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## More partial products

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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.**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. Example 1: 7 3 Blue 3 × 5 = 15 Yellow 3 × 2 = 6 Total = 21**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**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.**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 4 2**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 4 2 What if we need to find 2 × 4.7? Can we draw another small region on the right?**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 4 2 How wide will it be?**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 4 0.7 2 How wide will it be? (0.7)**More partial products**So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 4 0.7 2**More partial products**So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 So 2 × 4.7 = 9.4 4 0.7 2**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**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?**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**More partial products**So 2.1 x 4.7 = 8+1.4+0.4+0.07 = 9.87**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?**3**0.6 1 0.5 Partial products: 1.5 x 3.6 Does your drawing have these 4 regions? Did you have these lengths? Now find the 4 partial products.**Partial products: 1.5 x 3.6**3 0.6 1 x 3 = 3 1 0.5**Partial products: 1.5 x 3.6**3 0.6 1 x 3 = 3 1 0.5 0.5 x 3 = 1.5 Think: One half of 3 is 1.5, or 3 groups of 5 tenths is 15 tenths**3**0.6 1 x 3 = 3 1 x 0.6 = 0.6 1 0.5 0.5 x 3 = 1.5 Partial products: 1.5 x 3.6**3**0.6 1 x 3 = 3 1 x 0.6 = 0.6 1 0.5 0.5 x 3 = 1.5 0.5 x 0.6 = 0.3 Partial products: 1.5 x 3.6 Think: One half of 6 tenths is 3 tenths, or 5/10×6/10 = 30/100 = 3/10**3**0.6 1 x 3 = 3 1 x 0.6 = 0.6 1 0.5 0.5 x 3 = 1.5 0.5 x 0.6 = 0.3 Partial products: 1.5 x 3.6 So, the answer to 1.5 × 3.6 is the total of the 4 partial products: Add them up to find your total.**3**0.6 1 x 3 = 3 1 x 0.6 = 0.6 1 0.5 0.5 x 3 = 1.5 0.5 x 0.6 = 0.3 Partial products: 1.5 x 3.6 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**3**0.6 1 x 3 = 3 1 x 0.6 = 0.6 1 0.5 0.5 x 3 = 1.5 0.5 x 0.6 = 0.3 Partial products: 1.5 x 3.6 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.**Upper and lower bounds**• We can use the diagram to help us with estimation: • a lower bound and • an upper bound**3.6**1.5**3**1 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.**4**2 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.**3.6**1.5 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.**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?**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.**Can you pick Sam’s error?**Sam wrote 2.5 ×6.7 12.35**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?**6**0.7 2 x 6 = 12 2 x 0.7 = 1.4 2 0.5 0.5 x 6 = 3 0.5x 0.7= 0.35 Partial products: 2.5 x 6.7 There should be 4 partial products which need to be added (12 + 3 + 1.4 + 0.35 = 16.75)**6**0.7 2 x 6 = 12 2 x 0.7 = 1.4 2 0.5 0.5 x 6 = 3 0.5x 0.7= 0.35 Partial products: 2.5 x 6.7 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?**6**0.7 2 x 6 = 12 2 x 0.7 = 1.4 2 0.5 0.5 x 6 = 3 0.5x 0.7= 0.35 Partial products: 2.5 x 6.7 6 0.7 Sam only found 2 of these 2 x 6 = 12 2 0.5 0.5 x 0.7=0.35**6**0.7 2 x 6 = 12 2 x 0.7 = 1.4 2 0.5 0.5 x 6 = 3 0.5x 0.7= 0.35 Partial products: 2.5 x 6.7 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!