SFUSD Mathematics Core Curriculum Development Project · Students might have a misconception...

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.1: Analyzing Graphs, 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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Grade 8

8.1 Analyzing Graphs

Number of Days

Lesson Reproducibles Number of Copies

Materials

1–2 Entry Task CPM CCC3 Lesson 1.1.1 (2 pages)

1 per pair

2 Lesson Series 1 CPM CCC3 Lesson 1.1.3 (2 pages) Resource Page 1.1.3 HW: CPM CCC3 Lesson 1.1.3 Describe the Graph activity worksheet

1 per pair 1 per student CPM eBook 1 per student

Integer cards –6 to 6 in five colors Sticky dots in 5 colors Chart paper with coordinate graph (1 per class) Sidewalk chalk or 2 25’ lengths of rope Duct tape Colored pencils or markers

1–2 Apprentice Task Mathalicious: Sweet Tooth (3 pages) 1 per student Rulers 0 Lesson Series 2 No lessons in this lesson series 1 Expert Task On my way to School 1 per pair Poster paper (1 per group), markers or

colored pencils, rulers 1 Lesson Series 3 Graphs from Stories and Stories from Graphs 1 per pair Graph paper (1 per student) 1 Milestone Task Unit 8.1 Milestone Task (2 pages) 1 per student Rulers

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Unit Overview

Big Idea

The graph of a functional relationship between two quantities can be described qualitatively, and the qualitative features of a function that has been described verbally can be represented on a graph.

Unit Objectives

● Students will be able to interpret the behavior of a graph from one point to another. ● Students will be able to create and graph situations. ● Students will understand how changing the input of graph affects the output. ● Students will be able to describe a situation, given the graph of the situation.

Unit Description

Students will begin to interpret the significance of the coordinates of individual points and of continuous graphs in a given situation, understanding that a point conveys two pieces of information and that a continuous graph conveys trends.

CCSS-M Content Standards

Functions Use functions to model relationships between quantities. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In second grade, students make line plots, picture graphs and bar graphs to represent data. In third grade, students draw picture graphs and bar graphs to represent data sets with several categories. Students identify relationships between corresponding items, then form and graph ordered pairs of those corresponding items in fifth grade. In sixth grade, students use positive and negative numbers to represent quantities and graph points in all four quadrants of the coordinate plane. Students recognize and represent proportional relationships between quantities in seventh grade. They decide whether two quantities form a proportional relationship, identify the constant of proportionality, and explain what the coordinates of a point on a graph of the proportional relationship mean in terms of the situation.

Given a graph, students will describe the relationship between the variables; for example, one is increasing as the other decreases. Given a scenario, students will sketch a graph that models the features of the scenario.

In Algebra and Advanced Algebra, students will interpret key features of graphs of a function that models a relationship between two quantities in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Students will estimate the rate of change of a function from a graph. They will graph functions expressed symbolically and show key features of the graph and compare properties of two functions each represented in a different way. They will identify the effect on a graph of changing the equation of a function by adding or multiplying by a constant.

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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?

1–2 Days 2 Days 1-–2 Days 0 days 1 day 1 day 1 day

Total: 9 days

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Entry Task What Stories Can A Graph Tell?

Apprentice Task Sweet Tooth

Expert Task On My Way to School

Milestone Task Jason’s Trip to School

CCSS-M Standards

8.F.5 8.F.5 8.F.5 8.F.5

Brief Description of Task

Students are each given a piece of a graph to find their group. They create a story that could be represented by their graph.

Students plot graphs from tables based on a situation. They interpret and compare the graphs.

Students sketch graphs from a story.

Students create a graph from a scenario.

Source CPM Core Connections Course 3 Lesson 1.1.1

Mathalicious: Sweet Tooth SFUSD Teacher Created SBAC Math Practice Test Grades 6–8

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

8.F.5 8.F.5

Brief Description of Lessons

This series introduces students to graphing on the coordinate plane and understanding how to create a story from a graph.

There are no specific lessons for Lesson Series 2. However, additional resources for analyzing graphs are available in the Resources folder.

In this lesson, students describe a scenario from a graph and then sketch a graph based on a scenario.

Sources CPM CCC3 Lesson 1.1.3 http://illuminations.nctm.org/Lesson.aspx?id=2844

CPM CCC3 1-50, CL1-68, 4-53, 7-51

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Entry Task What Stories Can A Graph Tell?

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will create a story based on a graph.

CCSS-M Standards Addressed: 8.F.5 Potential Misconceptions:

● Students might think that the higher the graph, the longer the time period.

● Students might have a misconception regarding the relationship between distance and time on a graph.

Launch: You will need one copy of the Resource Pages for this lesson (5 pages) from the binder or printed from the digital version. Do Now: Students will be given a grid that has four ordered pairs on it. They are to plot the ordered pairs on the grid. Create four different templates and project the correct graph for students to self-check. Post rules for graphing ordered pairs. Before class, cut apart graphs on the Unit 8.1 Entry Task Resource page, and cut each graph into four pieces along the vertical lines. (See Unit 8.1 Entry Task Teacher Notes). Give each student one part of a graph and have them find the other students in the class who have the other three pieces of the graph. When students have found their matches, they are to sit together in teams and create a story that reflects the situation of the graph. (See Unit 8.1 Entry Task Student Worksheet.) During: Circulate as students discuss the sections of the graph and only answer team questions. Possible questions for groups: How did you decide? Can you prove that? Can anyone justify your team’s statement in a different way? What information did you decide to graph on your x-axis? Your y-axis? Give timely reminders for completion of tasks. Closure/Extension: Have students tape graphs to poster paper and write the team’s agreed upon story. Possible Day 2: Have teams present their stories.

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What Stories Can A Graph Tell?

How will students do this?

Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively.

Structures for Student Learning: Academic Language Support:

Vocabulary: axis/es, in/dependent variable, distance, time, variable, horizontal, vertical Sentence frames: I must remember that the number line that goes from right to left is the _______-axis. And the number line that goes up and down is the ______-axis. When graphing ordered pairs (points) we start with the ____ value in the ordered pair. I noticed that the x-value in an ordered pair tells you how far to move _________ or ___________ from the origin. The _____________ in an ordered pair tells you how far to move __________ or ___________ from the x-value.

Differentiation Strategies: • Assign parts of graphs to students making sure that those with special needs are teamed into heterogeneous groups.

Participation Structures (group, partners, individual, other):

• Students work in teams of four.

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Lesson Series #1 Lesson Series Overview: Students actively participate as coordinates in various graphs, and analyze the resulting graphs. They practice graphing and create stories that could be represented by a graph. CCSS-M Standards Addressed: 8.F.5 Time: 2 Days

Lesson Overview – Day 1 Resources

Description of Lesson: You will need to set up time and a designated location for this activity (suggestions: gym, outside, classroom). Students will each be given two integer cards from –6 to 6 in one of four colors and a data sheet (Lesson 1.1.3 Resource page). Teachers should take this time to go over the rules on behavior and what is expected of them outside the classroom. Make sure each student also has a pencil for sketching the “human graphs.” Bring students outside. Once students are outside, situate students facing the x-axis looking toward the positive y-direction. Start by calling out anyone with the “Red integer cards” to find their spot on the x-axis. Students will line up facing in the positive direction with their backs to the rest of the class. Give the first rule. Give them a minute to calculate problem. Then tell them to move to the resulting number forward or backward. Then recorders will draw the resulting sketch of the graph. This will be repeated until all colors are done. Then direct the students back to the classroom. On one large poster graph paper, have a coordinate axes up on the board. Again call up each group color to grab a sticky dot and place it on the graph paper. Repeat until all colors are on the graph paper. Closure/Extension: Point out issues that caused confusion during the Algebra Walk and how they were resolved. Based on how the activity went and the creation of the class graphs, point out the main mathematical ideas: each axis is a number line; the point (0,0) is called the origin; a point is named by x first and then y, giving rise to the term “ordered pair”; some equations give straight lines (linear) while others result in curves (non-linear); some lines go “uphill” (increasing) while others go “downhill” (decreasing); the steepness of the hill

CPM CCC3 Lesson 1.1.3 Resource Page 1.1.3

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varies for different equations, etc. Potential Misconceptions:

● Students may try to move horizontally instead of vertically. ● Students may move in a manner that does not line up with the rest of the other

points. ● Students may line up on the x-axis incorrectly, especially when given a negative

integer to start. ● Students may not understand that the axes are horizontal and vertical number

lines.


Description of Lesson: In this lesson students will review plotting points and labeling axes. Students generate a set of random points all located within the first quadrant. Students will plot and connect the points and then create a short story that could describe the graph. Students must ensure that the graph is labeled correctly and that someone could recreate their graph from their story. See instructional plan tab on website for teacher notes.

Instructional Plan: http://illuminations.nctm.org/Lesson.aspx?id=2844 Describe the Graph activity worksheet Describe the Graph example overhead

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Apprentice Task Sweet Tooth: How Much is a Piece of Halloween Candy Worth?



MathObjectives: ● Students will plot data from a story. ● Students will discuss qualitative features of .graphs and make

determinations based on those features ● Students will Use information about one graph to explain the

behavior of another. CCSS-M Standards Addressed: 8.F.5 Potential Misconceptions: Students may not understand the difference between marginal and total enjoyment. Also, the graphs should be discrete, but students may represent the story with a continuous graph.

In this task, students will compare the graphs of marginal enjoyment and total enjoyment of two siblings as they eat their Halloween candy. They will discuss how each sibling’s enjoyment changes, and how this affects how much candy they should eat. Launch: See Mathalicious Sweet Tooth Lesson Guide During: See Mathalicious Sweet Tooth Lesson Guide Closure/Extension: See Mathalicious Sweet Tooth Lesson Guide

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Sweet Tooth: How Much is a Piece of Halloween Candy Worth?


Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.


Vocabulary: marginal enjoyment, total enjoyment, continuous, discrete, increasing, decreasing Sentence frames: With each additional piece of candy the sister/brother eats, I notice that the graph … When the boy’s marginal enjoyment graph is positive, his cumulative enjoyment graph is … When the girl’s marginal enjoyment graph is flat, her cumulative enjoyment graph is ... If I were a parent, I would give the brother ___ pieces of candy because … If I were a parent, I would give the sister ___ pieces of candy because …

Differentiation Strategies:

• Pair English learners with fluent English speakers to facilitate the written descriptions. • See Mathalicious Sweet Tooth Lesson Guide for Guiding Questions to assist students who are struggling with interpreting the graphs, and Deeper

Understanding questions to prompt students to think about the relationships more deeply. Participation Structures (group, partners, individual, other):

• Begin the class with a whole group discussion of the relative enjoyment of eating one piece of candy after another. (See Mathalicious Sweet Tooth Lesson Guide).

• After students complete their individual graphs, compare the shapes of the shapes of the graphs as a class. • Students should then work on Act 1 and Act 2 in pairs.

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Expert Task

On My Way to School



Math Objectives: ● Students will sketch a graph that exhibits the qualitative features of a

function that has been described verbally. ● Students will label and scale axes appropriately.


● Students may draw a map of the journey, rather than graphing the distance traveled.

Launch: Have students work in groups of four. Provide each group with the On My Way to School worksheet describing the situation. Discuss as a whole class or in groups what makes sense to label the axes. (Although there are many ways to label the axes, it might make the most sense to put time on the x-axis and distance traveled on the y-axis.) Students create a graph to model the events in their story. Each group must add an additional transition to the story and include that in their graph. Students will make a poster of their graph. One variation you could include in this lesson is to create different scenarios to pass out to different groups, resulting in different posters. During: Circulate to make sure students are completing their graphs. Remind students to call for a teacher checkpoint after the graph is completed. Closure/Extension: Have a gallery walk or presentation of posters.

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On My Way to School


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics.


Vocabulary: graph, increase, decrease, speed, distance, time, miles, hours, interpret, sketch, axes, positive, constant Sentence frames: Time generally is our _______________ variable. Speed is generally our ______________variable.

Differentiation Strategies:

• If students are struggling, provide labels for the axes. • Break the problem into parts with a teacher checkpoint after each part. • Put appropriate vocabulary words on a word wall or on the board. • For students who want an extra challenge, suggest that they create and graph their own story.


• This task will work best as a group task using the worksheet provided.

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Lesson Series #3

Lesson Series Overview: This lesson has students describing a scenario from a graph in the first problems and then going the opposite way by sketching a graph based on the scenario. CCSS-M Standards Addressed: 8.F.5 Time: 1 Day


Description of Lesson: This lesson has students write a story or scenario based on the graphs. Then in the next problems students are directed to draw a graph based on a story. Notes:

• Problem #4 can be used as an exit ticket. • This is a great time to have a class discussion on what the term constant means

and what that looks like graphically.

Graphs from Stories and Stories from Graphs

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Milestone Task Jason’s Trip to School and Match the Graphs and Stories Sheet



Math Objectives: ● Students will sketch a graph that exhibits the qualitative features of a

function that has been described verbally. ● Students will label and scale axes appropriately.


● Students may draw a map of Jason’s trip rather than graph distance over time.

Launch: Do Now: Imagine walking across the classroom and back. What would the graph of the distance from your starting point as a function of time look like? Sketch the graph and share with an elbow partner. Distribute Milestone Task and have students read through the scenario and instructions. During: Circulate to make sure students are working independently. Closure/Extension: Review student work and re-teach if necessary.

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Jason’s Trip to School and Match the Graphs and Stories Sheet


Focus Standards for Mathematical Practice: 4. Model with mathematics. 6. Attend to precision.


Vocabulary: twice, graph, axis, appropriately Sentence frames: For the Do Now, if students are having trouble starting their graphs, provide the following sentence frames: As I walk away from my starting point, the graph looks like … As I walk back to my starting point, the graph looks like …

Differentiation Strategies: • If students are struggling, help them label the axes appropriately. Remind them of the graphs the class created in the Do Now.


• This is an individual task.

SFUSD Mathematics Core Curriculum Development Project · Students might have a misconception...

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