Advanced Multi-Step Problems

Activities to try

1. 1

   Set it up before you solve

   Why it works: When you represent the relationship first, the structure shows which computation will find the missing quantity: a ratio table or bar model makes clear what one part is worth and how many parts are needed, so you resist the pull to simply combine the given numbers.

   1. Read a ratio, proportion, or algebra problem, e.g. 'Flour and sugar are mixed in the ratio 3:2. A baker uses 750 g of flour. How much sugar? Then she scales the whole recipe by 1.5 — how much of each?'
   2. Before touching any numbers, label what is KNOWN and what is UNKNOWN in the problem
   3. Build a representation of the relationship first: a ratio table, a bar model, a double number line, or an equation with a letter for the unknown (e.g. 3 parts flour = 750 g, so 1 part = 250 g, sugar = 2 parts = 500 g)
   4. Only after the relationship is represented, carry out the computation to find the missing quantity
   5. State an estimate for the answer as a benchmark (e.g. scaling by 1.5 means each quantity ends up about half again — one and a half times — as large, so flour becomes roughly 1100-1150 g and sugar roughly 750 g)
2. 2

   Teach structure, not keywords

   Why it works: Keyword-to-operation matching is unreliable and collapses on multi-step problems; the structure of the problem, not its words, determines the operations and their order.

   1. Take a complex multi-step problem and resist choosing an operation from a cue word (e.g. 'altogether', 'left', 'share')
   2. Instead, map the problem onto a ratio table or equation so the operation choice follows from the structure
   3. Compare a keyword-driven plan with a structure-driven plan and identify where the keyword approach would choose the wrong operation or wrong order
   4. Justify the final set-up by explaining which quantity each step finds
3. 3

   Think-aloud partner: what's known, what's unknown, how are they related?

   Why it works: Monitoring your own solution by self-questioning — stating known/unknown, planning, estimating, and checking — keeps a long multi-representation solution on track and rehearses the antidote to the compulsion to calculate.

   1. Work through a ratio/algebra problem with a partner (real or in-role) who asks 'What is known? What is unknown? How are they related?' before allowing any computation
   2. Articulate the relationship aloud and describe the chosen representation before computing
   3. State an estimate aloud and justify it, then run an inverse check and say whether the answer is reasonable
   4. Explain what the answer means in context, including a negative or fractional result
   5. Reason aloud about a counterexample: 'Two painters together take 6 hours to paint a room; how long would one painter take alone — is it more or less than 6 hours, and why?' — decide whether you are reasoning about the situation or just combining the numbers