Unit 10 Area Model Factoring Research-based National Science Foundation-funded Learning transforms lives. TTA_U10_SB_DS_r6.indd 1 1/18/14 12:35 AM 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. x The process of discovering the factors that x2 can be multiplied to give a number or expression is called factoring. x x x x 1 1 1 1 x2 + 5x + 4 = (x + ____ )(x + ____) Unit 10 uses area models to make sense of factoring. You’ve already learned how to multiply two algebraic expressions using an area model: x x -2 x2 -14 7 (x + 7)(x – 2) = x 2 + _____ – 14 We can also use area models to un-multiply, to start with an algebraic expression and factor it, that is, find two other expressions whose product is that original expression. x x ____ ____ x2 3x 15 x2 + 8x + 15 = (x + ____ )(x + ____) 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 TTA_U10_SB_DS_r6.indd 2 1/18/14 12:35 AM Lesson 3: Factoring IMPORTANT STUFF Problems 1 and 2 ask about the equation (x – 2)(x – 5) = x2 – 7x + 10 and its area model. What calculation creates the 10 in x2 – 7x + 10? 1 x -2 x x2 -2x -5 -5x 10 -2 • ____ What calculation creates the -7x in x – 7x + 10? 2 2 -2x + ____ Problems 3 and 4 ask about the equation (x + 3)(x – 8) = x2 – 5x – 24 and its area model shown below. x -8 x 3 x2 3x -8x 3 What calculation creates the -24 in x2 – 5x – 24? 4 What calculation creates the -5x in x2 – 5x – 24? -24 List all the pairs of integers (positive or negative) whose product is 12. 5 6 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. 10 x2 + 7x + 12 = _______________________ x x x2 + 13x + 12 = _______________________ x ___ 12 TTA_U10_SB_DS_r6.indd 12 x x 12 x x2 + 8x + 12 = _______________________ ___ x2 ___ 12 11 x x 12 12 x2 – 7x + 12 = _______________________ ___ x2 x2 ___ 13 ___ ___ ___ x2 12 Unit 10: Area Model Factoring 1/18/14 12:35 AM Complete the model and finish Jay’s thought. 14 x2 + 3x + 2 = ( x x 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 ___. x2 The numbers are ___ and ___ so the factors must 2 ___ 15 )( be ________ and ________. List all the pairs of integers (positive or negative) whose product is 30. 16 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? 21 x2 – 13x + 30 = _______________________ Use an area model to factor. Complete each model and equation. 20 x2 + 11x + 30 = _______________________ x x x 27 Who Am I? • t+ u = 10 • tu = 21 • t> u Who Am I? • t+ u = 12 • tu = 36 Lesson 3: Factoring TTA_U10_SB_DS_r6.indd 13 23 x2 – 17x + 30 = _______________________ ___ x x2 ___ 30 ___ x2 + 31x + 30 = _______________________ 30 ___ t t u u ___ x2 x 30 ___ x2 x 30 x 24 x x2 ___ 22 ___ 25 Who Am I? • t+ u = 10 • tu = 25 t u 28 Who Am I? • t+ u = 11 • tu = 24 • t< u t u 26 Who Am I? • t+ u = 11 • tu = 30 • u > t t u 29 Who Am I? • tu = 81 • t+ u = 18 t u 13 1/18/14 12:35 AM STUFF TO MAKE YOU THINK 30 For some expressions, factoring with the area model doesn’t give any new information. For example, it’s not helpful to fill in the outside of this model 1 ____ because b2 and 4 have no common factors besides 1. b _____ 2 4 _____ b2 4 An expression like b2 + 4 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 expression. If so, complete the area model. If not, cross it out and label it prime. a ____ _____ _____ 2h 9 ____ _____ _____ 6p 60 _____ c ____ 4w2 ____ 8w _____ d ____ 8w2 ____ 3 g e ____ _____ _____ _____ 10m2 30m 80 _____ f _____ _____ c2 8c ____ ____ 31 b h _____ 30a2 7w ____ 24 3c _____ ____ 15a ____ 25b2 ____ 5c Here are four ways to set up the expression n2 + 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 ____ _____ _____ _____ n2 9n 14 _____ _____ ____ n2 9n ____ 0 14 _____ _____ b _____ _____ ____ n2 7n ____ n2 3n ____ 2n 14 ____ 6n 14 c 14 TTA_U10_SB_DS_r6.indd 14 d Unit 10: Area Model Factoring 1/18/14 12:35 AM 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. 32 34 Draw how you would arrange 5 roads so there are 6 intersections in the village. Factor the expression r2 + 5r + 6. ___ ___ N W E 33 Sketch a village map in which 7 roads form 10 intersections. 35 Factor the expression r2 + 7r + 10. S ___ r2 ( 6 ___ )( r2 ) 10 36 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 exactly 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. 38 ___ ___ 39 x2 ___ x2 + 712x + 11 = ( Lesson 3: Factoring TTA_U10_SB_DS_r6.indd 15 ___ ___ 11 2x 2x 2 ___ )( ) ___ 2x2 + 5x + 2 = ( 2 )( ) 15 1/18/14 12:35 AM Additional Practice Factor each expression. A a2 + 15a + 56 = ___ ___ a 7a ___ 56 ___ 2 ___ ___ C ___ x2 – 6x – 40 = ___ E ___ x2 ___ -10x D ___ I K x2 + 3x = ____ x ( ) ___ 1 ___ ___ ___ x2 ___ 92x H ___ ___ J -3x ___ 30 ___ y ___ x2 ___ L 6x 7x2 ___ k2 + 2k – 48 = 19 x2 – 14x + 33 = ___ ___ 1 ___ x2 – 13x + 30 = ___ 7x2 + 42x + 7y = b 4q x 42 x2 + 11 2 x + 19 = F q2 + 4q + bq = ___ xy -40 72 q There is no squared term here. x z ___ ___ ___ z2 + 17z + 72 = ___ G xy + 7x + 6y + 42 = B 33 r2 + 8r + 7 = Draw your own area models. 16 TTA_U10_SB_DS_r6.indd 16 Unit 10: Area Model Factoring 1/18/14 12:35 AM 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 ____ ____ _____ O ____ 10a2 ____ 6a P ____ _____ _____ ____ p2 2p ____ 10p 20 Q S N R ____ _____ _____ 2n2 16n _____ _____ _____ 3x2 24x 15 _____ _____ _____ ac 3a 4a2 Here are four ways to set up the expression w2 + 11w + 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 ____ _____ _____ _____ w2 11w 30 _____ _____ ____ w2 11w ____ 10 20 _____ _____ ii _____ _____ ____ w2 10w ____ w2 6w ____ w 30 ____ 5w 30 iii Lesson 3: Factoring TTA_U10_SB_DS_r6.indd 17 iv 17 1/18/14 12:35 AM Unit TEACHING GUIDE Area Model Factoring Unit 10 10 Teaching Guide June Mark E. Paul Goldenberg Mary Fries Jane M. Kang Tracy Cordner Un it 10 Are Fac a Mo tori del ng HEINEMANN Portsmouth, NH Research-based National Science Foundation-funded Learning transforms lives. TTA_U10_TG_DS_r6a.indd 1 3/11/14 10:26 1:48 PM 4/14/14 AM firsthand An imprint of Heinemann 361 Hanover Street Portsmouth, NH 03801-3912 www.heinemann.com Offices and agents throughout the world Education Development Center, Inc. 43 Foundry Avenue Waltham, MA 02453-8313 www.edc.org © 2014 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-0917958. Opinions expressed 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, except by a reviewer, who may quote brief passages in a review, and with the exception 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 Worktext may be colored incorrectly; the shading on the lower half of the model may be missing. (Compare the Worktexts 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 (x + 3)(x + 2) model. Transition to Algebra, Unit 10: Area Model Factoring Teaching Guide ISBN-13: 978-0-325-05324-0 Transition to Algebra Teacher Resources ISBN-13: 978-0-325-05790-3 Transition to Algebra, Unit 10 Student Worktexts 10-pack ISBN-13: 978-0-325-05312-7 Transition to Algebra Student Worktexts, 10 Sets of All 12 Units ISBN-13: 978-0-325-05791-0 Printed in the United States of America on acid-free paper 18 TTA_U10_TG_DS_r6a.indd 2 17 16 15 14 RRD 1 2 3 4 5 4/14/14 10:26 AM 10 Unit Area Model Factoring CONTENTS T4 Unit Introduction T7 T10 T13 T16 T17 T20 T22 T25 T26 T29 T31 Lesson 1: Division Undoes Multiplication Lesson 2: Area Model Inside Out Lesson 3: Factoring Student Reflections & Snapshot Check-in Lesson 4: Products, Sums, and Signs Lesson 5: Zero Product Property Lesson 6: Solving by Factoring Student Reflections & Unit Assessment Exploration: Area Model Cutouts Exploration: Signed Area Model Cutouts Activity: Area Model Puzzles RESOURCES T32 T33 T34 T35 T36 Finding Pairs Cutout (Lesson 5) Snapshot Check-in Snapshot Check-in Answer Key Unit Assessment 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 TTA_U10_TG_DS_r6a.indd 3 4/14/14 10:26 AM Unit 10 Area Model Factoring Research-based National Science Foundation-funded Learning transforms lives. 2/6/14 3:43 PM 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. UNIT 10 Area Model Factoring F actoring 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 explore area model puzzles that provide enough information •Factor monic quadratic trinomials by finding the two numbers with a given sum and product. about the factors and product to find the missing terms. This supports students in •Use the zero product property to solve quadratic equations. for factoring. The factoring problems in this unit focus primarily on expressing trinomial understanding the roles of and relationships between the pieces of the area model and their corresponding equations and in thinking flexibly with the model in preparation 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 (x + 3)(x – 4), where the FOIL (first, outer, inner, last) approach is often used, Factor x2 + 4x – 77 factoring can also be seen as several x -7 different processes. For instance, 2 -7x factoring x + 4x – 77 can appear x2 x altogether different from simplifying -77 11x 11 a rational expression like 9hx + 6h or 3h x 2 + 4x – 77 = (x – 7)(x + 11) solving an equation like x2 – 8x = 0. T4 Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE TTA_U10_TG_DS_r6a.indd 4 4/14/14 10:26 AM The area model provides a context in which students can unify their understanding of these seemingly different processes with a single tool. Simplify 3h 9hx + 6h 3h 3x 2 9hx 6h x2 – 8x = 0 x 9hx + 6h = 3h(3x + 2) 3h x -8 x2 -8x x 2 – 8x = x(x – 8) = 0 Either x = 0 or x – 8 = 0 Area Model Puzzles As Unit 4 did to some extent, 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 3m ____ other empty spaces around the puzzle until they’ve 6mj 14j ____ filled in all the blanks. This is, of course, neither a 35 5 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 exactly where to start. In a pure factoring problem, the puzzle might take a bit of experimentation 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, + 56, • TTA_U10_TG_DS_r6a.indd 5 2x2 + x, + 0, • 35, • Seeking and Using Structure In this unit, students use the 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 expressions and solving equations by factoring. Equations such as (x – 1)(x + 2) = 0 are viewed as two separate “chunks” of information: either x – 1 = 0 or x + 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 extend 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 context 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 next steps, and determine when the puzzle is solved. 2x, + 40, • 42, • 14, + 2, + 48, • 30, • MysteryGrid 0, 1, x, x2 Algebraic Habits of Mind x 1, + x2 + 1, + T5 4/14/14 10:26 AM 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 contexts for using the zero product property to solve systems of equations in Lessons 5 and 6. t Who Am I? u • t + u = 10 • tu = 21 • t > u • = + = = _____ Mental Mathematics: Factors, Products & Sums and Percentage Calculations 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 example, the expression x2 – 2x – 63 is a monic quadratic trinomial. 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’ experiences multiplying and dividing by 10, 2, and 5 in Units 1, 4, and 5, extending 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 contexts for mental mathematics in students’ lives. Explorations The two Explorations 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 x2, x, 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 TTA_U10_TG_DS_r6a.indd 6 4/14/14 10:26 AM Lesson at a Glance Lesson 3: Factoring Mental Mathematics (5 min) 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 exploring equations of the form x2 + ____ + 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 (10 min) •Students work through the first four problems in the Student Worktext 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 explore 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 expressions. Reflection and Assessment (10 min) Snapshot Check-in Unit 10 Related Activity: Area Model Puzzles (See page T31 and Student Worktext page 37.) Launch: The Roles of Sums and Products Allow time for students to consider PROBLEMS 1–4 in the Student Worktext. Problems 1 and 2 ask about the equation (x – 2)(x – 5) = x2 – 7x + 10 and its area model. 1 What calculation creates the 10 in x2 – 7x + 10? -2 • ____ 2 What calculation creates the -7x in x2 – 7x + 10? -2x + ____ x -2 x x -2x -5 -5x 2 10 Then bring the group together to discuss student responses. Next, draw this problem on the board. ? What if . . . What if students suggest something like this? x + 9x + 18 = _____________________ 2 x x ___ ___ x ___ x2 18 Discuss the form that the factors will take: (x + ____) (x + ____). 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 9x 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 explain their reasoning. x ___ x2 3 18 Draw students’ attention to the x term on the outside left. This term requires that any term in that row have a factor of x as well (unless the other factor had a term with an x 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 TTA_U10_TG_DS_r6a.indd 13 T13 4/14/14 10:26 AM Student Problem Solving and Discussion Allow time for students to work on the problems in the Student Worktext. Listen for students describing two things: • The coefficient of the x term is the sum of the two numbers in the factors. • The number (the constant term) at the end of the trinomial is the product of these same two numbers. If some students don’t complete the Thinking Out Loud box on page 13, spend some time analyzing the model using Jay’s thoughts: talk about the model, how it works, why it works that way, where there are factors, products, sums, and terms, and how they relate. Ask about this model and others on this page. Ask students to tell you which models they’ve understood, and then help them connect what they understand to this model. In PROBLEMS 5–13, students list all of the integer pairs that result in the product 12, then identify which pairs have various sums, and complete area models to factor trinomials of the form x2 + bx + 12, where b varies among the sums. Students then, in PROBLEM 14, complete a statement of reasoning about the process before working through a similar set of problems with a product of 30. Complete the model and finish Jay’s thought. 14 x2 + 3x + 2 = ( )( Thinking Out Loud ) Jay: I can tell from the equation that I need a pair of ___ x numbers whose product is 2 and whose sum is ___. x2 x The numbers are ___ and ___, so the factors must 2 ___ be ________ and ________. PROBLEMS 24–29 feature Who Am I? puzzles in which students search for numbers with a particular product and sum. 24 Who Am I? t u t Who Am I? 25 • t+ u = 10 • t+ u = 10 • tu = 21 • tu = 25 u • t> u Starting with an expression and using the area model to find the original factors is what has been called “un-multiplying.” Feel free to use this term as a way to help students see the interconnectedness between multiplication, division, and factoring. In the Stuff to Make You Think, students consider problems in which it is not helpful to factor, which arrangements of terms in area models are helpful, and several perpendicular street diagrams that use similar sum and product reasoning to that in the lesson. PROBLEMS 32–37 can be completed with manipulatives like toothpicks or popsicle sticks. Tough Stuff PROBLEMS 38 & 39 feature more difficult trinomial factoring, such as an expression with a fractional coefficient and a non-monic trinomial. ___ 38 ___ 39 x2 ___ ___ 11 ___ 1 x2 + 7 2 x + 11 = ( 2x 2x 2 ___ )( ) ___ 2 2x2 + 5x + 2 = ( Questions such as the following can support discussion: »» If you know that the area model for x + 11x + 24 will look like this (show the model at right), how can you determine what goes in the other two boxes? 2 T14 x ___ )( ) x ___ x2 24 Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE TTA_U10_TG_DS_r6a.indd 14 4/14/14 10:26 AM Listen for students who use the model to explain 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 x, 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 expression? Remind students that not all expressions are factorable and factoring often does require some trial and error. Present expressions like 5x2 + 35xy – 25x, x2 + 2x – 35, and x2 – 25. Whenever there is a common factor among all the terms in the expression (such as 5x in the first expression), 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 expressions x2 + 2x – 35 and x2 – 25, however, students may identify that there is no common factor, so using a 2 × 2 model allows for the x2 term and the constant term to be placed in different rows and columns. TTA_U10_TG_DS_r6a.indd 15 Lesson 3: Factoring T15 4/14/14 10:26 AM 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 expressions by a factor. •Completed area model puzzles with various pieces of information omitted. •Considered the use of factoring in simplifying rational expressions. •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 extended 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 TTA_U10_TG_DS_r6a.indd 16 4/14/14 10:26 AM Snapshot Check-in Name: Use the area model to write three equations: one using multiplication and two using division. 1 2x x -4 2x2 -8x 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. ___ ___ ___ v2 2v ___ 5v 10 3 5 4 v2 + ______ + 10 = __________________ List all the pairs of integers (positive or negative) whose product is -15. Use an area model to factor. Complete the model and equation. 6 x2 – 2x – 15 = _______________________ x x ___ x2 ___ -15 ©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 Snapshot Check-in T33 R5 ©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only. TTA_U10_TG_DS_r6a.indd 33 4/14/14 10:26 AM Snapshot Check-in Name: ANSWER KEY Use the area model to write three equations: one using multiplication and two using division. 1 2x x -4 2x2 -8x 2x(x – 4) = 2x2 – 8x 2x2 – 8x =x– 4 2x 2x2 – 8x x – 4 = 2x Draw an area model and use it to answer this division problem. 2 12w – 20 = ________________ 3w – 5 4 4 3w -5 12w -20 Fill in the missing information to complete this area model and the equation that goes with it. 3 v ___ 5 ___ 5 v ___ ___ 2 v2 2v 5v 10 4 v2 + ______ 7v + 10 = __________________ (v + 5)(v + 2) List all the pairs of integers (positive or negative) whose product is -15. 1 • -15 -1 • 15 3 • -5 -3 • 5 Use an area model to factor. Complete the model and equation. 6 (x – 5)(x + 3) x2 – 2x – 15 = _______________________ x 3 ___ x -5 ___ x 2 -5x 3x -15 (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 T34 Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE R6 ©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only. TTA_U10_TG_DS_r6a.indd 34 4/14/14 10:26 AM 10 Unit 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 expressed 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 maximizes practice and keeps students engaged. T37 TTA_U10_TG_DS_r6a.indd 37 4/14/14 10:26 AM Mental Mathematics • Activity 9 $90 $81 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. 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. •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! 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. 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. Input 10% Output (Input minus 10%) $80 $72 $20 $18 $10 $1 $9 $4 $3.60 $10 $9 $300 $30 $270 $200 $180 $400 $360 $500 $50 $450 $450 $405 $70 $63 $20 $2 $18 $25 $22.50 $75 $67.50 $80 $8 $72 $300 $270 $90 $81 $2000 $200 $1800 $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 TTA_U10_TG_DS_r6a.indd 46 4/14/14 10:26 AM