Multiplying with powers of ten, area models and partial products
الأسبوع 6 · الضرب في قوى العشرة ونماذج المساحة والنواتج الجزئية · 3 دروس
2
Area boxes and the distributive property
Discover Mathematics P5: Term 1, Unit 3, Lessons 1–4
GoalUse an area model and the distributive property to multiply a number by a two-digit number.
New words
area model · a rectangle split into boxes to show the parts of a multiplicationdistributive property · a × (b + c) = a × b + a × cexpanded form · a number written as the sum of its place values, such as 45 = 40 + 5
Explanation
Big multiplications are easier when we split a number into place values. The distributive property says that multiplying a sum is the same as multiplying each part and adding: 206 × 45 = 206 × (40 + 5) = 206 × 40 + 206 × 5 = 8,240 + 1,030 = 9,270. An area model shows this as a rectangle cut into boxes. Write one factor along the top and the other down the side, both in expanded form: 200 + 6 and 40 + 5. The four boxes hold 200 × 40 = 8,000, 6 × 40 = 240, 200 × 5 = 1,000 and 6 × 5 = 30. Add all four: 8,000 + 240 + 1,000 + 30 = 9,270. The distributive property also works with subtraction: 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686.
×
200
6
40
8,000
240
5
1,000
30
Worked examples
×
20
3
10
200
30
4
80
12
Multiply 23 × 14 with an area model. 23 = 20 + 3 and 14 = 10 + 4. The boxes are 20 × 10 = 200, 3 × 10 = 30, 20 × 4 = 80 and 3 × 4 = 12. Adding: 200 + 30 + 80 + 12 = 322. So 23 × 14 = 322.
Multiply 206 × 45 using the distributive property. 206 × 45 = 206 × 40 + 206 × 5. First 206 × 40 = 8,240 (206 × 4 = 824, then one zero). Then 206 × 5 = 1,030. The sum is 8,240 + 1,030 = 9,270. A bus ticket seller who sells 206 tickets a day for 45 days sells 9,270 tickets.
Work out 7 × 98 in your head. 98 is 2 less than 100. So 7 × 98 = 7 × 100 − 7 × 2 = 700 − 14 = 686.
Class activity
Draw the boxes. On squared paper draw a big rectangle for 27 × 13. Cut it with a vertical line into 20 and 7, and with a horizontal line into 10 and 3. Write the area of each box and add the four boxes. Check by using the distributive property, 27 × 13 = 27 × 10 + 27 × 3.
