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1 SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 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 M.1: Modeling (1st Semester), 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 M.1: Modeling (1st Semester), 2014–2015

Algebra

M.1 Modeling (1st Semester)

Number of Days

Lesson* Reproducibles Number of Copies

Materials

2 Task A: Datelines

Datelines Student Handout Modeling Task Write-Up

1 per student 1 per student

Computer and projector to show video from “Big Fish” on age differences

2 Task B: iPod dPreciation iPod dPreciation Student Handout iPod Press Release Modeling Task Write-Up

1 per student 1 per pair 1 per student

Computer and projector to show History of the iPod video

1 Task C: How Many Boomerangs?

How Many Boomerangs Task Card Peer Review Questionnaire - Student Modeling Task Write-Up (optional)

1 per pair 1 per student 1 per student

Computer and projector to video of how to throw boomerangs and how boomerangs are made

* Tasks may be done in any order.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Unit Overview

Big Idea

Linear and nonlinear functions and equations can be modeled in an applied world context.

Unit Objectives

● Students will be able to model mathematical concepts from prior units (such as linear and non-linear functions, equations and inequalities). ● Students will be able to critique, defend and verify their mathematical models.

Unit Description

Students apply mathematical content knowledge learned from the three prior units to tasks where they model applied world situations.

CCSS-M Content Standards

Numbers and Quantity 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.*

Algebra Creating Equations Create equations that describe numbers or relationships* A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Solve equations and inequalities in one variable A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve systems of equations A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* A.REI.12 Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Functions 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. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n – 1) for n ≥ 1. 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.* F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.* Analyze functions using different representations F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 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.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Building Functions

Build a function that models a relationship between two quantities F.BF.1 Write a function that describes a relationship between two quantities.* F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.* F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* Build new functions from existing functions F.BF.4 Find inverse functions. F.BF.4a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x = ̸1. 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.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent 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). F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 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.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In Units A.1, A.2 and A.3, students learn about linear and non-linear functions, equations and inequalities. Students learn how to differentiate between linear and non-linear functions and how to represent them in multiple ways.

In this unit, students will work on using linear and non-linear functions, equations and inequalities in an applied world setting.

In future modeling units in Algebra, Geometry and Algebra 2, students will deepen their mathematical modeling skills by applying them to more complicated modeling tasks and integrating the mathematics content that they are learning. Students will also deepen their understanding of modeling linear versus non-linear functions by specifically learning about quadratic functions in A.5, exponential functions in A.7 (Algebra 2).

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Task A Datelines

Task B iPod dPreciation

Task C How Many Boomerangs?

CCSS-M Standards

A.CED.1, A.CED.3, A.REI.3, A.REI.12, F.BF.4

A.CED.1, F.IF.8b, F.LE.1, F.LE.2, F.LE.5 A-CED.2, A-CED.3, A-REI.1, A-REI.6, A-REI.11

Brief Description of Task

Dating made easy. Mathematically choose your age-appropriate significant other. Involves systems of linear inequalities, feasible regions and inverses.

How does the value of an iPod change over time? Students will fit linear and exponential models, compare the two and then decide which model is more appropriate.

Business model problem for pricing a boomerang business involving systems of linear inequalities with discrete domain and range. This problem involves linear programming and feasible regions.

Suggested Days

2 2 1

Source Mathalicious: http://www.mathalicious.com/lessons/datelines (includes detailed lesson plan)

Mathalicious: http://www.mathalicious.com/lessons/ipod-dpreciation (includes detailed lesson plan)

Formative Assessment Lesson, Mathematics Assessment Projects, Shell Centre: http://map.mathshell.org/materials/download.php?fileid=1241 (includes detailed lesson plan)

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Task A

Datelines

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to write and manipulate linear equations and

inequalities in two variables. ● Students will be able to find the inverse of a linear function. ● Students will be able to graph a system of linear inequalities,

determine constraints with a problem’s context, and interpret the results.

● Students will be able to reason algebraically or numerically to determine wait times for couples outside the bounds of “Half Plus Seven.”

CCSS-M Standards Addressed: A.CED.1, A.CED.3, A.REI.3, A.REI.12, F.BF.4 Potential Misconceptions:

● The age 14 is the lower bound of the domain; ages below 14 can actually date people older than them.

● Students may have trouble figuring out the older age from the younger age. (Ask about order of operations or consider a specific example.)

● Students may have trouble with inequalities or lower bounds. (Ask them for examples that make the inequality true or are above the lower bound.)

● Students may be confused by what the axes represent (dater and datee. May be helpful to consider specific examples.)

● Technically, you can’t date more people as you get older; there is a wider range of people that you can date. Semantics.

Act I: Show Big Fish video about age differences: https://www.youtube.com/watch?v=20ZUUTZm3fI, starts at 1:35 with girl asking, “How old are you?” Ask students: “What do you notice? What do you wonder?” Act II: Distribute Student Handouts. First, students work in pairs to figure out the oldest and youngest ages a person can date using the “divide by 2, add 7” rule and its inverse. Next, students write and graph equations for the youngest and oldest person that someone can date. Then students shade in the acceptable dating region. Act III: Students figure out if certain celebrity couples fall within that region. If the couples don’t fall within that region, students figure out how long until they fall within that region. Students figure out that, once a couple is within that region, they don’t leave. When they are done, hand out the Modeling Task Write-Up handout. Students work individually to show what they know and what they learned from the task. While they can ask group members and you for help, the write-up helps show what students know as individuals. Explain that the Modeling Task Write-Up is a structure that will be used throughout this unit and in other math classes.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Datelines

How will students do this?

Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 4. Model with mathematics.

Structures for Student Learning: Academic Language Support:

Vocabulary: creepy (weird), total acceptable dating region or Romance Cone (the ages where someone is not too old and not too young to date someone else) Sentence frames: It is (weird/not weird) for a ____ year old and a ____ year old to date. (Could also use thumbs up/thumbs down.) If the older person is ___ years old, the younger person is ___ years old because ___. If the younger person is ___ years old, the older person is ___ years old because ___. The variable ___ is ___. The equation for the age of the (oldest/youngest) person this person can date is ____.

Differentiation Strategies: • If students are struggling, have them ask a partner or group member. • If the whole group is struggling, ask them to look at a single example (e.g., “If Joe is 17, what is the youngest person he can date?”). • Mathalicious has a detailed lesson plan, which can provide more information for differentiation.

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

• Students work in pairs and can ask other pairs at their tables if they need help. At the end of Acts II and III, do an explanation quiz and pick a student at random from each group to explain the group’s work so far.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Task B

iPod dPreciation

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to write, evaluate, and graph linear models for

iPod depreciation. ● Students will be able to write, evaluate, and graph exponential models

for iPod depreciation. ● Students will be able to compare and contrast linear and non-linear

depreciation models. ● Students will be able to compare and contrast non-linear depreciation

models for different products. CCSS-M Standards Addressed: A.CED.1, F.IF.8b, F.LE.1, F.LE.2, F.LE.5 Potential Misconceptions:

● Students may get confused when the linear model produces a negative value. This could be interpreted as the iPod being so junky that you need to pay someone to take it. It could also be interpreted as the shortcomings of a linear model.

● Students may interpret the x-axis of the graph as the year rather than the number of years that have elapsed since the product was released (i.e., x = 1 means 2002, the year after the iPod was released, rather than 2001).

Act I: First, show students the “History of iPod” video: http://youtu.be/kVX-Ko1NLAc Students should write 2 things they observe in the video. Next, give out handouts on Apple’s press release: http://www.apple.com/pr/library/2001/10/23Apple-Presents-iPod.html Finally, have students write 2-3 questions they want to know after watching the video and reading the press release. Act II: Distribute student handouts. First, students work in groups with some data to create a linear and non-linear depreciation model for the price of an iPod. Next, students work in groups to determine whether the linear or non-linear model is more appropriate. Finally, students fill out the Modeling Task Write-Up Card individually to show what they have learned. As a challenge, students can look up the current price of a used first generation iPod on eBay and then decide whether the linear or non-linear model is more appropriate for real life. (The price is actually around $99. This price is higher than either predicted value, probably because it’s a collector’s item. This anomaly might play nicely into the idea that modeling is imprecise and relies on many factor and assumptions.)

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

iPod dPreciation

How will students do this?

Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 4. Model with mathematics.

Structures for Student Learning: Academic Language Support:

Vocabulary: depreciation, non-linear, original iPod v. iPod Nano v. iPod Touch Sentence frames: We think the iPod is worth $___ because ___. We think the value of the iPod depreciates (goes down) by $___ each year because ___. We think the iPod was worth $___ in _(year)___ because ____. We think the (linear/non-linear) model is more reasonable because ___. We think the (original iPod/iPod Nano/iPod Touch) has maintained (kept) its value the best because ___.

Differentiation Strategies: • Mathalicious has a detailed lesson plan, which can provide more information for differentiation.

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

• Students work in partners or groups. • You should wait until the whole partnership/group is done before moving on to the next part of the task.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

Task C

How Many Boomerangs?

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to model a scenario with two variables. ● Students will be able to use a system of equations to optimize a

specific quantity. ● Students will be able to represent a system of equations in multiple

ways. CCSS-M Standards Addressed: A.CED.2, A.CED.3, A.REI.1, A.REI.6, A.REI.11 Potential Misconceptions

● Students may not know what boomerangs are. ● Students may not know where to start. (Have them re-read directions,

start with a table.) ● Students may consider part but not all of the problem, such as

considering just small boomerangs or one person’s (of Cath and Phil) work.

Act I: Show students a picture of a boomerang. Explain: ● When thrown, boomerangs travel in a roughly elliptical path and return

to the thrower. ● Cath and Phil are making boomerangs to raise money for charity.

Show students a YouTube video of a boomerang being thrown: https://www.youtube.com/watch?v=UNyK4oGkCDE Explain that Cath and Phil saw this video and said “Boomerangs are awesome!” Now they want to make boomerangs to make money for charity. Show students a video of boomerangs being made: https://www.youtube.com/watch?v=zl10s5xe2-4 Ask: What do you notice? What do you wonder? (Examples: What sizes are the boomerangs? How much do they cost?) *If you don’t have access to YouTube, you can print and show pictures. Act II: Hand out a task card for each group to read. Or use the How Many Boomerangs? Task Card Supplement. Students start by working individually until every student has a question or observation. Students work in groups to complete the problem using at least two representations (graph, table, pattern/situation, equations), then share through a gallery walk. Act III: Groups come back together to discuss what they learned from other groups before having a whole class discussion. Students could also fill out the Modeling Task Write-Up card prior to the discussion if this feels productive.

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit M.1: Modeling (1st Semester), 2014–2015

How Many Boomerangs?

How will students do this?

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

Structures for Student Learning: Academic Language Support:

Vocabulary: boomerangs. Be prepared to explain optimization, feasible regions, systems of equations.

Sentence frames: I think they should make _____ big boomerangs and ______ small boomerangs because _____. ________ big/small boomerangs is too many/too few because _____. ____ big boomerangs and ____ small boomerangs would make $_____ because ___.

Differentiation Strategies: • Ask students to consider one quantity first (e.g., how long it would take Phil to create small boomerangs), then slowly introduce the other quantities. Get

students started on a representation of their choice (e.g., graph, table). • If students need a challenge or you want to surface misconceptions, there is a detailed lesson plan on the MARS site that includes samples of student

work that can be used to surface misconceptions. Participation Structures (group, partners, individual, other):

• First, start students individually. Do a checkpoint with each group to make sure that each student has written an observation or a question (something that they can bring to their group).

• Once the group is checked off, start them working with their group to make a poster of their work. Continue to circulate and do checkpoints throughout. • When all groups are finished, everyone does a gallery walk with individual students organizing their thoughts and observations. Groups will reconvene.

Individuals will have time to organize their thoughts. When all group members are ready, the group will discuss what individuals saw and prepare to share out one thing from their group.

• Finally, groups will share out their observations with the whole class.