030i - Problem Solving with Expressions

Lesson 30 — Problem Solving and Consolidation: Algebraic Expressions

Strand: Algebra | Descriptor: AC9M7A02 | Duration: 45 minutes

Learning Intentions

• To consolidate forming, expanding, simplifying and evaluating expressions.
• To use algebra to justify that a general statement is always true.

Success Criteria

I can:

1. Form, expand and simplify an expression within a single problem.
2. Evaluate an expression, including with negative values.
3. Show that two expressions are equivalent.
4. Use algebra to prove a general claim, rather than testing examples.

Warmup

(6 minutes — “Equivalent or not?”, pairs)

Decide whether each pair is equivalent. Justify by expanding, and check with a value.

1. and
2. and
3. and
4. and
5. and

Answers: 1. Equivalent. 2. Not — . 3. Equivalent. 4. Equivalent, since . 5. Not — test : versus .

Note on Q5: students are not expected to expand at Year 7, but the numerical counterexample is well within reach and pre-empts a persistent later error.

Activities

Activity 1 — Mixed Consolidation (12 min)

Rotating stations or a graded worksheet. Four skill types mixed deliberately, so students must decide what each question needs.

Type A — Form. Write an expression:

1. less than four times
2. The cost of hours at 38$95$ fee
3. Twice the sum of and

Type B — Expand and simplify:

Type C — Evaluate:

7. when
8. when
9. when

Type D — Reverse:

10. An expression simplifies to . It was formed as . Find .

(Answers: 1. ; 2. ; 3. or ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. ; 10. ; matching gives , so , and then — so no value of works. A deliberate “no solution” item; see the discussion below.)

Discussion of Q10. Students should reach the contradiction themselves. Matching the constant term forces , but then the -coefficient is , not . Ask: what would the target expression need to be for a solution to exist? (For the result is .) Recognising that a problem has no solution — and proving it — is a genuine mathematical skill.

Activity 2 — Using Algebra to Prove (12 min)

The intellectual heart of the lesson.

The key distinction, stated explicitly:

Testing examples shows a statement works for those examples. Algebra shows it works for every value at once.

I do — the consecutive-numbers claim. “The sum of three consecutive whole numbers is always a multiple of .”

Testing ✓ and ✓ is evidence, not proof. Now the algebra:

Let the middle number be . The three numbers are , and .

Since is times a whole number, the sum is always a multiple of — for every starting value, without exception.

Ask: why choose the middle number as rather than the first? (The and cancel, making the algebra cleaner. Choosing the first gives — also correct, slightly messier.)

Pairs practice — prove each claim:

1. The sum of two consecutive numbers is always odd.
2. Adding to a number, doubling, then subtracting always gives double the original.
3. The sum of four consecutive whole numbers is never a multiple of .
4. If you take any number, add , multiply by , then subtract , you get three times the original.

Socratic scaffolding for Claim 3:

Prompt	Purpose
Understand: what must you show?	That the sum is never divisible by — a claim about all cases.
Would testing a few examples settle it?	No. Testing can disprove a claim, but cannot prove one.
Set up the algebra.	Let the first be : the four are .
Add them.	.
Is a multiple of ?	is; is not. So the total is more than a multiple of .
Write it convincingly.	— always a remainder of .
Looking back	Check with ✓

Answers: 1. , which is always odd. 2. ✓ 3. leaves remainder . 4. ✓

Activity 3 — Inquiry: the Number Pyramid (8 min)

Pairs.

In a number pyramid, each brick is the sum of the two directly below it.

        ?
      ?   ?
     x   3   y

1. Fill in the two middle bricks in terms of , and .
2. Write and simplify an expression for the top brick.
3. If and , find the top value.
4. If the top is and , find .

Socratic scaffolding:

Prompt	Purpose
Understand: what is the rule?	Each brick is the sum of the two below it.
Fill the middle row.	Left: . Right: .
Now the top.	.
Simplify.	.
Test with , .	. Check by building the pyramid: and , summing to ✓
Reverse for Q4.	, so .
Looking back — generalise	The top depends only on and the middle brick, not on their individual values.

Extension: Build a four-row pyramid with base , , , and find the top. (Answer: — the coefficients are the row of Pascal’s triangle, worth pointing out to interested students.)

Checks for Understanding

(7 minutes — exit ticket, collected)

1. Expand and simplify: .
2. Evaluate when .
3. Write and simplify an expression for the perimeter of a rectangle with width and length .
4. Reasoning. Show algebraically that the sum of two consecutive even numbers is always a multiple of but never a multiple of .
5. Reasoning. Ana says and are the same. Give a value of that disproves this.

Answers: 1. ; 2. ; 3. ; 4. Let the numbers be and ; their sum is , which is even but leaves remainder on division by ; 5. Any works — e.g. gives versus .

Common Misconceptions

Misconception	How to pre-empt it
Believing a few successful examples prove a general claim.	Make the distinction explicit in Activity 2 and return to it in every proof question.
Not distributing a subtraction across a bracket.	Require the fully expanded intermediate line, always.
Sign errors when evaluating with negatives, especially with squares.	Brackets around every substituted value, as established in Lesson 24.
Choosing an awkward variable for a proof (e.g. the first of three consecutive numbers).	Discuss why the middle term simplifies the algebra — good choices make proofs easier.
Giving up when a problem has no solution.	Activity 1 Q10 normalises this. Proving no solution exists is a valid, complete answer.
Simplifying to .	Test numerically with : versus .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Simplify , then find the value of making the result equal to .

Answer

E2 (AMC Junior style). The sum of five consecutive whole numbers is , where is the middle number. If the sum is , what is the largest of the five?

Answer

gives , so the numbers are to . The largest is .

E3 (Challenge). Prove that the difference between the squares of two consecutive whole numbers is always odd.

Answer

is even, so is always odd. (This is the same result met in Lesson 8’s square-number investigation.)

E4 (Challenge). In a number pyramid with base , , , the top is . If the top is , and , find .

Answer

E5 (Challenge). Show that is always even, for any whole number .

Answer

and are consecutive, so exactly one of them is even. A product with an even factor is even. Hence is always even — which is why in Lesson 26’s sum formula always gives a whole number.

Homework

1. Expand and simplify: (a) (b) (c) (d) .
2. Evaluate: (a) when (b) when (c) when (d) when , .
3. A rectangle has width and length . Write simplified expressions for its perimeter and area.
4. Write and simplify an expression for the perimeter of a triangle with sides , and .
5. Proof. Show algebraically that adding to a number, tripling, then subtracting always gives three times the original number.
6. Proof. Show that the sum of three consecutive even numbers is always a multiple of .
7. In a number pyramid, the base row is , , . (a) Write a simplified expression for the top brick. (b) If the top is and , find .
8. Reasoning. Ben claims ” is always bigger than .” Find two values of that disprove him.
9. Challenge. Prove that the difference between a two-digit number and the number formed by reversing its digits is always a multiple of . (You met this numerically in Lesson 3 — now prove it.)

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — , . Q4 — . Q5 — . Q6 — ; since is even, is even, so this is a multiple of . Q7 — (a) (b) . Q8 — gives ; gives . (Also any fraction between and .) Q9 — .