EL and SPED - Have them work with a partner. Participation Structures (group, partners, individual, other):

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Series #1

Lesson Series Overview: Students investigate proportional relationships in graphs and tables. Students apply various strategies to solve proportion problems. Students discover connections between all of the representations of a pattern: a graph, a table, a geometric presentation, and an equation. Students also look at different ways to represent the connections. CCSS-M Standards Addressed: 8.EE.5 Time: 9 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students review the concept of proportional relationships using graphs and tables. Then students compare rates in different representations of proportional relationships. Students will begin the lesson by making tables and graphs. Remind students about the characteristics of proportional relationships and if necessary have a class discussion. The teams will work on the problem together and move to the next problem with your approval. Students will compare the rate from a table to the rate from the graph to determine the proportional relationship. Notes: It is important to have a Task Manager to make sure that the team is on task, not talking to other teams and listening to the discussion to make sure everyone understands.

CPM CCC3 1.2.1

CW: Problems 1-41 through 1-45 HW: Problems 1-47 through 1-51

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Overview – Days 2–3 Resources

Description of Lesson: Students discover connections between all of the representations of a pattern: a graph, a table, a geometric presentation, and an equation. Students also use different representations to justify the connections. On day 1, give each team Resource Page 4.1.1 to analyze and discuss tile pattern and generate as many connections as possible between representations. On day 2, each team will present the connections between representations. They will describe the patterns they discover and explain how they find the same patterns in a different representation. Notes:

CPM CCC3 4.1.1 Resource Pages 4.1.1A, B, and C CW: Problem 4-1 HW: Choose from 4.1.1 Review & Preview Poster paper Markers or colored pencils

Lesson Overview – Day 4 Resources

Description of Lesson: Students generate linear algebraic rules for each figure number of a geometric pattern and its number of tiles. Students make connections between the growth of a pattern and its linear equation. Begin this lesson with a class discussion to sum up the investigations of multiple representations of patterns. Ask students to share a representation with the class: tile pattern, x and y tables, graphs, and equations (rules). Introduce the four representations: Graph, table, rule and pattern, to students on the board. They will connect all of the representations and find efficient ways to generalize the rule from a tile pattern using one of the representations. Make sure to emphasize the reversibility of these processes (see diagram in the lesson introduction). Create a Learning Log entry as a whole class. Notes:

CPM CCC3 4.1.2 Resource Page 4.1.2 CW: 4-12 through 4-16 HW: Choose from 4.1.2 Preview & Review Poster paper Marker or colored pencils, four colors per team General Team Roles Resource Page 1.1.2 (optional)

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Overview – Day 5 Resources

Description of Lesson: Students connect linear geometric patterns with patterns on a graph. They specifically work on how a geometric pattern grows and how the size of Figure Zero can be determined from information on a graph. Begin class by having one or two students read aloud from their Learning Log or just share out the connections they noticed or the question they have yet to answer. Then introduce problem 4-22 and distribute Resource Page 4.1.3 to students. In problem 4-22 students will use data from Lesson 4.1.2 to create x and y tables and graphs of those patterns. Make sure to remind them about the color-coding system they used in Lesson 4.2.1 as they create tables and graphs. When students have finished with 4-22, lead a whole discussion about their findings. Continue the Learning Log entry for today’s lesson. Notes:

CPM CCC3 4.1.3 Resource Page 4.1.3 CW: 4-22 through 4-24 HW: Choose from 4.1.3 Review & Preview Markers or colored pencils

Lesson Overview – Day 6 Resources

Description of Lesson: Students develop new connections between multiple representations of patterns and identify rules for these patterns using the y = mx +b form of a linear equation. Students apply their understanding of m (slope) as the growth factor and b as Figure Zero (constant) to write a rule from a graph and to create a pattern based on a linear rule. Review multiple representations of patterns from Day 5. Use the class Representations Patterns Web from Lesson 4.1.2 to show which connections are the most effective at this point: Table, graph, rule or pattern. Use problems 4-30 and 4-31 as a whole-class discussion and introduce the y = mx + b equation to students on the board. Make sure to indicate that all of the equations (rules) can be written in this form. Students continue to work on 4-32 to 4-34 to find more connections on the web. Notes:

CPM CCC3 4.1.4 CW: Problems 4-30 through 4-34 HW: Choose from 4.1.4 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Overview – Days 7–8 Resources

Description of Lesson: Students apply their understanding of growth, Figure 0 and connections between multiple representations in a problem and eventually generate a rule. They also apply their understanding of growth and Figure 0 to new contexts to generate complete representations. Begin Day 7 by reviewing the Representations of Patterns Web from Lesson 4.1.2. Then students will begin working on problem 4-42 Part (a) in teams to generate a pattern. These are challenging problems; each team will need to talk together, share ideas, question each other and justify their statements. While students are working in teams, circulate and encourage each team to work together to generate the complete pattern. Discuss students’ findings thus far as a whole class. Continue Day 8 with Part (b) of problem 4-42 right away. Then each team will create complete multiple representations to share with the class. Notes:

CPM CCC3 4.1.5 CW Day 7: Part (a) of problem 4-42 CW Day 8: Part (b) of problem 4-42 Homework: Choose from 4.1.5 Review & Preview General Team Roles Resource Page 1.1.2 (optional)

Lesson Overview – Day 9 Resources

Description of Lesson: Students apply their knowledge of m as the growth factor and b as Figure 0 or the starting value of a pattern to make graphs quickly without using an x and y table. Start the lesson by asking students this question, “How can we use our knowledge of y = mx + b to make graphs quickly?” Share the goal for this lesson with students: to develop strategies for graphing patterns directly from a rule without creating x y tables. In problems 4-54 through 4-56, students will attempt to graph from the rule only. Students should have some idea of using the y-intercept as the starting point of the graph and using a growth triangle to plot the next point. As teams work, circulate and listen to each team making connections with the patterns. Once teams have completed problem 4-56, have a whole-class discussion about their strategies they used on the board. Notes:

CPM CCC3 4.1.6 CW: Problems 4-54, 4-55, and 4-56 HW: Choose from 4.1.6 Review & Preview Graph paper for board display

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Apprentice Task

What Are the Connections? (CPM CCC3 4.1.7)

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives:

● Students will take a pattern and represent it as a table and a graph. ● Students will explore the connections between tables, graphs and

rules. CCSS-M Standards Addressed: 8.EE.5 Graph proportional relationships Potential Misconceptions

● Students may have difficulty going from the graph to the pattern and the table may be helpful here.

● Students may have difficulty with the use of the graphing technology. ● Students may have trouble finding the pattern with 79 tiles.

Launch: 4-64 Look at the table and graph for a tile pattern and draw on graph paper what this tile pattern might look like. Multiple answers are correct. During: 4-65 Revisit Growing, Growing, Growing Shown are tile patterns for figures 2-4. Extend the graph and table to include 1-5. Write a rule for the pattern. Closure/Extension: Students should share their work. As an extension you could ask students to come up with their own patterns with a graph and a rule.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

What Are the Connections? (CPM CCC3 4.1.7)

How will students do this?

Focus Standards for Mathematical Practice: 1. Students will make sense of problems involving moving between representations and persevere in solving these problems. Structures for Student Learning: Academic Language Support:

Vocabulary: representation, slope, unit rate, positive/negative change, steepness, constant of proportionality Sentence frames: If the line goes up left to right it has a _____ slope. If the line goes down left to right it has a ________ slope. The steepness of the line is called the _______ of the line.

Differentiation Strategies: Some students may have to use graphing technologies to help with the graphing. Participation Structures (group, partners, individual, other): Pairs or groups of three.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Series #2

Lesson Series Overview: Measure the steepness of a line for situations, tables and graphs. CCSS-M Standards Addressed: 8.EE.5, 8.EE.6 Time: 4 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students develop ways to find the equation of a line using different pieces of information about the line, such as two points that are on it. See CPM teacher notes for further information. Notes:

CPM CCC3 7.2.1 Resource Page 7.2.1

CW: Problems 7-35 and 7-36 HW: Choose from 7.2.1 Review & Preview

Lesson Overview – Day 2 Resources

Description of Lesson: Students learn to describe the rate of change of a line (slope) numerically as the ratio between the vertical change and horizontal change. See CPM teacher notes for further information. Notes:

CPM CCC3 7.2.2 CW: Problems 7-43 through 7-48 HW: Choose from 7.2.2 Review & Preview

Lesson Overview – Day 3 Resources

Description of Lesson: Students identify slopes from graphs, and recognize the effect of scaling on the steepness of a line. Notes: It is important to attend to precision in terms of the units they are using for comparison.

CPM CCC3 7.2.3 Resource Page 7.2.3 CW: Problems 7-55 through 7-60 HW: Choose from 7.2.3 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Expert Task

Stacking Cups

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will model a situation using

y=mx+b to solve a real world problem. ● Students will determine which part of the

cup represents b and m. CCSS-M Standards Addressed: 8.EE.6 Potential Misconceptions

● Students might not understand how the stack of cups is growing.

● Students might think this is a proportional situation.

● Students might be confused by the fact that the y-intercept is the height of the cup without the lip.

Launch: Show students an 8 oz. styrofoam cup and ask them to estimate how many cups should be stacked to equal your height. If you like, you could show them a picture of your head next to an equally high stack of cups. Give students a minute to write down their initial guess. Next give students your own height in cm and the height of the cup in cm and ask them to make another prediction. You will probably have at least several students who think they can just divide your height by the height of the cup. Have one or more of these students convince the class why they believe this to be correct, and then stack that number of cups to show that the stack is a lot shorter than you are. As a class discuss why that is, helping students to understand that stacking cups in the same direction, the increase in height is not proportional and is dependent on the height of the lip at the top of the cup. Give each group of students a ruler, some graph paper, and 6-10 cups, and explain that they will be using graphs, tables, and/or equations to solve this problem, and they will need to have a graph and equation. Give a timeframe. Consider a prize for the teams that come closest. During: Have groups work together on trying to come up with an answer to the problem. Reinforce that they will need to have a graph and equation that models the situation. Encourage them to draw and label a diagram of the situation. You may want to ask some of the following questions to guide student thinking:

1. How many parts of the cup are there? Two.

2. Which part of the cup represents the slope? the y-intercept? The lip. The base.

3. Is this situation proportional? Why or why not?

4. If I asked you to tell me how tall a stack of sixty cups would be, what would you do? Add the height of sixty lips to the height of the base.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

5. If I asked you to go backwards and tell me how many cups are in a 200-centimeter-tall stack, what would you do? Subtract the height of the base and then divide by the height of the lip.

6. Does it matter if you round to the nearest centimeter? Yes

Closure/Extension: When the time is up (or most teams have finished), depending on time, you might want to have a couple of teams present their work, focusing on how they approached the problem, not necessarily all of the individual calculations. Then, have the big reveal, showing how many cups it actually takes to equal your height--you should have about ¾ of the cups pre-stacked to save some time. Extension: Have some larger and smaller cups (24 oz and 6 oz, or different types) and ask students to predict how many would be needed to stack up to the same height. Discuss how the different measurements affect the graph. Alternate Extension: Show students the picture of the 2 cups. Have them measure and answer the following questions: Which cups will stack higher after 3 cups? After 100 cups? When will the smaller cups pass the larger cups? Lesson adapted from: http://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/ http://fawnnguyen.com/2013/02/05/20130204.aspx

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Stacking Cups

How will students do this?

Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. Structures for Student Learning: Academic Language Support:

Vocabulary: y-intercept, slope Sentence frames: The lip of the cup represents the _______ of the equation y = mx + b

Differentiation Strategies: Have a picture of cups stacked next to your head that are the same height (without showing all the way to the floor). Participation Structures (group, partners, individual, other): This task is done in groups of 2 to 4.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Series #3

Lesson Series Overview: Students will have identified proportional relationships in both tables and graphs. Now they will look at the connection between slope and graphs of a proportional relationship. CCSS-M Standards Addressed: 8.EE.5, 8.EE.6 Time: 3 Days

Lesson Overview – Day 1 Resources

Description of Lesson: Students investigate what the steepness of the slope represents in various situations. Notes: Positive rate of change (lines that go up from left to right) represents a positive slope and negative rates of change (lines that go down left to right) represent negative slopes.

CPM CCC3 7.2.4 CW: Problems 7-67 through 7-70 HW: Choose from 7.2.4 Review & Preview

Lesson Overview – Day 2 Resources

Description of Lesson: Understand the connection between slope and graphs of a proportional relationship. Notes: Illumination extension problem includes “line of best fit” term – this is a standard that will be addressed in a future 8th grade unit.

CPM CCC3 7.2.5 CW: Problems 7-77, 7-78, 7-80 HW: Choose from 7.2.5 Review & Preview Possible extension problems: http://illuminations.nctm.org/Lesson.aspx?id=947 http://schools.nyc.gov/NR/rdonlyres/B46D0228-1BB5-4E44-A3CB-3A5C81334461/140499/NYCDOE_G8_Math_PictureFrame_FINAL.pdf

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Lesson Overview – Day 3 Resources

Description of Lesson: Students will use a racer’s average speed for each leg of a triathlon to graph the racer’s progress for each segment on a single graph, then use the graph to estimate the average rate for the entire triathlon. Notes: Ask students how the racer’s speeds for each leg compare, and how the graph can be used to see the relative differences.

CW: Ironman Triathlon HW: Choose from 7.2.5 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Milestone Task

Milestone Task

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Describe and justify a proportional relationship. ● Determine the slope and describe its meaning in context. ● Write expressions from graphs and tables in the form y = mx and

y = mx + b. ● Analyze and compare slopes of the same line and of different lines. CCSS-M Standards Addressed: 8.EE.5, 8.EE.6 Potential Misconceptions: ● Students might think that the relationship in #1 is proportional because it has

a constant slope. ● Students might confuse slope for days and slope for weeks in #2. ● Students might have trouble labeling the graph vertices in #3. ● Students must find a decimal value for x in #3d. They might think it is

incorrect because it isn’t a whole number.

Launch: Help students quickly what they have learned in this unit. Explain that the Milestone Task will show what they have learned throughout the unit. Explain the procedure for the Milestone Task and how the students will go through it. Discuss grading criteria and expectations. During: While the task is designed to be an independent assessment, it is up to you whether or not students may work on some (or all) sections with their group. Students may also work for half of the time independently, and then discuss as a group to share answers and collaborate. Closure/Extension: Score the milestone assessment as you see fit. Grades will be given individually or as a group, based on how the assessment was given. Go over answers and results with students at the end of the unit as a wrap up. Clear up any questions or misconceptions based on student work. Close out the unit with big picture ideas of what students have learned (have them write their own) and end with appreciations.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.4: Linear Relationships, 2014–2015

Milestone Task

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics 5. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support:

Vocabulary: proportional relationship, mathematical reasoning, justify, slope, similar triangles, expression, equation Sentence frames: The graph represents a proportional relationship because ______________________. The graph does not represent a proportional relationship because _____________________. The slope of the relationship is _______________________. The slope represents _____________________________. It will take him _________________ days to sell....

Differentiation Strategies: • Allow some students to work in groups. • Provide sentence frames. • Give detailed instructions on some questions that require justification or reasoning. Participation Structures (group, partners, individual, other): • Although the milestone assessment is designed to be done independently, some (or all) of it may be given to a group. • Another option is to have students work on it independently for the first half of the class, and have groups convene for the second half to discuss and finalize answers. Assessments shall be graded on an individual or group basis, depending on how the assessment was given.