Red to blue, red to the whole
Name ratio order and reference, distinguishing part-to-part from part-to-whole.
Begin with the activity ↓Red to blue, red to the whole
With 2 red and 3 blue items, red to blue is 2 : 3. The whole has 5 items, so red to whole is 2 : 5. The same red quantity gives different ratios when the reference changes.
With 2 red and 3 blue items, red to blue is 2 : 3. The whole has 5 items, so red to whole is 2 : 5. The same red quantity gives different ratios when the reference changes.
A ratio records two quantities in a chosen order. “Red to blue” puts red first and blue second. Read 2 : 3 with the quantity names rather than adding its numbers.
Taking blue as the reference gives red divided by blue: 2 ÷ 3 = 2/3. The quotient expresses the amount of red per unit of blue. The ordered ratio 2 : 3 and the quotient 2/3 are related representations.
To ask what fraction of the whole is red, use all five items as the reference, not just the blue ones. Red is 2/5 of the whole and blue is 3/5. Their fractions add to one whole.
Reversing red to blue gives blue to red, 3 : 2, with quotient 3/2. Do not assume reversing order preserves the comparison. Rename the quantities, and ensure the reference is nonzero before dividing.
Here red and blue exhaust the whole, with no item counted in both groups. This lets us add the groups to find the whole. Next we scale both colors to see when their composition stays the same.
Listen one paragraph at a time
Hold on to this distinction
Using three as the whole leaves out the red items. First circle all five, then read red as 2/5 of that whole.
Change it and check
Change both counts and read red to blue beside red to whole. Explain why their denominators differ.
Red : Blue = 2 : 3 · quotient 2/3
Whole = 2 + 3 = 5
Red : whole = 2 : 5 · Red fraction of whole 2/5
The colors are disjoint and cover the whole. Blue stays at least one so the red-to-blue quotient is defined.
Try with paper and objects
Lay out 2 red and 3 blue paper pieces. Circle just blue, then draw a larger circle around the whole to show the two references.
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2/5 · fraction of the whole
The whole is ten: 4/10 = 2/5 = 0.4.
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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Check why scaling both quantities by the same factor preserves a ratio.