Mixes, scale drawings and models: say what is known and unknown, choose the ratio, solve, and answer in the units of the situation.
A builder mixes mortar as 1 part cement to 3 parts sand. She needs 20 litres of mix. How much of each does she need to buy?
Modelling means turning a real problem into maths you can solve, solving it, and turning the answer back into the real situation. Three steps keep you on track:
Formulate: What is known (the ratio, the total)? What is unknown? What is the relationship (parts of a ratio)?
Solve: Find one part, then each amount. Estimate first to know what to expect.
Interpret: Give the answer in the situation's own units, with a sensible amount of rounding, and check that it makes sense.
Ratios turn up in mixes (mortar, paint, fertiliser), scale drawings and maps (1 : 50), and models.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
How much sand?
Is the sand more or less than half of the mix? How do you know?
TryTry it on the board and compare the result with your prediction.
A mix of 20 litres
The same sharing as before: the context is new, and the units are litres.
Try it yourselfYour edits are saved in this browser.
Set the total to 20 litres. How much sand is needed?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Formulate first
A ratio such as 1 : 50 compares a length on the drawing with the real length.
A scale drawing uses the ratio 1:50. A wall is 3 cm long on the drawing. How long is the real wall, in cm?
Raj mixes fertiliser and water in the ratio 1 : 4 to make 2 L. Click the line where his working goes wrong.
ModifyChange the problem on purpose and see what happens.
New situations
Try it yourselfYour edits are saved in this browser.
Make 12 L of orange paint in the ratio 2 : 3. How much red is needed?
Use the board. This ticks when you get there.
A map has the scale 1:25000. Two towns are 4 cm apart on the map. How many metres apart are they really?
Lemonade is 1 part cordial to 5 parts water. How many millilitres of cordial are in a 1.2 L jug (1200 mL)?
In the 1 : 3 mortar mix of 20 L, what fraction of the mix is cement? (part to whole)
A model car is built at the ratio 1:18. The real car is 4.5 m (450 cm) long. How long is the model, in cm?
PractiseA few on your own, to check you have it.
Ratio models
Mortar is cement and sand in the ratio 1:4. How many kilograms of cement are in 30 kg of mortar?
A plan uses the scale 1:200. A room is 5 cm long on the plan. How long is the room, in metres?
A model ship is built at the ratio 1:50. The model is 24 cm long. How long is the real ship, in metres?
Review: share $48 in the ratio 3:5. What is the bigger share, in dollars?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Exactly 3 litres
Then write what is known, what is unknown, and a check that your two amounts add to the total, with units.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The first liquid is a parts, each of total ÷ (a + b).
Another hint
Try 1 : 2 with a total of 9 L.
Design a mix: choose a ratio and a total so that the first liquid is exactly 3 L.
Use the board. This ticks when you get there.
For teachers
Students model practical problems with ratios (mixes, scale drawings, models), say what is known and unknown before calculating, solve by finding one part or the scale factor, and answer in the situation's units (AC9M7M06, nodes 1 and 2). About 45 minutes.
I can say what is known and unknown in a ratio problem before I calculate.
I can solve mixing and scale problems with ratios.
I can give my answer with the right units and check that it makes sense.
Before class
Do the Run board yourself first. Students need sharing in a ratio and unit conversions between mm, cm, m and L, mL.
The one idea: ratio problems in measurement are the same structure as sharing in a ratio, plus scale (every 1 unit on the drawing is n units in real life). The modelling habit is the point: formulate, solve, interpret. Common mistakes are dividing by one term (Raj), multiplying when scaling down (and dividing when scaling up), and leaving the answer in the wrong unit.
Let students choose their own context for the Make task. A First Nations context can be added by the teacher: land and resource proportions, with local advice.
Running the lesson
Predict: students who chose 60 L multiplied the total by 3.
Run: ask whether the answer is a sensible size before checking the board.
Investigate: have students say the known, unknown and relationship out loud for the paint.
Modify: for the map, ask which unit the answer should be in and why.
Make: share the designs; ask what each mix is for.
The hard part (on purpose)
Starting to calculate before saying what is known and unknown. Make students write 'Known: … Unknown: …' before touching a number.
When students get stuck
A student divides 2 L by 4 for the fertiliser (Raj's slip). Ask: How many equal parts is the mix cut into? Fix: Add the parts first: 1 + 4 = 5, then divide.
A student multiplies for a model that should be smaller. Ask: Is the model bigger or smaller than the real thing? Fix: Decide the direction before calculating: smaller means divide.
A student answers 1000 for the towns on the map. Ask: 1000 what? Fix: Give the units: 1000 m, or 1 km.
Stretch and support
Stretch: A model of a building is 1 : 50. The real building is 12 m tall and its roof is a triangle with area 30 m². What is the model building's height, and the model roof's area? Is the area scaled the same way as the length? (24 cm and 120 cm²: area scales by 50 × 50.)
Support: Draw the bar first, label the total and the parts, then calculate.
Where it leads
This is the last Number lesson at this level; Year 8 extends to rates, simple interest and proportion.
Where it leads: similar shapes and scale factors for area and volume in Years 8 and 9, and rates.
What each step is for
how much sand?: Committing shows whether they find the parts first (4 parts, so 15 L), multiply the total by 3 (60 L) or divide the total by 3 (6.7 L).
a mix of 20 litres: The bar shows the whole as four equal parts, so the amounts are a share of the total, with units.
formulate first: Scale is a ratio too (1 : 50); Raj divides by the water's number instead of the parts. Writing known, unknown and relationship first prevents the slip.
new situations: Maps, mixes and models all use the same two moves, find one part or find the multiplier, then give the answer in the situation's units.
ratio models: A mix, a plan and a model, each needing the right units; the review keeps sharing in a ratio fresh.
exactly 3 litres: Choosing a ratio and a total so that one liquid is exactly 3 L shows the amount is its parts times one part, and many designs work.
Now you can
Say what is known and unknown in a ratio problem before I calculate.
Solve mixing and scale problems with ratios.
Give my answer with the right units and check that it makes sense.