2 red counters and 3 blue: the ratio of red to blue is 2 : 3 (part to part); red to all the counters is 2 : 5 (part to whole). Order matters.
A fruit punch is 2 cups of juice for every 3 cups of lemonade. Swap them and it tastes completely different. How do we write a recipe like that down?
A ratio compares two (or more) quantities of the same kind. "2 red to 3 blue" is written 2:3. The order matters: red first means 2 comes first.
There are two kinds of comparison:
Part to part: red to blue is 2:3.
Part to whole: red to all the counters is 2:5, because there are 2+3=5 counters in all. This is the same as the fraction 52.
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.
Red to blue
How many counters are in the bag altogether?
TryTry it on the board and compare the result with your prediction.
Build the bag
The board always shows both: part to part, and part to whole.
Try it yourselfYour edits are saved in this browser.
Make 3 red blocks for every 5 blue. What is the ratio of red to the whole?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Part to part, part to whole
Add the parts to find the whole.
A bag has 2 red and 3 blue counters. The ratio of red to all the counters is 2:?
Pip finds the ratio of apples to all the fruit. Click the line where the working goes wrong.
ModifyChange the problem on purpose and see what happens.
Change the bag
Try it yourselfYour edits are saved in this browser.
Show 4 dogs for every 3 cats. What is the ratio of dogs to the whole?
Use the board. This ticks when you get there.
At a school the ratio of walkers to bus riders is 3:2. How many parts are there altogether?
A shelter has 3 dogs and 5 cats. What fraction of the animals are dogs?
PractiseA few on your own, to check you have it.
Ratios
A box has 5 pens and 8 pencils. The ratio of pencils to pens is 8:?
In the same box, the ratio of pens to all the items is 5:?
A team won 7 games and lost 3. The ratio of wins to losses is 7:?
Review: estimate 6.8×31.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A three-colour mix
Then write the ratio of the first colour to the whole.
Try it yourselfYour edits are saved in this browser.
Need a hint?
3 + ? + ? = 10.
Another hint
Try 3 : 2 : 5 or 3 : 4 : 3.
Make a three-colour ratio with 10 parts in all, and 3 parts of the first colour.
Use the board. This ticks when you get there.
For teachers
Students build ratios from rows of equal blocks, write part-to-part and part-to-whole ratios in the order asked, and link part to whole to a fraction (AC9M7N08, node 1). About 40 minutes.
I can write a ratio of two quantities in a : b form, in the order asked.
I can tell part to part from part to whole.
I can write a part-to-whole ratio as a fraction.
Before class
Do the Run board yourself first. Students need fractions as parts of a whole (Equivalent fractions).
The one idea: a ratio compares quantities; the order in which they are named is the order of the numbers. The common mistakes are writing part to part when part to whole was asked (Pip), reversing the order, and using the whole (5) as a part.
Use the same bag all lesson so students see 2 : 3 and 2 : 5 about the same objects.
Running the lesson
Predict: students who chose 3 : 2 reversed the order; 2 : 5 compared to the whole.
Run: ask students to say both ratios aloud, e.g. 'three to five' and 'three to eight'.
Investigate: have students hold up the whole bag and a part to show which is which.
Modify: ask which ratio matches the fraction 3/8.
Make: collect ratios; check each adds to 10.
The hard part (on purpose)
Telling part to part from part to whole. Ask 'is the second number a part, or the whole bag?' every time before writing.
When students get stuck
A student writes 4 : 3 for apples to all the fruit (Pip's slip). Ask: How many pieces of fruit are there altogether? Fix: The second number is the whole: 4 : 7.
A student writes the ratio the wrong way round. Ask: Which is named first in the question? Fix: The first thing named is the first number.
A student says 3 : 5 and 3/5 are the same. Ask: What is the whole in each? Fix: 3 : 5 is part to part, 3/8 is part to whole.
Stretch and support
Stretch: A ratio of boys to girls is 3 : 4. What fraction of the group are girls? What if 2 more boys join, no more girls? Is the ratio still 3 : 4?
Support: Draw one block for each item and colour them; count before writing.
Where it leads
Next: Equivalent ratios, scaling a ratio up and down.
Where it leads: sharing in a ratio, rates, scale drawings, and proportion in Years 8 and 9.
What each step is for
red to blue: Committing to a ratio shows whether they write the order asked for (2 : 3), reverse it (3 : 2) or compare to the whole (2 : 5).
build the bag: Building 3 red for every 5 blue shows the ratio as rows of equal blocks, and the whole as the blocks added together.
part to part, part to whole: Part to whole adds the parts, so the second number is the total; Pip writes part to part when asked for part to whole. The order matters.
change the bag: Changing one number at a time, then turning part to whole into a fraction, links ratio to the fractions students know.
ratios: Writing a ratio in the order asked, part to whole, and a review of estimation.
a three-colour mix: A ratio can have three parts, and the whole is all of them added; making 10 parts shows many ratios give the same total.
Now you can
Write a ratio of two quantities in a : b form, in the order asked.