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1 SFUSD Mathematics Core Curriculum, Grade 9, Unit A.5: Quadratic 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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### Transcript of SFUSD Mathematics Core Curriculum Development Project · 2020. 2. 17. · 3 SFUSD Mathematics Core...

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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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Algebra 1

Number of Days

Lesson Reproducibles Number of Copies

Materials

2 Entry Task Parabola Investigation 1 per pair Poster paper and markers

2 Lesson Series 1 Presenting About a Feature Listening Guide to Quadratics Presentations (2 pages)

1 per pair 1 per student

Computer and projector for video

2 Apprentice Task Splash Water Balloon Contest

1 per pair 1 per student

8 Lesson Series 2 Vertex Form Exploration (3 pages) Vertex Form Notes How many x-intercepts (2 pages) Vertex Form Secret Problems Task Card Vertex Form Secret Problems Recording Paper (2 pages) What is an x-intercept? Factored Form Exploration (2 pages) What Do You Really Need (2 pages)

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

Graphing technology (1 per pair) Vertex Form Secret Problems cards

2 Expert Task Will It Hit the Hoop Golden Gate Bridge

1 per student 1 per pair

Graphing technology and related halfshot file (1 per student)

4 Lesson Series 3 Standard Form Investigation Venn Diagram Ideas for Your Venn Diagram

1 per pair 1 per student 1 per pair

Posters and Post-It notes

1 Milestone Task Fencing Lessons (2 pages) 1 per student Graph paper

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Unit Overview

Big Idea

Students will begin to explore non-linear change through the lens of quadratic functions. Quadratic relationships can be represented in graphs, tables, and equations, and different forms of each representation give different information about the key features of the relationship. Essential question: What do different representations tell me about a quadratic relationship?

Unit Objectives

● Students will know that quadratic functions can be represented by equations in multiple forms. ● Students will know key features of quadratics, including intercepts, shape including symmetry, max and min, and end behavior. ● Students will understand that one quadratic relationship can be represented by multiple representations (graph, table, equations in different forms). ● Students will understand that different forms of the quadratic equation give different important information about the relationship. ● Students will be able to identify and explain the vertex of a parabola given vertex form of a quadratic equation. ● Students will be able to identify and explain the x-intercepts of a parabola given factored form of a quadratic equation. ● Students will be able to identify and explain the y-intercept and whether a parabola opens up or down given standard form of a quadratic equation. ● Students will be able to write an equation in vertex or factored form given a graph of a parabola by identifying relevant features (x-intercepts, y-

intercepts, vertex, concavity). ● Students will be able to relate vertical/horizontal shift and vertical/horizontal stretch on a graph to corresponding changes in the equation. ● Students will be able to interpret key features of quadratics (intercepts, increasing and decreasing intervals, positive and negative portions, maximums

and minimums, symmetry, and end behavior) to solve problems in contexts. ● Students will be able to identify the meaningful domain and range of a quadratic function in contexts. ● Students will be able to estimate the rate of change of a quadratic function from a graph, or calculate it from a table. ● Students will be able to compare two different quadratic functions given in different representations (a graph and an equation for example).

Unit Description

Students will start by investigating quadratics generally and starting to identify the key features of a quadratic graph. Then students will investigate all the possible representations of quadratics. Students will explore each form of a quadratic equation, and decide how to best use the information each form gives. Students will also explore how changing the numbers in each equation transforms the graph. Finally, students will use their knowledge of the different equations and their connections to the graph to write equations in different forms from graphs.

CCSS-M Content Standards

A-SSE – Seeing Structure in Expressions Interpret the structure of expressions A.SEE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

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F-IF – Interpreting Functions 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.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

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.

F-BF – 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.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Build new functions from existing functions F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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

Students know how to plot coordinate points and graph linear equations. Students know how to find x- and y-intercepts using a graph and evaluating for zeros. Students know domain represents inputs, x-values, and independent values. Students know range represents outputs, y-values, and dependent values. Students know how to evaluate given a value for a function.

Students will begin to explore non-linear change through the lens of quadratics functions. Quadratic relationships can be represented in graphs, tables, and equations, and different forms of each representation give different information about the key features of the relationship. Students will learn to work with tables and graphs, and to identify and exploit the key features of quadratic functions in each. Students will also work with different forms of quadratic equations, and will understand when each is most useful.

Students will follow this unit with a study of quadratic equations, including factoring, the quadratic formula, and completing the square. Students will learn how to do all the algebra needed to get from one equation form to another. Looking further afield, students will apply what they learned about function by studying quadratics to other function families in Advanced Algebra. They will also be able to use their understanding of quadratics in Calculus when looking for roots, max, min, end behavior, etc.

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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 formative 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?

2 days 2 Days 2 Days 8 Days 2 Days 4 Days 1 Day

Total: 21 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

Entr

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Expert Task Will It Hit the Hoop and

Golden Gate Bridge

CCSS-M Standards

F.IF.4, F.IF.7, F.IF.9 F.IF.4, F.IF.5, F.IF.6, F.IF.7a, F.IF.8a, F.IF.9

A.SSE.2 F.IF.4, F.IF.7, F.IF.8, F.IF.9 F.BF.1

F.IF.4, F.IF.7, F.IF.7a, F.IF.9 F.BF.1

Students will evaluate quadratic equations to make tables, then graphs. Students will make observations about key features of the graphs. Students will start to recognize the common characteristics of quadratic functions.

Students will be given information about four water balloon launches, each a different representation. They will need to analyze the information given and create a table and a graph on the resource page for every launch.

Students will use what they have learned about factored and vertex form to write equations to represent the arc of a basketball shot. They will use their equations to prove whether the shot goes in the basket. Students will graph the cables of the Golden Gate Bridge from factored form, draw in the towers using the vertex, and find out if they can string lights across from tower to tower using what they have learned about factored form and the vertex.

Students model a situation involving a quadratic relationship and determine how to maximize the area for a fixed perimeter. Given one representation of a quadratic function, students determine other representations of the function.

Source Adapted from CPM Algebra Connections Lesson 8.2.1

CPM CCA Lesson 8.2.1

http://blog.mrmeyer.com/?p=8483 and materials adapted and created by SFUSD teachers

CPM CCA Lesson 11.3.4

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Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

F.IF.4, F.IF.9 A.SSE.2 F.IF.4, F.IF.7, F.IF.8, F.IF.9, F.BF.1 F.BF.3

F.IF.4, F.IF.5, F.IF.7, F.IF.8, F.IF.9

Brief Description of Lessons

Students will prepare presentations about a feature of a quadratic function based on their entry task posters. Students will listen to the presentations and generate notes about all the key features of quadratic functions.

Students will explore vertex and factored form and discover how the equations and graphs are connected. Students will use both technology and pencil and paper to explore.

Students will explore standard form and make decisions about when each form is most useful.

Sources

SFUSD teacher created based on CPM Algebra Connections 8.2.1

IMP Year 2 Fireworks CPM Algebra Connections Chapter 8

SFUSD teacher created

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Parabola Investigation

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Students will create tables and graphs for several quadratics equations. They will examine key characteristics of the graphs, and start to talk about the key characteristics of quadratics generally. Math Objectives:

● Students will know key features of quadratics, including intercepts, shape including symmetry, max and min, and end behavior.

● Students will be able to identify the key features of quadratics including shape, intercepts, vertex, symmetry, max and min, and end behavior.

● Students will understand that one quadratic relationship can be represented by multiple representations (graph, table, and equations in different forms)

CCSS-M Standards Addressed: F.IF.4, F.IF.7, F.IF.9 Potential Misconceptions:

● Algebra students typically struggle to evaluate quadratic equations, which is okay, this is a good chance to practice. Potential pitfalls include dealing with negatives at the same time as squared terms, dealing with squaring, and dealing with parentheses.

● Students may not make good choices of x-values for their tables, encourage them to use at least some negative x-values or they may not see complete parabolas. All graphs will show a complete parabola with intercepts on a graph with x and y from –10 to 10.

● Students may assume that at some point the sides become straight, or may not understand there is life outside of the graph at all. It would be helpful to try a few large values of x with them.

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Parabola Investigation

How will students do this?

Focus Standards for Mathematical Practice: 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: graph, table, equation, symmetry, vertex, intercepts, shape, end behavior, speed, characteristics Sentence frames: See EL task card at the end of the task document for an adapted version of the poster directions.

Differentiation Strategies: Students are working in pairs. We suggest creating pairs in which struggling students can get support around evaluating quadratics. The alternate version of the task card is includes EL scaffolding.

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

This task is best done in partners so that everyone has a chance to do some of the calculations and to make observations. Each partner should complete one table and graph, and then they can work together to complete the poster.

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Lesson Series #1

Lesson Series Overview: Lesson Series 1 is a short follow-up to the parabola investigation. Students will present in pairs or groups of four about an assigned parabola graph feature. CCSS-M Standards Addressed: F.IF.4, F.IF.9 Time: 2 days

Lesson Overview – Day 1 Resources

Description of Lesson: Put finishing touches on posters from the Parabola Investigations task. Put students into groups of four and have them share their posters with the other pair. Assign each group of four one feature. Students should use the remaining time to prepare and rehearse their presentation about their feature. Notes: There is a presentation scaffold that you can use or modify to the level of support your students need. ELL students should be encouraged to use the sentence frames from their poster in their presentation as well.

Lesson Overview – Day 2 Resources

Description of Lesson: Students present on their feature, and other students take notes. There is a listening guide for students included in the lesson series materials. At the end of the presentations, if you have the ability, show the parabolas video from Radiolab. You can show the whole thing, but we recommend to starting at 1:34. Sound isn’t necessary if you don’t have access to speakers. Ask students to make observations about what they are seeing. Explain that they are now going to study functions that will allow them to represent all the curves in the video. The key take-away should be that there are many things in the world that are not straight lines, and we need to know how to make equations, graphs, and tables for them, too.

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Notes: For the sake of mathematical accuracy, we wanted to note here that many of the curves in the video are not parabolas. This is not important for students to know at this stage of their understanding. It is most important for students to understand that there are many things that are not straight lines.

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What will students do?

Mathematics Objectives and Standards Framing Student Experience

Students will explore a real-world quadratic context using multiple representations (graphs, tables, equations, and sentences). Math Objectives:

● Students will identify connections between different representations of quadratics: an equation, a table, a situation, and a graph.

● Students will also connect the intercepts and vertex of a parabola to a situation.

CCSS-M Standards Addressed: F.IF.4, F.IF.5, F.IF.6, F.IF.7a, F.IF.8a, F.IF.9 Potential Misconceptions:

● Most students can easily understand the highest toss, but may get confused of the longest toss. Discuss that the length is the distance between launch and landing and not the farthest landing position on the field.

Launch: Students have been exploring the features of a quadratic, and they know that there are many real situations in which quadratics appear (from watching the video). Explain that they will be exploring an example of quadratics that happens in the real world. Also they will explore the connections between the different representations for quadratic functions. During: Students will analyze a situation to create tables and graphs for four different water balloon launches. Students will compare representations to determine the longest distance launched and the highest distance thrown (maximum). Students will find the x-intercepts, lines of symmetry, and vertices of each launch. Closure/Extension: Students will individually write in their journals, explaining the role of each key feature and what information it tells you about the graph. What representations did you use during your water balloon launches? What do the x-intercepts, vertices, and lines of symmetry of each parabola tell you about the water balloon launch?

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Water Balloon Contest

How will students do this?

Focus Standards for Mathematical Practice: 6. Attend to precision. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: vertex, line of symmetry, maximum, x-intercept, parabola, quadratic equation, multiple representations Sentence frames: Group roles sentence frames attached.

Differentiation Strategies: Students work in groups of four, each person with a group role, resource page - given blank coordinate graph and four tables. Participation Structures (group, partners, individual, other):

• Groups of four for majority of task. • Individual for closure.

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Lesson Series #2

Lesson Series Overview: This lesson series is focused on exploring vertex and factored form of the quadratic equation, and relating these forms to the graph. CCSS-M Standards Addressed: A.SSE.2, F.IF.4, F.IF.7, F.IF.8, F.IF.9, F.BF.1, F.BF.3 Time: 8 days

Lesson Overview – Days 1-2 Resources

Description of Lesson: Students will use a graphing calculator, desmos.com, geogebra.com, or other graphing software to explore the uses of a, h, and k in the vertex form y = a(x – h)^2 + k. By the end of the lesson students should know that:

• A positive a value causes the parabola to open up and vice versa. • A large a value causes the graph to compress, and a small one to stretch. • (h, k) is the vertex of the parabola.

The formal vocabulary of compression and stretching and reflection is not necessary here; students’ own terms are just fine. Notes: This task was adapted from IMP Year 2, Fireworks. This task can be completed by students in groups as written, or you can split students into three different groups and have one group explore each parameter. Then you can jigsaw and have students share out what they learned.

Graphing technology: Graphing calculators, www.desmos.com, www.geogebra.com, Geometer’s Sketchpad, or other graphing software Vertex Form Exploration

Lesson Overview – Day 3 Resources

Description of Lesson: First, have students take some notes to solidify what they learned about the different numbers in the vertex form of the equation. You can do this however you want, but a notes template is included in the materials if you want a scaffold. Then students will work on a group task to figure out how many x-intercepts an equation has, and also how to find the “a” in the vertex form using a point.

Vertex Form Notes How Many X-Intercepts

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Notes: If you use the scaffold, these notes will be added to later on, as students develop ideas about factored form and standard form, so don’t use the bottom two boxes.

Lesson Overview – Day 4 Resources

Description of Lesson: Students will solidify their new understanding of vertex form by doing a “secret problems” activity. See the lesson for directions on the “secret problems” structure. By the end of the day students should feel comfortable writing an equation for a graph and graphing from an equation. Graphing technology should be available for this activity. Notes: If you are uncomfortable with the secret problems structure, you can rearrange the “cards” and create a worksheet. The recording sheet can be used as is in that case.

Graphing technology Vertex Form Secret Problems

Lesson Overview – Day 5 Resources

Description of Lesson: Students will make examples of real-world situations in which x-intercepts are important. They will start to think about the usefulness of finding x-intercepts easily. Notes: These could be large posters that you refer back to later, or they could be regular size paper. The directions are text heavy for ELs, and could be modified considerably. A model would also help.

What is an X-Intercept?

Lesson Overview – Day 6 Resources

Description of Lesson: Students will explore factored form, and develop an understanding of how to find x-intercepts from the factored form. Notes: There is intentionally NO factoring taught here; they will learn about factoring in the next unit! The goal is just to know that if the equation is (x – 3)(x + 1), the intercepts are +3 and –1. Students may say things like “opposite” or “change the sign,” and this is a good level of observation.

Graphing technology Factored Form Exploration

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Lesson Overview – Day 7 Resources

Description of Lesson: Students will solidify their learning about factored form first by taking notes on their running notes paper (see Day 3 of this Lesson Series) about factored form. Then they will use this new knowledge to consider how much information they need to find the equation of a parabola.

What Do You Really Need

Lesson Overview – Day 8 Resources

Description of Lesson: Use this day to review and solidify knowledge thus far, or to get caught up if things have not been going as planned. By the time this lesson series ends (today) students should feel really good about their understanding of vertex and factored form and how they connect to graphs.

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Will It Hit the Hoop and Golden Gate Bridge

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will explore different quadratic forms (vertex

and factored) and understand which function will be most useful in a given situation.

● Students will be able to alter key parts of a function to fit a given situation.

● Students will be able to find the vertex given an equation.

● Students will use factored form to graph and problem solve.

CCSS-M Standards Addressed: A.SSE.2, F.IF.4, F.IF.7, F.IF.8, F.IF.9, F.BF.1 Potential Misconceptions:

● Students may not use correct values for the Will it hit the hoop? activity. Or may need more attempts than six to find the correct function.

● Students may use factored form rather than vertex form to find the correct function, which may be much harder to find.

● Students may not use the vertex to draw in towers.

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Will It Hit the Hoop and Golden Gate Bridge

How will students do this?

Focus Standards for Mathematical Practice: Structures for Student Learning: Academic Language Support:

Vocabulary: intercepts, increasing, decreasing, wide, narrow, symmetry, function, quantities, situation, vertex, factored form

Differentiation Strategies: There are EL versions of this task.

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

Groups

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Lesson Series #3

Lesson Series Overview: Students will explore standard form for the first time. They will use what they know about vertex and factored form, and start to understand when each form is most useful. CCSS-M Standards Addressed: F.IF.4, F.IF.5, F.IF.7, F.IF.8, F.IF.9 Time: 3 days

Lesson Overview – Days 1-2 Resources

Description of Lesson: This lesson parallels the initial quadratics investigation in the Entry Task and Lesson Series 1, though this time they will examine standard form y = ax2 + bx + c. The idea of the lesson is essentially the same as the Entry Task and Lesson Series 1; students will get two different quadratics written in standard form and will need to make tables and graphs. Then they will examine a number of different features of the graph, as well as writing equations in vertex and factored form for each. Students will summarize what information they can get from the standard form equation. This time students will also create a third graph that tests their theory about what the standard form is telling them. Students might want to graph more if their test doesn’t give their expected result. Some key ideas for students to figure out:

• The sign of the x2 term determines whether the parabola opens up or down. • The constant term is the y-intercept. • A large “a” is more narrow, a small “a” gives a wide parabola. • “b” is enigmatic.

Notes: Take your time with this lesson and let students develop their own ideas. This is a chance for them to really use the skills they have developed in connecting graphs and equations. There is an EL version of the task card included. The part where students are testing a theory could be deleted if you are running short on time.

Investigating Standard Form

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Lesson Overview – Day 3 Resources

Description of Lesson: You can debrief the standard form investigation lesson in a variety of ways:

1. You can assign each group a different feature to share out in a whole class presentation, as in Lesson Series 1.

2. You can do a gallery walk and have students take note of new ideas they didn’t have on their own. Students should end with a clear understanding of what they can glean about the graph from equations in standard form. Notes: EL students will probably need assistance in preparing a presentation. The scaffold for Lesson Series 1 (both for presenters and audience) could be a model.

Lesson Overview – Day 4 Resources

Description of Lesson: Students will take a day to tie up all that they know about standard form, vertex form, and factored form. They will do this by creating a Venn diagram as a group. (If your students are already familiar with Venn diagrams, skip this intro.) Start by showing an example of a Venn diagram and asking students what they think it is. Students usually catch on to these pretty quickly, but make sure everyone understands what it means before starting the main activity. You don’t want this to get in the way! You can do this on regular size paper at tables and have students write in ideas, but it will probably be a lot easier to access the task if the ideas are on post-its and students are working on poster sized paper. Large group whiteboards would also work. Possibly do a gallery walk to let students see how other students were thinking. Notes: This can be done at a variety of levels, based on what you think your students are ready for. I suggest choosing one (not trying to differentiate in class), though you are welcome to:

• Completely student generated elements (for strong English speakers who have been doing well). • Suggested ideas for elements (for strong English speakers who have been struggling in the unit). • Elements given (for EL students).

Venn Diagram

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What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Given one representation of a quadratic function, students determine

other representations of the function. ● Students will understand how the factored form of the function can

identify a graph’s roots. ● Students model a situation involving a quadratic relationship and

determine how to maximize the area for a fixed perimeter. ● Students will understand how the completed square form of the

function can identify a graph’s maximum or minimum point. CCSS-M Standards Addressed: F.IF.4, F.IF.7, F.IF.7a, F.IF.9, F.BF.1 Potential Misconceptions:

● Students have difficulty getting started. ● Students make incorrect assumptions about what the different forms

of the equation reveal about the properties of its parabola. ● Students make errors manipulating the equation.

Launch: Provide a warm-up that focuses on writing an algebraic expression based on a variable chosen to represent part of a situation. For example, a 17-inch piece of string is cut into two pieces. If one piece is x inches long, how long is the other piece? (17 – x) During: Give students plenty of graph paper. Rather than having students make a full-sized poster for Fencing Lessons, have them present all their representations on a piece of graph paper. Closure/Extension: Have students create their own quadratic situation.

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Fencing Lessons

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively.

Structures for Student Learning: Academic Language Support:

Vocabulary: parabola, factored form, standard form, vertex form, graph, vertex, maximum, minimum, y-intercept, x-intercepts, roots, symmetry, zeros, domain and range, increasing, decreasing, positive and negative Sentence frames:

Differentiation Strategies: Encourage students to start by making a table for Fencing Lessons, and then a graph and equation. The maximum can be found from the table or graph alone without an equation.

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

Individual