S1 Maths - Ratio, Rate and Proportion: Lesson Prep Guide for Tutors (three-part ratios, scale, speed, inverse proportion)

S1 Maths – Ratio, Rate and Proportion: Lesson Prep Guide for Tutors (three-part ratios, scale, speed, inverse proportion)

2026-11-14

Tutor lesson-prep guides · Secondary Maths

Faced with 3 to 5 sharing 40, students divide 40 by 5 and answer 24 and 40, because they treat a ratio as a fraction. Finding the value of one part first turns ratio, rate and proportion into a single opening move.

Objectives — By the end students can: (1) simplify 18 to 24 into 3 to 4, stating that both sides are divided by the same number; (2) split 40 in the ratio 3 to 5 into 15 and 25, stating that the number of parts comes first; (3) state that a scale of 1 to 20000 makes 4.5 cm on the drawing 0.9 km in reality, and that work taking 4 people 6 days takes 8 people 3 days, since an inverse proportion has a constant product.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Simplifying fractions	What divides both 18 and 24?	Six, leaving 3 and 4
Converting units	How many centimetres are in a kilometre?	One hundred thousand
Converting time	What is 45 minutes in hours?	Three quarters, that is 0.75 hours

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	What is one part?	Write 3 to 5 sharing 40 and ask what each side holds if one part is 8, collecting answers first	Say what they think and why
5-14 min	Simplifying a ratio	Divide 18 and 24 by 6 to get 3 to 4, then simplify a three-part ratio the same way	Simplify two ratios alone
14-24 min	Sharing by a ratio	Three to five gives 8 parts, 40 divided by 8 is 5, so 15 and 25; then 1 to 2 to 3 sharing 72	Write the parts, one part and each share
24-33 min	Scale and speed	At 1 to 20000, 4.5 cm becomes 90000 cm, which is 0.9 km; then 60 km/h for 45 minutes	Write units before calculating
33-41 min	Three practice items	Hand out a simplifying, a sharing and an inverse proportion item, watching for a missing parts total	Write the total number of parts first
41-45 min	Exit check	Three oral questions: the parts total, a scale conversion, and 4 people for 6 days becoming 8 people	Answer aloud with a reason

03 Teaching script (read it out)

Scene 1: Act 1: a ratio counts parts, it is not a fraction

“3 to 5 does not mean three fifths. It means three parts against five parts, eight parts together. So with 40 in total the first line is the parts total: 3 plus 5 is 8. 40 divided by 8 is 5, so one part is 5. Three parts is 15 and five parts is 25. Check it: 15 plus 25 is 40, and 15 to 25 simplifies to 3 to 5. Both checks pass, so the answer stands.”

Students repeat: add the parts first, then find one part

Scene 2: Act 2: a three-part ratio adds all three

“1 to 2 to 3 sharing 72 has 1 plus 2 plus 3, so 6 parts. 72 divided by 6 is 12, one part is 12, and the shares are 12, 24 and 36. The usual slip is adding only the first two, or treating the middle 2 as two items rather than two parts. Write three lines, parts, one part, each share, and nothing goes missing.”

Students repeat: three parts, add all three

Scene 3: Act 3: direct proportion multiplies, inverse keeps the product

“A scale of 1 to 20000 is direct: 1 cm on paper is 20000 cm in reality, so 4.5 cm becomes 4.5 times 20000, which is 90000 cm, and dividing by 100000 gives 0.9 km. Inverse proportion runs the other way: 4 people for 6 days is a total of 24 person-days. With 8 people, 24 divided by 8 is 3 days. Twice the people, half the days, and the product never changes, which is exactly how you tell inverse from direct.”

Students repeat: inverse proportion keeps the product constant

04 Question types and worked steps

Type	Example	Steps	Answer
Simplify a ratio	Simplify 18 to 24	(1) Find a common factor (2) divide both sides (3) write whole numbers (4) check nothing else divides	3 to 4
Two-part sharing	Share 40 in the ratio 3 to 5	(1) Add the parts (2) divide the total by the parts (3) multiply back (4) check the shares add up	15 and 25
Scale drawing	Scale 1 to 20000, drawing 4.5 cm	(1) Multiply by the scale (2) the result is in cm (3) divide by 100000 for km (4) state the unit	0.9 km
Inverse proportion	4 people need 6 days. How long do 8 people need?	(1) Find the total person-days (2) divide by the new number of people (3) check the direction (4) state the unit	24 divided by 8 is 3 days

Sanity check (teach this): Reasonableness checks: the shares must add back to the total, and 15 plus 25 is 40, so 15 and 40 is immediately wrong; a simplified ratio has no common factor left, and 3 to 4 shares only 1; a scale enlarges, so 4.5 cm becoming 0.9 km is sensible while 0.0009 km means dividing the wrong way; speed times time needs matching units, and 60 km/h times 0.75 hours is 45 km; doubling the workers must halve the days, so 8 people in 3 days is right and 12 days means direct proportion was used.

05 Independent practice (3 questions)

Question	Answer	What to watch
Squash and water in the ratio 2 to 7 make 900 mL. How much of each?	9 parts, one part 100, so 200 mL and 700 mL	Answering 200 and 900 means the parts were never added
At 60 km/h, how far do you travel in 45 minutes?	45 minutes is 0.75 hours, and 60 times 0.75 is 45 km	Multiplying 60 by 45 gives 2700
4 people need 6 days. How long do 8 people need?	24 person-days, and 24 divided by 8 is 3 days	An answer of 12 days treats it as direct proportion

06 Common mistakes and what to say

Mistake	Student writes	Say this
Using the ratio as a fraction	“3 to 5 of 40 is 40 times three fifths, so 24.”	“A ratio counts parts, not fractions. Add 3 and 5 to get 8 parts, divide 40 by 8 to get 5, so 15 and 25.”
Forgetting to add the parts	“1 to 2 to 3 sharing 72 means 72 divided by 3, so 24.”	“A three-part ratio adds all three: 1 plus 2 plus 3 is 6 parts, 72 divided by 6 is 12, so 12, 24 and 36.”
Skipping the unit conversion	“At 1 to 20000, 4.5 cm is 90000 km.”	“90000 is centimetres. Divide by 100000 for kilometres, which is 0.9 km.”
Treating inverse as direct	“4 people in 6 days means 8 people in 12 days.”	“More people means fewer days. The total is 4 times 6, which is 24 person-days, and 24 divided by 8 is 3 days.”

07 Homework (paper, section and question numbers)

Redo Act 1: share 40 in the ratio 3 to 5 and write the parts, one part and each share — Chapter practice, question 1
Do two scale conversions, one from the drawing to reality and one from reality to the drawing — Chapter practice, question 2
Preview: 3 people need 8 days. How long do 6 people need? Show the person-days. — Chapter practice, question 3

08 Exit check (last 5 minutes)

Question	Answer	Decision
In 3 to 5 sharing 40, how many parts are there?	Eight parts, so one part is 5 and the shares are 15 and 25	An answer of 24 means Act 1 again
On a 1 to 20000 scale, how long is 4.5 cm in reality?	0.9 km, after multiplying and converting	An answer of 90000 km means the unit was never converted
4 people need 6 days. How long do 8 need?	3 days, since the 24 person-days are fixed	An answer of 12 days means Act 3 again

09 Tutor FAQ

Why insist on finding one part first?

One part is the only quantity you can multiply. Once it is 5, three parts is 15 and five parts is 25, and every direct proportion question returns to that step.

My student works correctly but writes the wrong unit. Is that a real error?

Yes, because a wrong unit means they do not know what they measured. Treat it as an error and require units in every answer.

Is rate the same as ratio?

A rate compares two different quantities, such as kilometres per hour, while a ratio compares parts of the same total. S1 needs the working habit rather than the formal definitions.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.