Q1.
Here are some numbers arranged in some patterns. Find out the sum of the numbers in each of the below figures. Should we add them one by one or can we use a quicker way? Share and discuss in class the different methods each one of you used to solve these questions.
Answer
The quick way is always the same: count how many times each number appears, multiply, then add. That is much faster than adding one by one.
| Figure | What it contains | Quick working | Sum |
|---|---|---|---|
| a. | 12 forties and 10 fifties | 12 × 40 = 480; 10 × 50 = 500 | 980 |
| b. | an 8 × 8 grid of squares — 44 squares with 1 dot, 20 squares with 5 dots | 44 × 1 = 44; 20 × 5 = 100 | 144 |
| c. | 32 cells of 32 and 16 cells of 64 | 32 × 32 = 1024; 16 × 64 = 1024 | 2048 |
| d. | a 5 × 7 board of dice faces — 17 cards show 3 dots, 18 cards show 4 dots | 17 × 3 = 51; 18 × 4 = 72 | 123 |
| e. | 22 fifteens, 22 twenty-fives and 22 thirty-fives | 22 × (15 + 25 + 35) = 22 × 75 | 1650 |
| f. | 18 of 125, 8 of 250, 4 of 500 and one 1000 | 2250 + 2000 + 2000 + 1000 | 7250 |
Working shown in full
a. There are 3 rows of four 40s and 2 rows of five 50s.
(3 × 4) × 40 + (2 × 5) × 50 = 480 + 500 = 980
(3 × 4) × 40 + (2 × 5) × 50 = 480 + 500 = 980
b. The grid is 8 × 8 = 64 squares. 20 of them are the dark squares with 5 dots each; the other 44 have a single dot.
44 + (20 × 5) = 44 + 100 = 144
44 + (20 × 5) = 44 + 100 = 144
c. The top block is 4 rows × 8 columns = 32 cells of 32.
The lower block is 4 rows × 4 cells = 16 cells of 64.
(32 × 32) + (16 × 64) = 1024 + 1024 = 2048
The lower block is 4 rows × 4 cells = 16 cells of 64.
(32 × 32) + (16 × 64) = 1024 + 1024 = 2048
d. The board has 5 columns × 7 rows = 35 cards.
The first and last columns (14 cards) and the three cards of the 6th row show 3 dots → 17 × 3 = 51
The remaining 18 cards show 4 dots → 18 × 4 = 72
51 + 72 = 123
The first and last columns (14 cards) and the three cards of the 6th row show 3 dots → 17 × 3 = 51
The remaining 18 cards show 4 dots → 18 × 4 = 72
51 + 72 = 123
e. The design is symmetric and contains each of 15, 25 and 35 exactly 22 times.
22 × 15 + 22 × 25 + 22 × 35 = 22 × 75 = 1650
22 × 15 + 22 × 25 + 22 × 35 = 22 × 75 = 1650
f. 18 × 125 = 2250, 8 × 250 = 2000, 4 × 500 = 2000, 1 × 1000 = 1000
2250 + 2000 + 2000 + 1000 = 7250
2250 + 2000 + 2000 + 1000 = 7250
Why grouping is smarter: in figure (c) adding 32 one at a time needs 47 additions and invites mistakes. Grouping turns it into two multiplications and one addition — and it also shows a hidden fact: the 32s and the 64s contribute exactly the same amount, 1024 each.
Math Talk: different friends will group differently. In figure (a) some will do (40 × 12) + (50 × 10); others will add each row: 160 + 250 + 160 + 250 + 160 = 980. Both are correct — compare the methods in class.