033i - Equations with Brackets and Like Terms

Lesson 33 — Equations with Brackets and Like Terms

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

Learning Intentions

• To solve linear equations requiring simplification before solving.
• To expand brackets and collect like terms within an equation.

Success Criteria

I can:

1. Simplify each side of an equation before solving.
2. Expand brackets correctly within an equation.
3. Collect like terms on both sides.
4. Solve the resulting equation and verify the solution.

Warmup

(6 minutes — retrieval from Lesson 28, mini whiteboards)

Expand and simplify:

(Answers: ; ; ; ; .)

Bridging question: If I told you , could you solve it? What if I gave you instead — what would you do first?

Activities

Activity 1 — Explicit Instruction: Simplify, then Solve (14 min)

The extended protocol:

1. Expand any brackets.
2. Collect like terms on each side.
3. Gather variables on one side, numbers on the other.
4. Solve the resulting one- or two-step equation.
5. Check by substituting into the original equation.

Emphasise step 5. Always check against the original, not your simplified version — otherwise an expansion error goes undetected.

I do — brackets. Solve .

Check (in the original): ✓

Alternative route worth showing. Because the whole bracket is multiplied by , you may divide both sides by first: , so . Ask which route is quicker here, and when it would not be. (Dividing first is neat when the right side divides exactly; otherwise expand.)

I do — collecting like terms. Solve .

Check: ✓

I do — brackets on both sides. Solve .

Check: left ; right ✓

I do — subtracting a bracket. Solve .

Check: ✓

We do: ; ; .

Activity 2 — Independent Practice (10 min)

You do: Solve, showing every step and checking in the original.

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

Activity 3 — Inquiry: the Perimeter Equation (8 min)

Pairs. Applies equation-solving to geometry.

A rectangle has width cm and length cm. Its perimeter is cm.

1. Write an expression for the perimeter and simplify it.
2. Form an equation and solve for .
3. State the rectangle’s dimensions and find its area.
4. A second rectangle has width and length , and the same perimeter of cm. Find its dimensions. Does it have the same area?

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what is known?	The perimeter is again cm, but the sides are described differently.
Write the perimeter before simplifying.	.
Expand.	.
Form the equation.	.
Solve.	, so — not a whole number.
Is that a problem?	The descriptor expects natural-number solutions, so this signals a deliberately awkward case.
Adjust: what perimeter would give a whole answer?	gives .
With	Width , length ; perimeter cm, area .

Answers: 1. ; 2. , so ; 3. cm by cm, area ; 4. With perimeter the solution is not a whole number — see the scaffolding. Using cm instead gives cm by cm, area , so the areas differ despite similar perimeters.

Teacher note: Q4 is deliberately constructed to produce a non-integer solution. Rather than hiding this, use it: recognising that a modelled answer may be unrealistic or outside the expected number set is a genuine part of mathematical modelling.

Checks for Understanding

(5 minutes — exit ticket)

1. Solve: (a) (b) (c) .
2. Solve .
3. Reasoning. A student solves by writing . Identify the error and give the correct solution.
4. Check whether solves .
5. A rectangle has width and length , with perimeter cm. Find .

Answers: 1. (a) (b) (c) ; 2. , so ; 3. The must multiply both terms: , giving ; 4. Left ; right . Not equal, so is not the solution (in fact ); 5. , so .

Common Misconceptions

Misconception	How to pre-empt it
Expanding only the first term: .	Draw the arrow to both terms. Check numerically before proceeding.
Not distributing a minus across a subtracted bracket.	Model explicitly, with the sign of each product shown.
Solving before simplifying.	Enforce the five-step protocol in order; simplification comes first.
Checking against the simplified equation rather than the original.	State the rule explicitly and mark against it.
Losing a term when collecting across a long expression.	Underline or colour like terms before combining.
Assuming every equation has a whole-number solution.	Activity 3 Q4 confronts this directly.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Solve .

Answer

E2 (AMC Junior style). A triangle has sides , and cm, and a perimeter of cm. Find and the three side lengths.

Answer

Sides: cm, cm, cm. (Check: ✓)

E3 (Challenge). Solve . What do you notice?

Answer

Expanding gives — true for every value of . This is an identity, not an equation with a single solution. Every number is a solution.

E4 (Challenge). Solve . What happens?

Answer

Expanding gives , so — false. There is no solution. The two expressions can never be equal.

E5 (Challenge). A rectangle has width and length , with a perimeter of cm. Find its area.

Answer

Width cm, length cm, area .

Homework

1. Solve, showing working: (a) (b) (c) (d) .
2. Solve by collecting like terms: (a) (b) (c) .
3. Solve (brackets on both sides): (a) (b) (c) .
4. Solve: (a) (b) .
5. A rectangle has width cm and length cm, with perimeter cm. Find , the dimensions and the area.
6. A triangle has sides , and cm, with perimeter cm. Find and the sides.
7. Check whether solves .
8. Reasoning. Explain why you should check your answer in the original equation rather than a simplified version.
9. Challenge. Solve and explain your result.
10. Challenge. Two rectangles have equal perimeters. The first is by ; the second is by . Show that this is true for every value of .

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) (c) . Q4 — (a) (b) . Q5 — , so ; dimensions by cm; area . Q6 — , so ; sides , , cm. Q7 — left , right ; not equal, so no (the solution is ). Q9 — an identity, true for all . Q10 — first perimeter ; second . Equal for every .