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1 Unit 10 Area Model Factoring Research-based National Science Foundation-funded Learning transforms lives.

2 Dear Student, When we multiply two factors, we get their product. If we start with the product, we can undo the multiplication to find the factors. The process of discovering the factors that can be multiplied to give a number or epression is called factoring = ( + _ )( + _) Unit 10 uses area models to make sense of factoring. You ve already learned how to multiply two algebraic epressions using an area model: ( + 7)( 2) = We can also use area models to un-multiply, to start with an algebraic epression and factor it, that is, find two other epressions whose product is that original epression. _ 2 _ = ( + _ )( + _) Unit 10 shows how multiplication, division, and factoring are all related, how division and factoring are like opposites of multiplication, and how we can use area models to reason about all three. The Authors

3 Lesson 3: Factoring IMPORTANT STUFF Problems 1 and 2 ask about the equation ( 2)( 5) = and its area model. 1 What calculation creates the 10 in ? -2 _ 2 What calculation creates the -7 in ? -2 + _ Problems 3 and 4 ask about the equation ( + 3)( 8) = and its area model shown below What calculation creates the -24 in ? 4 What calculation creates the -5 in ? 5 List all the pairs of integers (positive or negative) whose product is Which pair has a sum of 7? 7 Which pair has a sum of 8? 8 Which pair has a sum of 13? 9 Which pair has a sum of -7? Use an area model to factor. Complete each model and equation = = = = Unit 10: Area Model Factoring

4 Complete the model and finish Jay s thought = ( ) ( ) 2 2 Thinking out Loud Jay: I can tell from the equation that I need a pair of numbers whose product is 2 and whose sum is. The numbers are and so the factors must be and. 15 List all the pairs of integers (positive or negative) whose product is Which pair has a sum of 11? 17 Which pair has a sum of -13? 18 Which pair has a sum of 31? 19 Which pair has a sum of -17? Use an area model to factor. Complete each model and equation = = = = Who Am I? t + u = 10 tu = 21 t > u t u 25 Who Am I? t + u = 10 tu = 25 t u 26 Who Am I? t + u = 11 tu = 30 u > t t u 27 Who Am I? t + u = 12 tu = Who Am I? t u t u t + u = 11 tu = 24 t < u 29 Who Am I? tu = 81 t + u = 18 t u Lesson 3: Factoring 13

5 STUFF TO MAKE YOU THINK For some epressions, factoring with the area model doesn t give any new information. b For eample, it s not helpful to fill in the outside of this model _ 1 b because b 2 and 4 have no common factors besides An epression like b that cannot be factored (that is, has no factors other than itself and 1) is called prime, just like a number whose only factors are itself and 1! Determine if it makes sense to factor each epression. If so, complete the area model. If not, cross it out and label it prime. a b c 2h 9 _ 6p 60 _ 4w 2 _ 8w d 8w 2 _ 3 e 10m 2 30m 80 f g _ c 2 8c _ 3c 24 h _ 30a 2 7w _ 15a _ 25b 2 _ 5c 31 Here are four ways to set up the epression n 2 + 9n + 14 in a model. Three of the ways don t help or don t work when you try to fill out the outside. Cross out the three that don t help or don t work, complete the one that does, and write an equation to match it. a n 2 9n 14 b _ n 2 9n _ 0 14 c d _ n 2 7n _ 2n 14 _ n 2 3n _ 6n Unit 10: Area Model Factoring

6 Imagine you have been asked to help design a village. The people only want roads that travel northsouth, or east-west, so that their intersections form right angles like this: and not like this:. All of the north-south roads must cross all the east-west roads. W N E 32 Draw how you would arrange 5 roads so there are 33 Sketch a village map in which 7 roads 6 intersections in the village. form 10 intersections. S 34 Factor the epression r 2 + 5r Factor the epression r 2 + 7r r 2 6 ( )( ) r Suppose the people said they wanted 12 intersections but didn t care how many roads they had. How many different possible arrangements could you make? Draw the maps. What is the smallest number of roads that will still give them 12 intersections? 37 With eactly 10 roads, what is the smallest number of intersections there can be? What is the greatest number of intersections? TOUGH STUFF Factoring doesn t always work out into neat and tidy factors. You may have to try a lot of different ways = ( ) ( ) = ( ) ( ) Lesson 3: Factoring 15

7 Additional Practice Factor each epression. A a a + 56 = B y y + 42 = a 2 7a y There is no squared term here C = D = _ ( ) E z z + 72 = F = z G q 2 + 4q + bq = q b H y = 6 y 4q 7 2 I = J = K k 2 + 2k 48 = L r 2 + 8r + 7 = Draw your own area models. 16 Unit 10: Area Model Factoring

8 For these problems, only the inside of the area model is filled in. Find a way to complete the outside of the model and use your work to write at least one equation (using multiplication or division) that is represented by the area model. M _ 9y 45 N _ 2n 2 16n O 10a 2 P _ 6a Q _ p 2 2p R ac 3a 4a 2 _ 10p 20 S Here are four ways to set up the epression w w + 30 inside an area model. Three of the ways don t help or don t work when you try to fill out the outside. Cross out the three that don t help or don t work, complete the one that does, and write an equation to match it. i w 2 11w 30 ii _ w 2 11w _ iii _ w 2 10w _ w 30 iv _ w 2 6w _ 5w 30 Lesson 3: Factoring 17

9 TEACHING GUIDE Area Model Factoring Unit 10 Teaching Guide Unit 10 June Mark E. Paul Goldenberg Mary Fries Jane M. Kang Tracy Cordner Unit 10 Area Model Factoring HEINEMANN Portsmouth, NH Research-based National Science Foundation-funded Learning transforms lives.

10 firsthand An imprint of Heinemann 361 Hanover Street Portsmouth, NH Offices and agents throughout the world Education Development Center, Inc. 43 Foundry Avenue Waltham, MA by Education Development Center, Inc. Co-Principal Investigators and Project Directors: E. Paul Goldenberg and June Mark Development and Research Team: Tracy Cordner, Mary Fries, Mari Halladay, Jane M. Kang, and Josephine Louie Contributors: Cindy Carter, Susan Creighton, Jeff Downin, Doreen Kilday, Deborah Spencer, and Yu Yan Xu This material is based on work supported by the National Science Foundation under Grant No. ESI Opinions epressed are those of the authors and not necessarily those of the Foundation. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, ecept by a reviewer, who may quote brief passages in a review, and with the eception of reproducible pages, which are identified by the Transition to Algebra copyright line, and may be photocopied for classroom use only. Dedicated to Teachers is a trademark of Greenwood Publishing Group, Inc. Due to a printing error, the area model on the front cover of the Student Worktet may be colored incorrectly; the shading on the lower half of the model may be missing. (Compare the Worktets covers to the Answer Key cover, which has the correct shading.) If your students have mis-printed copies, you may wish to discuss and have students color the model to complete an ( + 3)( + 2) model. Transition to Algebra, Unit 10: Area Model Factoring Teaching Guide ISBN-13: Transition to Algebra Teacher Resources ISBN-13: Transition to Algebra, Unit 10 Student Worktets 10-pack ISBN-13: Transition to Algebra Student Worktets, 10 Sets of All 12 Units ISBN-13: Printed in the United States of America on acid-free paper RRD

11 Unit 10 Area Model Factoring CONTENTS T4 Unit Introduction T7 Lesson 1: Division Undoes Multiplication T10 Lesson 2: Area Model Inside Out T13 Lesson 3: Factoring T16 Student Reflections & Snapshot Check-in T17 Lesson 4: Products, Sums, and Signs T20 Lesson 5: Zero Product Property T22 Lesson 6: Solving by Factoring T25 Student Reflections & Unit Assessment T26 Eploration: Area Model Cutouts T29 Eploration: Signed Area Model Cutouts T31 Activity: Area Model Puzzles RESOURCES T32 Finding Pairs Cutout (Lesson 5) T33 Snapshot Check-in T34 Snapshot Check-in Answer Key T35 Unit Assessment T36 Unit Assessment Answer Key T37 MENTAL MATHEMATICS Factors, products & sums and percentage calculations T38 Finding factor pairs T39 Finding factor pairs with negatives T40 Product and sum T41 Finding factors using products and sums T42 Finding factors using sums and products T43 10% of a number T44 20% of a number T45 Finding 5% and 15% of a number T46 10% off T47 10% more T48 20% off T3

12 2/6/14 3:43 PM Unit 10 Area Model Factoring UNIT 10 Area Model Research-based National Science Foundation-funded Learning transforms lives. Factoring Learning Goals By the end of Unit 10, students should be able to: Understand both division and factoring as ways of undoing multiplication. Understand how a single area model can represent multiplication, division, or factoring depending on which pieces are given and which pieces are sought. Factor monic quadratic trinomials by finding the two numbers with a given sum and product. Use the zero product property to solve quadratic equations. Factoring can be a difficult concept for introductory algebra students to learn and master. This unit presents factoring as a kind of un-multiplying, building on the ideas of Unit 4: Area and Multiplication, which introduced the area model as a tool for organizing multiplication of numbers and polynomials. In this unit, students first consider division with area models, working with a given area (the product of the multiplication being undone) and one side length of the area model (a factor) to find the other length (the other factor of the original multiplication). Students then connect the models to corresponding multiplication and division equations and eplore area model puzzles that provide enough information about the factors and product to find the missing terms. This supports students in understanding the roles of and relationships between the pieces of the area model and their corresponding equations and in thinking fleibly with the model in preparation for factoring. The factoring problems in this unit focus primarily on epressing trinomial products as two binomial factors. Students also practice finding sums for pairs of numbers with a given product and build logic about the zero product property and its role in solving equations by factoring. Factoring and Area Models As in distribution, where multiplying 2(n + 3) can appear to students to be a different process from multiplying ( + 3)( 4), where the FOIL (first, outer, inner, last) approach is often used, factoring can also be seen as several different processes. For instance, factoring can appear altogether different from simplifying 9h + 6h a rational epression like 3h or solving an equation like 2 8 = 0. Factor = ( 7)( + 11) T4 Transition to Algebra Unit 10: Area Model Factoring TEACHING GUIDE

13 The area model provides a contet in which students can unify their understanding of these seemingly different processes with a single tool. Simplify 3 2 3h 9h 6h 9h + 6h 3h 9h + 6h 3h = 3h(3 + 2) Area Model Puzzles As Unit 4 did to some etent, this unit presents area model problems in the form of puzzles. Parts of the product are shown, parts of the factors are shown, and students need to treat these problems much the way they d treat any puzzle by first looking to see what empty space(s) they can fill in with certainty. Then, they can chase other empty spaces around the puzzle until they ve filled in all the blanks. This is, of course, neither a case of pure division, where the product and one of the factors are completely known, nor a case of pure factoring, where only the product is known and one has no knowledge about the factors. In pure division, the puzzle is simpler: one knows eactly where to start. In a pure factoring problem, the puzzle might take a bit of eperimentation before chasing the empty spaces around, but the process of using logic together with the known structure of the area model to find the remaining pieces is the same. Lesson 2 introduces area model puzzles to focus attention on the roles of different parts of the area model and how these pieces relate within a given model and to the problem at hand. Factoring Puzzles MysteryGrid puzzles requiring numerical factoring appear in the Stuff to Make You Think and Tough Stuff problems in Lessons 1 and 2; then in these same sections of Lesson 4, students confront MysteryGrid puzzles with polynomial elements and clues that require factoring. MysteryGrid 5, 6, 7, 8 12, + 48, 2 8 = = ( 8) = 0 Either = 0 or 8 = 0 _ 5 3m 6mj MysteryGrid 0, 1,, 2 2, , + _ 14j 35 Seeking and Using Structure In this unit, students use the Algebraic Habits of Mind familiar structure of the area model and its relationship to multiplication to strengthen and build an intuitive understanding of the necessary computational skills for factoring algebraic epressions and solving equations by factoring. Equations such as ( 1)( + 2) = 0 are viewed as two separate chunks of information: either 1 = 0 or + 2 = 0 (or both). This relies on an understanding of the zero product property and supports solving quadratic equations by factoring. Using Tools Strategically As students etend their use of area models to the division and factoring of polynomials, they have to be strategic about their placement of the given information in the model. Students also use tables to organize their search for pairs of numbers with a given sum and product. Puzzling and Persevering All of the puzzles in Unit 10 MysteryGrids, Who Am I? puzzles, Mystery Number puzzles, and area model puzzles are designed to engage students in mathematics related to factoring. These puzzles offer a natural contet for perseverance in factoring, as students must look around, make sense of given information, identify a reasonable starting place, consider potential solving strategies, identify reasonable net steps, and determine when the puzzle is solved. 56, 30, 40, 2, + 42, 0, 1, + 14, + 35, 2 + 1, + T5

14 Who Am I? puzzles in the Additional Practice of Lessons 2 and 4 and in the Important Stuff of Lesson 3 require students to consider pairs of numbers with a given product and sum. Mystery Number puzzles, such as the problem with butterflies shown here, appear as contets for using the zero product property to solve systems of equations in Lessons 5 and 6. Who Am I? t + u = 10 tu = 21 t > u = + = t u = A monic quadratic trinomial is a polynomial with three terms (trinomial) with a highest power of two (quadratic) and a coefficient of 1 on the leading term (monic). For eample, the epression is a monic quadratic trinomial. Mental Mathematics: Factors, Products & Sums and Percentage Calculations There are two strands of Mental Mathematics for this unit. The first strand connects directly to the mathematical content of Unit 10; students consider the factor pairs of a given number, calculate the product and sum for pairs of numbers, and identify which pair of numbers will yield a given product and sum. These activities support the factoring of monic quadratic trinomials presented in the lessons. The second strand of Mental Mathematics in this unit builds on students eperiences multiplying and dividing by 10, 2, and 5 in Units 1, 4, and 5, etending these ideas to calculations with percentages. Students calculate 5%, 10%, 15%, and 20% of given numbers and later also percent off, which requires subtraction. These ideas are likely to be relevant contets for mental mathematics in students lives. Eplorations The two Eplorations in this unit, Area Model Cutouts and Signed Area Model Cutouts, support greater understanding of factoring with quadratic polynomials through a hands-on activity. Students arrange 2,, and square unit paper cutouts into rectangles and record their results using area models and equations. Related Activity In the Related Activity, Area Model Puzzles, students build and share their own area model puzzles, reinforcing the structure and function of the area model and giving students an opportunity to create their own mathematics. T6 Transition to Algebra Unit 10: Area Model Factoring TEACHING GUIDE

15 Lesson 3: Factoring PURPOSE This lesson introduces factoring monic quadratic trinomials by identifying which calculations result in which pieces of the product and by considering how finding sums and products relates to factoring. By eploring equations of the form 2 + _ + 12 and finding that there is only one possible factor pair for a given middle term, students prepare for the product/sum search involved in factoring. Mental Mathematics Begin each day with five minutes of Mental Mathematics (pages T37 T48). In today s lesson, students will see how to use both products and sums in factoring quadratic trinomials. Launch: The Roles of Sums and Products Allow time for students to consider PROBLEMS 1 4 in the Student Worktet. Lesson at a Glance Mental Mathematics (5 min) Launch: The Roles of Sums and Products (10 min) Students work through the first four problems in the Student Worktet and discuss the roles of sums and products in factoring trinomials into binomial pairs. Student Problem Solving and Discussion (20 min) Allow time for students to work through the Important Stuff and eplore additional problems. Students identify pairs of numbers with a given product and then pairs with given sums, and they use this information to factor several epressions. Reflection and Assessment (10 min) Snapshot Check-in Unit 10 Related Activity: Area Model Puzzles (See page T31 and Student Worktet page 37.) Problems 1 and 2 ask about the equation ( 2)( 5) = and its area model. 1 What calculation creates the 10 in ? -2 _ 2 What calculation creates the -7 in ? -2 + _ Then bring the group together to discuss student responses. Net, draw this problem on the board.? What if... What if students suggest something like this? = Discuss the form that the factors will take: ( + _) ( + _). Remind students that not all trinomials are factorable, but for now, we will be looking only at ones that are. Ask students to identify the calculations in the model that result in the 18 and 9 terms, and work together to identify pairs of numbers with a product of 18 and a sum of 9. Write the factors out, and verify that these are the correct factors by completing the model. As students offer suggestions for filling in the empty spaces, challenge them to eplain their reasoning. Draw students attention to the term on the outside left. This term requires that any term in that row have a factor of as well (unless the other factor had a term with an in the denominator, but this is not common in introductory algebra). Then ask students to identify the term that must go at the top of the rightmost column. Likely, someone will notice that the 3 actually belongs there. Lesson 3: Factoring T13

17 Listen for students who use the model to eplain why the two missing pieces of the factors will have a product of 24 and a sum of 11. Students should be able to describe that because the area of the lower right rectangle is 24, the two side lengths will have a product of 24 and because the remaining two pieces of area will each have one factor of, the coefficients of these two like terms will have to sum to 11.»» How do the Mental Mathematics activities we ve been doing in this unit relate to factoring? Listen for students who identify the connection between finding the pair of numbers with a given product and sum and the thinking required for factoring.»» How can we determine what size area model will best help us factor an epression? Remind students that not all epressions are factorable and factoring often does require some trial and error. Present epressions like y 25, , and Whenever there is a common factor among all the terms in the epression (such as 5 in the first epression), it is helpful to use a 1 something model (in this case, 1 3) to factor out the common factor; then additional factoring can take place from there as needed. In the epressions and 2 25, however, students may identify that there is no common factor, so using a 2 2 model allows for the 2 term and the constant term to be placed in different rows and columns. Lesson 3: Factoring T15

18 CHECK IN Student Reflections & Snapshot Check-in Ask students to reflect on their learning: What are some things you ve learned so far in this unit? What questions do you still have? Assess student understanding of the ideas presented so far in the unit with the Snapshot Check-in on page T33. Use student performance on this assessment to guide students to select targeted Additional Practice problems from this or prior lessons as necessary. So far in Unit 10, students have: Translated area models into algebraic equations showing multiplication and division. Used area models to divide algebraic epressions by a factor. Completed area model puzzles with various pieces of information omitted. Considered the use of factoring in simplifying rational epressions. Used area models to factor monic quadratic trinomials. Students have been developing the following Algebraic Habits of Mind: Puzzling and Persevering Students have worked through area models inside out, using available clues to work out the missing information. They have also solved Who Am I? number puzzles that require thinking similar to factoring finding two numbers with a given product and sum. Using Tools Strategically Students have etended their use of the area models to support their thinking about division and factoring. Seeking and Using Structure Students have connected the structure of the information provided in or found using area models to the structure of algebraic equations involving multiplicative operations with polynomials. T16 Transition to Algebra Unit 10: Area Model Factoring TEACHING GUIDE

19 Snapshot Check-in Name: Use the area model to write three equations: one using multiplication and two using division Draw an area model and use it to answer this division problem. 2 12w 20 4 = _ Fill in the missing information to complete this area model and the equation that goes with it. 3 v 2 2v 5v 10 4 v = 5 List all the pairs of integers (positive or negative) whose product is -15. Use an area model to factor. Complete the model and equation = by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only. Unit 10: Area Model Factoring Snapshot Check-in R5 Snapshot Check-in 2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only. T33

20 Snapshot Check-in Name: ANSWER KEY Use the area model to write three equations: one using multiplication and two using division ( 4) = = = 2 Draw an area model and use it to answer this division problem. 2 12w 20 4 = _ 3w 5 3w w -20 Fill in the missing information to complete this area model and the equation that goes with it. 3 v v 2 2v 5 v 2 5v v (v + 5)(v + 2) v = 5 List all the pairs of integers (positive or negative) whose product is Use an area model to factor. Complete the model and equation = ( 5)( + 3) (Factors can be written in either order.) 2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only. Unit 10: Area Model Factoring Snapshot Check-in R6 Transition to Algebra Unit 10: Area Model Factoring TEACHING GUIDE T by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.

21 Unit 10 Mental Mathematics Factors, products & sums and percentage calculations The two mental mathematics activities in this unit are not related to each other but are both useful in high school mathematics. Students work with number pairs corresponding to specific products and sums, which supports the product-sum search in factoring quadratic trinomials. The structure of these activities differs in format from the previous ones in that inputs and outputs comprise multiple pieces of information. Then students review common percentages. Students make mental calculations, such as finding 20% off prices epressed in dollar amounts. The percentage activities serve multiple purposes: they re-animate skills and associations students have already developed, and they give the opportunity for the lively, fast-paced back-and-forth that students may have been missing in the first five activities. In nearly all of these mental mathematics activities, students enact a function : an input-output rule is established at the outset, and students give the output for each input they hear. Each function rule focuses on a key mathematical idea or property (e.g. complements or the distributive property) that students begin to feel intuitively. After introducing the day s task, the teacher deliberately does not reiterate the task but says only the input numbers for students to transform. Minimizing words lets students focus on the numerical pattern of the activity, helping them perceive the structure behind the mathematics. A lively pace maimizes practice and keeps students engaged. T37

22 \$90 \$81 Mental Mathematics Activity 9 10% off PURPOSE Students keep mental track of two processes: they find 10% of a given number and then they subtract it from the original number. 10% off is a familiar sight for students and one worth understanding. Do not pressure students to stop verbalizing the value for 10%. Encourage them, always, to whisper intermediate steps to themselves if they find that helps. Competent mathematicians will do the same! The easiest way to calculate a 90% off sale is to take 10% of the price! Introduce: When you see a sale that says 10% off, do you know what that means? It s not the same as 10% of. Let students respond. Sure, you look at the price, figure out 10% of it, and then subtract that from the original price. Today, you ll take 10% off every price I name. So, if I say \$200, what would you say? Start by first saying softly to yourself what 10% of that is. Right, \$20. Now subtract that from \$200. Right, \$180. What if I said \$400? You d say... yes, \$40, then \$360. You ve got it. Let s keep going! About this sequence: Calculating 10% off is different from anything students have been asked to mentally calculate before, so both steps consist mostly of inputs that are deliberately easy calculations. Step 1: Have students find 10% off each price you give. Use multiples of 10 and 100, and encourage students to verbalize (softly) their calculation of 10% before calling out their final answer. Input 10% Output (Input minus 10%) \$10 \$1 \$9 \$300 \$30 \$270 \$500 \$50 \$450 \$20 \$2 \$18 \$80 \$8 \$72 \$2000 \$200 \$1800 Step 2: Encourage students to keep whispering the interim step (finding 10%) to themselves, but have them say only the final result (10% off) out loud. Include some values that are not multiples of 10. \$80 \$72 \$20 \$18 \$4 \$3.60 \$10 \$9 \$200 \$180 \$400 \$360 \$450 \$405 \$70 \$63 \$25 \$22.50 \$75 \$67.50 \$300 \$270 \$90 \$81 \$4000 \$3600 \$170 \$153 \$5 \$4.50 \$220 \$198 \$150 \$135 \$6.99 \$6.29 \$230 \$207 \$1.99 \$1.79 T46 Transition to Algebra Unit 10: Area Model Factoring TEACHING GUIDE

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