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Whole Numbers

Whole Numbers. Naming Them and Using Them: Place Value and Operations— Big Ideas and Scenarios [Tapping the Textbook—Small (2013)] {And with ideas based on Van de Walle & Folk (2008) also}. Number…. What do we use ‘number’ for? For a variety of purposes Principal among them---

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Whole Numbers

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  1. Whole Numbers Naming Them and Using Them: Place Value and Operations— Big Ideas and Scenarios [Tapping the Textbook—Small (2013)] {And with ideas based on Van de Walle & Folk (2008) also}

  2. Number… • What do we use ‘number’ for? • For a variety of purposes • Principal among them--- • Number tells us about “quantity” • For example--- • “Muchness”: How much? • “Manyness”: How many?

  3. Let’s Just Make Sure… • Question: What are whole numbers? • Do they include numbers like ½ or 4.7? • Do they include numbers like 4; 932; and 60 001(or is it 60,001?) • Do they include 0 (zero)? • Do they include numbers like -153 and -1? • You are strongly encouraged to read chapters 8 to 10 (Small, 2013) • [chapters 6 to 8 (Small, 2009)]

  4. Three Big Ideas • Chapter 9: (A Sense of Quantity with larger Whole Numbers) • The position of the digits [Name those “base ten” digits!] in numbers determines what they represent—which size group they count. This is the major principle of place-valuenumeration. • Patterns are inherent in our numeration system because each place value is 10 times the value of the place to the right. [Is there only base 10??] • A number has many different ‘forms.’ The groupings of ones, tens, and hundreds, etc., can be taken apart in different ways. • For example, 256 can be 2 hundreds, 5 tens, and 6 ones, but also 1 hundred, 14 tens, and 16 ones. • Taking numbers apart and then combining them in flexible ways is a significant skill for computation. • We can refer to this as “composing” and “decomposing”

  5. Place Value and Manipulatives • Base Ten Blocks… • Stop now and unpack the materials in the kit. • Why do you think they are called “base ten” ? • [For example: 2000 = 2 x 1000 = 2 x 103 -- a ‘base’ of 10 and an exponent or power of 3] • How are they structured as a set? • Pick any three digit number • Represent it using base ten blocks in two different ways. • [decompose and compose in two different ways]

  6. Manipulatives Websites • http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html • http://www.learningbox.com/Base10/BaseTen.html • http://nlvm.usu.edu/

  7. More Place Value and Manipulatives • Base Ten Blocks… • They are known as a “proportional model” representing the base-ten number system. • Why do you think that is? • “Money” –coins—as a limited base ten model: proportional or non-proportional? Explain. • Think of a number like “eleven”… • What might be a clearer way to express 11? • (nicknames and ‘real’ names) maybe ‘ten-one’? • [Primary through early junior and as needed, or differentiation]

  8. Another Big Idea • Chapters 8 & 10 (Early Operation Concepts, and Computation with Whole Numbers) • Excellent chapters for gaining an understanding of addition, subtraction, multiplication, and division • First, addition and subtraction. They are related. • For example, think of the numbers 9, 17, and 26. What relations can you see among these numbers? • We can think of 17 + 9 = 26; 9 + 17 = 26; 26 – 9 = 17 and 26 – 17 = 9 as “companion statements” • Take a few minutes to model these operations with base ten blocks. [Commutative property? Inverse operations?] • See also Grade 3 Patterning and Algebra!

  9. This is a grade 3 lesson, from Patterning and Algebra (see previous PJ Math Concurrent Session #9)

  10. Here’s a Classroom Scenario: An Upper Primary Grade (2, or 3) or Early Junior Grade (Like Grade 4) • Reviewing and Exploring on paper and with base ten materials… • Let’s begin here with the relationships between addition and subtraction • and later we’ll look at the relationship between multiplication and division… • Watch for both Number Sense and Numeration and Patterning and Algebra elements

  11. Addition and Subtraction • You might ask the students: • Think about 14 + 8 = _____ and 22 – 14 = ____ • Then---Write two other companion statements or equations for addition and subtraction for this same set of numbers. • Have students represent these with manipulatives. • Ensure it makes sense before moving on. • Then increase the level of challenge: • If 328 – 71 = 257, write the remaining three companion statements for addition and subtraction using these three numbers. • [Why do you think it might be useful for children to think about addition and subtraction operations in this way?] • Addend, minuend, subtrahend, sum, difference—wha…?

  12. Continuing with Our Classroom Scenario: Early Junior grade… • Reviewing and Exploring on paper and with base ten materials… • Now…the relationship between multiplication and division… • But First… • Model 4 x 7 with base 10 blocks • [ Think also array model or area model] • Model 32 ÷ 4 with base 10 blocks • And now—model 12 x 23 with base 10 blocks.

  13. A Companion Big Idea • Still with Chapter 10 (Computations)… • Multiplication and division are related. • [e.g., two factors multiplied produce a product --- 3 x 6 = 18] • Division names a missing factor in terms of the known factor and product. • [e.g., 3 x ?? = 18 or, 18 ÷ ?? = 3] • Think, “one undoes the other.” • (What are ‘factor,’‘product,’‘dividend,’‘divisor,’‘quotient’?)

  14. Multiplication and Division • Think about multiplication and division as companion operations. • Ask your junior level students… • If 7 × 4 = 28, what are the remaining three companion statements or equation? • Have your students model the four equations with base ten materials. • See also Grade 4 Patterning and Algebra! • Then… • If 2 926 ÷ 418 = 7, write the remaining three companion statements. • But, Hey! Who cares?? Give me a reason to care!

  15. A Very Big Idea • Chapter 10 continued (Whole-Number Computation) • “Invented” strategies are flexible methods of computing that vary with the numbers and the situation. The success of these strategies requires that they are understood by the person using them—hence, the term invented. Strategies may be invented by a peer or the class as a whole; they may even be suggested by the teacher. However, they must be constructed by the student. (VdW & F) • Small (2013) [and (2009)] has numerous important things to say about invented strategies. • (The math curriculum refers to these as “student-generated algorithms.” What are algorithms?) • Primary grades, Junior grades

  16. Classroom Scenario: Junior Grade • Solving addition and subtraction of four digit numbers, using student-generated algorithms, and standard algorithms • Grade 4… early in the year review with smaller numbers… (picking up on Primary level work...) • [Larger numbers for later grades.]

  17. Addition and Subtraction Algorithms… • Given the problem of finding 257 + 71 … • One child said, “I did it this way on paper- 250 + 70 = 320 320 + 7 + 1 = 328” • Was she correct? • Is this an acceptable way to perform this addition? Explain. • Invented strategies can be approached through teaching through problem solving—suggest how.

  18. Addition and Subtraction Algorithms… • Another student performed the addition in the following way: 257 + 71 2128 • This answer is obviously incorrect. What misconceptions about the algorithm might he hold? • How might you help him? • Might B-10 blocks help with possible place value difficulties?

  19. Classroom Scenario… • An encounter with a grade 6 student performing subtraction in a way you’ve never seen before… (likely)! • But . . . this could also be a grade 4 or grade 5 student . . .

  20. Addition and Subtraction Algorithms… • Suppose a student in your Grade 6 class performed the subtraction as follows: 328 2 12 8 - 71- 7 1 2 5 7 (2 first, then 5, then 7) • Is his answer (257) correct? • Is his (expanding, left to right) strategy acceptable? Explain your position.

  21. Another Classroom Scenario… • Grade 5…part way through the year… • You decide to pose to the class some conceptual questions about the standard algorithm for multiplication to see what they understand… • …And then ask them to use their own “invented” strategy to multiply the same two, two-digit numbers…

  22. Multiplication Algorithms 27 X 16 162  Where does the ‘6’ come from? (Show it) 270  Why is the ‘zero’ here? 432 • Create or “invent” a different pencil and paper strategy to perform this multiplication. • Is place value important in this problem?

  23. Scenario…For Us Here, Now… • Here’s thinking about multiplication and addition together… giving them meaning • Another way you might help students struggling with multiplication… 

  24. More Multiplication… • Explain how multiplication may also be understood as repeated addition. Investigate this at your table now with an example. • Write 12 x 8 as a repeated addition in two ways. • Is the answer the same in both cases? • Try using B-10 blocks (or chips) to solve, also. • Write 653 x 4 as a repeated addition. • Which way did you choose to perform it? • If a person can just add, why bother to learn to multiply?

  25. And Now Division… • If Multiplication can be thought of as “repeated addition,” what might be said about the related operation of Division? • Stop now and perform the following division as a “repeated subtraction”… 128 ÷ 16 = ? • Explain how you determined the answer. • Try performing this division with B-10 blocks

  26. A “Can You Believe It?!” Scenario • Your encounter with another grade 6 student, Mary, who says, “I really like division by repeated subtraction. Ms. Carson showed us it last year. This is how I solved the problem you gave us…” Looking at the next slide… • Do you see it? • What are you going to do now? • Are you upset with Ms. Carson? • Let’s have a look.

  27. More Division…From Mary… • Look for the repeated subtraction in the following problem. Try to explain what’s happening. • Problem: 2 478  42 • 42) 2478 | 10 • 420 | • 2058 | 20 • 840 | • 1218 | 20 • 840 | • 378 | 5 • 210 | • 168 | 4 • 168 | • 0 Therefore 2 478  42 = 59

  28. But back in Grade 5… (Scenario) • A student struggles with the standard division approach…

  29. Division Algorithms • Suppose Harry Putter, in your Grade 5 class, when asked to calculate 356 ÷ 4, did the following: • What procedural and/or conceptual difficulties might he be experiencing? • How might you help him?

  30. And Yet Another Big Idea • Chapter 10: (Estimating with Whole Numbers) • Multi-digit numbers can be built or taken apart in a wide variety of ways (composing and decomposing). When the parts of numbers are easier to work with, these parts can be used to make estimates for calculations, rather than using the exact numbers involved. For example, 36 is 30 and 6 or 25 and 10 and 1. 483 can be thought of as 500 – 20 + 3.

  31. Classroom scenario…go mental! • (Although they started to work with 2-digit by 2-digit multiplication in grade 5. . .) You feel that your grade 6 class needs practice with mental math and estimation activities… • You form small groups and present the following…

  32. Mental Math & Estimation • Let’s go back to this multiplication problem… 27 X 16 162 270 432 Devise a mental mathematics strategy to accurately calculate this answer. Make it your own! Devise another strategy to arrive at an estimate of this answer. Explain both your strategies.

  33. Your Own Classroom Scenario • Situation: You are a grade 4 teacher. It is near the beginning of the year. • You are eager to have the students really start to experience ‘learning through problem solving.’ [TTPS] • You also know they need more experience with operations such as adding large whole numbers (up to four digits). • You think about a lesson that will integrate both the TTPS environment and opportunities to develop addition skills (including their own ideas for algorithms). • In your table groups, outline a three-part lesson that you think will incorporate both your objectives. (10 minutes MAX!) • BRIEFLY share with class.

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