Represent one- and two-step problems with the unknown anywhere

Learning goal: Solve addition and subtraction stories within 100 by representing the relationship, locating the unknown and explaining each step. A number sentence should match the story before it is used to calculate. Words such as more or left are clues to meaning, not automatic instructions to add or subtract.

Consider a joining story: 27 books are on a cart and 16 more are added. The unknown is the result, so write 27 + 16 = □. The total is 43 books. Now change the question: 27 books are on the cart, some are added and the result is 43. Write 27 + □ = 43. The unknown is the added amount, 16, even though the story still describes joining.

An unknown can come first: some books are on the cart, 16 are added and 43 are there afterward. Write □ + 16 = 43. Finding 43 − 16 gives the starting 27. Subtraction helps solve this addition relationship. Choosing an operation from the story's action alone would miss the role of the unknown.

Separating stories also have different unknown positions. If 52 markers are available and 18 are used, 52 − 18 = □ asks how many remain: 34. If 34 remain and 18 were used, □ − 18 = 34 asks for the starting amount: 52. If 52 were available and 34 remain, 52 − □ = 34 asks how many were used: 18.

A part-and-whole story does not need movement. A box contains 19 blue tiles and 23 yellow tiles, so 19 + 23 = 42 tiles. If the total and one color are known, find the missing part. A comparison concerns two amounts: Mina has 38 tiles and Jo has 25, so Mina has 13 more. The difference is not another pile added to both people's collections. A comparison can hide a different unknown: if Mina has 38 tiles, which is 13 more than Jo, then Jo has 38 − 13 = 25. The word more appears, yet subtraction finds the smaller amount. If Jo has 25 and Mina has 13 more, 25 + 13 = 38 finds the larger amount.

Two-step problems connect relationships. In the illustrated story, 27 books are on the cart, 16 are added and 12 are borrowed. First find 27 + 16 = 43; then find 43 − 12 = 31. Label 43 as an intermediate total and 31 as the final amount. The three views show one changing cart, not three independent collections whose totals should be combined.

Before calculating, retell the question in your own words and name the counted unit. After calculating, check that the result answers that question. If asked how many were borrowed, reporting the number remaining would answer a different question even if the arithmetic was correct. Use objects, bars or simple marks when they clarify the relationships; decorative details are unnecessary.

Worked example 1: Unknown change. Start with 27 + □ = 43. Count from 27 to 43 or subtract 43 − 27. The added amount is 16 books. Check by adding 27 and 16 to recover 43.

Worked example 2: Comparison. Mina's 38 tiles can be split into 25 matching Jo's amount and 13 extra. Write 25 + □ = 38 or 38 − 25 = □. Both describe the same difference, 13 tiles.

Worked example 3: Two steps. For the cart story, 43 is only the amount after adding. Borrowing 12 happens afterward, so 43 is not the final answer. Subtract 12 to obtain 31 books and explain both actions.

Guided practice: Change the location of the unknown while keeping a related number family. Model result-unknown, change-unknown and start-unknown joining and separating stories, then a part-whole and comparison example. Ask learners to explain the equation before solving it.

Independent tasks: Solve one unfamiliar story of each relationship across several sessions and a two-step story. Include an equation with a symbol for the unknown, an appropriate labeled model and a check. Do not judge understanding only by whether a learner uses the same method as the model.

Feedback: If a keyword drives an unsuitable operation, return to known quantities and the question. If an intermediate answer is reported as final, trace the remaining event. Credit a different valid strategy when its reasoning and units match the story.