Two-Step Story Problems: What the Second Step Is For
The third-grade team at Riverside Elementary is scoring Friday's two-step problem: 'A farm stand sold 9 pounds of apples on Saturday. On Sunday it sold 3 times as many pounds of apples as it did on Saturday. How many pounds did the stand sell in all?' Four exit tickets were pulled for the team meeting. Amari wrote 9 × 3 = 27 and stopped. Bella wrote 9 + 3 = 12. Carlos wrote 9 × 3 = 27, then 27 + 9 = 36. Destiny wrote 9 × 3 = 27, then 27 × 2 = 54. The team must decide what each response shows about the student's thinking, which responses actually answer the question that was asked, and what to ask each student on Monday. The coach reminds them that the standard is about representing a two-step problem with equations and solving it, not just producing a number.
1. For each of the four exit tickets, name the misconception or partial understanding it shows and state whether the student answered the question asked.
Read the worked answer
All four students got 9 × 3 = 27, so the multiplication fact is not the problem. The problem is what the second step is for.
| Student | Work | What it shows | Answers the question? |
|---|---|---|---|
| Amari | 9 × 3 = 27 | Solved step 1 correctly and stopped. Likely reads the last number found as 'the answer.' | No - 27 is Sunday only |
| Bella | 9 + 3 = 12 | Reads '3 times as many' as an additive comparison ('3 more'). | No |
| Carlos | 9 × 3 = 27; 27 + 9 = 36 | Both steps, correctly labelled: Sunday = 27, total = 27 + 9. | Yes |
| Destiny | 9 × 3 = 27; 27 × 2 = 54 | Found Sunday, then doubled Sunday instead of adding Saturday once. 54 = 27 + 27, so she counted Sunday twice. | No |
Carlos is the only complete response. Worth naming for the team: 4 × 9 = 36 is also a valid model, because Sunday is three groups of 9 and the total is four groups of 9. Destiny's work is close to that idea but misapplied.
2. Write the one follow-up question you would ask Amari and the one you would ask Destiny, and explain what each question should reveal.
Read the worked answer
To Amari: 'Your 27 is right - what does the 27 count?'
This is not a correction question; it is a locating question. If Amari says 'Sunday's apples,' he has the first step and simply has not tracked that a second quantity is being asked for. If he says 'all the apples,' he has collapsed the two steps and needs the situation re-read with the two questions separated: How many on Sunday? then How many in all? The wording of his answer tells you which reteach he needs.
To Destiny: 'Why did you multiply by 2 at the end?'
This reveals whether she is treating Saturday and Sunday as two equal groups (a 'two things happened, so double it' reflex) or whether she lost track of which quantity she had already found. Her numbers give it away: 54 = 27 + 27, so she added Sunday to Sunday. A productive follow-up is to have her build the two days with two colors of counters and physically combine 27 and 9, then compare that to what she computed.
3. Bella's ticket (9 + 3 = 12) is the most common error in the class. Design a short follow-up problem and representation that would let her see the multiplicative comparison, and explain why your choice works.
Read the worked answer
Drop the numbers to something she can build: 'Sam has 4 marbles. Jo has 3 times as many marbles as Sam. How many marbles does Jo have?' Represent it as a tape diagram - one bar of 4 for Sam, and three equal bars of 4 for Jo.
Why this works: Bella's additive answer, 4 + 3 = 7, does not land anywhere sensible on the diagram. There is no place where 3 is a quantity of marbles; 3 is the number of bars, not the size of anything. Building Jo as three copies of Sam's bar gives 4 + 4 + 4 = 12, which is 4 × 3. The diagram makes the multiplier visible as a count of groups, which is exactly the idea her equation is missing.
Then extend before changing the numbers: 'How many marbles in all?' adds a fourth bar, giving 16, and connects directly to Carlos's structure (find the compared amount, then combine). Keep the numbers small so the reasoning, not the computation, is the work.