1. Unmultiplication (Division) © Math As A Second Language All Rights Reserved next Taking the Fear out of Math #3 Using Tiles

2. next © Math As A Second Language All Rights Reserved In the same way that the term “unadding” leads to a better understanding of the relationship between adding and subtracting, the term “unmultiplying” leads to a better understanding of the relationship between multiplying and dividing. Our Point of View

3. next © Math As A Second Language All Rights Reserved To find the answer to a “question” such as… Our Point of View …we multiply 4 and 3 to obtain 4 × 3 = 12. 1 next note 1 When taught to look for “key” words, students are often told to interpret “and” as indicating addition. However, “and” is also a conjunction, and when we say “Find the product of 4 and 3, the term “product” is telling us to multiply 4 and 3. So while it is correct to say that 4 + 3 = 7, it is incorrect to say that the product of 4 and 3 is 7 when in fact it is 12. Find the product of 4 and 3 (or in terms of fill-in-the-blank 4 × 3 = ____).

4. next © Math As A Second Language All Rights Reserved On the other hand to find the answer to a question such as… Our Point of View 4 × ____ = 12 …we unmultiply 12 by 4 to obtain 12 ÷ 4 = 3. next In words, 4 × ____ = 12 is asking us to find the number we must multiply by 4 to obtain 12 as the answer.

5. © Math As A Second Language All Rights Reserved Using more user-friendly vocabulary, the question is asking us “How many 4’s do we have to add to obtain 12 as the sum?”, or, in slightly more formal language, “12 is what multiple of 4? 2 ” Our Point of View note 2 A multiple is the product of a given number and any whole number. The multiples of 4 would be 4, 8, 12, 16, 20, 24, 28, 32, 36, 40…

6. next © Math As A Second Language All Rights Reserved If we represent the numbers by tiles and use our idea of rectangular arrays, we count by 4’s until we get to 12. In other words, each row of tiles represents 4, and 4 + 4 + 4 = 12. 123456789101112

7. next © Math As A Second Language All Rights Reserved On the other hand, looking at the columns in the above array, we see that there are 4 columns, each with 3 tiles. In other words, we have to add four 3’s in order to obtain 12 as the sum. In the language of unmulitplying (dividing), this means that 12 ÷ 4 = 3. next 1234 5678 9101112

8. next © Math As A Second Language All Rights Reserved The Mathematical Definition of Division 12 ÷ 3 is the number (the quotient) by which we must multiply 3 (the divisor) in order to obtain 12 (the dividend) as the product. In other words, 12 ÷ 3 is the answer to 3 x _____ = 12.

9. next More generally, if b and c are numbers, b ÷ c is the number by which we must multiply c in order to obtain b as the product. In other words, b ÷ c is the answer to _____ × c = b. 1 © Math As A Second Language All Rights Reserved note 1 This is an excellent way to motivate the “invention” of fractions. For example, it is possible to divide $12 equally among 5 people, but the answer is not a whole number. Since 2 × 5 = 10, 2 is too small to be the correct answer, and since 3 × 5 = 15, 3 is too great to be the correct answer. And since there are no whole numbers between 2 and 3, the correct answer must be a fractional number of dollars (and, in fact, the exact answer is $2.40).

10. next In summary, as shown below, each multiplication fact leads to two division facts. For example… © Math As A Second Language All Rights Reserved 3 rows each with 4 tiles or 3 × 4 = 12 or 12 ÷ 3 = 4 4 columns each with 3 tiles or 4 × 3 = 12 or 12 ÷ 4 = 3 next 1234123412341 2 3 1 2 3 1 2 3 1 2 3

11. In a similar way the diagram below illustrates the “chain”… 5 × 6 = 30 next © Math As A Second Language All Rights Reserved next 30 ÷ 5 = 630 ÷ 6 = 5 1 7 13 2 8 14 3 9 15 4 10 16 5 11 17 6 12 18 192021222324 252627282930 12345623451 2 3 4 5 23456

12. next 5 rows each with 6 tiles or 5 × 6 = 30 or 30 ÷ 5 = 6 © Math As A Second Language All Rights Reserved next 1234561234561234561234561234561 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 columns each with 5 tiles or 6 × 5 = 30 or 30 ÷ 6 = 5

13. © Math As A Second Language All Rights Reserved All of the discussion in this presentation can be illustrated in terms of four questions based on the following information… You buy 6 pens, each of which cost $5. next

14. © Math As A Second Language All Rights Reserved Question #1 next You buy 6 pens, each of which cost $5. How much did you pay for the pens? In this question, we were given both numbers and multiplied them to find their product. In this case, we are spending $5 six times. So we multiply to obtain… 6 × 5 dollars = 30 dollars.

15. next © Math As A Second Language All Rights Reserved Question #2 next You spend $30 to buy 6 equally-priced pens. How much did each pen cost? In terms of unmultiplying, we wanted to find out what we had to multiply 6 by in order to get 30 as the product. In this case, one way to represent the answer is by… 30 (dollars) ÷ 6 (pens) = 5 (dollars per pen)

16. next © Math As A Second Language All Rights Reserved Question #3 next You spend $30 to buy pens that cost $5 each. How many pens didyou buy? In terms of unmultiplying, we wanted to find out what we had to multiply 5 by in order to get 30 as the product. In this case, we can represent the answer in the form… 30 (dollars) ÷ 5 (dollars per pen) = 6 (pens)

17. next © Math As A Second Language All Rights Reserved Question #4 next You spend $30 to buy equally-priced pens. How many pens did you buy, and how much did each pen cost? This is the situation in which factorization is needed. There are as many answers as there are divisors (factors) of 30. In this case, we can represent the answer in the form… 30 (dollars) ÷ ? (dollars per pen) = ? (pens)

18. next © Math As A Second Language All Rights Reserved Question #4 next More specifically, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. The possible answers are… 1 pen @ $30 2 pens @ $15 3 pens @ $10 5 pens @ $6 6 pens @ $5 10 pens @ $3 15 pens @ $2 30 pens @ $1

19. next © Math As A Second Language All Rights Reserved A Closing Note In later grades, the children will represent 12 ÷ 3 in the form… Some students run into trouble by reading the above division problem as 3 ÷ 12 because in the above array, the 3 is to the left of 12 and they tend to read from left to right. Thus, they might write such things as 3 ÷ 12 = 4. 312

20. next © Math As A Second Language All Rights Reserved A Closing Note To avoid this type of misinterpretation, in some cultures rather than write… they write it in the form… 312 4 3 4 thus, preserving the same order as in 12 ÷ 3.

21. next © Math As A Second Language All Rights Reserved A Closing Note The main point is that it is crucial for the students to understand that 3 ÷ 6 is not the same as 6 ÷ 3. Namely, it makes a difference in how much pie each person gets, if we divide 3 pies equally among 6 people or 6 pies equally among 3 people.

22. next This completes our present discussion. In our next presentation, we will show how the tiles can be used to introduce some elementary number theory. © Math As A Second Language All Rights Reserved 15 ÷ 33 × _ = 15 next