1 · Letters stand for numbers
A variable is a letter that stands for a number. In n + 5, if n is 3, the value is 8; if n is 10, it is 15.
4n means 4 × n. So when n = 6, 4n = 24 and 4n − 3 = 21.
2 · Writing expressions from words
"7 more than a number" is n + 7. "7 less than a number" is n − 7: the 7 is taken away from n, so it comes second.
"Twice a number, then 3 more" is 2n + 3. "Twice the sum of a number and 3" is 2(n + 3); the word "sum" means the adding happens first.
3 · Evaluating an expression
Put the number in for the letter, then follow the order of operations. When x = 4: 3x + x × x = 12 + 16 = 28.
x × x is 4 × 4 = 16, not 4 + 4. Write the number in before you work it out.
4 · One-step equations
An equation is a balance: both sides are equal. n + 9 = 23: take 9 off both sides, n = 14.
6n = 54: divide both sides by 6, n = 9. Check by putting the answer back in.
5 · Two-step equations
3n + 5 = 26. Undo the steps in reverse order: first subtract 5 (3n = 21), then divide by 3 (n = 7).
Check: 3 × 7 + 5 = 26. A common slip is stopping at 21, which is 3n, not n.
6 · Input and output rules
A rule turns each input into an output. Look at how the output changes when the input goes up by 1.
Here each +1 in the input adds 3 to the output, so the rule starts "× 3". Then 3 × 1 = 3, but the output is 5, so add 2. The rule is 3n + 2, and input 10 gives 32.
7 · The nth term of a pattern
7, 11, 15, 19, … adds 4 each time. To reach the 20th number from the 1st, you add 4 a total of 19 times, not 20.
So the 20th number is 7 + 19 × 4 = 83. Count the gaps, not the numbers.
8 · Patterns that multiply
3, 6, 12, 24, … doubles each time, so the next number is 48. The gaps (3, 6, 12) grow, which tells you it is not adding the same amount.
Some patterns alternate: 2, 5, 4, 7, 6, 9, … goes +3, −1, +3, −1. The next is 8.
9 · Symbol puzzles
"■ + ■ + ▲ = 17 and ■ + ▲ = 11." Compare the two lines: the first has one extra ■ and is 6 more, so ■ = 6. Then ▲ = 11 − 6 = 5.
Look for two lines that differ by one symbol. The difference in totals is that symbol.
10 · Consecutive numbers
Consecutive numbers come one after another: 14, 15, 16. Three of them add to 3 times the middle one.
"Three consecutive numbers add to 57." The middle one is 57 ÷ 3 = 19, so they are 18, 19, 20.
11 · Sum and difference
"Two numbers add to 30, and one is 8 more than the other." Take away the extra 8: 30 − 8 = 22. Split it evenly: 11 and 11. Then give back the 8: the numbers are 11 and 19.
Check: 11 + 19 = 30, and 19 − 11 = 8.
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