Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Grades 3–5

Expressions, unknowns and rules

Name: ______________________

Date: ______________________

You will learn to

Evaluate expressions using brackets and the order of operations (multiplication and division before addition and subtraction)
Find the unknown in equations such as $3 \times \square = 24$ or $\square + 17 = 40$ , explaining the method with inverse operations, and check the answer
Explain that "=" means "has the same value as", so an equation can have the unknown on either side
Describe input–output rules in words and with expressions such as $2 \times n + 1$ , complete tables from a rule, and work backwards from an output
Explain that a letter in an expression stands for a number

1.
Work out $5 + 4 \times 3$ .
2.
Work out $(8 + 4) \div 2 \times 3$ .
3.
Find the unknown: $\square + 17 = 40$ .
4.
Find the unknown: $45 = \square \times 5$ .
5.
Tendai thought of a number. He doubled it, then added 6, and got 20. What was his number?
6.
Which rule fits every row of this table?

input $n$ 1 2 3 4

output 5 8 11 14
- A. $n + 4$
- B. $3 \times n + 2$
- C. $3 \times n + 5$
- D. $5 \times n$
7.
Where should the brackets go to make this true?

$2 + 3 \times 4 - 1 = 19$
- A. $(2 + 3) \times 4 - 1$
- B. $2 + 3 \times (4 - 1)$
- C. $(2 + 3) \times (4 - 1)$
- D. $2 + (3 \times 4) - 1$
8.
At a stall, one pencil costs $p$ shillings and one rubber costs $r$ shillings. What does $3 \times p + 2 \times r$ mean?
- A.3 pencils and 2 rubbers
- B.The cost, in shillings, of 3 pencils and 2 rubbers
- C.5 things altogether
9.
In Lean, ↔ means "exactly when", ∀ x : Nat means "for every whole number $x$ " and ∃ x means "there is a number $x$ ". Which statement says that 23 is the only answer to $\square + 17 = 40$ ?
- A.23 + 17 = 40
- B.∀ x : Nat, x + 17 = 40 ↔ x = 23
- C.∃ x : Nat, x + 17 = 40

input $n$	1	2	3	4
output	5	8	11	14

Answer key — Expressions, unknowns and rules

1.
Multiplication comes before addition: $5 + 4 \times 3 = 5 + 12 =$ 17. (With brackets, $(5 + 4) \times 3 = 27$ — a different expression.)
2.
Brackets first: $8 + 4 = 12$ . Then, from left to right, $12 \div 2 = 6$ and $6 \times 3 =$ 18.
3.
Undo "add 17" by subtracting 17: $\square = 40 - 17 =$ 23. Check: $23 + 17 = 40$ ✓.
4.
"45 has the same value as some number times 5." Undo "multiply by 5" by dividing: $\square = 45 \div 5 =$ 9. Check: $9 \times 5 = 45$ ✓. It does not matter that the unknown is on the right of the equals sign.
5.
Undo the steps in reverse order. Undo "add 6": $20 - 6 = 14$ . Undo "double": $14 \div 2 =$ 7. Check: $7 \times 2 + 6 = 14 + 6 = 20$ ✓. (Undoing in the original order — halving first — gives $10 - 6 = 4$ , which does not work.)
6.
Each time the input goes up by 1, the output goes up by 3, so try $3 \times n$ : that gives 3, 6, 9, 12 — always 2 less than the table. So the rule is $3 \times n + 2$ . Check every row: $5, 8, 11, 14$ ✓.
7.
Working out each option: $(2 + 3) \times 4 - 1 = 20 - 1 = 19$ ✓; $2 + 3 \times (4 - 1) = 11$ ; $(2 + 3) \times (4 - 1) = 15$ ; $2 + (3 \times 4) - 1 = 13$ . So the answer is $(2 + 3) \times 4 - 1$ .
8.
The letters stand for numbers: $p$ is the price of one pencil and $r$ the price of one rubber. So $3 \times p + 2 \times r$ is the cost, in shillings, of 3 pencils and 2 rubbers. For example, if $p = 10$ and $r = 5$ , it is $30 + 10 = 40$ shillings.
9.
∀ x : Nat, x + 17 = 40 ↔ x = 23 says: for every whole number $x$ , $x + 17 = 40$ holds exactly when $x = 23$ . The direction "→" says every answer must be 23 (there are no others); the direction "←" says 23 really is an answer. 23 + 17 = 40 is only the check, and ∃ x, x + 17 = 40 only says that some answer exists. In the lesson's Lean file this is add_unknown.