027i - Forming Algebraic Expressions

Lesson 27 — Forming Expressions from Word Descriptions

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

Learning Intentions

• To formulate algebraic expressions using constants, variables and operations.
• To translate written descriptions into algebraic notation accurately.

Success Criteria

I can:

1. Identify the variable in a worded description and define it clearly.
2. Translate the key words of each operation into algebraic symbols.
3. Distinguish an expression from an equation.
4. Write expressions in correct algebraic convention.

Warmup

(6 minutes — vocabulary sort, pairs)

Sort these words under the four operations: sum, product, difference, quotient, more than, less than, times, shared between, increased by, twice, reduced by, of, total, per, decreased by.

Answers: — sum, more than, increased by, total. — difference, less than, reduced by, decreased by. — product, times, twice, of. — quotient, shared between, per.

Warning to flag now: ” less than ” is , not . The order reverses. This trips up more students than any other translation.

Activities

Activity 1 — Explicit Instruction: the Translation Toolkit (12 min)

Step 1 — always define the variable. Before writing anything, state: “let be the number”.

Step 2 — translate phrase by phrase.

In words	In algebra
more than
less than
times
divided by
Twice
Half of
The sum of and
The product of and
squared
more than twice
less than half of
Subtract from

The order-reversal cases — model these slowly:

Ask students to test each with : they give , and . The difference is real, not pedantic.

Expression versus equation — define both.

• An expression has no equals sign: .
• An equation states that two expressions are equal: .

An expression can be evaluated; an equation can be solved.

You do: Translate into algebra, letting be the number.

1. more than the number
2. The number decreased by
3. Five times the number
4. The number divided by
5. Twice the number, plus
6. less than three times the number
7. The number subtracted from
8. Half the number, minus
9. The square of the number
10. Four more than the number, all doubled

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

Activity 2 — Expressions from Contexts (10 min)

Move from abstract phrasing to real situations.

I do. “Ravi is years older than Sam.” Let Sam’s age be .

Now: “In years’ time, how old will Ravi be?”

We do:

1. A pen costs p6$ pens.
2. A rectangle’s length is cm more than its width . Write an expression for the length, and one for the perimeter.
3. A number is shared equally between people. Write an expression for each share.

You do:

1. Mia has m$18$. How much is left?
2. A box holds apples. How many apples in boxes?
3. Tom is years old. How old was he years ago?
4. A concert ticket costs c$43$ tickets?
5. A rope of length metres is cut into equal pieces. How long is each?

(Answers: ; ; ; or ; .)

Activity 3 — Inquiry: Think of a Number (10 min)

Pairs. A genuine “why does this work?” investigation.

Try this trick on a partner:

1. Think of a number.
2. Add .
3. Double the result.
4. Subtract .
5. Halve it.

You end up with your starting number. Why does it always work?

Socratic scaffolding:

Prompt	Purpose
Understand: what is the claim?	The final answer always equals the starting number, whatever it was.
Have you tested it?	Try , then , then . It works each time.
Does testing prove it always works?	No — it only shows it works for those numbers.
Devise a plan	Do the trick with a letter instead of a number.
Carry out step by step	Let the number be . Write the expression after each instruction.
Step 1
Step 2
Step 3
Step 4
Step 5
Looking back	The algebra shows it for every number at once — this is a proof, not just evidence.

Extension: Have pairs design their own trick that always returns the starting number, and prove it algebraically. (A reliable recipe: any sequence of operations followed by their exact inverses in reverse order.)

Checks for Understanding

(5 minutes — exit ticket)

1. Write an expression for: (a) more than (b) less than (c) three times (d) divided by .
2. Write an expression for ” less than twice a number “.
3. A book costs b4$7$ postage.
4. Reasoning. Explain why ” less than ” is and not . Test both with .
5. Which of these is an equation, not an expression: , , ?

Answers: 1. (a) (b) (c) (d) ; 2. ; 3. ; 4. Starting from and going lower gives . With : (correct) but (wrong); 5. .

Common Misconceptions

Misconception	How to pre-empt it
” less than ” written as .	Flag in the warmup, model in Activity 1, and test with a numerical value every time.
Confusing an expression with an equation.	Define both explicitly, and ask “is there an equals sign?” for every item.
Writing for ” doubled”.	Enforce number-before-letter convention from Lesson 23.
Forgetting brackets: ” more than , all doubled” written as .	Ask what is being doubled — the whole thing. Test both forms with : but .
Using the letter as a label (” means six boxes”).	Return to Lesson 23’s convention: is the number of apples in a box.
Treating and as interchangeable.	Test both with : versus .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Anya is years old. Her brother is twice her age, and her mother is years older than the brother. Write an expression for the mother’s age, and find it when .

Answer

Mother’s age . At : years.

E2 (AMC Junior style). A number is multiplied by , then is added, giving . Write an equation and find the number.

Answer

E3 (Challenge). Think of a number, add , multiply by , subtract , then divide by the original number. Show algebraically that the answer is always (provided the starting number is not zero).

Answer

The restriction is needed because dividing by zero is undefined.

E4 (Challenge). A rectangle has width and length cm more than twice the width. Write expressions for the length, the perimeter and the area.

Answer

Length ; perimeter ; area .

E5 (Challenge). Three consecutive whole numbers have a sum of . Write an expression for the smallest of them.

Answer

Let the middle number be . Then , so and the smallest is .

Homework

1. Write an expression for each, letting be the number: (a) more than the number (b) the number less (c) seven times the number (d) the number divided by (e) the number subtracted from .
2. Write an expression for: (a) more than three times (b) less than half of (c) twice the sum of and (d) the square of , plus .
3. A T-shirt costs t55$9$1003$ shirts.
4. Leila is years old. Write an expression for her age (a) in years (b) years ago (c) when she is twice her current age.
5. A rectangle has width cm and length cm more than the width. Write expressions for its length and its perimeter.
6. State whether each is an expression or an equation: (a) (b) (c) (d) .
7. Reasoning. Explain the difference between and . Evaluate both when .
8. Challenge. Think of a number, double it, add , halve the result, then subtract the original number. Prove algebraically that the answer is always .

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) . Q4 — (a) (b) (c) . Q5 — , . Q6 — (a) expression (b) equation (c) expression (d) equation. Q7 — but ; brackets change what is doubled. Q8 — .