Scaling an equivalent ratio
Check why scaling both quantities by the same factor preserves a ratio.
Begin with the activity ↓Scaling an equivalent ratio
Make one group of 2 red and 3 blue items. Two identical groups give 4 red and 6 blue. The ratios 2 : 3 and 4 : 6 are equivalent because both quantities share the same scale factor.
Make one group of 2 red and 3 blue items. Two identical groups give 4 red and 6 blue. The ratios 2 : 3 and 4 : 6 are equivalent because both quantities share the same scale factor.
The total changes from five to ten while the composition stays the same. Red to blue has equal quotients 2/3 and 4/6, and red fractions 2/5 and 4/10 agree too. Equal quantities and equivalent ratios mean different things.
Adding the same number follows a different rule. Add two to both quantities and get 4 : 5. Since 4/5 differs from 2/3, the ratio changed. A constant difference of one does not guarantee an equivalent ratio.
To scale down, divide both quantities by the same nonzero number. Dividing 6 : 9 by three gives 2 : 3. When dividing actual objects into groups, also check whether the resulting counts are whole numbers.
Keep quantity names and order fixed when comparing the quotients. Do not mix red-to-blue with blue-to-red. Multiplying both quantities by zero gives 0 : 0 with an undefined quotient, so zero is excluded as a scale factor.
Switch between multiplying by a common factor and adding a common number below. Compare the ratio and red fraction to see which rule preserves composition. Next we make the reference quantity one.
Listen one paragraph at a time
Hold on to this distinction
Both sides double in 2 : 3 → 4 : 6. Adding two instead gives 4 : 5. The same additive change does not ensure an equivalent ratio.
Change it and check
Choose a number and compare multiplication with addition. Predict equivalence before checking the result and reason.
2 : 3 → 4 : 6
Equivalent ratio: both scale factors agree.
Red fraction of whole: 2/5 → 2/5
The base ratio is 2 : 3, with positive integer factors from 1 to 4. Additive comparisons use increments from 0 to 4.
Try with paper and objects
Copy a group of 2 red and 3 blue pieces three times. Write 6 : 9, then separate the groups to recover 2 : 3.
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12 · items
Red scaled from 2 to 8 by four, so blue is 3 × 4 = 12.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
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Keep quantity names and units while making the reference one.