Prepare instruction - Piedra Vista High Schoolpvhs.fms.k12.nm.us/teachers/malonzo/08EF901C-00757F35.12/AlgebraI... · Prepare instruction ... KDW iV D PDWKePDWiFDl IXnFWion" $IWer

Advice for Instruction | 3. Functions | Prepare instruction

Prepare instruction

Goals and objectives Topic at a glance Prerequisite skills Resources Language support

Goals and objectives

After completing the topic Functions, students will be ableto

demonstrate understanding of the concept of afunction as a systematic relationship between twovariables;identify independent and dependent variables infunctions, and describe the domain and range of afunction;recognize that sequences are functions whosedomains are subsets of the integers;represent functions using words, tables, symbols, andgraphs;use function notation to represent functions and to determine specific values of either the domain orrange.

Topic at a glance

In previous mathematics work, students have informally investigated the concept of a function as a systematicrelationship between two variables, and this concept will be revisited throughout Algebra I. The topic Functions

builds on this previous work to formalize this concept for students and add to the tools they will use throughoutthe course to represent functional relationships.

The topic also includes explicit tasks and strategies that build students’ academic literacy through mathematicalcontent. The literacy tasks included in this topic build on the work of the Literacy Design Collaborative, aninitiative funded by the Bill and Melinda Gates Foundation to support the teaching of literacy in the content areas.More information about the work of the Literacy Design Collaborative can be found athttp://www.literacydesigncollaborative.org.

Plan nine 45‐minute blocks for this topic.

Description Resources Suggested assignment

Block 1 formalizes the concept of a function andintroduces students to function notation. This block alsointroduces students to the writing prompt that will bethe focus of Block 6.

Overview

Exploring "Functionnotation"p1‐3

SAS1

SAS2

SAS2Q2, 3‐6, 7‐9

In Block 2 students continue to refine theirunderstanding of functions as they investigate theconnection between sequences and functions.

Exploring "Functionnotation"p5‐13

SAS2Q2, 18a‐b, 19a‐c, 20a‐d,and 21a‐d

SAS2

Block 3 deepens student understanding of functionconcepts through a real‐world context, reviews the useof tables to represent functional relationships, andconnects the process captured in generating a table tothe development of expressions that lead to generalfunction rules. Even though students have had priorexperiences with this idea, the work in this blockprovides opportunities for students to continue to refinetheir skills in generalizing functional relationships usingsymbolic representations.

Exploring "Modeling withfunctions"

SAS3

SAS3Q11a‐c, 12, and 13a‐d

Block 4 deepens students' understanding of the use ofgraphing as a tool for representing and understandingfunctional relationships. The suggested assignment forthis block gives students the opportunity to experiencea short writing task comparing two functionalrelationships.

Exploring "Graphs"

SAS4

SAS4Q18‐22

Block 5 provides an opportunity for students to applywhat they have learned about functions in a newsituation.

Guided practiceMore Practice

Summary

Block 6 provides time for students to analyze twopostage stamp vending machines to decide if theyrepresent functional relationships. Students also beginwork on a written response paper to demonstrate theirunderstanding of the concept of a function, drawing ontheir work with the two stamp vending machines and onother evidence presented in this topic.

SAS5

SAS6

SAS7

Quick write and outline ofresponse paper

Block 7 provides time for students to continue to refinetheir written response paper. In Block 7, studentsdevelop their initial drafts.

SAS5

SAS6

SAS7

Continue to work on paper

Block 8 provides time for students to continue to refinetheir written response papers. In Block 8, studentsconduct peer reviews and finalize their claim outside ofclass.

SAS5

SAS6

SAS7

Finalize written responsepaper

Block 9 provides time for a topic‐level assessment.Automatically scoredConstructed response

None

While the topic does not require it, it does provide a good opportunity for your students to continue to developfacility with graphing technology. If you decide to incorporate graphing calculators into the lessons, rememberthat this may be one of only a handful of experiences for many of your students with a graphing calculator. Forthat reason, you may want to allow extra time for exploration and questions.

Prerequisite skills

To be successful with the material in this topic, students should understand the following concepts:

Order of operationsOperations with rational numbers

Identifying independent and dependent variables in a functional relationshipDomain and range of a functionSolving two‐step equations by inspectionPlotting points in the coordinate plane

Resources

LESSON RESOURCES

Computer with projection deviceGraphing calculatorsPostable chart or grid paper (see Block 4)Computer lab (Block 5)

RELATED RESOURCES

Agile Mind Glossary

Language support

The activities in this topic provide opportunities for students to talk with each other as they practice themathematical vocabulary that is central to the concepts.

All students should become proficient in using the core vocabulary of function, variable, constant, independentvariable, domain, dependent variable, range, sequence, and term. These terms are defined in the Agile Mindglossary. Some non‐native speakers may also struggle with collateral vocabulary such as beverage, vendingmachine, chores, horizontal, vertical, and flower arrangement. Language notes are provided to help with someof the core vocabulary words; these notes can serve as a model for you to look for and address other potentiallanguage issues.

Advice for Instruction | 3. Functions | Deliver instruction

Deliver instruction (Block 1)

Agile Mind materials Opening the lesson Framing questions Lesson activities Suggested assignment

Agile Mind materials

Functions:

OverviewExploring “Function notation”Student Activity Sheet 1: Teacher, Student

Student Activity Sheet 2: Teacher, Student

Opening the lesson

Introduce the lesson by asking students to think about a vending machine.

Framing questions

How does a vending machine work, when the machine works correctly?

Lesson activities

Literacy strategy. Page 2 of the Overview and pages 2 and 3 of the Exploring “Function notation” review anumber of key terms: function, variable, domain, range, independent variable, and dependentvariable. If you have ELL students, many of these terms may pose special challenges because of multiplemeanings outside of mathematics. You may find the "Language notes" that are provided as these terms areintroduced helpful as a basis for brief class discussions that could benefit all of your students as theymaster the vocabulary of this topic.

Overview

Page 1

Play panel 1 of the animation. Pause the panel during the action of the "C" button, immediately after thesecond skim milk is delivered. Ask:

What drink will be delivered next? How do you know?Press play again to complete the action of panel 1. Spend some time discussing what is meant by “a uniquebeverage type” as stated in the caption. Point out that, even though two buttons (C and D) delivered thesame beverage, each individual button delivered only one type of beverage.Play panel 2 of the animation. Pause the panel during he action of the “B” button, immediately after thethird apple juice is delivered. Ask:

What drink will be delivered next? How do you know?Press play again to allow the action of button “B” to finish, and then press pause again. Ask:

Is this what you expected?Does button “B” deliver a unique beverage type?

Finish playing the animation. Spend some time comparing the actions of the two beverage machines. Besure that you use the two vending machines as a way to emphasize the notion of "uniqueness" as it relatesto functions.

Pages 2-3

Define function as on page 2. Then, ask students to apply the definition to explain why the “Refreshments”machine illustrates the definition and the “Cold Drinks!” machine does not. [SAS 1, questions 1 and 2]

After students have shared their analysis of the two vending machines, ask:What change could you make to the “Cold drinks” machine so that it would represent a function?

Use page 3 as needed to summarize the discussion.

Page 4

Have students work in small groups to list at least 3 examples of dependent relationships such as thoseshown on this page. [SAS 1, question 3]

After a few minutes, ask each group to share 3 of their "function statements" and record them on a classchart.Ask students to create their own definitions of a function. [SAS 1, question 4]

Exploring “Function notation”

Pages 1-2

Introduce function notation. Spend some time emphasizing the connection between the verbal andsymbolic representations of the relationship. [SAS 2, question 1]

Discuss the different meanings of variable as on page 2.Literacy strategy. Use the language note on page 2 to bring in additional discussion of the ways in whichthe term variable is used outside of mathematics. Connect those examples to the two ways in whichvariable is used in a mathematical sense on this page.Literacy strategy. Students must be able to read and write function notation as they describemathematical functions. Spend time on page 2 reinforcing the notation with students. Ask students to readeach of the statements written in function notation and put the statements into words. For example, thefirst statement could be read as “Receiving a container of orange juice is a function of pressing the A code”or "Receiving a container of orange juice depends on pressing the A code."

Page 3

Review the terms independent and dependent variable and relate the terms to the vending machinescenario.Review the terms domain and range and connect these terms to the vending machine scenario.Direct students’ attention to the “function statements” they generated from page 4 of the Overview.Conduct a class discussion of which functions more clearly or strongly represent dependent relationships.Ask the students to try to represent some of the relationships using function notation.

What is the independent variable?What is the dependent variable?How would you describe the domain of your function?How would you describe your function's range?

Literacy task. Discuss with students the importance of good definitions. Ask students to write a shortanswer response giving a definition, with support, in response to the prompt below. Student responsesshould be written in complete sentences and use precise mathematical language. Students can completethis task as homework if necessary.

What is a mathematical function? After studying the vending machine examples, write a definitionthat communicates precisely what it means for a mathematical relationship to be a function. [SAS

2, question 2]

Literacy strategy. To help students in formulating their definitions, give them examples and non‐examplesof precise mathematical definitions to illustrate the characteristics of a high quality definition. Forexample, defining a rectangle as a four‐sided figure is not precise. (This is a definition for a generalquadrilateral.) A good definition of a rectangle might be: "A four‐sided figure with a right angle at eachvertex."

Further questions

How could the function associated with the vending machine be written using "if‐then" statements or"cause‐and‐effect" statements?How does one part of the function associated with the vending machine cause or depend on the other part?

Suggested assignment

Have students complete their definitions of a mathematical function. [SAS 2, question 2]

Have students complete the puzzle on page 4 of “Function notation." [SAS 2, questions 3-6]

Student Activity Sheet 2, questions 7‐9





Functions:

Exploring “Function notation”Student Activity Sheet 2: Teacher, Student

Opening the lesson

Remind students of the square pool scenario using the text and image at the top of page 5 of “Function notation.”

Framing questions

What makes a relationship a function?How do you know the relationship between the length of the pool and the number of tiles in the border is afunction? [SAS 2, question 10a]

Lesson activities

Exploring “Function notation”

Pages 5-6

Give students time to individually think about and respond to the framing questions. Be sure students writean individual response to discuss question 10a on their activity sheets. Discuss the framing questions, usingthe pool example to continue to deepen students’ understanding of a function.Remind students about their previous work with function notation, using the animation on page 1 of thisExploring as necessary. Ask:

How would you represent the relationship between the length of the pool and the number of tilesin the pool using function notation? [SAS 2, question 10b]

Debrief students’ responses using the reveal at the bottom of page 5, as needed. Spend some timediscussing the fact that, even though the letter chosen to represent the function may vary, the functiondoes not change. It still maps the each element of the domain to the same element of the range.Classroom strategy. To help students understand this point, ask them to construct tables using eachdifferent representation of the function. They should quickly see that the tables are identical.Give students a minute or two to read the text at the top of page 6. Ask:

What is f(10) for this function?

What does f(10) mean in the context of the problem situation? [SAS 2, question 11]

Debrief students’ responses using the reveal on page 6 as needed.

Page 7

Play panels 1 and 2 of the animation. Then have students complete the table on their activity sheets todescribe the pattern. [SAS 2, question 12]

Play the remaining panels of the animation to check students’ visual depictions.

Pages 8-9

Give students a moment to compare their table with the one on page 8. Ask:Is the relationship between the figure number and perimeter of the figure a function? Why or whynot? [SAS 2, question 13]

Use the reveal on this page as needed to debrief this question. Ask:How could you represent this relationship using function notation?

Give students time to complete the puzzle on page 9. [SAS 2, question 14]

Debrief the puzzle by inviting different students to come to the class computer and fill in each line of thepuzzle, explaining their thinking as they do so. Ask:

Suppose these terms were the only values of the function. What is the function’s domain?Suppose this pattern continued indefinitely. What is the domain of the function in this case?

Pages 10-11

Define the terms sequence and term as on page 10. Emphasize that each of the terms in this sequencerepresents the perimeter of a specific figure. Ask:

How would you write f(4) using this notation?How would you write f(3) using this notation?How would you write f(2) using this notation?How could you use the pattern to determine the nth term in the sequence (the perimeter of the

nth figure) from the previous term in the sequence?Classroom strategy. This question asks students to make use of recursive thinking. In later lessons,students will develop both recursive and general formulas for arithmetic and geometric sequences. Fornow, focus on helping students verbalize these recursive relationships so that they will be able to handlethe symbolic representations more easily in later lessons.Give students time to complete the puzzle on page 10. [SAS 2, question 15]

Classroom strategy. Because the domain of this sequence includes some of the terms in the sequence,students may become confused. To combat this, you may need to precede each of the questions above byreferring students to the puzzle they just completed, so that they will recall what each of the expressionsin the questions above is equal to.After students have finished, invite different students to the class computer to fill in different parts of thepuzzle. Require students to explain the reasons for their answers, and encourage them to includeconnections between domain and range values as part of their explanations. Ask:

Can you write f(1) using recursive notation? Why or why not?Give students time to discuss this question in pairs. Students should see that they must also include thedefinition of a “starting value” and so the complete definition of this function would be as follows:

f(1) = 3, f(n) = f(n − 1) + 1Summarize this discussion by introducing the recursive definition of the function as on page 11. Ask:

How could you write this function using function notation, without defining it recursively? [SAS 2,

question 16]

Use the reveal on this page to debrief, as needed.

Pages 12-13

Introduce the Fibonacci sequence using the animation on this page. Give students time to respond to thequestions posed in the animation.Classroom strategy. Spend some time discussing the domain as given on page 13. While the tree branchingscenario lends itself to including 0 in the domain, to indicate the beginning of the first year, students maywish to restrict the domain to the positive integers. Classically, the Fibonacci sequence is defined as havingan initial term of 0, thusly:

f(0) = 0, f(1) = 1

f(n) = f(n − 1) + f(n − 2)Keep the discussion focused not on the specific “correctness” of the domain that students state, but ratheron the importance of the domain in defining this function.Give students time to respond to the remaining questions on page 13. [SAS 2, question 17]

Check that students have continued to solidify their understanding of the concept of a function by havingthem share their written definitions of a mathematical function with a partner. Students should critiquetheir partner’s definition.Literacy strategy. An important part of developing precision in communication is giving and receivingfeedback about specific communications. Encourage students to be specific in giving feedback and to askclarifying questions in receiving feedback.

Further questions

How could the function associated with the rose offers be written using "if‐then" statements or "cause‐and‐effect" statements?How does one part of the function associated with the rose offers cause or depend on the other part?


Student Activity Sheet 2, question 2: have students revise their definitions for “function” in response to thefeedback they received.Student Activity Sheet 2, questions 18a‐b, 19a‐c, 20a‐d, and 21a‐d





Functions:

Exploring “Modeling with functions”Student Activity Sheet 3: Teacher, Student

Opening the lesson

Describe the soccer team's fundraiser as you show page 1 of “Modeling with functions.”

Framing questions

What would a successful project look like?How will the factors in the list influence the success of the project?

Lesson activities

Exploring “Modeling with functions”

Discuss the framing questions. Students should suggest that a successful project makes money, and thediscussion should then be focused on how each of the factors in the list would impact the money‐makingpotential of the project. Ask:

Which factors in this list seem to be most directly linked to how much money the soccer team willmake?

Page 2

Use the animation on page 2 to illustrate the two proposals. For each proposal, ask:What are the variables in this offer?Is there a functional relationship between these two variables? Why or why not?

Classroom strategy. Investigating functions through repeated addition is a great way to build on whatstudents already know in order to learn something new. Revisit this idea many times when learning how torepresent functions symbolically. It reinforces the idea of “patterns to rules” and gives students a strategyfor recognizing patterns and writing expressions, equations, and inequalities.Technology tip. The graphing calculator can be used to find answers using the “repeated addition” modelthat is presented in this Exploring. This can free students from possible confounding arithmetic errors sothey can focus on the mathematics of the lesson.

Pages 3-4

Use the puzzle on page 3 to reinforce students’ understanding of the dependency relationship illustrated bythe rose situation. [SAS 3, questions 1-3]

To debrief, invite different students to the class computer to drag tiles and explain their thinking. Then,ask:

How much would it cost to order 32 roses from Roses-R-Red?How much would it cost to order 32 roses from the Flower Power?How many roses can you order from Roses-R-Red for $37.35?How many roses can you order from Flower Power for $81.50?

Give students 5 or 10 minutes to discuss and answer these questions. Ask:Did you use repeated addition to answer these questions? If not, how did you answer them?

After the students have (hopefully) realized how difficult it will be to use the addition method for largenumbers of roses, direct students to begin building a table using a multiplication strategy. [SAS 3, question

4]

Play panel 1 of the animation on page 4, and ask students to answer the question about 32 roses. Studentscan respond independently, in small groups, or as a class. Ask a student to report her answer and how shefound the answer to the class. Ask others in the class to comment on the solution technique or to shareother techniques. Then play panel 2 to confirm the class’ work.Language strategy. It is recommended that students record their own answers to the questions as they areposed, but as a result of peer interaction either through whole‐class discussion or at least with a partner.These interactions allow students to share their thinking and learn from others. They are also good ways forall students, and especially ELL students, to build their mathematical vocabulary.Ask students to solve the problem shown in the final caption on panel 2. Once all students have finished,ask them to share their answers and their thinking with a partner. Then have a class discussion on thedifferent techniques students used to answer the question. Play panel 3 to confirm their work.

Pages 5-6

Ask students to turn to a partner and describe the general process they would use to find the cost for anynumber of roses. Review what is meant by expression and then ask students to write an expression thatrepresents the process. [SAS 3, question 5] Use these questions to prompt students who may be struggling:

In every line of the process column so far, what stays the same? What changes?If you were to add more lines to the table for other numbers of roses, how would the new entriesin the process column look? What would be the same? What would change?How could you use numbers and variables to write an expression to represent the general process?

Classroom strategy. Emphasize to students that the expression they write is a mathematical description ofthe general process they have already described in words. Focus students on the process column andencourage them to duplicate the constants and operations as they write the expression. This will help themidentify the variable in the expression.Ask students to use their expressions to write a function rule that relates the cost to the number of roses.Play the animation on page 5 to check student work and to reinforce the connection between the processcolumn and the expression.Ask:

How could you write your function rule using function notation?Show page 6 to illustrate the use of function notation in this situation.Classroom strategy. Students often think that there is only one way of expressing function notation, by

using "f(x)." This example provides another opportunity to help students understand that the choice offunction name can be made based on the scenario the function models. To help students becomecomfortable with this idea, invite them to propose different ways to represent this same function infunction notation. They should quickly see that the letter used to name the function is less important todescribing the functional relationship than is the expression that captures the operations on theindependent variable.

Ask students to explain why the cost of roses from Roses‐R‐Red is a function of the number of rosesordered. [SAS 3, question 6]

Pages 7-8

Have students work in pairs to complete the table to determine costs for roses from Flower Power. [SAS 3,

question 7]

Use the animations on pages 7 and 8 to validate their work.Ask students to explain why the cost of roses from Flower Power is a function of the number of rosesordered. [SAS 3, question 8]

Pages 9-10

Give students time to answer the first question on page 9. [SAS 3, question 9a] Ask:

Should the 30 be substituted for x or for r(x)? How do you know?Have a student share his or her answer and strategy for finding it. Then, ask:

Is 30 a value for the domain or the range of the function?Is $42.50 a domain value or a range value?

Give students time to answer the second question on page 9. [SAS 3, question 9b] Ask:

How is this second question different from the first question? Should the 100 be substituted for xor for r(x)? How do you know?

Have a different student share his or her answer and strategy for finding it.Classroom strategy. As students explore the functional relationships presented in these two questions andthe two questions on the next page, they will apply the order of operations as they evaluate function rules.They will also encounter equations that arise naturally from these function rules in response to questionsabout the situation. The intent of these last types of questions is not to have students spend time on theformal steps of equation solving. Instead, students should be allowed to use more informal strategies for"undoing" to solve these equations as they solidify their understanding of the relationship between twovariables.Give students time to answer the two similar questions on page 10. [SAS 3, question 10] Ask:

How does the cost of 30 roses from Roses-R-Red compare to the cost of 30 roses from FlowerPower?How does the number of roses the soccer team can order for $100 compare at the two shops?

Further questions

Will the cost of the roses ever be the same for the two flower companies? If so, when?Which flower company offers a better deal if the soccer team thinks it will sell 200 roses? Why?


Student Activity Sheet 3, questions 11a‐c, 12, and 13a‐d





Functions:

Exploring “Graphs”Student Activity Sheet 4: Teacher, Student

Opening the lesson

Review with students the work they did previously with the tables and rules for Roses‐R‐Red and Flower Power.

Framing questions

How can we compare the cost of flowers at the two shops using the tables?How can we compare the cost of flowers at the two shops using the rules?How can we use graphs to help us compare the cost of flowers at the two shops?

Lesson activities

Exploring “Graphs”

Classroom strategy. The advice for this block supports facilitation of small group work for all activities.However, encourage students to individually respond to the pertinent questions on their student activitysheets before working on the group responses to those questions; this will encourage individual studentaccountability and engagement with the group.

Pages 1-2

Ask the students to work in small groups to determine the independent and dependent variables. Askstudents to use this information to decide on appropriate axes labels in preparation for graphing each setof data. [SAS 4, question 1]

Once the groups have decided how to label their grids, use the puzzle on page 2 to verify that decision.Spend a little time discussing variations on the labels shown in the puzzle (cost, cost in dollars, number ofroses, number ordered, roses ordered) and let the class talk about the advantages and disadvantages ofeach label.Have each group sketch a graph grid on chart paper in preparation for graphing the data.

Pages 3-4

Give the groups time to discuss the questions about domain and range on page 3. Emphasize that thisdiscussion will help them decide what values need to be represented on their graphs. [SAS 4, question 2]

Once the groups have discussed the questions, ask them to share their thinking with the class. Then, ask

them to complete their graphs on the Roses‐R‐Red and Flower Power data and post the charts. [SAS 4,

question 3]

Conduct a gallery walk of the graphs, and then begin a brief discussion, using the reveals on page 4 asneeded. Use these questions to guide the discussion:

Are all the graphs accurate?What is the same about the graphs?What is different about the graphs?How do the graphs reflect the domain and range of the Roses-R-Red situation?How do the graphs reflect the domain and range of the Flower Power situation?

Have each group retrieve its charts.

Pages 5-8

Direct the groups to draw lines through their plotted data points as shown on page 5. [SAS 4, question 4]

Give each group a few minutes to compare the graphs, and then ask the groups to share their thoughts withthe class. Do not close this discussion, but use it as a bridge to page 6.Give each group a few minutes to discuss how they would complete the puzzle on page 6. [SAS 4,

questions 5-8] Then, allow different groups to give an answer for one of the slots. Be sure that they justifytheir answers by referring to their own graphs.Follow a similar process with the puzzles on pages 7 and 8. [SAS 4, questions 9-15]

Pages 9-10

Introduce the new scenario using the initial panels of the animation on page 9. Give students time toanalyze the different representations of the different functions to answer the question in panel 3. [SAS 4,

question 16]

Ask students to share their answers and their thinking. Encourage them to explain how they used the tworepresentations to determine their answers. Use page 10 as needed to support this discussion.

Page 11

Pair students and ask them to determine whether each relationship graphed is a function. [SAS, question

17]

After students have discussed the question, invite different students to share their answers and theirthinking.

Classroom strategy. Students may fail to recognize that f(x) = 10 still satisfies the definition of a function

as a relationship in which every x‐value has only one corresponding y‐value. It may be helpful at this pointto introduce the geometry of the definition through the "vertical line test" for a function. Be sure youconnect this test to the definition so that students do not lose sight of the “big picture” of the concept of afunction and begin to rely solely on graphical strategies that are disconnected from the concept.

Further questions

How are the graphs of the functions that model the problem situations and the graphs of the mathematicalfunctions different?What information does a table give you about a functional relationship?What information does a graph give you about a functional relationship?Which do you prefer using to solve problems: tables, rules, or graphs? Why? What are some advantages anddisadvantages of each method?


Student Activity Sheet 4, questions 18‐22





Functions:

Guided practice

Opening the lesson

Explain that the soccer team needs to order new shirts. Show a transparency of only the top section of page 1 ofthe Guided practice to give students the pricing information about the two shirt companies.

Framing questions

How can we compare the cost of T‐shirts from the two companies using the tables?How can we compare the cost of T‐shirts from the two companies using the rules?How can we use graphs to the cost of T‐shirts from the two companies?

Lesson activities

Classroom strategy. The Guided practice provides another opportunity for students to explore functionalrelationships using tables, rules, and graphs. It is recommended that you distribute hard copies of theGuided assessment to each student and that you allow the students to work in pairs on the questions.

Guided practice

Page 1

Give students time to complete the table. Then, invite different students to the class computer to dragtiles to fill in each blank and give complete explanations for the answers chosen.

Page 2

Give students time to complete the puzzle and then discuss their answers. Drag the tiles in response tostudent input to check their work.

Page 3

Give students time to complete this puzzle to write a function rule for Quality Clothes.Classroom strategy. If you notice students struggling with this task, direct them back to the table on page1. Focus them on the process column, and suggest they try to write an expression that represents the

process they see in the process column. Be sure they can identify what stays the same and what changes at

each line of the process column; this will help them write the general expression.Invite different students to the class computer to show their function rules.Classroom strategy. This puzzle allows students to different, yet equivalent rules. Take this opportunity toreinforce the field properties by asking students to explain why different rules are equivalent. Some rulesmay be equivalent through application of the symmetric property of equality; others may be equivalentbecause they contain equivalent expressions. Students should be able to identify uses of the commutativeand associative properties in these equivalent expressions.

Pages 4-6

Give students time to complete the questions on these three pages. As they work, circulate around theroom to clarify or redirect. For struggling students, connect the work on these pages back to the work onpages 1‐3.Invite different students to the class computer to solve the puzzles as a check of student work.Classroom strategy. Use the puzzle on page 6 as a second opportunity to reinforce the field properties.Challenge students to find and justify as many equivalent rules as they can.

Page 7

Give students time to respond to this question. Discuss student answers. Be sure students understand therole of each variable as they translate their original rule into function notation. Ask:

What are some other ways you could represent the rule for Quality Clothes using functionnotation?What are some other ways you could represent the rule for T-Riffic using function notation?

Pages 8-9

Give students time to respond to the questions on these pages.Discuss student answers, being sure to ask students to share their strategies.

Pages 10-11

Show page 10. Ask:What is the domain of the function that models the cost from Quality Clothes? What is the range?How do these questions help you in constructing your graph?

Have students make the graphs described on pages 10 and 11. Use the reveals as needed to check theirgraphs. Ask:

What is the same about the graphs?What is different about the graphs?How does the Quality Clothes graph reflect the domain and range of the situation?How does the T-Riffic graph reflect the domain and range of the situation?

Page 12

Give the students time to complete the puzzle on this page.Invite different students to the class computer to explain their solutions and drag tiles to show theiranswers.Literacy strategy. This puzzle begins to introduce some descriptive language that students will continue touse as they continue to explore graphs in this course. Spend time here to make sure all students understandthe meaning of each term so that they are comfortable in using them.

Page 13

Give the students time to complete the puzzle on this page.

Invite different students to the class computer to explain their solutions and drag tiles to show theiranswers.After the puzzle is complete, ask students to discuss any difficulties they had in using the graphs to answerthese questions. Ask if any of them went back to the function rules to check their answers.

Further questions

How are the graphs of the problem situations and the graphs of the function rules different?What information does a table give you about a functional relationship?What information does a graph give you about a functional relationship?Which do you prefer using to solve problems: tables, rules, or graphs? Why? What are some advantages anddisadvantages of each method?


More practiceSummary





Functions:


Student Activity Sheet 6: Student


Opening the lesson

Remind students of the vending machine example they saw in the Overview. Present the stamp machine scenario,using Student Activity Sheet 5.

Framing questions

What makes a relationship a function?

Lesson activities

This block provides an opportunity for students to solidify their understanding of what makes a relationshipa function and functional notation through the development of a written product. This task builds students’proficiency with these standards for mathematical practice.

Construct viable arguments and critique the reasoning of others.Attend to precision.

The task also gives students the opportunity to build proficiency with these standards for reading.Read closely to determine what the text says explicitly and to make logical inferences from it; citespecific textual evidence when writing or speaking to support conclusions drawn from the text.Interpret words and phrases as they are used in a text, including determining technical,connotative, and figurative meanings, and analyze how specific word choices shape meaning ortone.Read and comprehend complex literary and informational texts independently and proficiently.

The task also gives students the opportunity to build proficiency with these standards for writing.Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoningand relevant and sufficient evidence.Produce clear and coherent writing in which the development, organization, and style areappropriate to task, purpose, and audience.Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a newapproach.Draw evidence from literary or informational texts to support analysis, reflection, and research.Write routinely over extended time frames (time for research, reflection, and revision) and shorter

time frames (a single sitting or a day or two) for a range of tasks, purposes, and audience.

Literacy strategy. Refer students to the definition of a function at the top of the activity sheet. Askstudents to pair read the definition and discuss it’s meaning. Ask:

How does this definition compare to the definition from page 2 of the Overview?How does this definition compare to the definition you wrote in Student Activity Sheet 2, question2?What key words are in the definition and what do they mean?What does it mean for a quantity to “depend” on another quantity?How can you relate “dependence” to the vending machine scenario?

Give students time to respond to the Reflection question [SAS 5, Reflection question].Have students work through the questions 1–7 on the activity sheet to solidify their understanding of afunction. [SAS 5, questions 1–7]

Use questions 8–10 to help students begin to draft an argument that can be used in their written responseto the prompt given below. [SAS 5, questions 8-10]

Literacy task. Present students with the following writing prompt [SAS 6], being clear that they will beworking towards a response to the prompt over the next few class periods.

What makes a relationship a function? After reading and studying the topic Functions, write aresponse paper in which you define “function” and explain if each of the two stamp machinesrepresents a functional relationship. Support your discussion with evidence from the text.In a quick‐write, ask students to write their first reaction to the task prompt. Give students time toread the prompt with a partner, circle any words they are unfamiliar with, and discuss any clearunderstandings they have based on the prompt.Ask students to write in their own words a brief explanation of what the task is asking them to do.Introduce the rubric to students. [SAS 7] Ask students to translate the rubric into their own words.

Further questions

What change could you make to the second stamp machine to make it a function?


In a quick‐write, have students give a brief overview of their response paper, describing how the paper willbe constructed and what the central argument of the paper will be.Have students create an outline of their response paper including key elements from their study of the twostamp machines. Students should order their outline in a logical way.





Functions:




Opening the lesson

This block is intended for continued development of a written response paper.

Framing questions


Lesson activities

Have students write an initial draft of their response paper including the opening, a comparison of the two stampmachines using precise mathematical language, and an ending to include a conclusion about whether the stampmachines represent functions or not. For any functional relationship, the draft should include function notation todescribe the relationship.







Functions:




Opening the lesson


Framing questions


Lesson activities

Have students perform a peer review of another student's draft response paper. The peer review should includecomments about the opening, the logic of the argument in the draft, the use of function notation, and the overallstrength and accuracy of the argument presented.


Ask students to finalize their written response to the writing prompt “What makes a relationship a function?”

Students should revise their draft, obtaining a 2nd round of review if necessary. Students might choose tohave a different peer review their draft or they may choose to read their essay aloud for an adult (a parentor another teacher) for adult feedback.Students should finalize the draft, applying finishing touches, including visuals, neatness, formatting andcopy editing. The final draft should demonstrate the use of strategies that enhance the readability andappearance of the work for presentation.





Functions:

Automatically scoredConstructed response

Opening the lesson

This block is intended for a topic level assessment.

Framing questions


Lesson activities

This block is intended for a topic level assessment. The Automatically scored and Constructed response questionscan be used for such an assessment.



Prepare instruction - Piedra Vista High Schoolpvhs.fms.k12.nm.us/teachers/malonzo/08EF901C-00757F35.12/AlgebraI... · Prepare instruction ... KDW iV D PDWKePDWiFDl IXnFWion" $IWer

Documents