023i - Introducing Variables and Formulas

Lesson 23 — Introducing Variables and Everyday Formulas

Strand: Algebra | Descriptor: AC9M7A01 | Duration: 45 minutes

Learning Intentions

• To understand a variable as a symbol standing for a quantity that can change.
• To recognise and use variables to represent everyday formulas algebraically.

Success Criteria

I can:

1. Explain what a variable represents in a given formula.
2. Write an everyday rule as a formula using variables.
3. Use correct algebraic convention: rather than , rather than .
4. Identify the subject of a formula and the variables it depends on.

Warmup

(6 minutes — pattern to rule, pairs)

Display a growing pattern of tables and chairs:

Tables	1	2	3	4	5
Chairs	4	6	8	10	?

1. How many chairs for tables?
2. How many for tables?
3. How many for tables? Explain how you worked it out without extending the table.
4. Describe your rule in words.

Answers: ; ; . Rule: double the number of tables and add two.

Bridging question: Writing “double the number of tables and add two” every time is slow. Could we write it more briefly?

Activities

Activity 1 — Explicit Instruction: what a Variable is (10 min)

Definition. A variable is a letter used to represent a quantity that can take different values. A formula is a rule connecting variables, written as an equation.

From the warmup, letting be the number of tables and the number of chairs:

Test it: gives ✓ ; gives ✓

Language to insist on. The letter stands for the number of tables, not for “tables”. This distinction prevents the classic error of reading as “two tables”.

Algebraic conventions — state each explicitly and enforce from now on:

Convention	Write	Do not write
Multiplication sign omitted
Number before letter
Division as a fraction
Coefficient of one is implied
Letters in alphabetical order

Common formulas students already know — reframe them algebraically:

Discussion: In , which letter is the subject of the formula? (The one on its own: .) What does the formula let you find, and what must you already know?

Activity 2 — Writing Formulas from Words (12 min)

I do. “A plumber charges a 60$85$ per hour.”

Define the variables first — always.

Let be the number of hours worked and the total cost in dollars.

Emphasise the two-part structure: something that changes with (the rate), plus something fixed (the one-off fee). Ask which term would change if the job took twice as long.

We do:

1. A taxi charges 4$2kC$ = cost.)*
2. A rectangle’s perimeter, given its length and width.
3. The total number of legs on spiders.

You do: Write a formula for each, defining your variables.

1. A gym charges 25$15$ per month.
2. A phone plan costs 30$0.20$ per text message.
3. The total number of wheels on cars and bicycles.
4. The number of chairs needed if each of groups has students plus one teacher.
5. The change from 50n$12$ each.

(Answers: ; ; ; ; .)

Activity 3 — Inquiry: Matchstick Patterns (10 min)

Pairs, matchsticks or toothpicks.

Build a row of squares sharing edges:

□   □□   □□□   □□□□

1. Count the matches needed for , , and squares. Record in a table.
2. Predict the number for squares, then check.
3. Write a formula connecting the number of squares to the number of matches .
4. A friend writes . Is this the same as your formula? How can you tell?

Socratic scaffolding:

Prompt	Purpose
Understand: what is changing?	The number of squares; the matches follow from it.
Record your data. What do you notice?	— each square after the first adds .
Can you see why three?	The first square costs ; each new one adds a top, a bottom and one new right side — reusing the existing side.
Devise a formula from that structure	.
Can you write it more simply?	.
Test it	gives . Build it to check.
Compare with your friend’s version	— the same formula, seen differently.
Looking back	Different ways of seeing a pattern give different-looking but equivalent formulas.

Discussion: Which structural description matches directly? (One match at the far left, then a group of three for every square.) Have students draw the grouping on their matchstick row.

Checks for Understanding

(5 minutes — exit ticket)

1. Rewrite using correct convention: (a) (b) (c) (d) .
2. A cinema charges 14Cn$ tickets.
3. In the formula , what does each letter represent, and which is the subject?
4. A hire company charges 40$18Cd$ days.
5. Reasoning. Explain why should be read as “twice the number of tables” rather than “two tables”.

Answers: 1. (a) (b) (c) (d) ; 2. ; 3. is area (the subject), length, width; 4. ; 5. A variable stands for a number, not an object. Reading it as an object leads to errors — for example, with means , not “two tables”.

Common Misconceptions

Misconception	How to pre-empt it
Reading as “two tables” (the letter as a label, not a number).	Always define variables in full: “let be the number of tables”. Insist on this phrasing in every answer.
Writing or .	Enforce the convention table from Activity 1 in all written work from this lesson onwards.
Believing a variable has one fixed secret value.	Substitute several different values into the same formula to show it varies.
Confusing the variable with its unit, e.g. writing dollars inside the formula.	Units belong in the variable definition, not the formula itself.
Putting the fixed fee where the rate belongs: .	Ask “which amount is charged once, and which repeats?” before writing.
Assuming means the two-digit number "".	In algebra, always means .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). In a matchstick pattern of triangles in a row sharing edges, triangle uses matches, use , use . Write a formula for triangles, and find how many matches triangles need.

Answer

. For : matches.

E2 (AMC Junior style). A plumber charges ChC = 90h + 55$325$. How long did it take?

Answer

E3 (Challenge). Two students describe the same pattern of dots. Ali writes ; Bree writes . Are they the same formula? Justify.

Answer

Yes — identical. They simply saw the pattern grouped differently.

E4 (Challenge). A pattern of hexagons in a row uses matches for one hexagon and more for each additional one. Write a formula, then find how many hexagons can be made with matches.

Answer

. Setting gives hexagons exactly.

Homework

1. Rewrite using correct algebraic convention: (a) (b) (c) (d) (e) .
2. Write a formula for each, defining your variables clearly: (a) The cost of pens at 3C$20$8csPs$100n$22$ each.
3. In the formula , state what each variable represents and which is the subject.
4. A matchstick pattern of pentagons in a row uses matches for one and more for each additional one. Write a formula for in terms of .
5. A phone plan costs 25$0.15Ct$ minutes are used.
6. Reasoning. Yusuf writes the formula for “a 50$30C = 50h + 30$. Explain his error and give the correct formula.
7. Challenge. A pattern uses tiles for the first shape and more for each shape after. (a) Write a formula. (b) How many tiles for the th shape? (c) Which shape number uses tiles?

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) (e) . Q4 — . Q5 — . Q6 — the 50$30C = 30h + 50T = 4n + 3634n + 3 = 95n = 23$.