Mathematics • Year 10 • Unit 2 • Lesson 7

Mixed Algebraic Techniques — Skill Drill

Build fluency with the three big moves from Lesson 7: expand brackets first, distribute the minus across every term, then collect like terms. One worked example, one guided trace with blanks, then eight independent problems graded from foundation to extension.

Build · I Do / We Do / You Do

1. I do — fully worked example

Expand, distribute the minus, and collect like terms. Each step has a reason underneath so you can see why, not just what.

Problem. Simplify (x + 4)(x + 2) − (x + 3)(x + 1).

Step 1 — Expand both products separately (FOIL).

(x + 4)(x + 2) = x² + 2x + 4x + 8 = x² + 6x + 8

(x + 3)(x + 1) = x² + x + 3x + 3 = x² + 4x + 3

Reason: brackets block collection. Both products must be fully expanded first.

Step 2 — Rewrite the subtraction.

(x² + 6x + 8) − (x² + 4x + 3)

Reason: the minus applies to EVERY term in the second bracket. Do not forget any.

Step 3 — Distribute the minus.

= x² + 6x + 8 − x² − 4x − 3

Reason: every sign inside the second bracket flips: +x² → −x², +4x → −4x, +3 → −3.

Step 4 — Collect like terms.

x² − x² = 0, 6x − 4x = 2x, 8 − 3 = 5 → = 2x + 5

Reason: x² cancels (a clue that the working is on track). Linear and constant terms combine.

Step 5 — Check by substituting x = 1.

Original: (5)(3) − (4)(2) = 15 − 8 = 7. Answer: 2(1) + 5 = 7. ✓

Answer: 2x + 5.

Stuck? Revisit lesson § "Distribute the Minus" — write the sign change out on its own line.

2. We do — fill in the missing steps

Same method as Section 1, but with the working faded. Fill in each blank. 5 marks

Problem. Expand and simplify 2(x + 5) + (x − 3)(x + 4).

Step 1 — Expand the first part (distribute the 2):

2(x + 5) = ____ x + ____

Step 2 — Expand the quadratic product using FOIL:

(x − 3)(x + 4) = x² + ____ x − ____ x − ____ = x² + ____ x − 12

Step 3 — Add the two expanded pieces:

2x + 10 + x² + x − 12

Step 4 — Collect like terms:

x²-term: ____ , x-terms: 2x + x = ____ x , constants: 10 − 12 = ____

Step 5 — Write the final answer:

Simplified = ____________________

Stuck? Revisit lesson § "Expand First" — brackets must be removed before you can collect terms.

3. You do — independent practice

Show your working on each line. The first four are foundation (single expansion or simple sum). The middle two are standard (two pieces plus collection). The last two are extension.

Foundation — single skill

3.1 Expand and simplify 3(x + 2) + 4(x + 5). 1 mark

3.2 Expand and simplify (x + 3)(x + 5). 1 mark

3.3 Expand and simplify (x − 2)(x + 6). 1 mark

3.4 Simplify (x² + 5x + 6) − (x² + 2x + 1). Watch the minus. 1 mark

Standard — combine tools

3.5 Expand and simplify (x + 1)(x + 4) + 3(x − 2). 2 marks

3.6 Simplify (x + 5)(x + 1) − x². 2 marks

Extension — push your thinking

3.7 Simplify (x + 5)(x + 2) − (x + 4)(x + 3). Show every line: expand both, distribute the minus, collect. 3 marks

3.8 A student writes (x + 6)(x + 1) − (x + 4)(x + 2) = x² + 7x + 6 − x² + 6x + 8 = 13x + 14. Identify (in one sentence) the error, then write the correct simplified expression. 2 marks

Stuck on 3.8? Revisit lesson § "Spot the Trap" — did the student flip every sign in the second bracket?

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (faded 2(x + 5) + (x − 3)(x + 4))

Step 1: 2(x + 5) = 2x + 10.
Step 2: (x − 3)(x + 4) = x² + 4x − 3x − 12 = x² + 1x − 12 (i.e. x² + x − 12).
Step 4: x²-term: x²; x-terms: 2x + x = 3x; constants: 10 − 12 = −2.
Step 5: Simplified = x² + 3x − 2.

3.1 — 3(x + 2) + 4(x + 5)

= 3x + 6 + 4x + 20 = 7x + 26.

3.2 — (x + 3)(x + 5)

FOIL: x² + 5x + 3x + 15 = x² + 8x + 15.

3.3 — (x − 2)(x + 6)

FOIL: x² + 6x − 2x − 12 = x² + 4x − 12.

3.4 — (x² + 5x + 6) − (x² + 2x + 1)

= x² + 5x + 6 − x² − 2x − 1 = 3x + 5. x² cancels — sanity check that the minus was distributed.

3.5 — (x + 1)(x + 4) + 3(x − 2)

(x + 1)(x + 4) = x² + 5x + 4. 3(x − 2) = 3x − 6. Sum: x² + 5x + 4 + 3x − 6 = x² + 8x − 2.

3.6 — (x + 5)(x + 1) − x²

(x + 5)(x + 1) = x² + 6x + 5. Subtract x²: x² + 6x + 5 − x² = 6x + 5.

3.7 — (x + 5)(x + 2) − (x + 4)(x + 3)

(x + 5)(x + 2) = x² + 7x + 10.
(x + 4)(x + 3) = x² + 7x + 12.
Subtract: (x² + 7x + 10) − (x² + 7x + 12) = x² + 7x + 10 − x² − 7x − 12 = −2.
Check at x = 1: (6)(3) − (5)(4) = 18 − 20 = −2. ✓

3.8 — Identify the error

The student forgot to distribute the minus to all three terms of the second bracket — they only flipped the x² and left +6x and +8 unchanged when the line should read "− x² − 6x − 8".
Correct working: (x + 6)(x + 1) − (x + 4)(x + 2) = (x² + 7x + 6) − (x² + 6x + 8) = x² + 7x + 6 − x² − 6x − 8 = x − 2.