036i - Consolidation Linear Equations

Lesson 36 — Consolidation and Check: Linear Equations

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

Learning Intentions

• To consolidate all methods for solving one-variable linear equations.
• To select an efficient method and verify every solution.

Success Criteria

I can:

1. Solve one-step, two-step and multi-step equations accurately.
2. Handle brackets, like terms and variables on both sides.
3. Form an equation from a context and solve it.
4. Verify every solution by substitution.

Warmup

(6 minutes — sorting by difficulty, pairs)

Sort these into one-step, two-step and multi-step, without solving. Then solve the one-step ones.

Answers: One-step — (a) , (d) , (f) . Two-step — (b). Multi-step — (c), (e).

Discussion: How did you decide, just by looking? (Count how many operations separate the variable from being alone.)

Activities

Activity 1 — Mixed Practice by Type (14 min)

Graded worksheet or stations. Students must identify the type before solving.

Set A — one-step

Set B — two-step

Set C — brackets and like terms

Set D — variables on both sides

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. ; 10. ; 11. ; 12. ; 13. ; 14. ; 15. ; 16. .)

Efficiency discussion. Q9 and Q11 can be solved either by expanding or by dividing both sides by the bracket’s multiplier first. Have students try both on Q11 and decide which they prefer. (Dividing by gives immediately.)

Activity 2 — Contextual Consolidation (12 min)

Pairs. Full five-step protocol required.

Problem 1. A hire company charges 65$28$233$. How many days?

Problem 2. Four consecutive whole numbers add to . Find them.

Problem 3. A rectangle’s length is cm more than its width. Its perimeter is cm. Find its area.

Problem 4. Leo has 80$6$24$8$ per week. After how many weeks do they have equal amounts?

Problem 5. A number is doubled, then is added; the result equals five times the number, less . Find the number.

Socratic scaffolding for Problem 5:

Prompt	Purpose
Understand: what does “the result equals” tell you?	Two expressions are equal — this is an equation.
Define the variable.	Let be the number.
Translate the left side.	”Doubled, then added”: .
Translate the right side.	”Five times the number, less ”: .
Form the equation.	.
Which side for the variables?	The right, which has more — avoiding negatives.
Carry it out	, so and .
Check	Left: . Right: ✓

Answers: 1. , so days. 2. Let the first be : , so ; the numbers are . 3. , so cm, cm, area . 4. , so and weeks (both have 56n = 8$.

Activity 3 — Inquiry: Build and Swap (8 min)

Pairs, then swap with another pair.

1. Choose a whole number between and — this is your secret solution.
2. Build an equation with that solution by starting from your number and applying the same operations to both sides, at least three times.
3. Write only the final equation on a card.
4. Swap cards with another pair and solve theirs.
5. Check each other’s answers against the secret numbers.

Worked demonstration — building from :

Final card: . (Check: and ✓)

Socratic prompts if pairs get stuck:

Prompt	Purpose
What does “apply the same operation to both sides” preserve?	The truth of the statement — and therefore the solution.
Does the order of your operations matter?	Not for validity, but it changes how hard the final equation looks.
How could you make yours harder without changing the answer?	Add a variable term to both sides, or introduce a bracket.
How do you know your equation is correct before swapping?	Substitute your secret number back in.

Extension: challenge pairs to build one equation with brackets on both sides that still has their secret solution.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Solve: (a) (b) (c) .
2. Solve .
3. A number is multiplied by and is subtracted, giving . Form an equation and solve it.
4. Verify whether solves .
5. Reasoning. Describe, in your own words, the general strategy for solving any linear equation.

Answers: 1. (a) (b) (c) ; 2. , so ; 3. , so ; 4. LHS ; RHS . Not equal — is not the solution (it is ); 5. Simplify each side, gather variables on one side and numbers on the other, undo operations in reverse order, then check by substitution.

Common Misconceptions

Misconception	How to pre-empt it
Choosing a method before identifying the equation’s type.	Require students to classify each item before solving.
Expanding when dividing first would be simpler.	Discuss efficiency explicitly in Activity 1.
Undoing operations in the applied order rather than reverse.	Return to the function-machine image from Lesson 32.
Sign errors when gathering variables.	Advise moving to the side with the larger coefficient.
Skipping the check under time pressure.	Keep it compulsory — it is faster than re-solving.
Forgetting to answer the contextual question in a sentence.	Marked as part of the five-step protocol.

Enrichment — Competition-Style Problems

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

Answer

, so and .

E2 (AMC Junior style). Explain why the sum of four consecutive whole numbers can never be .

Answer

Four consecutive numbers starting at sum to . Setting gives , and is not divisible by — so is not a whole number. No such four numbers exist.

More generally, always leaves a remainder of on division by , so the sum must be more than a multiple of . Since exactly, it fails this test. (Compare , which does work: .)

E3 (Challenge). Solve .

Answer

E4 (Challenge). A rectangle has length and width . Its perimeter is cm. Find its area.

Answer

Length cm, width cm, area .

E5 (Challenge). Find a value of for which and are equal. Then explain what happens if the second is changed to .

Answer

gives , so . With : gives , so . Changing the constant simply shifts the solution.

Homework

1. Solve: (a) (b) (c) (d) .
2. Solve: (a) (b) (c) (d) .
3. Solve: (a) (b) (c) (d) .
4. Solve: (a) (b) (c) (d) .
5. A courier charges 18$4$70$. What was the mass?
6. Five consecutive whole numbers add to . Find them.
7. A rectangle’s length is cm more than its width, and its perimeter is cm. Find its area.
8. Kai has 150$9$70$11$ per week. After how many weeks are their amounts equal?
9. Verify whether solves .
10. Reasoning. Explain why has infinitely many solutions but has none.
11. Challenge. The equations and share a solution. Find .

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) (d) . Q5 — , so kg. Q6 — middle , so . Q7 — , so cm, cm, area . Q8 — , so weeks (both 114= 20= 20x6 = 75x = 30x = 618 + k = 25k = 7$.