031i - Solving One-Step Equations

Lesson 31 — Solving One-Step Linear Equations

Strand: Algebra | Descriptor: AC9M7A03 | Duration: 45 minutes

Learning Intentions

• To understand an equation as a statement of balance between two expressions.
• To solve one-variable linear equations requiring a single inverse operation.

Success Criteria

I can:

1. Explain what it means to solve an equation.
2. Identify the inverse of each operation.
3. Solve equations of the form , , and .
4. Keep the equation balanced by doing the same to both sides.

Warmup

(6 minutes — the balance model, physical or drawn)

Draw a balance scale with blocks on the left and blocks on the right.

1. Is it balanced? What happens if I add blocks to the left only?
2. What could I do to restore balance?
3. If I take block from each side, is it still balanced?
4. If I double what is on each side, is it still balanced?

The principle to state and keep returning to:

Whatever you do to one side, you must do to the other. That is the only rule.

Activities

Activity 1 — Explicit Instruction: Inverse Operations (10 min)

Definitions.

• An equation states that two expressions are equal: .
• To solve it is to find the value of the variable that makes it true.
• That value is called the solution.

Inverse operations — build the table with the class:

Operation	Inverse
Add	Subtract
Subtract	Add
Multiply by	Divide by
Divide by	Multiply by

The strategy: identify what has been done to the variable, then undo it — on both sides.

I do — addition. Solve .

Check: ✓

Setting out — insist on this from the start. Write the operation being applied to the right of each line, e.g. ” both sides”. Line up the equals signs vertically. Finish with a check.

I do — multiplication. Solve .

Check: ✓

We do: ; ; ; .

Activity 2 — Independent Practice with All Four Types (12 min)

You do: Solve, showing both lines of working and a check.

(Answers: ; ; ; ; ; ; ; ; ; .)

Note on Q5 and Q6. These produce and . Students often distrust these answers. Both are perfectly valid solutions — check them and move on.

Note on the descriptor. AC9M7A03 specifies equations with natural number solutions, so every answer in this lesson is a whole number. Fractional and negative solutions come later in the course.

Activity 3 — Inquiry: the Balance Puzzle (12 min)

Pairs. Use pan-balance diagrams rather than symbols at first.

Each diagram shows a balanced scale. Bags of the same colour hold the same number of marbles. Find how many marbles are in one bag.

Puzzle A. Three identical bags balance loose marbles.

Puzzle B. One bag plus loose marbles balances loose marbles.

Puzzle C. Two bags plus marbles balance marbles.

Puzzle D. Five bags balance two bags plus marbles.

Socratic scaffolding for Puzzle D:

Prompt	Purpose
Understand: what is the unknown?	The number of marbles in one bag.
What is different about this puzzle?	There are bags on both sides.
Can you simplify before solving?	Remove two bags from each side — the scale stays balanced.
What is left?	Three bags balance marbles.
Now solve.	, so .
Write it algebraically.	.
Looking back — check	Left: . Right: ✓
Generalise	Removing the same thing from both sides is just another balanced move.

Answers: A — , . B — , . C — , . D — .

Bridging note: Puzzles C and D are two-step equations. Students solve them here informally with the balance; Lesson 32 formalises the method.

Checks for Understanding

(5 minutes — exit ticket)

1. Solve, showing your working: (a) (b) (c) (d) .
2. What is the inverse of “multiply by “?
3. Reasoning. A student solves by writing . Identify the error and correct it.
4. Check whether is the solution to . Show your check.
5. Write an equation whose solution is , using multiplication.

Answers: 1. (a) (b) (c) (d) ; 2. Divide by ; 3. The student applied the same operation instead of the inverse — to undo "" you subtract, giving ; 4. ✓ so yes; 5. Any correct example, e.g. or .

Common Misconceptions

Misconception	How to pre-empt it
Applying the same operation instead of the inverse ().	Build the inverse table explicitly and require students to name the inverse aloud before writing.
Operating on one side only.	Return to the balance model whenever this appears. Write the operation next to both sides.
Believing or cannot be a solution.	Include such items deliberately (Q5, Q6) and check them together.
Confusing with .	Ask “is the multiplying or adding?” before choosing the inverse.
Not checking, so errors go undetected.	Make the check a compulsory final line from this lesson onwards.
Reading as needing division.	The variable has already been divided; the inverse is to multiply.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). If , what is the value of ?

Answer

, so . (A two-part question — solve, then answer what was actually asked.)

E2 (AMC Junior style). A number is divided by and the result is . What is the number?

Answer

E3 (Challenge). Five identical bags balance three identical bags plus marbles. How many marbles in one bag?

Answer

E4 (Challenge). If and , find .

Answer

, so and .

E5 (Challenge). Find a whole number such that .

Answer

Check: ✓

Homework

1. Solve, showing working and a check: (a) (b) (c) (d) (e) (f) .
2. Solve: (a) (b) (c) (d) .
3. State the inverse of each: (a) add (b) divide by (c) subtract (d) multiply by .
4. Write an equation with solution using (a) addition (b) multiplication (c) subtraction (d) division.
5. Check whether the given value solves the equation, showing your check: (a) for (b) for (c) for .
6. Reasoning. Explain why solving requires adding to both sides rather than subtracting.
7. Challenge. Four identical boxes balance one box plus kg. Find the mass of one box.
8. Challenge. If , what is ?

Answers: Q1 — (a) (b) (c) (d) (e) (f) . Q2 — (a) (b) (c) (d) . Q3 — (a) subtract (b) multiply by (c) add (d) divide by . Q5 — (a) yes, (b) no, (c) yes, . Q7 — , so and kg. Q8 — , so .