071i - Equivalent Ratios and Scaling

Lesson 71 — Equivalent Ratios and Scaling

Strand: Number | Descriptor: AC9M7N08 | Duration: 45 minutes

Learning Intentions

• To generate equivalent ratios and use them to scale quantities up and down.
• To share a quantity in a given ratio.

Success Criteria

I can:

1. Generate equivalent ratios by multiplying or dividing all parts.
2. Scale a recipe or mixture up or down while preserving its ratio.
3. Share a quantity in a given ratio using the “parts” method.
4. Find the whole, or the other part, when one part is known.

Warmup

(6 minutes — retrieval and cordial, mini whiteboards)

1. Simplify: (a) (b) (c) mL : L.
2. Cordial is mixed (syrup : water). If I use cups of syrup, how much water keeps the taste the same?
3. If I use cups of water, how much syrup?

Answers: 1. (a) (b) (c) ; 2. cups; 3. cups.

The idea to name: keeping the taste the same means keeping the ratio the same — multiply both parts by the same factor.

Activities

Activity 1 — Explicit Instruction: Scaling with the Factor Method (12 min)

I do — scaling up. A recipe for pancakes uses flour and milk in the ratio (cups). I want to use cups of flour.

So cups of milk. The factor is found from the known pair: .

I do — scaling down. Concrete is mixed cement : sand . A small job needs only bucket of cement.

Scaling can produce non-whole parts — that is fine for continuous quantities like sand, but not for things counted in wholes.

The unitary method — the all-purpose fallback. Find the value of one part first.

I do. Orange paint mixes red : yellow . A painter uses L of red. How much yellow?

We do:

1. Ratio ; the first part is . Find the second.
2. Ratio ; the second part is . Find the first.
3. A fruit punch uses juice : soda . Scale for cups of soda.
4. Ratio ; the middle part is . Find the others.

(Answers: ; ; cups of juice; and .)

Activity 2 — Sharing in a given Ratio (14 min)

The parts method, modelled in full.

I do. Share 602 : 3$.

Check: ✓ and ✓ — both checks, every time.

I do — three-way share. Share g of trail mix between nuts, seeds and fruit in ratio .

I do — the difference case. Two friends share money in ratio . The first receives 14$ more than the second. How much does each get?

The three cases to distinguish — what does the given number represent?

Given	Divide by
the whole	the sum of the parts
one share	that share’s number of parts
the difference between shares	the difference of the parts

You do:

1. Share 454 : 5$.
2. Share kg in the ratio .
3. In a class the ratio of girls to boys is and there are girls. How many boys? How many students?
4. Two numbers are in ratio and differ by . Find both.

(Answers: and ; ; boys, students; and .)

Activity 3 — Inquiry: the Golden Recipe (8 min)

Pairs.

A muffin recipe for muffins uses cups flour, cups milk and cup sugar.

1. Write the flour : milk : sugar ratio in whole numbers.
2. Scale the recipe for muffins.
3. Tara has only cups of flour and plenty of everything else. What is the largest batch she can make?

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what limits the batch?	The flour — the scarcest ingredient relative to need.
How much flour per muffin?	cups per muffins cup each.
How many muffins from cups?	muffins.
Check with the scale factor instead.	Factor ; muffins ✓
Complete the recipe.	Milk cups; sugar cups.
Looking back	Both routes — per-unit and scale-factor — give the same answer. Choose whichever the numbers favour.

Answers: 1. (); 2. factor : cups flour, cups milk, cups sugar; 3. muffins.

Checks for Understanding

(5 minutes — exit ticket)

1. The ratio is scaled so the first part becomes . Find the second part.
2. Share 843 : 4$.
3. Paint mixes blue : white . A painter uses L of white. How much blue?
4. Two amounts in ratio differ by . Find both.
5. Reasoning. When sharing in a ratio, why must you check both the total and the ratio of your answer?

Answers: 1. ; 2. 36$482072404044843:43040$ is in ratio but has the wrong total. Each check catches a different kind of error.

Common Misconceptions

Misconception	How to pre-empt it
Dividing the whole by a part instead of the sum of parts.	The three-case table; ask “what does the given number represent?” first.
Adding the scale factor instead of multiplying.	is , not . Cross-check: is the new ratio equivalent?
Scaling only one part of the ratio.	Every part is multiplied by the same factor — the cordial context makes taste the test.
Treating the difference case like the whole case.	Model the difference case explicitly with its own worked example.
Assuming shares must be whole numbers.	buckets is legitimate for continuous quantities.
Giving one share when both were asked for.	Sentence answers naming each person or part.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Share 1325 : 6$. What is the larger share?

Answer

One part ; the larger share is 72$.

E2 (AMC Junior style). The angles of a triangle are in the ratio . Find the largest angle.

Answer

One part ; the largest angle is . (Connects to Lesson 46.)

E3 (Challenge). In a bag, red : blue . After more red marbles are added, the ratio becomes . How many blue marbles are there?

Answer

Blue is unchanged. Let blue . Before: red . After: .

blue marbles (and red went from to ; check ✓).

E4 (Challenge). Three friends share a prize in ratio . The largest share is 91$ more than the smallest. Find the total prize.

Answer

Difference in parts ; one part 18.20= 12 \times 18.20 = $218.40$.

E5 (Challenge). A cm ribbon is cut into three pieces in ratio . The longest piece is then cut in half. List all four piece lengths.

Answer

One part cm: pieces , , . Halving the : final pieces , , , cm.

Homework

1. Find the missing part: (a) (b) (c) .
2. Share: (a) 702 : 5$1203 : 51441 : 3 : 4$.
3. A salad dressing mixes oil : vinegar . (a) How much vinegar with tbsp of oil? (b) How much oil with tbsp of vinegar?
4. Green paint mixes yellow : blue . A painter has used L of blue. (a) How much yellow was used? (b) How much paint in total?
5. Two numbers are in ratio and differ by . Find both.
6. A recipe for biscuits uses g flour and g butter. Scale it for biscuits.
7. The sides of a rectangle are in ratio and its perimeter is cm. Find its dimensions and area.
8. In an orchard, apple : pear trees . There are pear trees. How many trees altogether?
9. Reasoning. Ratio , total : a student divides and . Diagnose the error and give the correct shares.
10. Challenge. Share 180\dfrac12 : \dfrac13 : \dfrac16$. (Hint: clear the fractions first.)

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) . Q3 — (a) tbsp (b) tbsp. Q4 — (a) L (b) L. Q5 — and . Q6 — factor : g flour, g butter. Q7 — semi-perimeter ; one part ; sides and ; area . Q8 — one part ; total trees. Q9 — the whole is divided by the sum of parts: ; shares and . Q10 — : ratio ; one part 30$90, $60, $30$.