Linear equations and expressions
Building algebraic expressions from words, substituting into formulas, and solving one-variable equations
About three to four lessons of 45 to 60 minutes
How do you describe a rule for ANY number, not just one?
"Think of a number, double it, then add 5." You could test that rule one number at a time forever, or you could write it once, for every number at once, as 2n + 5. That single move, from a specific number to a variable standing for any number, is what algebra actually is.
Once a rule is written as an expression, three things become possible: you can substitute any value in to get an answer instantly, you can build up more complex formulas from simple ones, and you can solve an equation to find exactly which number makes a statement true. This unit builds all three skills, starting with turning words into algebra.
- A phone plan: $20 plus 10c per textwritten as cost = 20 + 0.10t, a formula you substitute into
- Total ticket cost for a groupprice times number of people, an expression built from words
- Distance = speed x timea real formula you substitute known values into to find the unknown
- "I'm thinking of a number..." puzzlessolving 3n + 4 = 19 finds the exact number being thought of
What students will be able to do
Students will translate a worded description into an algebraic expression, substitute given values into a formula to evaluate it, and solve a one-variable linear equation with a whole-number solution, checking the answer by substitution.
- I can write an algebraic expression from a worded description, using a variable, constants and the correct operations.
- I can substitute a given value for a variable into a formula and evaluate the result.
- I can solve a one-variable linear equation by undoing operations in reverse order.
- I can check a solution by substituting it back into the original equation.
Standards this unit teaches
- AC9M7A02Australian Curriculum v9 (ACARA)Build algebraic expressions
Write algebraic expressions from words using constants, variables, operations and brackets.
- AC9M7A01Australian Curriculum v9 (ACARA)Substitute into formulas
Use variables to write everyday formulas and substitute values into a formula to find an unknown.
- AC9M7A03Australian Curriculum v9 (ACARA)Solve linear equations
Solve one-variable linear equations with whole-number solutions and check each answer by substitution.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Variable
- a letter that stands for a number that can change or is not yet known
- Expression
- a combination of numbers, variables and operations with no equals sign, such as 2n + 5
- Equation
- a statement that two expressions are equal, using an equals sign, such as 2n + 5 = 19
- Formula
- an equation that relates two or more variables in a general, reusable rule, such as distance = speed x time
- Substitute
- to replace a variable with a specific number and then evaluate the result
- Solve
- to find the value (or values) of a variable that make an equation true
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Turning words into algebra
ConcreteEvery worded rule has a number that is not fixed, that number becomes the variable. Read the words carefully and translate each part in order: a starting amount, an operation, and how it depends on the variable.
"5 more than double a number" becomes 2n + 5: double the number is 2n, and 5 more than that is + 5. "3 less than a number divided by 4" becomes n/4 - 3. Order matters: "3 less than X" means X - 3, not 3 - X.
Write an expression for: 'a taxi charges $4 plus $2 for every kilometre travelled'.
- Identify the variable: let k = the number of kilometres travelled.
- The fixed part ($4) is added no matter what: start with 4.
- The variable part is $2 per kilometre, so it is 2 x k, written 2k.
- Combine: total cost = 4 + 2k.
Answer: Total cost = 4 + 2k (in dollars, where k is the number of kilometres).
- Why does '3 less than a number' translate to n - 3, and not 3 - n?
- What is the variable in a worded problem, and how do you spot it?
2. Substituting into formulas
PictorialA formula is a reusable rule connecting variables, like distance = speed x time. Substituting means replacing each variable with a known number, then following the order of operations to evaluate the result.
For the taxi formula, cost = 4 + 2k, substituting k = 7 (a 7 km trip) gives cost = 4 + 2(7) = 4 + 14 = 18 dollars. Multiplication happens before addition, exactly as the order of operations requires.
A formula for the perimeter of a rectangle is P = 2l + 2w. Find P when l = 9 and w = 5.
- Substitute the values: P = 2(9) + 2(5).
- Multiply first: 2(9) = 18 and 2(5) = 10.
- Add: 18 + 10 = 28.
Answer: P = 28.
- Why must multiplication happen before addition when substituting into 2l + 2w?
- If a formula has two variables, how many values do you need before you can evaluate it?
3. Solving one-variable equations
AbstractSolving an equation means finding the value of the variable that makes it true. The method is to undo each operation in reverse order, doing the same thing to both sides so the equation stays balanced.
For 3n + 4 = 19, the variable has had 3 multiplied and then 4 added, in that order. To undo it, reverse the order: subtract 4 first, then divide by 3. Subtracting 4 from both sides gives 3n = 15; dividing both sides by 3 gives n = 5.
Solve 5n - 8 = 27, then check the answer.
- Undo the subtraction first (reverse order): add 8 to both sides. 5n - 8 + 8 = 27 + 8, so 5n = 35.
- Undo the multiplication: divide both sides by 5. n = 35 / 5 = 7.
- Check by substituting n = 7 back into the original equation: 5(7) - 8 = 35 - 8 = 27. It matches.
Answer: n = 7.
- Why do you undo operations in the REVERSE of the order they were applied?
- What does 'checking' a solution actually involve?
Common misconceptions and how to address them
Misconception'3 less than a number' means 3 - n.
Why it happens: Students translate words left to right instead of identifying which quantity is being subtracted FROM which.
How to address it: Read for the subject: 'less than' always subtracts from the number mentioned first in the ENGLISH sense of the phrase's target, so '3 less than n' is n - 3. Practise rephrasing as 'n, minus 3' before writing it algebraically.
MisconceptionWhen solving an equation, you undo operations in the SAME order they appear.
Why it happens: Students apply operations left to right instead of reversing the order for solving.
How to address it: Solving reverses the build-up: if n was multiplied then had something added (to build 3n + 4), undo addition first, then undo multiplication. Say it as 'last operation applied, first operation undone'.
MisconceptionYou only need to do an operation to the side with the variable.
Why it happens: Students focus on isolating the variable and forget the equation must stay balanced.
How to address it: Whatever you do to one side, you must do to the OTHER side too, or the equation stops being true. Show both sides of the equation being adjusted every single step.
Guided practice (with answers)
1. Write an expression for '7 more than three times a number, n'.
Answer: 3n + 7, because 'three times a number' is 3n, and '7 more than' that is + 7.
2. Write an expression for 'a number, n, divided by 2, then decreased by 5'.
Answer: n/2 - 5, because dividing by 2 comes first, then 5 is subtracted from the result.
3. The formula for the area of a triangle is A = (b x h) / 2. Find A when b = 10 and h = 6.
Answer: 30, because A = (10 x 6) / 2 = 60 / 2 = 30.
4. Solve 4n + 3 = 23.
Answer: n = 5, because subtracting 3 from both sides gives 4n = 20, then dividing by 4 gives n = 5.
5. Solve 2n - 9 = 11, and check your answer.
Answer: n = 10, because adding 9 to both sides gives 2n = 20, then dividing by 2 gives n = 10. Check: 2(10) - 9 = 20 - 9 = 11, which matches.
Independent practice worksheets
Practise building expressions, substituting and solving with computed, never-wrong answer keys.
Differentiation
- Start with expressions using only one operation ('5 more than n') before combining two operations.
- Use a function-machine diagram (input, operation boxes, output) to make substitution and solving visually concrete.
- Provide sentence frames for translating words: '___ more than ___' = '+', '___ less than ___' = 'reversed -', 'times' = 'x'.
- For solving, always write out both sides of the equation at every step, never skip to the answer.
- Introduce equations with the variable on both sides, such as 3n + 2 = n + 10.
- Ask students to write their own worded problem that translates to a given target equation.
- Explore formulas with three or more variables, substituting for all but one and solving for the remainder.
- Investigate why some worded phrases are ambiguous (e.g. 'twice a number plus 3' vs 'twice a number and 3 more') and how brackets remove the ambiguity.
Assessment: exit ticket
A three-question exit ticket sampling building expressions, substitution, and solving.
1. Write an expression for '4 less than double a number, n'.
Answer: 2n - 4, because 'double a number' is 2n, and '4 less than' that is - 4.
2. A formula is C = 5 + 3n (cost in dollars for n items). Find C when n = 6.
Answer: 23, because C = 5 + 3(6) = 5 + 18 = 23.
3. Solve 6n - 5 = 31, and check your answer.
Answer: n = 6, because adding 5 to both sides gives 6n = 36, then dividing by 6 gives n = 6. Check: 6(6) - 5 = 36 - 5 = 31, which matches.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 building expressions (section 1), Lesson 2 substitution (section 2), Lesson 3 solving equations (section 3), Lesson 4 mixed practice plus the exit ticket.
- This unit assumes comfort with the order of operations, since substitution and solving both depend on evaluating expressions correctly.
- Language to repeat: build an expression by translating each word in order; solve an equation by undoing the LAST operation first, working backward.
- Curriculum note: AC9M7A02 (Australian Curriculum v9) covers building expressions, AC9M7A01 covers substituting into formulas, and AC9M7A03 covers solving linear equations, all at Year 7. This unit sequences all three together since each depends on the last.
- Present and print both work: use the Print button for a clean handout, or work the taxi/perimeter examples live on the board, inviting students to substitute their own values.