035i - Forming and Solving Equations from Context

Lesson 35 — Forming and Solving Equations from Context

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

Learning Intentions

• To translate a worded problem into a linear equation.
• To solve the equation and interpret the answer in context.

Success Criteria

I can:

1. Define a variable clearly for a worded problem.
2. Form an equation that captures the relationships described.
3. Solve the equation and check the solution.
4. Answer the question that was actually asked, with units.

Warmup

(6 minutes — expression or equation? pairs)

For each, decide whether the situation gives an expression or an equation, and write it.

1. “A number increased by .”
2. “A number increased by equals .”
3. “Three times a number, less .”
4. “Three times a number, less , is .”

Answers: 1. Expression, . 2. Equation, . 3. Expression, . 4. Equation, .

Key distinction to name: an equation appears when the words give you a known total — “equals”, “is”, “comes to”, “altogether”. That is your signal to solve rather than just describe.

Activities

Activity 1 — Explicit Instruction: the Modelling Cycle (12 min)

The five-step protocol:

1. Define the variable, precisely and with units.
2. Form the equation from the relationships described.
3. Solve the equation.
4. Check by substituting into the original equation.
5. Answer the question in a sentence, with units.

Step 5 matters. Students routinely solve correctly and then fail to answer what was asked — giving the variable when the question wanted a total, or vice versa.

I do — the taxi. “A taxi charges a 5$3$41$. How far was the trip?”

Define: let be the number of kilometres travelled.

Check: ✓

Answer: The trip was km.

I do — an age problem. “Maya is years older than Sam. Together their ages total . How old is each?”

Define: let be Sam’s age in years. Then Maya’s age is .

Check: ✓

Answer: Sam is and Maya is .

Note the pattern: define the smaller or simpler quantity as the variable, then express the other in terms of it. Choosing well makes the algebra easier.

We do:

1. “A number is multiplied by and is subtracted, giving . Find the number.”
2. “Three consecutive whole numbers add to . Find them.”
3. “A rectangle’s length is cm more than its width. Its perimeter is cm. Find its dimensions.”

Activity 2 — Contextual Problems (14 min)

Pairs. Every solution must show all five steps.

Problem 1 — Concert. Tickets cost 24$9$225$. How many tickets were bought?

Problem 2 — Sharing. Three friends share 96$6$ more than Ali. How much does each receive?

Problem 3 — Savings. Priya has 45$12$105$8$ per week. After how many weeks will they have the same amount?

Problem 4 — Angles. Two angles on a straight line sum to . One is more than twice the other. Find both.

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: how many unknowns are there?	Three amounts — but they are all linked.
Which should be the variable?	Ali’s share, since both others are described relative to Ali.
Define it.	Let be Ali’s share in dollars.
Express the others.	Ben: . Cara: .
Form the equation.	.
Simplify and solve.	, so and .
Is that sensible?	Yes — money can be in dollars and cents.
Answer fully.	Ali 22.50$45$28.50$.
Check	✓

Answers:

1. , so and tickets.
2. Ali 22.50$45$28.50$.
3. , so and weeks (both have 81$).
4. Let the smaller be ; then , so and . The angles are and .

Discussion of Problem 2. The answer is not a whole number of dollars. Ask whether that matters. (No — money is not restricted to whole dollars. But if the problem had been about people or objects, a non-whole answer would signal a problem.)

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A number is multiplied by and is added, giving . Form an equation and solve it.
2. A plumber charges 70$45$295$. How many hours did it take?
3. Two consecutive whole numbers add to . Find them.
4. A rectangle’s length is cm more than twice its width. Its perimeter is cm. Find its dimensions.
5. Reasoning. Explain why defining the variable clearly, with units, is the most important first step.

Answers: 1. , so ; 2. , so and hours; 3. , so ; the numbers are and ; 4. , so cm and cm; 5. Without a clear definition you cannot tell what your answer means — whether is kilometres, hours or dollars — and you cannot check whether it is reasonable.

Common Misconceptions

Misconception	How to pre-empt it
Not defining the variable, then losing track of what the answer represents.	Make the definition line compulsory and marked.
Answering with the variable when the question asked for something else.	Step 5 of the protocol: re-read the question before writing the final sentence.
Choosing the harder quantity as the variable.	Advise defining the quantity that others are described relative to.
Forming an expression instead of an equation.	Look for the “known total” signal words. No equals sign means nothing to solve.
Discarding a non-whole answer as automatically wrong.	Depends on context — money can be fractional; people and tickets cannot.
Not checking, so a sign slip survives.	Substitution check compulsory, into the original equation.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The sum of three consecutive whole numbers is . What is the largest?

Answer

Let the middle be : , so . The numbers are , and the largest is .

E2 (AMC Junior style). A father is three times as old as his son. In years he will be twice as old. How old is the son now?

Answer

Let the son be . Then the father is .

The son is and the father is . (Check in 12 years: ✓)

E3 (Challenge). A rectangle’s length is cm more than its width. If both dimensions increase by cm, the perimeter becomes cm. Find the original dimensions.

Answer

Let the width be ; the length is . After increasing:

Original dimensions: cm by cm.

E4 (Challenge). In a class of students there are more girls than boys. How many of each?

Answer

Let be the number of boys: , so and . There are boys and girls.

E5 (Challenge). A shop sells pens at 3$712$56$. How many of each?

Answer

Let be the number of pens; notebooks .

pens and notebooks. (Check: ✓)

Homework

1. Form and solve an equation for each: (a) A number multiplied by , then subtracted, gives . (b) A number divided by , then added, gives . (c) Twice a number, plus , gives .
2. A gardener charges 55$38$283$. How long did it take?
3. Two consecutive whole numbers add to . Find them.
4. Three consecutive whole numbers add to . Find them.
5. A rectangle’s length is cm more than its width, and its perimeter is cm. Find its dimensions and area.
6. Ali, Ben and Cara share 140$20$ more than Ali. Cara gets twice as much as Ali. How much does each get?
7. Two angles on a straight line sum to . One is less than three times the other. Find both.
8. Sam has 60$15$130$5$ per week. After how many weeks do they have the same amount?
9. Reasoning. Explain why, in “Ben gets twice as much as Ali”, it is easier to let be Ali’s share than Ben’s.
10. Challenge. A rectangle’s length is three times its width. If the width increases by cm and the length decreases by cm, the perimeter is unchanged. Explain why, using algebra.

Answers: Q1 — (a) , (b) , (c) , . Q2 — , so hours. Q3 — and . Q4 — . Q5 — , so cm, cm, area . Q6 — Ali 30$50$60x + (3x - 40) = 180x = 55°55°125°60 + 15w = 130 - 5ww = 3.54\tfrac{b}{2}P = 2w + 6w = 8wP = 2(w+4) + 2(3w-4) = 8w+4-4$ twice, cancelling.