Bar models depict relationships between quantities. They do not replace reading comprehension or serve as keyword-matching templates. Equal units represent equal amounts.
1. Part–whole
One missing part
Emma has 40 marbles. Together, Emma and Noah have 68. How many belong to Noah?
Answer: Noah has 28 marbles.
2. Additive comparison
Two amounts, one difference
Emma has 12 more marbles than Noah. Together they have 68.
Subtract the unmatched 12, then divide what remains into equal shares.
Answer: Noah has 28; Emma has 40.
3. Multiplicative comparison
“Three times as many”
Sarah has three times John's stickers. Together they have 72.
Answer: John has 18; Sarah has 54. Algebra: x + 3x = 72.
4. Fraction of a whole
Partition into equal units
A class has 35 students. Three-fifths are present.
Answer: 21 students are present.
5. Ratio from a known total
Share 75 in the ratio 2:3
For every two equal units of A, there are three equal units of B.
Answer: A receives $30; B receives $45.
6. A fraction of the remainder
The whole changes mid-problem
Of 240 notebooks, 3/5 are sold, then 1/4 of the remaining notebooks are sold.
Answer: 72 remain. The second fraction is of 96, not of 240.
7. Difference of ratio parts
Ratio 3:5 with a known difference
The difference is 28. The amount represented by 5 parts exceeds that represented by 3 parts by 2 equal units. One unit is 28 ÷ 2 = 14.
Answer: 42 and 70.
8. Before-and-after with a constant total
The invariant is the total
Two children have money in the ratio 2:3. The second gives the first $20 and the ratio becomes 3:2. The total remains five equal units; the first child gains one part.
Answer: Initially $40 and $60, total $100.
Model-drawing checklist
- Label the total and all known and unknown amounts.
- Keep equal units drawn to equal lengths; show the unmatched portion in comparisons.
- For fractions, ensure all diagrams refer to the same-sized whole.
- Name the whole again after a sale, transfer, or other change.
- Convert the drawing to an equation, compute, and check against the story.
Common failure: Automatically add or subtract when a problem says “more.” The relationship, not one keyword, determines the operation.