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1 SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 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, Algebra 1, Unit A.1: Modeling with Functions, 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, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Algebra 1

A.1 Modeling with Functions

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Comparing Age and Height of Boys and Girls Team Task Card Individual Worksheet

1 per pair 1 per student

3 Lesson Series 1 Multiple Representations Task Card Multiple Representations Sort Cards Bike Rentals Task Card

1 per pair 1 per pair 1 per pair

2 Apprentice Task Comparing Cell Phone Plans Task Card (2 pages) 1 per pair Construction paper and markers

4 Lesson Series 2 CPM CCA Lesson 1.2.4 (3 pages) HW: CPM CCA Lesson 1.1.1, 1-4, 1-5, 1-7 What is the Function? Task Card (2 pages) HW: Function Notation HW Function Jumble Task Card Function Jumble Resource Page HW: CPM CCA Lesson 1.1.2, 1-14 and 1-18 Is My Scatterplot a Function? Task Card Is My Scatterplot a Function? Cards (3 pages, one sided) Is My Scatterplot a Function? Student Sheet HW: CPM CCA Lesson 1.2.5, 1-78 and 1-80

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

2 Expert Task What's Your Prediction? Student Page (3 pages) Team Data Collection Page

1 per student 1 per pair

Graph paper

2 Lesson Series 3 Axes Sorting (2 pages, one sided) What’s Wrong with this Graph? Snakes Graph Snakes Worksheet (2 pages)

1 per pair 1 per student 1 per pair 1 per student

1 Milestone Task Journey to the Bus Stop 1 per student

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Unit Overview

Big Idea

Represent and interpret functions as an input-output relationship between quantities, with a focus on graphs and tables.

Unit Objectives

● Students will be able to move between tables, graphs, equations, and situations. ● Students will be able to identify if a relationship is a function given a graph, table, or situation. ● Students will be able to qualitatively describe different functions by analyzing tables, graphs, and equations and justifying their decisions. ● Students will be introduced to variable appropriate function notation, e.g., h(t) for height and time. ● Students will be able to choose and identify appropriate domains and ranges for functions that model data (without technical notation).

Unit Description

Students will start by comparing two graphs of functional relationships. Then, students will strengthen their understanding of tables, graphs, and equations that model situations. Using these skills, students will interpret and compare three different functional relationships to answer a question. Next, students will be introduced to the definition of a function, and be exposed to the ideas of input, output, and function notation. Students will be challenged to apply their new understand of functions by analyzing real-world data that they collected. Finally, students will consider how to appropriately choose the scale, domain, and range of graphs and tables to match the given situation. The milestone task requires students to synthesize their understanding of functional relationships in different representations.

CCSS-M Content Standards

Quantities ★ Reason quantitatively and use units to solve problems. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ N-Q.2 Define appropriate quantities for the purpose of descriptive modeling. ★ N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ★ Interpreting Functions Understand the concept of a function and use function notation. F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Interpret functions that arise in applications in terms of the context. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★ F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ Analyze functions using different representations. F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Linear, Quadratic, and Exponential Models ★ Construct and compare linear, quadratic, and exponential models and solve problems. F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. ★

F-LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. ★ F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). ★ Interpret expressions for functions in terms of the situation they model F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. ★ ★ A star indicates a modeling standard

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

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

In eighth grade, students know how to identify a function and construct models that represent linear relationships, including rate of change and initial values; students will be able to compare linear to other functions next year (reference 8.F4).

Students strengthen their understanding and interpretation of tables and graphs as ways of representing functions that model different situations. Students also gain a more technical understanding of functions, domain, and range.

In the next unit students will write and work more with equations to represent linear functions. Later this year, students will work with multiple representations of quadratic functions. In Algebra 2, students will work with multiple representations of exponential, square root, trigonometric, and other functions that model situations.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 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 3 days 2 days 4 days 2 days 2 days 1 day Total Days: 15

Lesson Series 1

Lesson Series 2

Lesson Series 3

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Entry Task Comparing Height and Age of Boys

and Girls

Apprentice Task Comparing Cell Phone Plans

Milestone Task Journey to the Bus Stop

CCSS-M Standards

N-Q.1, F-IF.4 N-Q.1, N-Q.2, F-IF.2, F-IF.4, F-IF 5, F-LE.5

N-Q.1, N-Q.2, N-Q.3, F-IF.1, F-IF.4, F-IF.5

N-Q.1, N-Q.2, N-Q.3, F-IF.4, F-IF.5, F-IF.9, F-LE.2, F-LE.5

Students are given graphs and tables to compare girls’ and boys’ average heights between ages 9 and 18. This will allow teachers to see the skills and strengths students are bringing into the unit, especially related to interpreting, describing, and comparing tables and graphs.

Students interpret given information about cell phone plans to determine which is better for different customers. They go on to write recommendations for different types of customers and use tables and graphs to support their explanation. Some students may also use equations.

Students choose two quantities to research and make predictions. Then they collect data on those quantities in the class and verify their predictions. Finally students use their new understanding of function to decide if the data they collected represents a function.

Students work individually to interpret, describe, and compare a journey to the bus stop represented as a table with a journey to the bus stop represented as a table. They then demonstrate their learning from this unit by writing a story that includes specific details about time, distance, speed, and when each person reached the bus stop.

Sources SFUSD teacher created Data from http://www.fpnotebook.com/endo/exam/hghtmsrmntinchldrn.htm) Graphs created with Desmos

Adapted from Comparing Cell Phone Plans www.matheducationpage.org

What’s Your Prediction? SFUSD teacher created

Journey to the Bus Stop Adapted from the Shell Center and OUSD

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

F-IF.4, F-IF.9, F-LE.2, F-LE.5

N-Q.1, F-IF.1, F-IF.2, F-IF.4, F-IF 5, F-LE.1b N-Q.1, N-Q.2, F-IF.1, F-IF. 4, F-IF.5

Brief Description of Lessons

Using tables and graphs, students will interpret what is happening in a situation.

Students will be introduced to and begin using the definition of a function to determine whether or not a relationship is a function, based on a given situation, table, or graph. Students will also begin to use function notation to evaluate, create tables, create graphs, and work with input-output pairs. Finally, students will use what they have learned to interpret and describe the relationship shown in scatterplots, in preparation for the Expert Task.

Students will consider the ideas of domain and range, and will work on choosing appropriate scales to best show off data.

Sources

http://www.mrmeyer.com/graphingstories1/graphingstories3.mov Multiple Representations Sort Cards Multiple Representations Task Card

Adapted from CPM Core Connections Algebra 1.1.1A Team Sort

Bike Rentals Adapted from UCSMP Algebra Chapter 5 Test, Form C

CPM CCA Lesson 1.2.4 Homework: CPM CCA 1.1.1, 1-4, 1-5, 1-7 What is the Function? Task Card

Adapted CPM CCA Lesson 1.1.1, 1.2.3 Homework: Function Notation

Adapted CPM CCA Lesson 1.1.2,1.2.3, 1.2.4 Function Jumble

Adapted  from  CPM  CCA  1.1.1  and  1.1.1C  Is My Scatterplot a Function?

SFUSD teacher created

Axes Sorting What’s Wrong with this Graph?

SFUSD teacher created Snakes Worksheet (1 student) Snakes Graph (1 per pair)

Adapted from MARS 2003 Course 1

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Comparing Age and Height of Boys and Girls

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to interpret, describe, and compare data

presented in tables and graphs to answer questions. CCSS-M Standards Addressed: N-Q.1, F-FIF.4 Potential Misconceptions

● Students may not realize that the graph does not start at the origin ● Students may compare different heights without attention to the age

(i.e., comparing the first variable without paying attention to the second variable)

Launch: Ask the class: Have you ever come back to school and noticed that a classmate grew a lot taller over the summer? Have students briefly discuss what they know about how quickly people grow at different ages. Do you grow the same amount each year? Does everyone grow at the same rate? Do boys and girls grow at the same rate? During: Comparing Age and Height of Boys and Girls Part 1: Team Task Pass out the team task card (one per pair) and give students 5 minutes to look at the graphs in teams of four and discuss what they see. Then, pass out the individual worksheets (one per student) and direct students to work in teams to come up with and record 10 observations. Monitor team discussions, and ask guiding questions as needed to help students make sense of the data provided:

• What do you notice about the graph? The scale? Each axis? • What do you notice about the table? The units? • What do you notice about the ages? What about the heights?

Part 2: Individual Task Direct students to work individually to answer questions about the data represented on the tables and graphs. Monitor student work, and ensure that students are justifying their answers with explanations. These explanations may be informal, but should communicate the student’s thinking. Identify and prepare a few students to share out their answers. For students who struggle with this task, these guiding questions may help: • Which part of the table can you use to help you answer this question? • Which part of the graph can you use to help you answer this question? • Which two points are you comparing right now?

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Closure/Extension: Debrief how graphs and tables represent information differently, and can be used to compare different sets of data. Choose a few volunteers to share out their explanations with the full class, to show different examples of how people interpreted and described the data they compared.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Comparing Age and Height of Boys and Girls

How will students do this?

Focus Standards for Mathematical Practice: 1. Reason quantitatively. 4. Model with mathematics.

Structures for Student Learning: Academic Language Support:

Vocabulary: increasing, decreasing, input, output, rate of change (slope), function, scale, x-axis, y-axis, scatterplot, value Please translate these vocabulary words into all known languages. Sentence frames: I notice that _________________________________ because ___________________________________________. At age ____ the (graph/table) shows that (boys/girls) __________________ because ___________________________________________. The (boys/girls) grew (quickly/slowly/not at all) at age _____, which the graph shows by _________________________.

Differentiation Strategies: Provide sentence frames, guiding questions, and word bank as needed. Participation Structures (group, partners, individual, other): Groups of 3-4 students (can use assigned roles)

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Lesson Series #1

Lesson Series Overview: Using tables and graphs, students will interpret what is happening in a situation. CCSS-M Standards Addressed: F-IF.4, F-IF.9, F-LE.2, F-LE.5 Time: 3 days

Lesson Overview – Day 1 Resources

Math Objectives: • Students will create a graph and table to represents a situation. (F-LE.2)

Description of Lesson: In this lesson series it is appropriate to do openings/warm ups that review graphing points and reading points from graphs. It is also appropriate to review evaluating an equation (linear or other simple equations) to make a table. Intro: Explain that this unit is all about how graphs and tables can help us to understand situations. Remind students about how the graphs and tables for the heights allowed us to see what was happening better than words could have. We don’t usually have a graph in the real world though, so we have to create them. Today we will create a graph and table that represents a situation. Do the Dan Meyer Elevation vs. Time task. For more details see lesson notes.

http://www.mrmeyer.com/graphingstories1/graphingstories3.mov   See Lesson Notes for Elevation vs. Time (no student handout)

Lesson Overview – Day 2 Resources

Math Objectives: • Students will match different representations (tables, graphs, equations, and

situations) of the same linear function together and justify their reasoning. (F-IF.9, F-LE.5)

Description of Lesson: Remind students that we are working on using graphs and tables to better understand situations. Today they will find tables, graphs, and equations that match situations. You

Multiple Representations Sort Cards (1 per group) Multiple Representations Task Card (1 per pair)

Adapted from CPM Core Connections Algebra 1.1.1A Team Sort

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

can start by saying that there are many situations in the world that make straight lines. These situations are called ‘linear’. You could have students write this in a toolkit or reference page. Today’s activity will be about situations that are all linear. Give the task card and one set of sorting cards to each group of 4 with a pair of scissors. Students should cut up their set of cards and then follow the task card. As the teacher, you will want to check in with groups when they get to #3. A good technique would be to randomly choose one person in the group to explain the answer to one of the questions. You do NOT need each group to answer all of these questions, and you can judge for yourself how much to push them to explain. If they are not ready, tell the group you will return, but they need to help that student prepare to explain. Walk away and give them some time to get ready. The color-coding can serve as an extension as groups are finished, so that you can complete all of the checkpoints.

Lesson Overview – Day 3 Resources

Math Objectives: • Students will interpret a function represented in a graph. (F-IF.4, F-IF.9)

Description of Lesson: You can launch by saying that the students are getting really good at making connections between graphs, tables, and equations, and they are ready to see three different situations all on the same graph. This makes it possible to compare the different situations, which is a really good thing when you want to save money. You can decide whether you want students to work in groups or pairs for this task. Some good possible discussion questions for closure: What does it mean that Mike’s Bikes is a straight horizontal line? What does it mean when the lines cross? Is there one best choice? What does the y-intercept mean for these graphs (it is not shown on the graph)? (More of a challenge)

Bike Rentals  Task  Card  Adapted from UCSMP Algebra Chapter 5 Test, Form C

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Comparing Cell Phone Plans: Choosing a Plan

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: • Students will be able to interpret given rate information to solve a

problem. • They will use tables, graphs, and/or equations to support their

explanation. CCSS-M Standards Addressed: N-Q.1, N-Q.2, F-IF.2, F-IF.4, F-IF 5, F-LE.5 Potential Misconceptions

• Students may need help getting started because this task is open ended. Allow students to struggle and encourage them to use what they know (trying out numbers, making tables, etc.)

• Students may not be able to say why they know which plan is best. Ask students to explain why and encourage them to use their data, tables, graphs in their explanation.

• See Hints in the Lesson Plan

Launch: In the web-based unit resources there are slides to go with the student worksheet. If you show the introductory slides, stop at the question "What information do you need?" Solicit ideas from the students about what information is relevant. Show the final slide, and/or hand out the student worksheet to launch the exploration. Make sure students understand that while the initial question is about specific cases, they are to find a general answer to the question of what range of minutes makes each plan the least expensive option. During: We recommend that students work on part 1 in a group of 4, and that you use a participation quiz to support good groupwork. This is explained in depth in the SFUSD Math Teaching Toolkit. Students have had a few class periods where you (the teacher) have described norms and values of this math class and have been able to point out things you are looking for. Today, explain to the students that part of their grade for the day will be based on how well they work together. The flier on day 2 is more appropriate as pair work or individual work, but you (the teacher) can choose what will be best for your class. See the lesson plan for further support during the task. Closure/Extension: As described in the lesson plan, you can show a copy of the graph with all three plans, and have a discussion. The key question is to have students think about why it is useful to look at a graph, instead of just hearing about the numbers. There are other interesting extension discussion questions that get at the motive of the company vs. the customer.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Comparing Cell Phone Plans: Choosing a Plan

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Construct viable arguments. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. Structures for Student Learning: Academic Language Support:

Vocabulary: rate, per, recommendation, type, Sentence frames: _______ should choose the ________ plan because ___________. Between _____ and _____ minutes you should choose _________ because ___________. A ______ (type of person) should choose ____________ because ___________. As you can see on the graph (table), __________ is cheaper for ______ to ____ minutes.

Differentiation Strategies:

• Some hints are provided in the lesson notes that you can give to teams that are struggling. • You can give lots of help and suggestions for how to set up the graph and tables, since this is early on in the unit. Try to ask questions instead of just

telling folks what to do. • The task itself allows for students to use at least two ways to explain, but they can use more. This choice in itself is differentiation.

Participation Structures (group, partners, individual, other): Part 1: Groups of 4, participation quiz Part 2: Pairwork or individual for the flier

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Lesson Series #2

Lesson Series Overview: Students will be introduced to and begin using the definition of a function to determine whether or not a relationship is a function, based on a given situation, table, or graph. Students will also begin to use function notation to evaluate, create tables, create graphs, and work with input-output pairs. Finally, students will use what they have learned to interpret and describe the relationship shown in scatterplots, in preparation for the Expert Task. CCSS-M Standards Addressed: N-Q.1 F-IF.1, F-IF.2, F-IF.4, F-LE.1b Time: 4 days

Lesson Overview – Day 1 Resources

Math Objectives: ● Students will be able to distinguish between functions and non-functions from

situations, tables, and graphs. (F-IF. 1) ● Students will be able to give an informal definition of the term function. (F-IF. 1)

Description of Lesson: Students will learn about the definition of a function, using the example of a soda vending machine, and then applying this understanding to determine if tables and graphs are functions. You can launch by saying: We used graphs and tables to compare cell phone plans, which is very helpful as long as the cell phone plans are predictable. Today, we are going to look at some other relationships and whether we can Some misconceptions students might have: Students might look at repeated y-values instead of repeated x-values to determine if the relation is a function. Students might think all lines are functions. To close use a Think-Ink-Pair-Share. Have students individually write an informal definition of function, with examples and non-examples. Then, have students share these with a partner, and finally ask for volunteers to share out with the class. Together, come up with a class definition of a function to write down (can put “function” into a vocabulary toolkit/glossary). Notes:

• The Cola Machine is also in unit 8.7. Depending on if students have seen this or

Lesson: The Cola Machine (CPM CC Algebra 1.2.4) • Use the full lesson, but not the Review and Preview problems Homework: CPM CC Algebra 1.1.1 problems 1-4, 1-5, 1-7

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

not, it may be review, but a good realistic situation to remind students after a summer off.

• You may want to carefully work through the soda machine problem yourself before teaching this lesson.

• You do not need to introduce the vertical line test yet, so that students do not rely only on graphs to determine if relations are functions.

Lesson Overview – Day 2 Resources

Math Objectives: ● Students will be able to use function notation to evaluate, create a table, create

a graph, and determine input-output pairs. (F-IF. 2) Description of Lesson: Students will learn about and begin to use function notation through the idea of a “function machine.” They will then evaluate, create a table, create a graph, and determine input-output pairs, using function notation. Students should work in teams of four with assigned roles to complete the What is the Function? Task. Each student individually records the team’s ideas and work on their own paper. You may choose to use a participation quiz or explanation quiz/checkpoint to support student engagement and increase individual accountability in this task. Students can complete any problems they do not finish during the next lesson. Some misconceptions students might have: Students mistake f(x) notation for multiplying (f)(x). For example, saying f(3) = 3f. Students confuse inputs (x) with outputs (f(x)). Students not realizing that f(x) replaces y on familiar input-output tables and graph axes. To close you can ask for volunteers to share out a summary of what function notation is and how it works. As a class, record these notes about this vocabulary (“function notation”) and notation (“f(x)”) into a vocabulary toolkit/glossary.

“What is the Function?” Task card (1 copy per pair) Adapted from these Sources: CPM CC Algebra 1.1.1 problem 1-2 CPM CC Algebra 1.2.3 problems 1-53, 1-55, and 1-56 Teacher-created question (called problem 1-54)

Teacher version’s lesson notes for CPM CCA 1.2.3. Homework: Function Notation

Taken from CPM CCA Lessons 1.1.2, 1.2.3, and 1.2.4 Review and Preview 1-22, 1-57, and 1-70)

Lesson Overview – Day 3 Resources

Math Objectives: ● Students will use their understanding of functions defining one output for each

input to put functions in an order that will produce the desired output value. (F-IF.1, F-IF.2)

Function Jumble Resource Page (cut into cards) Adapted from CPM CCA 1.1.1C

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Description of Lesson: Students will work together to use what they know about input-output pairs and function notation put four different functions in an order that will achieve a desired function. This is an informal introduction to the composition of functions, and gives students another way to think about functions. You can also direct students to complete any problems that they did not finish from the “What is the Function?” task To launch you can say: Today we will use what we know about functions to figure out how to get a certain output. We will also go back to complete our work from yesterday. During the task, Resource Manager should call the teacher for a “checkpoint quiz” when everyone in the team can explain. Randomly call on a different student to explain each part of the team’s work. You can ask about how they chose the order of the functions, how they worked together, their answer for 1-3a, and their answer for 1-3b. If a student cannot yet explain the team’s work, direct them to discuss and prepare more, and call you again when everyone in the team is prepared. After students pass the checkpoint quiz, you can direct them to complete the work they did not finish from the “What is the function?” team task from the previous lesson. Some misconceptions students might have: Students may misunderstand that the output from one function becomes the input for the next function (in this informal introduction to composition of functions).

Function Jumble Teacher Answers Homework: CPM CC Algebra 1.1.2 problems 1-14, 1-18

Lesson Overview – Day 4 Resources

Math Objectives: ● Students will identify if a scatterplot represents a function and why. (F-IF.1) ● Students will describe the relationship between two changing quantities in a

scatterplot. (N-Q.1, F-IF.4, F-LE.1b) Description of Lesson: Students will work together to use what they know about input-output pairs and the definition of a function to decide whether ten scatterplots are functions or not. They will then describe the relationship each function shows, and decide on a possible real-world relationship these might represent. Launch: Give an example scatterplot related to your context (e.g., the number of students in our classroom and the number of empty seats in our classroom). As a class, discuss what the scatterplot shows and how they know that. Encourage students to use

Is My Scatterplot a Function? Task Card (1 per pair) Is My Scatterplot a Function? Cards (1 per pair, cut up) Is My Scatterplot a Function? Student Sheet (1 per student) Homework: CPM CC Algebra 1.2.5 problems 1-78 and 1-80

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

precise vocabulary terms (such as increasing/decreasing, function, and input/output). Pass out scatterplots (one set of cards per group of four) with different units that represent functions and relations. Students group functions and relations based on definitions and experience with input/output from The Cola Machine. Students will describe the relationship in words (e.g., As the number of students at our high school increases, the number of teachers increase at a constant rate; there is no relationship between age and hair color of students). Students should work in groups of four, with assigned roles, and the Resource Manager should call the teacher over for each checkpoint to check their work before moving on. At each checkpoint, randomly call on students to explain the group’s thinking about a particular graph. If a student is unable to answer, give the group more time to prepare, and have them call you over again once they are ready. Use the provided sentence frames, on the student sheet, to organize student work and scaffold the necessary language. Some misconceptions students might have: Students look at repeated y-values instead of repeated x-values to determine if the scatterplot is a function. Students confuse increasing and decreasing (read a graph from right to left instead of left to right). Closure/Extension: Debrief how a scatterplot is different than a function that models a relationship that exists within a scatterplot. Models are ideal representations that do not always map onto real-world data perfectly.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to identify two quantities to research. ● Students will be able to formulate predictions about the relationship

between the quantities. ● Students will be able to collect data. ● Students will be able to create a graph and table with reasonable

scale that shows their data. ● Students will be able to verify their predictions. ● Students will be able to explain which quantity is input and which is

output and why. ● Students will be able to decide if the data they collected represents a

function. CCSS-M Standards Addressed: N-Q.1, N-Q.2, N-Q.3, F-IF.1, F-IF.4, F-IF.5 Potential Misconceptions

● Students may confuse cause and effect (though this doesn’t matter for relationships with no correlation).

● Students may not be consistent with units. ● Students may have trouble scaling their data, and may make scales

that don’t show off the data well. ● Students may connect the points to make a line, even if there is no

correlation. ● The last part is the most difficult, and students will probably struggle

with the difference between the function created by just their own data, and the relationship they chose.

Launch: Remind students of the Entry Task that looked at height over time, and ask students to think of other quantities that might be related. Choose one student suggestion, and as a class write this as an example, and formulate a research question together as a class. Discuss the difference between quantitative and qualitative (we can ask about number of siblings but not the color of your eyes). Turn it over to groups to come up with their own question. During: See lesson notes for more details. For presentations, look for the sample Presentation Rubric. Teacher can use this for scoring. It is also good practice to give a copy per group so that groups can use this as guidelines when creating their presentations. Closure/Extension: See lesson notes for more details. The presentations serve as closure of the data collection and analysis piece. The final part about functions allows students to show what they learned about functions and how they can apply it to a new situation. This is NOT an optional extension; all students should do this after the presentations.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

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: positive correlation, negative correlation, no correlation, scatterplot, increasing, decreasing, quantity, prediction, true, false, input, output, function, relationship Sentence frames:

• Included on worksheets

Differentiation Strategies: • Sentence frames are provided within the lesson plan, but are optional. • Do not provide axes for students because this is something they should be struggling with at this point, but you can ask supporting questions to help

them choose a scale. • If you have EL students, you might want to provide sentence frames to use for the presentations, or help students choose appropriate sentences from

their work. Participation Structures (group, partners, individual, other):

• Groups of four • Checkpoints are included to make sure groups stay together • You can also use a participation quiz with a focus on justification and sticking together as a team (emphasize the mathematical practices).

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Lesson Series #3 Lesson Series Overview: Students will consider the ideas of domain and range, and will work on choosing appropriate scales to best show off data. CCSS-M Standards Addressed: N-Q.1, N-Q.2, F-IF.1, F-IF.4, F-IF.5 Time: 2 days

Lesson Overview – Day 1 Resources

Math Objectives: • Students will match a situation to an appropriate table and scaled axes. (N-Q.1) • Students will explain how to correct the scale of a graph, connecting tables to

graphs. (N-Q.1, N-Q.2) Description of Lesson: Remind students that it was difficult (or if it was not difficult, ignore this) to scale their graphs for their data they collected. Explain that they will learn more about how to choose a good scale. Both Part 1 and Part 2 are short and can be completed in one day. Part 1 – Axes Sorting (1 per group)– Have cards cut up so that each graph, table, and situation are separate. Students work in groups to sort the graphs, tables, and situations that go together. When a group is finished, quiz a random individual on how they know the graphs should go together. If their understanding seems weak, tell the group they need more time to prepare, walk away and come back when they are ready. The situations are designed to be slightly confusing – some of the data points appear in more than one situation, so students should make arguments particular to the situations. Part  2  –  What’s Wrong with this Graph? (1 per student) –  When  a  group  has  successfully  explained  their  thinking,  move  them  on  to  the  graph  mistakes  activity.  They  can  talk  in  their  group,  but  should  write  their  own  answers.  The  four  graphing  mistakes  from  left  to  right  and  top  to  bottom  are  a  range  that  is  too  large,  a  domain  that  is  too  small,  an  x-­‐axis  that  doesn’t  start  with  0,  and  a  range  that  starts  counting  by  5  and  switches  to  counting  by  10. In the closure today you can introduce the words domain and range informally as the

Axes Sorting (cut up both sets of cards, they are already mixed up) (1 per pair) What’s Wrong with this Graph? (1 per student) Both of these are SFUSD teacher created

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

numbers that are good for x, and the numbers that are good for y. It would be good to have a short discussion about how to choose a good scale for a graph. A good question to ask: What are some characteristics of a really good graph? (all the numbers fit, there isn’t a lot of empty space, the numbers go by a consistent number, the axis starts at 0, students may say other things)

Lesson Overview – Day 2 Resources

Math Objectives: • Students will interpret situations represented as scatterplots. (F-IF.4) • Students will continue to make sense of the concept of a function. (F-IF.1) • Students will relate domain and range of the function from a graph (F-IF.5)

Description of Lesson: This lesson gives students another opportunity to interpret scatterplots. In particular they get an opportunity to think about the definition of a function, and to discover a useful application of domain and range. Because the unit is wrapping up, there are many opportunities in this task for students to write their thinking. You can launch by asking students for reasons that we might care about domain and range. You can explain that today they will see a way that they can use domain and range to solve a problem. Students should talk in groups but write their own responses on their paper. Groups should only get two copies of the graph to encourage them to work together. A good closure activity would be to discuss question 5 to help solidify understanding of domain and range. Notes: The challenge is meant as an extension for any student, but not necessary for students to complete. Students may say things like Species B’s head is larger relative to its body. They may draw sketches of two snakes of the same length, one with a larger head.

Snakes Worksheet (1 per student) Snakes Graph (1 per pair)

Adapted from MARS 2003 Course 1

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Individual Milestone Task: Journey to the Bus Stop

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: • Students will be able to interpret, describe, and compare a situation

represented as a table with a situation represented as a graph. CCSS-M Standards Addressed: N-Q.1, N-Q.2, N-Q.3 F-IF.4, F-IF.5, F-IF.9 F-LE.2, F-LE.5 Potential Misconceptions: • Students may misinterpret who starts the journey first and who

reaches the bus stop first. • Students may not recognize the interval where Gabriela walks back

towards school.

Launch: Today, you will show what you’ve learned about tables and graphs that model different situations. During: Provide each student with the milestone task (one copy per student). Answer student questions and monitor progress on the task. Provide individual support (e.g., word bank, sentence frames, etc.) for English Learners and students with IEPs, as needed. Closure/Extension: (Optional) Choose a few volunteers to share their stories aloud.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.1: Modeling with Functions, 2014–2015

Individual Milestone Task: Journey to the Bus Stop

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments.

Structures for Student Learning: Academic Language Support:

Vocabulary: function, increasing, decreasing, faster, slower, constant rate, positive, curve, negative, line, interval, intercept, slope, scale, units, variables Sentence frames: I see from the (graph/table) that ________________________ because _________________________ After ________ seconds, _________________________________________________________ I can tell (Gabriela/Qiwen) was walking towards ___________________ because ______________________ (Gabriela/Qiwen) walked fastest from (time) to (time), which is shown on the (table/graph) by _________________________________ (Gabriela/Qiwen) stopped from (time) to (time), which is shown on the (table/graph) by _________________________________

Differentiation Strategies: Provide word bank and/or sentence frames as needed. Participation Structures (group, partners, individual, other): Individual work for individual milestone assessment.