SFUSD Mathematics Core Curriculum Development Project · Bivariate data can be modeled and...

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.8: Bivariate Data, 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

Unit 8.8: Bivariate Data

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Hand Span and Height 1 per pair Rulers, graph paper 5 Lesson Series 1 CPM CCC3 Lesson 7.1.2 (3 pages)

CPM CCC3 Lesson 7.1.3 (5 pages) CPM CCC3 Lesson 7.3.1 (2 pages) CPM CCC3 Lesson 7.3.2 (2 pages)

1 per pair 1 per pair 1 per pair 1 per pair

Graph paper Poster paper, markers

1 Apprentice Task Used Cars 1 per pair Graph paper 3 Lesson Series 2 Math Links p. 1-3; Laptop Battery Charge

(4 pages) Math Links p. 4-6 (3 pages) Birds’ Eggs (2 pages) Math Links p. 7; What’s Your Favorite Subject? (2 pages)

1 per student 1 per student 1 per student 1 per student

Calculators Graph paper

1 Expert Task Music and Sports 1 per pair Calculators 3 Lesson Series 3 Bivariate Data Survey Project task card (2 pages) 1 per pair Graph paper, rulers, calculators 1 Milestone Task Bivariate Data (2 pages) 1 per student Graph paper, rulers, calculators

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Unit Overview

Big Idea

Bivariate data can be modeled and interpreted to find associations between variables, to draw conclusions and to make predictions.

Unit Objectives

● Students will be able to construct a scatterplot for bivariate measurement data, and understand what kinds of data can be modeled with a scatterplot. ● Students will be able to interpret the association between variables represented in a scatterplot: Is there an association? Is it strong or weak?

Is it linear or nonlinear? Is it positive or negative? ● Given a linear association, students will be able to informally fit a straight line to data represented in a scatterplot, and determine an equation in

y = mx + b form that describes this line. ● Students will be able to interpret the slope and y-intercept of the line of best fit, and use the equation to make predictions. ● Students will be able to use two-way tables to summarize bivariate categorical data by displaying frequency and relative frequency for the rows and

columns of the two-way table. Students understand what kinds of data can be modeled with a two-way table. ● Students will be able to identify patterns of association in a two-way table.

Unit Description

Students investigate patterns in quantitative and categorical bivariate data. Initially, they look for patterns in quantitative data by using scatterplots. They graph height vs. hand span, and decide whether there is an association between the variables, and, if so, whether it is strong or weak, positive or negative, linear or nonlinear and whether or not there are outliers. Students informally position a line of best fit and determine the equation of the line, using it to make predictions. Students then investigate relationships between sets of categorical data, organize the data into two-way tables, calculate relative frequencies for the rows and columns, and use these to describe patterns of association. As a culminating project, students create a survey question, investigate their question, design an appropriate representation for their data and interpret their data. Finally, given a set of data, students determine which representation is appropriate, create a display for their data and describe any patterns of association that they see.

CCSS-M Content Standards

Investigate patterns of association in bivariate data. 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

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8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In elementary grades, students learn ways to display and interpret data. In sixth grade, students investigate what makes a good statistical question. They use dot plots, histograms and box plots to display data. They look at measures of center such as mean and median, measures of spread such as interquartile range, and measures of variability such as mean absolute deviation, both by looking at the shape of a plot and by calculation. In seventh grade, students produce data using random sampling methods and investigate the theoretical and experimental probability that might predict relative frequency. They compare two populations by comparing the shape and measurements of data distributions. In eighth grade, prior units explore linear relationships.

Students model bivariate data and interpret their models to find associations between variables, draw conclusions and make predictions. They represent bivariate measurement data on the plane as a scatterplot and they observe relationships between the variables in terms of strong vs. weak, positive vs. negative and linear vs. nonlinear. Students model linear relationships with a line of best fit and understand that the slope represents the rate of change in the context of the relationship between the variables. Students use two-way tables to investigate possible relationships between bivariate categorical variables.

Students in high school learn how to interpret correlation coefficients that can be used to more precisely describe strong and weak association. Students learn how to model nonlinear associations with nonlinear functions. They use probability distributions to build an understanding of the normal distribution and how it can be used to measure sampling error. They learn that standard deviation measures variation from the mean.

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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 day 5 days 1 day 3 days 1 day 3 days 1 day

Total Days: 15

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Entry Task Hand Span and Height

Apprentice Task Used Cars

Expert Task Music and Sports

Milestone Task Bivariate Data

CCSS-M Standards

8.SP.1 8.SP.1, 8.SP.2, 8.SP.3 8.SP.4 8.SP.1, 8.SP.2, 8.SP.3, 8.SP.4

Brief Description of Task

Students use a scatterplot to compare height vs. hand span and describe patterns that they see in the data.

Students create a scatterplot for a set of data, then draw and interpret a line of best fit.

Students will use a two-way table to determine whether there is an association between playing music and participation in sports.

Students will create and interpret a scatterplot and a two-way table.

Source Illustrative Mathematics CPM CCC3 problem 9-50 Illustrative Mathematics CPM CCC3 Parent Guide Georgia Department of Education

Lesson Series 1 Lesson Series 2 Lesson Series 3

CCSS-M Standards

8.SP.1, 8.SP.2, 8.SP.3 8.SP.4 8 SP.1, 8.SP.2, 8.SP.3, 8.SP.4

Brief Description of Lessons

Students use scatterplots to decide whether there is a relationship between quantitative variables, determine the direction and strength of that relationship, model the relationship informally with a line of best fit and use the equation of the line to make predictions.

Students investigate categorical vs. numerical data and display categorical data in two-way tables. They calculate relative frequencies for the rows and columns, and use these to describe patterns of association.

Students design their own surveys, formulating the survey question, collecting and analyzing data, and representing it appropriately in a scatterplot or a two-way table.

Sources

CPM CCC3 Chapter 7 MathLinks Grade 8, Student Packet 10 Illustrative Mathematics MARS

SFUSD Teacher Created CPM CCC3 Parent Guide Illustrative Mathematics Georgia Department of Education

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Entry Task

Hand Span and Height

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will construct a scatterplot from collected numerical data. ● Students will use visual cues to describe the relationship between two

variables. CCSS-M Standards Addressed: 8.SP.1 Potential Misconceptions

● Students might think that hand span and height have to be measured in the same units, and try to convert the data.

● Students might think that all points must fall exactly on a straight line.

Launch (5 min): Do Now: Display two linear graphs, one positive and one negative, and two stories. Which story matches which graph? During: Provide cm rulers and graph paper; prepare height measuring stations with yardsticks or measuring tape. TIME LIMITS for each section: • 10 minutes for gathering data (one person from each group goes up to

report all 4 group members’ data). If you are using iPads, consider creating a Google Form to collect and display the data.

• 15 minutes for making the graph (1 graph per group). If they haven’t figured out how to scale the axes after 4 minutes, provide a hint card.

• 5 minutes for answering questions b and c. Closure/Extension: 5 minute debrief: Discuss questions b and c. If some groups put hand span on the x-axis and some put it on the y-axis, discuss how the graphs are similar and how they are different.

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Hand Span and Height

How will students do this?

Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. Structures for Student Learning: Academic Language Support:

Vocabulary: Linear, nonlinear, positive, negative, association, outlier Sentence frames: I think there is a ____________________ , _______________________relationship between hand span and height because _______________________.

Differentiation Strategies: As this is an Entry Task and you are assessing current understanding, no differentiation is needed. Participation Structures (group, partners, individual, other): This is a group task.

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Lesson Series #1

Lesson Series Overview: Students use scatterplots to decide whether there is a relationship between quantitative variables, determine the direction and strength of that relationship, model the relationship informally with a line of best fit and use the equation of the line to make predictions. CCSS-M Standards Addressed: 8.SP.1, 8.SP.2, 8.SP.3 Time: 5 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students create scatterplots and identify whether there is a relationship between two variables. They draw a line of best fit and use it to make predictions. See the CPM teacher guide for detailed teaching instructions. Notes: Skip problem 7-14. 7-16 is a great problem, but not essential if you are worried about time. Sometime during this lesson series, have a fast finisher create posters based on Math Notes 7.1.3 and 7.3.2 for display. These should be left up for reference for the rest of the unit. Math Notes 7.1.3 and 7.3.2 are provided in Day 5 of this Lesson Series.

CPM CCC3 Lesson 7.1.2 Core problems: 7-13 and 7-15 Graph paper Suggested HW: Choose from Review & Preview

Lesson Overview – Days 2–3 Resources

Description of Lesson: Students develop their understanding of association and look at the direction of association. They create scatterplots and use them to make predictions, where possible, and identify when it is not possible to make a prediction. See the CPM teacher guide for detailed teaching instructions. Notes: If you didn’t get to 7-16 on Day 1, use a modified version of it (2 scatterplots instead of 4) for a Do Now. 7-23: Do Now (10 minutes): Don’t expect everyone to answer all 5 questions; give optional sentence frames; talk to science teacher about work on writing hypotheses so you can make connections. (continued on next page)

CPM CCC3 Lesson 7.1.3 Graph paper Optional: What is the Relationship? Suggested HW: Choose from Review & Preview

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7-24: Whole class (10 minutes): Students discuss what kind of graph would be useful and why; how to scale the axes. 7-25: Group work (15 minutes): Assign one data set to each group, on poster; make sure all group members can explain their work. 7-26: Group work. Groups start by analyzing their own graph, then analyze other groups’ graphs as they finish. Optional: “What is the Relationship?”, a graphic organizer that teams can use to record their thinking, is included. Day 2 Launch: Based on the first group graph to be finished, discuss 7-26 parts a, b, c, d, and e. (Option: have different teams report on each part.) Groups finish 7-26 for as many graphs as their group has time for. 7-27: With 15 minutes left of class, move on to 7-27 if you haven’t already. 7-28: Optional extension. Debrief (5 min): Use 7-28 as discussion points.


Description of Lesson: Students determine the equation of their line of best fit using two points, and then they use the equation to make and justify predictions. They interpret the meaning of the y-intercept and slope of the line of best fit in context. See the CPM teacher guide for detailed teaching instructions. Notes: Core Problems: 7-87 and 7-88 In 7-87, even though time comes first in the table, time goes on the y-axis because she is trying to predict how long her ride will take. The length of time depends on the distance of the ride. Students might get confused about 7-88b. Optional: Check in with groups after 7-88a so you can address 7-88b. 7-88c: It’s okay if their slope is close to but not exactly 4.8, and the y-intercept could also vary depending on how they drew the line of best fit. 7-89: The problem uses positive numbers to represent negative changes, so be aware of this confusion if you use this problem.

CPM CCC3 Lesson 7.3.1 Graph Paper Suggested HW: Choose from Review & Preview

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Description of Lesson: Students apply their knowledge about lines of best fit to analyze a situation. They describe strength of an association between two variables on a scatterplot, and they describe an association using form, direction, strength and outliers. See the CPM teacher guide for detailed teaching instructions. Notes: Put a time limit on 7-97 and 7-98. Pull groups together after 20 minutes to finish together so that you can get to 7-100, the new learning in this lesson. Emphasize strong, moderate, and weak in the example graphs. Groups work on parts a and b. 7-100a: Note that “form” in this context means “linear or nonlinear.” Optional 7-101: Class poster that summarizes how to describe scatterplot associations (linear/nonlinear, positive/negative/no association, strong/moderate/weak, clusters & outliers) Put up the posters based on Math Notes 7.1.3 and 7.3.2. These should be left up for reference for the rest of the unit. Note: See Resources folder for additional materials on scatterplots, including the Barbie Bungee activity.

CPM CCC3 Lesson 7.3.2 Math Notes 7.1.3 & 7.3.2 Poster paper and markers Suggested HW: Choose from Review & Preview

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Apprentice Task

Used Cars (CPM CCC3 Problem 9-50)



Math Objectives: ● Students will be able to create a scatterplot and fully describe

associations. ● Students will be able to draw and interpret a line of best fit.

CCSS-M Standards Addressed: 8.SP.1, 8.SP.2, 8.SP.3 Potential Misconceptions Students may not understand what question B is asking. You can contrast a scatter plot with a box plot or a circle graph. Push students for their next level of understanding; it’s okay if they don’t understand this question fully. Possible answers could be:

● Because it has two variables. ● Because it has two variables that are not necessarily in a dependent-

independent relationship. (Association vs. causation) ● Because we’re trying to find out if there is an association between two

variables.

Launch: Do Now/Discussion: Why would someone display data in a scatterplot? When you look at a scatterplot, what are some ways that you can describe any associations you see? (You may want to point out the posters based on Math Notes 7.1.3 & 7.3.2.) During: Students work in pairs or teams of 4. Provide graph paper. As students work, make sure that each member of a group can explain at least part of their solution. Closure/Extension: Pick 2-3 teams to share their work on the Apprentice Task, making connections and comparisons between different answers.

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Used Cars (CPM CCC3 Problem 9-50)


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

Structures for Student Learning: Academic Language Support:

Vocabulary: Scatterplot, association, linear, nonlinear, positive, negative, strong, moderate, weak, line of best fit, slope, y-intercept Sentence frames: Age and cost have a _____ ______ association. As the age of the car __________, the cost of the car _____________ (increases, decreases).

Differentiation Strategies: Allow students to reference the posters based on Math Notes 7.1.3 & 7.3.2. Allow students to discuss their answers to a and b with other teams if necessary. Review y = mx + b through the Do Now or on a previous day. Participation Structures (group, partners, individual, other): Pairs or groups

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Lesson Series #2

Lesson Series Overview: Students investigate categorical vs. numerical data and display categorical data in two-way tables. They calculate relative frequencies for the rows and columns, and use these to describe patterns of association. Note: Using proportional reasoning in a two-way table to describe associations may be very challenging for students; this idea will be picked up again in high school. CCSS-M Standards Addressed: 8.SP.4 Time: 3 days


Description of Lesson: Students distinguish between categorical and numerical data, and construct questions to collect categorical data. Use the MathLinks teacher guide for ideas on how to teach the lesson. Notes:

MathLinks: Grade 8 (Student Packet 10) p. 1, 2, 3 Teacher Guide p. 1 & 2 Suggested HW: Laptop Battery Charge (provide graph paper)


Description of Lesson: Students use a Venn diagram to organize categorical data and summarize it in a frequency table. Use the MathLinks teacher guide for ideas on how to teach the lesson. Notes: p. 4: Students can work alone or in pairs on #1-9, but #10 and #11 should be discussed in groups and/or with the whole class. Students should understand that the total across and the total down should both equal the same number. Questions 10 & 11 are essential questions to discuss and/or write about. The teacher guide for #11 answers the question “HOW are the answers different?” A better answer for “WHY are the answers different” might be that #4 is out of 20 students and #8 is out of 10 students. Question 12 is optional if time. (continued on next page)

Calculators MathLinks: Grade 8 (Student Packet 10) p. 4, 5, 6 Teacher Guide p. 3 & 4 Suggested HW: Birds’ Eggs

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p. 5: #7 and #8 are tricky. p. 6 should be done in partners or groups. Note the discussion questions at the end of the Teacher Guide page 4.


Description of Lesson: Students use percents to create relative frequency tables from a two-way frequency table and recognize whether or not there is an association between the variables. Notes: p. 7: Do Now: When looking at ¼ = 25% in the table, ask: Where does the 1 come from? (1 student with no chores had a curfew.) Where does the 4 come from? (4 students had no chores.) What does the 25% mean? (Of the students with no chores, 25% had a curfew.) (This should be a whole class discussion. You may want to repeat this with other entries in the tables.) While students are working, ask them to draw comparisons between Table 1 and Table 2, and between these two tables and the table on p. 6. “Drawing Conclusions: Associations” is optional if time, and draws out another strategy for finding associations. Note: See the Resources folder for a worksheet on recognizing association in relative frequency tables.

Calculators MathLinks: Grade 8 (Student Packet 10) p. 7 Teacher Guide p. 5 Drawing Conclusions: Associations (optional) Suggested HW: What’s Your Favorite Subject?

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Expert Task

Music and Sports



Math Objectives: ● Students will use a two-way table and the resulting relative

frequencies to determine whether there is an association between playing music and participating in sports.

CCSS-M Standards Addressed: 8.SP.4 Potential Misconceptions

● Students may not know how to label the sides of the two-way table. ● Students may not know how to transfer the data from the table in

number 1 to a two-way table. ● Students may try to draw conclusions about association without using

proportional reasoning (for example, they would use the raw numbers instead of relative frequencies).

NOTE: If not many students at your school play a musical instrument, consider using a different categorical variable such as playing video games, owning pets, or going to bed before 10 pm. Launch: Introduce the problem and collect data for the table. If you are using iPads, consider creating a Google Form to collect and display the data table. During: Provide calculators. Students work in groups on #2 through #4. Make sure that each member of a group can explain at least part of their solution. Closure/Extension: If some groups found relative frequencies for #3 without making tables, consider having a group that did make tables display them. If none of the groups made relative frequency tables for the rows and columns, consider making them as a whole class. Facilitate a class discussion about #4. Ask students to write their answers clearly, using evidence from the table. (SMP 3 and 6)

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Music and Sports


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

Vocabulary: two-way table, relative frequency, association

Sentence frames: “Of the student who do/do not play ______, ______% do/do not play ______.” “I think there is/is not an association because ______.”

Differentiation Strategies: Allow students to use their notes from Lesson Series 2. Provide a two-way table with the labels completed. Participation Structures (group, partners, individual, other): This is a group task.

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Lesson Series #3

Lesson Series Overview: Students design their own surveys, formulating the survey question, collecting and analyzing data, and representing it appropriately in a scatterplot or a two-way table. CCSS-M Standards Addressed: 8.SP.1, 8.SP.2, 8.SP.3, 8.SP.4 Time: 3 days


Description of Lesson: Survey Project Parts 1, 2, & 3 Students work in pairs to decide on a numerical or categorical survey question and collect data. Each pair will then decide how to represent their data. Ideally, students will be finished with their data collection (Part 3) by the end of the first class period. See Teacher Notes for detailed teaching instructions. Notes:

Bivariate Data Survey Project task card Bivariate Data Survey Project rubric for display Graph paper, rulers, calculators For Do Nows, Exit Tickets, and/or Homework, choose from 8.8 LS3 Problems


Description of Lesson: Survey Project Parts 4 & 5 Students represent data using either a scatterplot or two-way table. For scatterplots, students describe associations by form, strength, and direction, and identify any outliers. For two-way tables, students describe associations by examining the relative frequencies. See Teacher Notes for detailed teaching instructions. Notes:

Bivariate Data Survey Project task card Bivariate Data Survey Project rubric for display Graph paper, rulers, calculators For Do Nows, Exit Tickets, and/or Homework, choose from 8.8 LS3 Problems

SFUSD Mathematics Core Curriculum Development Project · Bivariate data can be modeled and...

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