Extension & Challenge — Equalities & Inequalities — Equalities & Inequalities: Balancing the Four Operations

Part A — Balancing the Scales

A balance scale is a picture of an equation. When the pans are level, both sides have the same value, so we write =. When one side is heavier we use > (greater than) or < (less than) — the open mouth always faces the larger amount.

Three balance scales. A compares 7 plus 5 with 15 minus 3. B compares 6 times 3 with 20. C compares 4 times 4 with 10 plus 5. — Three balances, A, B and C. Work out each side before deciding which way it tips.

1 For each balance above, work out the value of both sides, then write =, < or > in the box.

A: 7+5 15−3

B: 6×3 20

C: 4×4 10+5

2 A bag of identical marbles plus a 3 g weight exactly balances an 11 g weight. Write an equation and find the mass of the bag of marbles.

3 Two identical apples balance a 200 g weight. How heavy is one apple? Then predict what would balance three apples.

4 Keep it balanced. A balance is currently equal. You add 4 to the left pan. What must you do to the right pan so it stays balanced? Explain the rule in your own words.

Part B — True, False & Open Sentences

A number sentence can be true, false, or open (it has a missing number, so we cannot decide until we fill the gap). Mathematicians test sentences and find values that make open ones true.

1 Write True or False for each sentence. If it is false, change the right-hand side to make it true.

(a) 7 × 8 = 54

(b) 100 − 45 < 60

(d) 5² > 4 × 6

2 Find a whole number that makes each open sentence true.

(a) □ + 7 > 12

(b) 2 × □ = 18

3 List all the whole numbers that make this inequality true: 5 < n ≤ 9.

4 Prove it. If a = b, explain why a + 5 must equal b + 5. Use the idea of a balance to help your explanation.