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Multiples and factors

Can use a multiplication fact table to reduce by finding the common factors. Students explained procedurally find number in table and then look to see what numbers are left and up and use them for common multiples or factors to reduce. Ask students to explain if it will always work and how they know that they are sure. Multiplication table is a list of multiples and factors for 1-10 or whatever. If you can find numbers in the same column or row, then they have a common factor. E.g. 21/28 both are in the row or column with 7 (3*7 , 4*7). If you take two numbers that you want to find a common multiple and find one on the side and the other at the top, you will a common multiple at the intersection. When you multiple two numbers the product will always be a common multiple of the factors, but not necessarily the least common multiple.  

 

 

Suggests more problems and have students put examples in their journals, (25 * 9, 26 * 9, 46 * 5) use the strategy, draw arrays, and share with partner. Eventually students will invent a rule. "Split one factor into tens and ones and multiply each part by the other factor." Before students can use the array as a model they must construct it. Can use Cuisenaire rods and squares, tiles, grid paper, Multilink cubes, and eventually it should become a mental model. Extending the distributive property 49 x 7 = (50 x 7) - 7 48 x 7 51 * 7 98 x 32 = (100 x 32) - (64) Create a model, name the strategy. E.g. "friends more or less" 37 x 84 = (30 x 80) + (7 x 80) + (30 x 4) + (7 x 4) Create a model, name the strategy. E.g. "The ugly one " Using the associative property (30 x 6) 3 x 10 x 6, I added 10 six times = 60, then 60 three times = 180, Create a model 10 x 6= 3 x 6 x 10, 3 x 6 = 18 and added zero, Really add?, No I put it on… 4 x 7 10 x 7 40 x 7 4 x (7 x 10) associative property 60 x 8 Suggest students create more problems and put examples in their journals. Doubling and halving Directly associated with the associative property, but because it is so important, highlight it separately. 2 x 24 4 x 24 8 x 24 8 x 12 4 x 12

50 x 42, (100 x 42 = 4200), 4200/2 = 2100 or 100 * 21 = 2100 3 1/2 x 14, 7 x 7 2 x 24, any of the following? (48 * 1), (24 * 2), (12 * 4), (6 * 8), (3 * 16), (1 * 48) 4 x 24, 2 x 48, 8 x 12, 16 x 6, 32 x 3

Generalize to reciprocal

18 x 3 1/3, take 1/3 of 18 and 3 x 18, then 6 + 54 = 60

or 18 x 3 1/3, divide and multiply by 3 to get 6 x 10 = 60

3 x 3 1/3

.8 x 30, multiply and divide by 10 to get, 8 x 3, 24 20 x 9 = 20 x 10 - 20 (add one group to multiply and subtract it later) Money 25 X 9 = 8 * 25 + 1 * 25 (8 quarters is 200 and one more = 225), or (ten quarters is 250 and one less is 225) How many 25's does it take to make 300? 4, 6, 8, 10, 12 What is the value of a 4 x 4 array of quarters? 16 x 25, 4 x 4 x 25, 4 x 100 Using fractions After students have constructed a very good understanding of fractions 1/2, 1/4, 3/4 try: 75 x 80 = 3/4 x 80 = 60 x 100 = 6,000 1/4 x 80 .25 x 80 25 x 80 1/2 x 60, .5 x 60 50 x 60 Illustrate with models 3 1/2 x 14, double and half or multiply and divide by two to get 7 x 7 Using the open array with division Teaching division as goes into is insufficient. Teaching division by isolating numbers is confusing and often makes little sense to children. Children choose to build up to the whole rather than subtract from the whole. The open array can represent repeated addition and subtraction from the whole and the relationship of division to multiplication. Use an open array with reducing (Cuisenaire rods) 24/6 48/2 (an array would have 4 columns and 12 rows, each cell would be 1/12, each column would be 1, and the whole array would be 48/12, and each row would be 4/12 of 1/3) 48/6 96/12

Show the relationship of division to multiplication

Ask students for problems and make two lists similar to the following:

Division:

48 / 12; 4

48 / 6; 8

48 / 3; 16

48 / 1.5; 32

Multiplication:

4 * 12; 48

8 * 6; 48

16 * 3; 48

32 * 1.5; 48

These are difficult relationships and take much time for students to learn. However, it is essential they do if they are to understand these operations and for algebraic reasoning.

The division algorithm is dependent on the distributive property of multiplication.

To divide 275 / 25 using the traditional algorithm we take (25 * 10) + (25 * 1); 250 * 1

When you use the distributive property it isn't necessary to decompose the numbers into exact factors of the divisor ( 25 * 11) as done with the long division algorithm. Sometimes it will be easier to us place value:

275 / 25

(200 + 75) / 25

(200 / 25) + (75 / 25)

8 + 3; 11

Or

275 / 25 think 25 * 12 is 300

... 300 - 25 is 275 & 12 - 1 is 11

Use arrays, matrices, grids, base ten blocks, Cuisenaire ..., Multilink cubes, Meter stick, Money… to show these representations

1224/24, Use halving 612/12, 306/6, then 51 Or make it friendly: 1200/24 + 24/24, then halve to get 600/12 and divide to get 50, That makes 50 + 24/24 = 50+1 = 51 Or halving 1224/24, 612/12, 306/6, 153/3, then divided by three and got 51. Story problems to solve, model, chart, and represent with symbols. How many ways can you group ____ How many ways can you make an array ____ How many ears in groups of students (1-10)? There are ten kids on the playground that want to swing. There are two swings how many groups would be swinging? How many leaves on groups of three leaf clovers? (1-10)? Pizza party so that each person gets 4 pieces of pizza. How many slices will we need (1-10)? Three children each want 5 cookies. How many cookies do they want? Ratio table     Division Algorithm by piling-up. 1 20 20 12 498 240 258 240 18 12 6