Multiplying Binomials Lesson Plan (Access Version)

8/9/2019 Multiplying Binomials Lesson Plan (Access Version)

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Access St rategies:Throughout the document you will see icons calling out use of the access strategies for English Learners, Standard English Learners, andStudents With Disabilities.

Access Strategy Icon Descript ion

Cooperative and Communal LearningEnvironments

Supportive learning environments that motivate students to engagemore with learning and that promote language acquisition throughmeaningful interactions and positive learning experiences to achievean instructional goal. Working collaboratively in small groups, studentslearn faster and more efficiently, have greater retention of concepts,and feel positive about their learning.

Instructional Conversations Discussion-based lessons carried out with the assistance of morecompetent others who help students arrive at a deeper understandingof academic content. ICs provide opportunities for students to use

language in interactions that promote analysis, reflection, and criticalthinking. These classroom interactions create opportunities forstudents conceptual and linguistic development by makingconnections between academic content, students prior knowledge,and cultural experiences

Academic Language Development The teaching of specialized language, vocabulary, grammar,structures, patterns, and features that occur with high frequency inacademic texts and discourse. ALD builds on the conceptual

knowledge and vocabulary students bring from their home andcommunity environments. Academic language proficiency is aprerequisite skill that aids comprehension and prepares students toeffectively communicate in different academic areas.

Advanced Graphic Organizers Visual tools and representations of information that show the structureof concepts and the relationships between ideas to support criticalthinking processes. Their effective use promotes active learning thathelps students construct knowledge, organize thinking, visualizeabstract concepts, and gain a clearer understanding of instructional

material.

2007 University of Pittsburgh FUNDED BY THE JAMES IRVINE FOUNDATION Institute for Learning Adapted 2009 LAUSD Secondary Mathematics 2


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Academic Language Goals of the Lesson:

Develop academic vocabulary to be used in the descriptions

Describe algebraic patterns orally or in writing

Explain the process used in solving the task, orally or in writing

Assumption of PriorKnowledge:

Experience using AlgebraTiles - naming andunderstanding the pieces aswell as representing, addingand subtracting polynomials

Distributive Property ofmultiplication overaddition/subtraction

The area model ofmultiplication

Academic Language:

Binomial

Trinomial

Polynomial

Factor

Product

ConstantCoefficients

Distributive Property

Model

Counter-Example (see p. 12)

Terms

Conjecture

Materials:

Task

Graph paper

Plain paper to recordalgebra tile models

Transparencies or chart

paperAlgebra Tiles or AlgebraPieces (with edgepieces)

Overhead Algebra Tilesor Pieces (optional)

Follow-Up Lessons:

Procedures fo rMultiplyingBinomials,FOIL and boxmethods, and howthey relate to usingalgebra tiles

Special products ,i.e., squaringbinomials and thedifference of twosquares.

Connections to the LAUSD Algebra 1, Unit 4, Instructional Guide

Understand monomials and polynomials andperform operations on them (including factoring)

and apply to solutions of quadratic equations

2.0, 10.0, 11.0, 14.0, 15.0

Perform operations on monomials andpolynomials

Factor 2nd degree polynomials over the integers Use the zero-product rule and factoring as well as

completing the square to solve simple quadratics

Solve application problems using the abovetechniques



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Phase EXPLORE PHASE: Support ing Students Exploration of the Task

STRUCTURE

E

XPLORE

Use this struc ture for both Investigation 1 and Investigation 2.

PRIVATE THINK TIME

Ask students to work individually for about 5 minutes (depending on your class) on the initial problem (Investigation 1: #1#2; Investigation 2: #1 and #2) so that they can make sense of the problem for themselves.

Each student should create his or her own model and sketch to discuss with their group.

Circulate around the classroom and clarify confusions.



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Phase Investigation 2: Questions #6, #7, #86) Use your algebra tiles to find the products of the expressions below. Sketch each of your tile models and clearly label the factors and products of

each one. A) (x+ 5)(x+ 1) B) (3x+ 2)(2x+ 4)7) Use your answers to question 6 above to look for patterns relating the factors and products. List all the patterns that you notice.

8) Jessica wants to multiply the binomials (15x+ 12)(3x+ 10) but because the numbers are larger she does not have enough algebra tiles. Use thepatterns you found above to explain how Jessica can multiply binomials withoutusing the tiles.


Possible Solutions Possible Questions Misconceptions/Errors

Questions to AddressMisconceptions/Errors

E

XPLORE

EXPLOR

E

These questions extend the patterns studentsnoticed in #2, #3, #4 and #5.

Depending on students understanding of #2#5, you might want to have the classdiscussion of #2#5 before having studentswork on #6#8.

#8: Look for students who made diagrams to

find the products, e.g.,

15x 12

3x

10

#8: Look for students who used diagrams orsymbols to justify Jessicas conjecture.

Ask questions such as:

How can you use the patternsthat you found in #2 through #5to answer #6 and #8?

#8: Can you rewrite thebinomials so that they bothcontain addition? How can yourewrite a subtraction problemas an addition problem?

#8: Review the distributiveproperty discussion from thebeginning o f the lesson (GettingStarted) and Investigation 1.

See discussion ofQuestions #1#5and the Sharing,Discussing, and

Analyzingdiscussion for #6#8.

See discussion of Questions#1#5 and the Sharing,Discussing, and Analyzingdiscussion for #6#8.

45x2 36x

150x 120


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Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goals

of the lesson, and possible Student Responses that demonstrate understanding.

Phase


Sequencing of StudentWork

Rationale and Possible Questions and Student ResponsesMathematical Ideas

SHAR

E

DISCUS

S

AND

ANA

LYZE

Order of discussion:

Class discussion starts with #3.(#1 and #2 should have beenaddressed during the Explore

phase.)

#3. Have a different grouppresent their solution to a, b, andc. Students should:

Show their model

Show their intermediateproducts

Show their final product(i.e., like terms combined)

The factors, intermediateand final products for eachproblem should bedisplayed so that studentscan refer to them in thediscussion of #5.

Showing the intermediate products (all fourterms) makes it easier to see the patterns.It also will help students see how thedistributive property is used to find theproducts.

The goal is to make explicit the relationshipbetween the terms in each binomial andthe four intermediate products; i.e., that tofind the product of two binomials, eachterm in one of the binomials is multiplied byboth terms in the other binomial.Each group may give only part of the largerpattern (e.g., you multiply the x termstogether to get the x2term); however, thecombined patterns from the class shouldproduce the entire pattern.

Show us your model of (x + 3)(x+2). Where are the factors?Where is the product?

We made a rectangle with x+3 on the top and x+2 on theside. Then we filled in the four areas. We got x2+ 3x + 2x+6, which equals x2+ 5x + 6.

What factors did you multiply to get x2

? To get 3x? To get 2x?To get 6?

x and x.; 3 times x; x times 2; 3 times 2.

What pattern did you notice? What justification do you have?Why do you think so? How do you know? Does your patternwork in all of our examples so far?

Answers will vary.

Does anyone have a counter-example?

Answers will vary.

Sharing, Discussing, and Analyzing, Investigation 2, Questions #1-#6


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Phase Sequencing of StudentWork

Rationale andMathematical Ideas

Possible Questions and Student Responses

SHARE

DISCUSS

AND

ANAL

YZE

#5. Groups should describehow they determined thefactors from the diagram.

Here, students combine the patterns theynoticed in #3 and #4 into a procedure formultiplying binomials.

This lays the ground work for factoring.

What patterns did you use to find the product?

Answers will vary.

If students do not show the intermediate products, ask How didyou get your x term?

Answers will vary.

How does your model support your predictions?

Answers will vary.

How did you determine the factors? How are the factors shownin the diagram?

I looked at the top edge and it was 2x + 3 in length. Then Ilooked at the side edge. It was x + 1 in length.

How is the trinomial related to the diagram? Where are the threeterms shown?

The blue shows the x2term, the green shows the x term, andthe yellow shows the units.

There are three terms in the product, but the diagram shows fourareas. How is this possible?

We put the greens together because they are both x terms.


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Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goalsof the lesson, and possible Student Responses that demonstrate understanding.

Sharing, Discussing, and Analyzing, Investigation 2: Questions #7, #8, #9

Whole class discussion. As students

present solutions other students

(and the teacher) ask questions


The purpose of this sharing/discussing is to make explicit the conclusions from Investigation 2.

Phase Sequencing of StudentWork

Rationale andMathematical Ideas

Possible Questions and Student Responses

SHARE

DIS

CUSS

AND

ANALYZE

Order of discussion:Whole-class discussion of eachquestion.

#8. Have at least two groupspresent their solutionsonean algebraic solution; theother, a solution based on adiagram. (See Explore notes onp. 9.)

Other students ask themquestions about theirsolution.Elicit any differentapproaches to the problem.Be sure that any alternatesolutions that you noticedwhile students worked on theproblem are highlighted.

If no one in your classcreates a diagram to solvethe problem, introduce thediagram as an alternatesolution created by otherstudents (last year, otherclass, etc.)

Here, students are generalizing therelationships they discovered with themodels to create a formal algebraicprocedure that can be used to multiply anytwo binomials.

Students further generalize their procedureto apply to binomials that involve negativenumbers and subtraction.

How did you find the product without using algebra tiles?

Answers will vary. Different patterns will be used.

How did your work with the tiles help you solve the problem?

Answers will vary.

Diagram questions: How are the sections in your diagramrelated to the tile models? How are they related to the factorsand the product?

15x 12

3x

10

45x2 36x

150x 120


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CLOSUREHave students reflect on the mathematics of the lesson; find links to math that they have explored before; think of tasks that might berelated to the big ideas of the lesson. In particular, have students reflect on how the algebra tile models helped them create a meaningful

procedure that they can use to multiply binomials.

It is important for students to step back and reflect on the ideas that surfaced and to situate their learning within past experiences, and tothink forward to ways that they might build on these ideas in future tasks. This helps them to focus on the interconnectedness ofmathematical ideas.

These connections are particularly important when students are learning new procedures. Traditionally, procedures like multiplyingbinomials have been taught rotely, with few, if any connections to the underlying mathematics or to other, related procedures. By

developing the underlying rationale for the procedures through connections to the mathematics, to other representations, and otherprocedures (e.g., multiplying whole numbers), students are better able to remember the procedure, apply it to new situations, and recreateit if they forget it. In this case, students will be better prepared to reverse the procedure to factor polynomials and multiply otherpolynomials.

NEXT STEPS FOR SUCCESSThe next lesson should extend these ideas and helps students to solidify their understanding of multiplying binomials by analyzing twopopular procedures for multiplying binomials: the Box and FOIL and discussing how these methods are related to the work done in this

lesson using the algebra tiles.

Multiplying Binomials Lesson Plan (Access Version)

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