Word problems
A word problem needs extraction before any method applies. Extraction is a set of decisions, done in any order: which question the story is, which numbers go together and which way round, whether a number is extra or a total, which numbers aren’t needed. Each decision has one correct option and its traps: the mistakes students actually make. A trap is the correct reading with that one decision swapped, so the build computes the wrong answer it leads to, and the first steps that give it away (steps no correct reading starts with). That lets an app diagnose an extraction mistake from a student’s answer, and often from their first step.
Extraction concepts
| Concept | What the student decides | Cues | Typical mistake |
|---|---|---|---|
Recognize the question type question_type |
Decide which question the story is: missing value, compare, better buy, proportion check, percent of, add fractions. | at the same rate / speed; how many … for …; which is the better buy / deal; who is better / sweeter / faster; same taste / shade; % of, % off, tip, tax; in all, altogether (with fractions) | Solving a different question than the one asked (sale price instead of savings). |
Pull out each quantity quantity |
Write each number with its unit and what it counts: 4 spoons (of sugar), 6 glasses (of lemonade). | — | Keeping the number but losing what it counts, so it gets used in the wrong place. |
Spot numbers written as words implicit_number |
Turn words into numbers. | a dozen = 12; a pair = 2; half = 1/2; twice = 2 ×; a quarter = 1/4; a / an / each = 1 | Reading ‘2 dozen’ as 2. |
Make units match unit_conversion |
Convert so both sides of a ratio use the same units. | minutes ↔︎ hours; cents ↔︎ dollars; inches ↔︎ feet | Mixing hours on one side with minutes on the other. |
Find the pair that forms the ratio rate_pair |
Identify which two quantities belong together. | per; for every; each; to make; out of; for; stands for; in (150 miles in 3 hours) | Pairing the wrong two numbers. |
Keep the same order on both sides correspondence |
If the known ratio is sugar : lemonade, the unknown ratio must be sugar : lemonade too. | — | Flipping one side: 4/6 = 18/x instead of 4/6 = x/18. |
Ignore numbers that don’t matter distractor |
Recognize numbers the question doesn’t need. | — | Using a distractor in place of a real quantity. |
Name the unknown unknown |
Say what’s asked for and in what unit, before computing. | — | Computing a related number that isn’t the one asked for. |
‘More’ vs ‘in all’ delta_vs_total |
Decide whether a number is an increase or a new total, and whether the answer should be. | more; additional; another; in all; altogether; total; already; left | Answering the total when the extra amount was asked for, or the reverse. |
Scale, don’t add multiplicative |
Proportional quantities grow by the same factor, not by the same amount. | — | ‘18 more glasses, so 18 more spoons.’ |
Which way wins direction |
In a comparison, decide whether the bigger or smaller ratio is better, and which ratio (sugar per glass, or glasses per spoon) the story is about. | cheaper; sweeter; better rate; faster; stronger | Picking the bigger number when the smaller one wins, or comparing the flipped ratio. |
Finish the story follow_up |
A step after the core computation: subtract what’s already used, find the price after a discount. | — | Stopping after the core computation. |
Answer in the story’s terms answer_in_context |
Name the person or offer, give units, write 19/12 as 1 7/12. | — | Answering ‘first’ or a bare number. |
| Word problem | Is a | Decisions |
|---|---|---|
| Cereal boxes | Which is the better buy? | Recognize the question type, Ignore numbers that don’t matter, Find the pair that forms the ratio, Which way wins |
| Free throws | Compare two fractions | Recognize the question type, Find the pair that forms the ratio, Which way wins |
| Jacket on sale | Find a percent of a number | Recognize the question type, Name the unknown |
| Lemonade: how many in all | Find the missing value in a proportion | Recognize the question type, Find the pair that forms the ratio, ‘More’ vs ‘in all’, Finish the story, Scale, don’t add |
| Lemonade: more sugar | Find the missing value in a proportion | Recognize the question type, Find the pair that forms the ratio, Keep the same order on both sides, ‘More’ vs ‘in all’, Scale, don’t add |
| Map scale | Find the missing value in a proportion | Recognize the question type, Find the pair that forms the ratio, Name the unknown, Keep the same order on both sides |
| Same shade of paint? | Are two ratios proportional? | Recognize the question type, Scale, don’t add |
| Pancakes by the dozen | Find the missing value in a proportion | Recognize the question type, Find the pair that forms the ratio, Spot numbers written as words, Keep the same order on both sides |
| Sharing pizza | Add two fractions | Recognize the question type, Answer in the story’s terms |
| Whose lemonade is sweeter? | Compare two fractions | Recognize the question type, Which way wins, Find the pair that forms the ratio, Scale, don’t add |
| Train in minutes | Find the missing value in a proportion | Recognize the question type, Find the pair that forms the ratio, Make units match, Keep the same order on both sides |
| Walking to school | Find a percent of a number | Recognize the question type, Find the pair that forms the ratio, Ignore numbers that don’t matter |