058i - Problem Solving with Equivalent Forms

Lesson 58 — Problem Solving and Consolidation: Equivalent Representations

Strand: Number | Descriptor: AC9M7N04 | Duration: 45 minutes

Learning Intentions

• To consolidate converting, comparing and ordering rational numbers.
• To choose the most useful representation for a given problem.

Success Criteria

I can:

1. Convert fluently between fractions, decimals and percentages.
2. Compare and order rational numbers in mixed forms.
3. Choose the most convenient form for a calculation and justify the choice.
4. Solve practical problems involving equivalent representations.

Warmup

(6 minutes — “which form would you use?”, pairs)

For each task, say which form you would work in, and why.

1. Splitting a pizza between people.
2. Comparing two test results: out of and out of .
3. Calculating off a 60$ jacket.
4. Recording a long jump of metres and centimetres.
5. Describing how much of a mL glass is left when mL remains.

Sample answers: 1. Fractions — thirds are exact, decimals are not; 2. Percentages — a common basis for comparison ( vs ); 3. Decimals — is a quick calculation; 4. Decimals — m is standard for measurement; 5. Any, but a fraction or percentage communicates the proportion most clearly.

The point to name: the three forms are equivalent, but they are not equally convenient. Choosing well makes problems easier.

Activities

Activity 1 — Mixed Consolidation (12 min)

Graded worksheet. Students identify what each item requires before working.

Set A — convert.

1. to a decimal and a percentage.
2. to a fraction in lowest terms.
3. to a decimal and a fraction.
4. to a decimal.

Set B — simplify.

Set C — order.

8. , , ,
9. , , ,

Set D — between.

10. Find a fraction between and .
11. Find the number halfway between and .

(Answers: 1. , ; 2. ; 3. , ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. ; 10. e.g. ; 11. .)

Activity 2 — Applied Problems (14 min)

Pairs. Each answer needs the working, the form chosen, and a sentence.

Problem 1 — Test comparison. Three students sit different tests: Ana scores , Ben scores , Cara scores . Rank them and state the ranking method.

Problem 2 — Survey. In a survey of students, walk to school, catch the bus, and cycle. The rest are driven.

(a) How many use each method?

(b) What fraction are driven?

Problem 3 — Sale. A 80\tfrac14$9520%$. Which sale price is lower?

Problem 4 — Recipe. A recipe needs cup of flour. Ravi has of a cup. Does he have enough? How much more does he need, as a fraction?

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: what is the whole?	students.
The three proportions are in different forms. What first?	Convert to one form — percentages are natural here.
Convert.	; bus ; .
Total so far?	.
So what fraction are driven?	, which is .
Now the numbers. How do you find of ?	.
Carry out the rest.	Bus , cycle , driven .
Check	✓

Answers:

1. As decimals: , , . Ranking: Ana, Ben, Cara. Converting to a common form — decimals or percentages — allows direct comparison.
2. (a) Walk , bus , cycle , driven . (b) .
3. Shoes 1: 6095 \times 0.8 = $76$. The first is lower.
4. , so no. He needs of a cup more.

Activity 3 — Inquiry: Which is the Better Deal? (8 min)

Pairs.

Three shops sell the same 120$ headphones:

• Shop A: off.

• Shop B: off.

• Shop C: pay of the marked price.

1. Find each sale price.
2. Which is cheapest?
3. Rank the discounts from largest to smallest without calculating the prices.

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what is being compared?	The size of the discount, not the price paid.
What discount does each shop give?	A: . B: . C: pays , so the discount is .
These are in different forms. What now?	Convert to decimals: , , .
Rank them.	, so A, C, B.
Does that match your prices from Q1?	Yes — the biggest discount gives the lowest price. ✓
Looking back	Watch the wording: Shop C states what you pay, not what you save.

Answers: 1. A: 80120 \times 0.7 = $84120 \times 0.68 = $81.6033.\dot{3}%32%30%$).

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Convert to a decimal and a percentage.
2. Simplify .
3. Order from smallest: , , , .
4. A 70\tfrac15$. Find the sale price.
5. Reasoning. Ana scores and Ben scores . Who did better, and how did you decide?

Answers: 1. and ; 2. ; 3. ; 4. 56\tfrac{21}{25} = 84%83%$. Converting to a common form makes the comparison direct.

Common Misconceptions

Misconception	How to pre-empt it
Comparing mixed forms without converting.	Always convert to one form first — usually decimals.
Confusing the discount with the price paid.	Shop C in Activity 3 states the fraction paid. Read carefully.
Subtracting a percentage from a fraction directly.	Convert both to the same form before any arithmetic.
Assuming a bigger percentage discount always means a lower price.	Only if the original prices are the same. Compare final prices when they differ.
Forgetting to simplify a final fraction.	Lowest terms required in every answer.
Treating and as the same.	. The denominator matters.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A shirt costs 5020%10%$ off the new price. What is the final price?

Answer

Note this is not the same as off, which would give 35$. Successive discounts do not simply add.

E2 (AMC Junior style). In a class, of the students play soccer and play netball. The remaining students play neither. How many students are in the class?

Answer

So play neither, and of the class is students:

E3 (Challenge). Which is larger: or ?

Answer

As decimals: and . So is larger. (Alternatively cross-multiply: and , and .)

E4 (Challenge). A price increases by , then decreases by . What is the overall change?

Answer

The price returns exactly to its original value — no overall change. A surprising and instructive result.

E5 (Challenge). Of a school’s students, are in Year 7, are in Year 8, and the remaining are in Year 9. How many students are there in total?

Answer

, so Year 9 is . Then , giving students.

Homework

1. Convert: (a) to a decimal and percentage (b) to a fraction (c) to a decimal and fraction (d) to a decimal (3 d.p.).
2. Simplify: (a) (b) (c) .
3. Order from smallest to largest: (a) , , , (b) , , , .
4. Find a fraction between (a) and (b) and .
5. In a survey of people, prefer tea, prefer coffee, and prefer neither. The rest prefer juice. (a) How many prefer each? (b) What percentage prefer juice?
6. A 150\tfrac15$14015%$. Which sale price is lower?
7. Ana scores , Ben scores , Cara scores . Rank them.
8. Reasoning. Explain why off followed by off is not the same as off.
9. Reasoning. Explain which form you would use to compare and , and why.
10. Challenge. Of a club’s members, play tennis, play squash, and the remaining play neither. How many members are there?

Answers: Q1 — (a) , (b) (c) , (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) . Q4 — (a) e.g. (b) e.g. . Q5 — (a) tea , coffee , neither , juice (b) . Q6 — 120140 \times 0.85 = $119\tfrac{34}{40} = 0.850.860.850.840.8 \times 0.9 = 0.7228%76%77.5%\tfrac{3}{10} = 30%30%0.3n = 60n = 200$ members.