Why this matters for FAST: Ratios are foundational to proportional reasoning, which is heavily tested on FAST. Students must recognize ratio relationships, express them in multiple forms, and apply them to real-world situations.
Students think "3 to 5" is the same as "5 to 3." This is WRONG! The order in a ratio is critical and determined by the context.
"The ratio of boys to girls is different from the ratio of girls to boys! If there are 3 boys and 5 girls, boys to girls is 3:5, but girls to boys is 5:3. Always read the question carefully to know which quantity comes first!"
Students confuse the ratio of red to blue (part-to-part) with the ratio of red to total (part-to-whole).
"If I have 3 red and 5 blue marbles, the ratio of red TO blue is 3:5 (part-to-part). But the ratio of red TO total is 3:8 (part-to-whole, since 3+5=8). Read carefully: is it comparing parts, or part to total?"
Students see 3/5 and automatically think it means "3 divided by 5" rather than "3 to 5."
"When we write a ratio as a fraction like 3/5, we're saying '3 for every 5' or '3 to 5.' It CAN be read as division too, but in ratio contexts, think of it as a comparison. Context tells us which meaning to use!"
Show 4 red counters and 6 blue counters. Ask: "How can we compare these two groups? What if I said there are 4 red FOR EVERY 6 blue?" Introduce this as a RATIO.
"A ratio compares two quantities. We can write the same ratio THREE different ways: 4 to 6, 4:6, or 4/6. All three mean the same thing - 4 for every 6!"
Three Ways to Write a Ratio
4 to 6 | 4:6 | 4/6
"4 for every 6" or "4 compared to 6"
Example: 3 cats and 5 dogs
Part-to-Part: Cats to dogs = 3:5
Part-to-Whole: Cats to all animals = 3:8
Part-to-Whole: Dogs to all animals = 5:8
"Part-to-part compares one group to another group. Part-to-whole compares one group to the TOTAL. Always identify which type the question asks for!"
Equivalent Ratios: Multiply or divide BOTH parts by the same number
2:3 = 4:6 = 6:9 = 8:12
(multiply both by 2, then 3, then 4)
12:8 = 6:4 = 3:2
(divide both by 2, then by 2 again)
Work through these together:
"A bag has 6 red and 10 blue marbles. What is the ratio of red to total marbles?"
A) 6:10 B) 6:16 C) 10:6 D) 16:6
Correct answer: B) 6:16 (or simplified 3:8). This is part-to-whole: 6 red out of 16 total (6+10=16).
For struggling students: Use concrete manipulatives like counters. Focus on part-to-part ratios before introducing part-to-whole. Use color-coding to help track which quantity is first.
For advanced students: Challenge with multi-step ratio problems or have them create their own real-world ratio scenarios. Introduce comparing ratios to find which represents a better deal.
For home: Send Parent Activity sheet. Families can find ratios in recipes, sports statistics, and mixing drinks/paints.