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More partial products Recall that we can use a drawing of a rectangle to help us with calculating products. The rectangle is divided into regions and we determine “partial products” which are then added to find the total.

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More partial products Recall that we can use a drawing of a rectangle to help us with calculating products. The rectangle is divided into regions and we determine “partial products” which are then added to find the total. 3 7 Blue 3 × 5 = 15 Yellow 3 × 2 = 6 Total = 21 Example 1:

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More partial products Example 2: The total number of squares can be found from 54 × 23. One of the ways to calculate 54 × 23 is to divide the rectangle into 4 regions Orange: 50 x 20 = 1000 Yellow: 4 x 20 = 80 White: 50 x 3 = 150 Blue: 4 x 3 = 12 Total: 1242 So 54 × 23 = 1242

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More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4.

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More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4

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More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 What if we need to find 2 × 4.7? Can we draw another small region on the right?

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More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 How wide will it be?

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More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 How wide will it be? (0.7) 0.7

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More partial products So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 2 4 0.7

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More partial products So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 2 4 0.7 So 2 × 4.7 = 9.4

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More partial products This is much easier to see if we use grid paper Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4, So 2 × 4.7 = 9.4

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More partial products What about 2.1 × 4.7? Can we draw an extra region underneath? What size will it be? Should we leave it as one region or make two regions?

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More partial products What about 2.1 × 4.7? Here there are two more regions coloured green and orange. Green: 0.1 × 4 = 0.4 Orange: 0.1 × 0.7 = 0.14

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More partial products pinkyellowgreenorange 2 x 42 x 0.70.1 x 40.1 x 0.7 81.40.40.07 So 2.1 x 4.7 = 8+1.4+0.4+0.07 = 9.87

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Another example Here is one for you to try: Step 1: Sketch a rectangle and label the sides with 1.5 and 3.6 Step 2: Draw lines to make 4 regions. Step 3: What are the lengths of the sides of these regions?

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Partial products: 1.5 x 3.6 30.6 0.5 1 Does your drawing have these 4 regions? Did you have these lengths? Now find the 4 partial products.

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 Think: One half of 3 is 1.5, or 3 groups of 5 tenths is 15 tenths

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 Think: One half of 6 tenths is 3 tenths, or 5 / 10 × 6 / 10 = 30 / 100 = 3 / 10

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 So, the answer to 1.5 × 3.6 is the total of the 4 partial products: Add them up to find your total.

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 So, the answer to 1.5 × 3.6 is the total of the 4 partial products: 3 + 1.5 + 0.6 + 0.3 = 5.4

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Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 This is an alternative method to use to multiply decimals. Even if you don’t use this diagram and the four partial products it has other uses.

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Upper and lower bounds We can use the diagram to help us with estimation: a lower bound and an upper bound

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1.5 3.6

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1 3 The purple rectangle is sitting on top of the green rectangle, and is definitely smaller, so 3 (from 1 x 3) is a lower bound for the size of the green rectangle.

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2 4 The orange rectangle is on top of the green rectangle and it is definitely larger, so 8 (from 2 x 4) is an upper bound for the size of the green rectangle.

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1.5 3.6 So we are sure that the area of the green rectangle is more than 3 and less than 8. So, we can reject any answer which is not between 3 and 8. This is a good strategy for mental estimation, before we do an exact calculation.

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Practice Consider 42 x 28 Without actually finding the answer, can you give: a lower bound for the answer? an upper bound for the answer?

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Practice Consider 42 x 28 Without actually finding the answer, can you give: a lower bound for the answer? -for example, 800 (40 x 20) an upper bound for the answer? for example, 1500 (50 x 30) So if we calculate the answer and it is not between 800 and 1500, then we know it is wrong.

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Can you pick Sam’s error? Sam wrote 2.5 × 6.7 12.35

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Can you pick the error? Sam wrote 2.5 × 6.7 12.35 Using our method of upper and lower bounds, we could predict the answer is between 12 and 21 (i.e. between 2 × 6 and 3 × 7). As 12.35 does lie between 12 and 21, we cannot reject it for this reason. Can you find the correct answer? Can you see what Sam has done incorrectly?

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Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 There should be 4 partial products which need to be added (12 + 3 + 1.4 + 0.35 = 16.75)

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Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 There should be 4 partial products which need to be added (12 + 3 + 1.4 + 0.35 = 16.75) Did Sam find these 4 partial products?

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Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 60.7 0.5 2 2 x 6 = 12 0.5 x 0.7=0.35 Sam only found 2 of these

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Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 So, even if you do not use the rectangular region and the partial products to actually do the calculation (you might prefer another method), it is a helpful diagram to make sure that you don’t use Sam’s method!

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