Planting the Seeds of Algebra, 3-5: Explorations for the Upper Elementary Grades

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Monica Neagoy, Francis (Skip) Fennell & Keith Devlin

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    Foreword by a Mathematics Educator

    She's done it again! Monica Neagoy has authored a second book that deftly presents important foundations of algebra while celebrating mathematics through carefully crafted explorations, all of which include student and teacher vignettes and comments about the mathematics they have learned and are teaching. Wow. When I read this book I felt like I was in a classroom!

    Monica Neagoy's gift is in her ability to “tell a story” while she deeply and carefully explores really important, and challenging, mathematics. This “story” digs deep into mathematics topics in grades 3–5 (and beyond) that are foundational to algebra as it is developed at the elementary school level and extended through the middle and high school grades.

    Classroom teachers, particularly those for whom this book is intended, need this book. They will deepen and extend their mathematical understandings from the discussions provided in each chapter and leave each discussion with a treasure trove of activities that they can use in their own classrooms. But this book is so much more than a professional contribution for mathematics content and pedagogy. It's a resource that will be used—regularly—as in daily. Monica's Mottos themselves could provide the topic for grade level or building learning community discussions. Consider discussing “Address the Whys, Not Just the Hows,” “Instill a Sense of Wonder,” or “Foster Friendliness With Numbers.”

    Finally, as most of the country addresses and moves forward in implementing the Common Core State Standards for Mathematics, a key issue will be the establishment of curricular opportunities that focus on important mathematics, are coherent across both developmental levels of students and mathematical topics and expectations, and, indeed, that challenge students to dig deeper! This book does that. Read it! Enjoy the story.

    Dr. Francis (Skip) FennellMcDaniel College, Westminster, MD Past President of the National Council of Teachers of Mathematics (NCTM) L. Stanley Bowlsbey Professor of Education and Graduate and Professional Studies Project Director, Elementary Mathematics Specialists and Teacher Leaders Project (ems&tl)

    Foreword by a Mathematician: Algebra: What's the Big Deal?

    In a world where we all carry powerful computing devices around with us in our pockets, the need to be skilled in arithmetic has greatly diminished. Instead, today we have to be skillful users of those computing machines. That has greatly ramped up the importance of algebra. For algebra is, to all intents and purposes, the language computers use.

    Unfortunately, the education system has not kept pace with the changes in our lives brought about by the computing revolution. Algebra teaching today is largely unchanged from the ninth century, when the Arabic-speaking mathematician al-Khwārizmī wrote the world's first algebra textbook. (The term al-jabr in the book's title, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqābala, is where we get the name algebra.) Back then, and right up to the 1960s, one of the keys to a successful and hopefully prosperous life was mastery of arithmetical and algebraic tools for solving numerical problems. But today, cheap computers we carry around in our pockets do arithmetic computations for us, compute the values of algebraic formulas, and solve algebraic equations. What's more, they do so faster and more accurately than we can. People no longer need to acquire that mastery.

    That does not mean that arithmetic and algebra are no longer important. Indeed, they are used much more today than at any other time in history. It's just that the grunt work is now done by machines. That shifts the focus people must bring to arithmetic and algebra. Today, we have to be accomplished users of those computers. This is why mathematicians and math teachers have started using the terms “arithmetical thinking” and “algebraic thinking” (and more generally, “mathematical thinking”). Those terms mean the kind of thinking required in order to solve problems—real problems that arise in today's world—with the aid of arithmetic and algebra, frequently using computers. Modern computer systems also eliminate the need to craft artificial problems that can be solved using paper-and-pencil methods. Today, it is possible for learners to work on real problems—though it still makes sense for beginners to start with simple ones.

    Algebraic thinking, the focus of this book, requires far more conceptual understanding than the basic algebraic skills of times past. Today, we don't (at least, we shouldn't) teach algebra with the goal of fast, accurate (and generally mindless) manipulation of symbolic expressions. Our machines can do that, far faster and more reliably than we can. Rather, what today's world requires is people who can look at a real-world problem and understand it from an algebraic perspective, see what steps are required to solve it, and then set about finding a solution—using a computer where appropriate. Put simply, our computers can do almost everything that was the focus of most algebra teaching of the past. Give a computer a problem from an old algebra book, and it will give you a correct solution. But that's all the computer can do. It does not understand the world. It actually does not understand anything. It can just follow rules. To use a computer to solve a real-world problem, a person has to first turn that problem into an algebraic expression or equation that a computer can accept, and then take the computer's solution and interpret it in real-world terms. That requires not only knowing what the rules are that the computer will follow but also understanding them—the part the computer cannot do.

    That is why the focus in this book is on understanding problems in an algebraic way, and understanding the methods of algebra. This is a very different way of teaching algebra than today's parents experienced (and today's teachers, for that matter). As a result, both parents and teachers may initially find some things unfamiliar. But if you use the book as your guide, you should do okay. People who learned algebra in the past would frequently be heard to say, “I can do algebra, but I never really understood it.” Back in earlier times, you could often get by with that level of mastery. But those days are gone, never to return. In today's world, if your mastery of algebra is purely procedural (i.e., manipulation of symbolic expressions), you will lose out every time to a faster, cheaper, more accurate computer. In fact, once you get past that initial strangeness, you will make a surprising discovery. Once you approach algebra from an understanding perspective, it actually turns out to be a whole lot easier than trying to master all those rules for moving symbols around! So much so, that there is no reason why algebra (i.e., algebraic thinking) cannot be taught in the elementary grades. Indeed, that's when we should start teaching it!

    Remember, in the 21st century, no one is going to be hired for being able to do something a computer can do. (Better, faster, and cheaper wins every time.) Instead, our students—indeed, everyone—need to master something the computer cannot: understanding algebra! This book will help you, your children, and your students do just that.

    —Dr. Keith DevlinStanford University, CA Co-Founder and Executive Director of the Stanford University's H-STAR Institute Co-Founder of the Stanford Media X Research Network The Math Guy on NPR's Weekend Edition

    Acknowledgments

    Writing this second book has been yet another edifying journey. The process took me to private places of the mind, spirit, and heart as I dug deeply into my multiple teaching and learning experiences; it also took me to public places—different countries, schools, and classrooms—where I connected with teachers and teacher leaders who directly or indirectly inspired me, enriched me, and enlightened me. Collectively, they helped me shape the goals and content of Planting the Seeds of Algebra, 3–5.

    I thank my parents who instilled a value for mathematics, my schoolteachers who nurtured my love for mathematics, my university professors who opened windows onto new learnings, and my favorite authors and researchers who shed new light on old ideas. I especially thank the myriad teachers and administrators of the elementary schools—public, private, independent, and international—I have worked with over the past 25 years. Their questions, comments, and insights always prompted me to go deeper and search further. Their desire to excite children about mathematics nurtured my own. I will not risk listing all their names for fear of leaving out an important one. For this reason, I hope they will understand—as they know well who they are—that my gratitude to them is endless.

    That said, I would be remiss not to specifically acknowledge a few individuals who helped me bring this book to life: I thank Cathy Hernandez, who first contacted me to write for Corwin; associate editor Desirée Bartlett and her assistants Ariel Price and Andrew Olson, who supported my work with patience and consideration through the development stage; and Melanie Birdsall, who diligently and graciously guided the production stage of the book. In short, I thank the entire Corwin team. A special thanks also goes to my wonderful copy editor Cate Huisman.

    On a more personal note, I have been blessed with a supportive and thoughtful husband, who has lived this book with me through its countless iterations. I am profoundly grateful to Didier, a writer and artist in his own right, my intellectual companion and loving mate, and the first editor of all my writings. Merci infiniment!

    Last, I wish to express my gratitude to all teachers who believe that children are capable of sophisticated and creative mathematical thinking and who guide them to delve deeper into the wondrous world of mathematics. They will help change the face of school mathematics in our culture!

    Publisher's Acknowledgments

    Corwin gratefully acknowledges the contributions of the following reviewers:

    Stacey Bennett Ferguson, Fourth-Grade Teacher North Bay Elementary School Bay Saint Louis, MS

    Dr. Nancy J. Larsen, Sixth-Grade Teacher Coeur d'Alene Charter Academy Coeur d'Alene, ID

    Dr. Joyce Sager, Inclusion Math Teacher Gadsden City High School Gadsden, AL

    About the Author

    Photo by Tim Coburn

    Monica Neagoy, international consultant and popular keynote speaker, has a contagious passion for mathematics. In addition to authoring books, over the course of a 25-year mathematics career she has done national program directing, teacher professional development, math specialist training, national and international conference speeches, live television courses, video creation and hosting, math apps conception, live math shows, including her popular MathMagic Show, and last but not least has helped the game of SWISH become a reality for Thinkfun. Whether in the United States or abroad, whether presenting in English, French, or Spanish, whether working with teachers, students, or parents, Monica's life-long goal is to infuse people with a fascination for the beauty, power, and wonder of mathematics.

    The seeds of Dr. Neagoy's love for mathematics took root on the beaches of Saipan, in the Northern Mariana Islands, where her father wrote problems in the sand for his daughters to solve. Pleasure, beauty, and mathematics were intertwined for her from the age of four. Her career in mathematics teaching began at Georgetown University. Later, she was invited to become a program director at the National Science Foundation, where she directed K–12 national math projects. As an independent consultant, she has served numerous school districts and systems across the country and notable organizations, including the Carnegie Institution of Washington, PBS TeacherLine, MATHCOUNTS, and the American Association for the Advancement of Science. Since 2004, her consulting has expanded to serve Europe, the Middle East, and Africa as well.

    Dr. Neagoy has had a parallel career in theatre as actor, choreographer, and stage director, notably with the professional LE NEON Theatre, which she codirected, in the Washington, DC, area. Her exposure to many cultures, mastery of several languages, double career in the arts and the sciences, and mindfulness training through yoga practice and teaching provide her with a unique perspective on the learning and teaching of mathematics. Her passions for mathematics and the arts converge in her popular keynote presentation, The Mathematics of Beauty and the Beauty of Mathematics.

    Dr. Neagoy combined her love for the arts and the sciences in the creation of a series of innovative mathematics videos for teachers and students—about 60 in all, including Discovering Algebra With Graphing Calculators for Discovery Education and Mathematics: What's the Big Idea? for the Annenberg Channel (now available on Annenberg Learner). Lately, she has been collaborating with two colleagues in the conception and creation of math apps for teachers and students.

    Dr. Neagoy was educated in the French school system, grades 1–12, in Asia and the United States. She has a BS in mathematics and philosophy from Georgetown University, an MA in pure mathematics from Catholic University, and a PhD in mathematics education from the University of Maryland.

    Acknowledgements

    To Didier, my partner on stage and in life who shares my ineffable joy of creation, I dedicate this book.

  • Appendix

    Answers to Nontrivial Problems in Chapter 3
    Beyond Circling Circles: More Problems to Explore

    Here you will find the answers and/or explanations to the nontrivial questions from Chapter 3.

    I. Next Steps
    4. Observations

    The follow-up exploration sparked off more questions:

    • It looks like the points line up. Why?

      The tips of all vertical strings line up because the ratio of vertical string (circumference) to horizontal string (radius), in each of the seven circles, is the same and equal to “a little more than 3,” or π.

    • It looks like (0, 0) lines up with the rest. Why?

      If the radius of a circle is zero then its circumference is also zero. Thus, the point (0, 0) is on the line. Note: In the upper grades students will learn that the quotient 0/0 is undefined.

    • What might other points on the line represent?

      Other points, (C, D), on the line represent the diameter and circumference values of infinitely many other possible circles, which we did not measure but can imagine.

    III. For your Students' Learning
    1. Fix One and Vary the Other
    Fixed Area

    Figure 3.A shows the graph of the varying perimeter, P, of a rectangle whose fixed area is 48 square units, as a function of its length, L. The blue points correspond to the (L, P) integer-value data pairs in Figure 3.B. The other points on the red curve correspond to all imaginable (L, P) data pairs with non-integer values. We see that the perimeter is the smallest for rectangles 6-by-8 or 8-by-6. These rectangles are the most compact. (Note: we can get smaller perimeters if we choose a rectangle with length between 6 and 8, but that's beyond the scope of grades-3-5 mathematics, though students could grasp the idea.) The perimeter is the largest for rectangles 1-by-48 or 48-by-1. These rectangles are the most elongated ones.

    Discuss with students how in some situations rectangles 6-by-8 and 8-by-6 can refer to different objects—namely when position is meaningful—and in others, such as in holding a rectangular box of chocolates, 6-by-8 and 8-by-6 can refer to the same object.

    Figure 3.A Graph of Perimeter (P) as a Function of Length (L)

    Fixed Perimeter

    Figure 3.C shows the graph of the varying area, A, of a rectangle whose fixed perimeter is 36 linear units, as a function of its length, L. The blue points correspond to the (L, A) integer-value data pairs in Figure 3.D. The other points on the red curve correspond to all imaginable (L, A) data pairs with non-integer values. We see that the area is the smallest for rectangles 1-by-17 or 17-by-1. These rectangles are the most elongated ones. (Note: we can get smaller areas if we choose a rectangle with length between 0 and 1 or between 17 and 18, but that's beyond the scope of grades-3-5 mathematics, though students could grasp the idea.) The area is the largest for the one and only special rectangle called a square. Indeed, when length equals width equals 9, the area attains the maximum value of 9 × 9, or 81 square units. This rectangle is the most compact.

    Figure 3.B Data Values: L is the Independent Variable and P is the Dependent Variable

    Compare and contrast the two sets of findings with students. For example, as a rectangle becomes more compact, the perimeter decreases but the area increases. Discuss the units of perimeter versus area and model them with a piece of string versus a square tile. Perhaps more challenging would be to observe and comment on the symmetry or nonsymmetry of the two graphs.

    Real-World Context: Family Reunion

    The number of people in this case can be thought of as the perimeter: each person sitting along one side of a unit table represents a linear unit of measure for the perimeter. Students will hopefully apply their learning from the previous problems: the more compact the table, the smaller the perimeter. The maximum and minimum perimeters are 50 and 20 (Figure 3.E).

    Figure 3.C Graph of Area (A) as a Function of Length (L)

    Figure 3.D Data Values: L is the Independent Variable and A is the Dependent Variable

    Figure 3.E Maximum and Minimum Number of People

    Figure 3.F Possible Student Suggestions

    Given the opportunity to explore different arrangements and justify their ultimate choice, students become more creative and begin to think outside the box. Welcome creativity! Figure 3.F shows two options that students may decide on. Make sure their arguments are convincing to their peers (conviviality, number of people, aesthetics, practicality, and so on). Discuss other arrangements and propose some yourself as well.

    Challenge

    Depending on the nibble, the perimeter could decrease, increase, or remain equal (Figure 3.G). The area, however, would always decrease no matter how big or small the size of the nibble.

    6. A Pattern of Non-Overlapping Triangles inside Polygons

    The number of non-overlapping triangles (T) in a dodecagon (a polygon with 12 sides) is ten. We conclude from our observation of the pattern of polygons with consecutive numbers of sides, n (3, 4, 5, 6, 7…) that as the number of sides n increases, the number of triangles T is always 2 less than n. Algebraically, we can write this relationship as T = n - 2 (Figure 3.H). This relationship will come in handy in middle school when students will be asked to figure out the sum of the measures of the interior angles in an n-sided polygon. Since an n-sided polygon has n - 2 non-overlapping triangles inside of it, and since the sum of the measures of the interior angles in any triangle is 180°, the answer is (n - 2) x 180°.

    Figure 3.G Depending on the Nibble, the Perimeter Could Decrease, Increase, or Remain Equal

    Figure 3.H The Number of Non-Overlapping Triangles in an N-Sided Polygon is N - 2

    7. How does it Change?

    The relationships students will discover in this exploration will help them make sense of the notion of dimension. It will also help them understand how perimeter, area, and volume change no matter what the shape of the figure or solid. As a first example, have students use a piece of string to model the 1-dimensional object, a rectangle made of unit tiles to model the 2-dimensional object, and a rectangular prism made of unit cubes to model the 3-dimensional object. Help them organize their findings in a chart or table to better discern the pattern. After doubling the dimensions of all three objects, try tripling, then quadrupling the objects' dimensions. Make a larger chart and compare all findings (Figure 3.I). What do they notice? Can they put it in words? Once students learn exponential notation (which begins in grade 5), this exploration will help them better understand that if all dimensions of an object are multiplied by a factor n (be it integer or not):

    • The length of a 1-dimensional object is multiplied by n1 or n
    • The area of a 2-dimensional object is multiplied by n2
    • The volume of a 3-dimensional object is multiplied by a factor of n3

    Figure 3.I This Chart Helps Students Discover how Length, Area, and Volume Change as the Dimensions of an Object are Multiplied by a Factor of 2, 3, 4, and So On.

    9. Practicing with Numeric and Algebraic Expressions
    • I will multiply 5 by a factor of four, for the four equal sides of length 5. The numeric expression is 4x5, where four is the multiplier. If the side length is any number of centimeters s, the algebraic expression for the perimeter is: P - 4 x s, which in later years will be simplified to P = 4s.
    • I can write several equivalent numeric expressions for the perimeter using 5, 7, and the addition symbol, including 5 + 5 + 7 + 7 or 5 + 7 + 5 + 7; or using 5, 7, and both the addition and multiplication symbols, which is more sophisticated: 2 x (5 + 7). If the side lengths are any numbers of centimeters l and w, the algebraic expression for the perimeter can be written is several equivalent ways, including P = l + l + w + w, P = l + w + l + w, and P = 2 x (l + w).
    10. Visualizing Unit Conversion
    Linear Measure

    There are 10 times more millimeters than centimeters, so 3 cm = 30 mm.

    Area Measure

    There are 100 times more square millimeters than square centimeters, so 12 sq. cm = 1,200 sq. mm.

    Answers to Nontrivial Problems in Chapter 6
    Beyond Fancy Fences: More Problems to Explore

    Here you will find the answers and/or explanations to the nontrivial questions from Chapter 6.

    I. Next Steps
    3. Growing Patterns IV

    See worksheet for Growing Patterns IV on page 198.

    III. For your Students' Learning
    1. Function Table Puzzles

    4. Folding to the Sky

    Young students, and even older ones, have a hard time grasping the power of exponential growth. The first few powers of 2 or 3 or 10 are clear, but as the number of factors—or the value of the exponent—increases, they lose sense of how big the numbers truly are. A wonderful film that conveys the power of the powers is the now classic film directed by Charles and Ray Eames and titled Powers of Ten. Click on this YouTube link to view it with your students: https://www.youtube.com/watch?v=0fKBhvDjuy0

    6. Operation Patterns
    Investigating
    • The first pattern 10 + 10 → 11 + 9 → 12 + 8 → 13 + 7 … is constant and each sum equals 20. Depending on students' number sense, some will continue beyond the sum “20 + 0” and venture into negative addends, which is a natural outgrowth of this pattern: 20 + 0 → 21 + -1 → 22 + -2 → 23 + -3 … To look at these sums from another perspective, place the consecutive pairs of addends on the number line and ask students to observe their relative positions. They will notice that the consecutive pairs of addends are always equidistant from 10, with one on either side of 10. Even the first two addends, 10 and 10, have an equal distance of zero from 10!
    • The second pattern 10 × 10? 11 × 9 → 12 × 8 → 13 × 7 … is constructed with the same pairs of numbers but with a different operation: multiplication in lieu of addition. The sequence of products gradually decreases, as each product is smaller than the preceding one. When the second factor becomes zero, the product will too. You can delve deeper: (1) Have students notice the “jumps down” or decrements between consecutive products [1, 3, 5, 7… or -1, -3, -5, -7… depending how you phrase the question] and (2) Accompany them in their exploration if they venture beyond 20 × 0 into the negative zone.
    9. Practicing with Equivalent Expressions
    Practice

    Figure 6.A 5-By-5 Square Frame Made of Square Tiles

    Figure 6.B 5-Square Row Made of Craft Sticks

    Numeric expressions for the number of tiles in the border of a 5-by-5 square (Figure 6.A) include but are not limited to: 4 × 3 + 4, 4 × 4, 5 + 5 + 3 + 3, 2 × 5 + 2 × 3, and 2 x (5 + 3). Discuss the thinking behind each of these and other potential answers by asking students to model for others how they came up with their numerical expressions. Also discuss the equivalence among the different expressions. Numeric expressions for the number of toothpicks needed to make a 5-square row (Figure 6.B) include but are not limited to: 5 + 5 + 6, 2 × 5 + 6, 3 + 3 + 3 + 3 + 3 + 1, 5 × 3 + 1. Same comments as above.

    Answering the same questions for a 10-by-10 square of tiles, and a 10-square row of toothpicks, requires replacing all “5s” with “10s,” and observing if the other factors and addends need adjusting or not.

    Generalizing for any number beyond 5 or 10 gives the following expressions, among other possible ones:

    Number of tiles in the border of an n-by-n squareNumber of toothpicks needed to make a 5-square row
    4 × (n - 2) + 4n + n + (n + 1)
    4 × (n - 1)2 × n + (n + 1)
    n + n + (n - 2) + (n - 2)n × 3 + 1
    2 × n + 2 × (n - 2)3n + 1

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Answers to Nontrivial Problems in Chapter 9
    Beyond Multiplication Musings: More Problems to Explore

    Here you will find the answers and/or explanations to the nontrivial questions from Chapter 9.

    III. For your Students' Learning
    1. The Marvels of the Multiplication Table

    Figure 9.A “Why are they Equal?” Puzzles

    Why are They Equal?

    Create puzzles requiring students to identify multiplication properties or characteristics to figure out why two cells contain equal numbers:

    • Why are the yellow cells equal (8 × 3 =3 × 8)? Because multiplication is commutative. For any two numbers x and y, x x y = y x x. These product pairs are equidistant from the main diagonal (populated by the perfect squares). A nice way of rendering this fact concrete for children is to ask them to fold their square multiplication table along the main diagonal. They will notice that all pairs of superimposed cells contain the same products. Ask them why this is the case.
    • Why are the green cells equal (8 × 5 = 4 × 10)? Because of the doubling/having, or times2/divided by two property: From left to right, 8 has been halved and 5 has been doubled. We saw that the cumulative effect of is change of 1/2 × 2 = 1, and 1 is the identity element for multiplication. So the two products are equal.
    • Why are the blue cells equal (9 × 4 = 3 × 12)? The same reason as in number ii above except that here the changes are tripling/thirding or times three/divided by three: from left to right, 9 has been thirded to give 3 and 4 has been tripled to get 12. The cumulative change is 1/3 × 3 = 1, so the two products are equal.
    • Generalizing: For any product a x b, dividing one factor by any n (or multiplying it by 1/n) and multiplying the other factor by n produces no change. Note:n cannot be 0.
    Finding Equivalent Fractions

    When explaining to students why fraction 3/7 is equivalent to 12/28, we often say that both numerator and denominator of 3/7 have been multiplied by the same factor—in this case 4—to obtain 12/28 (Figure 9.B). While this is correct, we need to delve more deeply:

    • We've multiplied numerator and denominator of 3/7 by 4
    • The combined effect is multiplying by 4/4
    • For any n ≠ 0, n/n = 1 (in particular for n = 4)
    • The identity element of multiplication is 1, so 3/7 = 12/28. Here is the proof: 12/28 = 3/7 × 4/4 = 3/7 × 1 = 3/7
    3. Multiplying by 10

    One way of showing why a number n multiplied by 10 is a new number composed of the digits of n with a zero tagged on to the right is to use base-10 blocks. Begin with any number, say 23. Ask students to build the number with base-10 blocks. They will pick two 10-rods and three unit cubes. Next, line up these blocks horizontally in a row, forming a long, 23-unit rod (Figure 9.C).

    Figure 9.B Equivalence in the Multiplication Table

    Next, visualize a rolled-up window shade and imagine the length of the shade is 23 units long. In your mind's eye, “pull down” the shade until you've stretched it to 10 times as big in surface area (Figure 9.D)

    We notice that the two 10-rods have been stretched to two 100-flats, and the three ones to three 10-rods. Looking at this change from the place-value perspective, multiplying the number 23 by ten results in shifting both digits to the next highest place values (Figure 9.E). As a follow-up exercise, ask students to multiply a 3-digit number, say, 123, by 10 and observe what happens. Finally, ask them to generalize.

    Figure 9.C Number 23 Represented with Base-10 Blocks

    Figure 9.D Geometric Visualization of 23 Multiplied by 10 with Base-10 Blocks

    5. Differentiating Addition and Multiplication
    Examining Structure

    Figure 9.E Multiplying 23 by 10 Results in Shifting Both Digits to the Next Highest Place Values

    Looking at the structure of each of the four expressions below, and without evaluating them, find the ones equivalent to 9 x (8 + 7):

    • (7 + 8) x 9 is equivalent to 9 x (8 + 7) because both addition and multiplication are commutative operations.
    • (9 × 8) + 7 is not equivalent to 9 x (8 + 7) because applying the distributive property to the expression 9 x (8 + 7) would result in (9 × 8) + (9 × 7).
    • (9 × 8) + (9 × 7) is equivalent to 9 x (8 + 7) as explained immediately above
    • (9 + 8) x (9 + 7) is not equivalent to 9 x (8 + 7) as it is an erroneous application of the distributive property.
    Solving by Thinking
    Equivalent Sums
    5 + 100 = 10 + 95Explanation: “Plus 5 (5 → 10)/minus 5 (100 → 95)”
    16 + 28 = 30 + 14Explanation: “Plus 2 (28 → 30)/minus 2 (16 → 14)”
    Equivalent Products
    5×100 = 10×50Explanation: “Times 2 (5 → 10)/divided by 2 (100 → 50)”
    15×28 = 30×14Explanation: “Times 2 (15 → 30)/divided by 2 (28 → 14)”
    6. Maximizing Products without Computing

    Question: Which is greater, 94 × 72 or 92 × 74? There are several ways of answering this question without computing but rather by thinking algebraically.

    • Using the distributive property: We can write the two products as follows, by decomposing one of the two factors into a sum:
      • 94 × 72 = (92 + 2) x 72 = 92 × 72 + 2 × 72
      • 92 × 74 = 92 x (72 + 2) = 92 × 72 + 92 × 2

      The common partial products, 92 × 72 (highlighted in green) are equal. Since 92 × 2 is greater than 2 × 72, the second product is greater. Using Shena's rectangle method, ask students to draw a picture modeling these two expressions. The difference between the two products will become more apparent.

    • Using number sense: In 92 × 74, there are two more 92s than in 92 × 72; in 94 × 72, there are two more 72s than in 92 × 72. Using these facts, and the fact that 72 and 74 are very close in size, as are 92 and 94, we conclude that 92 × 74 is greater than 94 × 72.
    • Using the pattern of products (from Chapter 6, section III, problem 6): We saw that as pairs of factors move further and further apart from each other, the products become smaller and smaller: 74 × 92 < 73 × 93 < 72 × 94.
    9. Magic Tricks and Multiplication
    Trick 1

    Multiplying a three-digit number abc by 7 then by 11 and lastly by 13 is the same as multiplying the number by 1001 or by 1000 + 1: abc x (1000 +1) = abc x 1000 + abc x 1 = abc000 + abc = abcabc.

    Trick 2

    Let aaa be the 3-digit number (N) with equal digits. The number aaa can be written as 100a + 10a + a or 111a. The sum of the three digits (S) can be written as 3a. So dividing N by S is equivalent to dividing 111a by 3a. The answer is 111 divided by 3 or 37.

    Chapter 1 Worksheet

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Chapter 4 Worksheets

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

    Chapter 7 Worksheet

    Visualizing the Partial Products Algorithm for Multiplication as an Area

    XPlace 13 along this edge (in base-10 blocks)
    Place 21 along this edge (in base-10 blocks)

    Copyright © 2015 by Corwin. All rights reserved. Reprinted from Planting the Seeds of Algebra, 3–5: Explorations for the Upper Elementary Grades by Monica Neagoy. Thousand Oaks, CA: Corwin, www.corwin.com. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.

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