subtraction problems. Reversing the action in change minus or change plus situations shows the connection between subtraction and addition. For example, if Whitney had 9 dinosaurs and gave away 3 dinosaurs, how many dinosaurs did Whitney have left? This problem can be formulated with the subtraction equation, 9 − 3 = ? Starting with the dinosaurs Whitney has left, if she gets the 3 dinosaurs back, she will have her original 9 dinosaurs, which can be expressed with the addition equation ? + 3 = 9. Subtraction problems can thus be reformulated in terms of addition, which connects subtraction to addition.

In put together situations, there are two parts, A and B, which together make a whole amount, C. These situations are formulated in a natural way with an addition equation, A + B = C.

Change plus, change minus, and put together problems in which either A or B (the start quantity, the change quantity, or one of the two parts) is unknown involve an interesting reversal between the operation that formulates the problem and the operation that can be used to solve the problem from a more advanced perspective. For example, consider this “change unknown” problem: “Matt had 5 cards. After he got some more cards, he had 8. How many cards did Matt get?” This problem can be formulated with the addition equation 5 + ? = 8. Although young children will solve this problem by adding on to 5 until they reach 8 (perhaps with actual cards or other objects), older children and adults may solve the problem by subtracting, 8 − 5 = 3, which uses the opposite operation than the addition equation that was used to formulate the problem.

Comparison situations concern precise comparisons between two different quantities, A and C. Instead of simply saying that A is greater than, less than, or equal to C, the situation concerns the exact amount by which the two quantities differ. If C is B more than A, then the situation can be formulated with the equation A + B = C. If C is B less than A, then the situation can be formulated with the equation A − B = C. To consider this precise difference, B, requires one to conceptually create a collection that is not physically present separately in the situation. This difference is either that part of the larger collection that does not match the smaller collection, or it is those objects that must be added to the smaller collection to match the larger collection. Of course, these matches can be done by counting and with specific numbers rather than just by matching. Note that these situations are called additive comparison situations even when formulated with subtraction (A − B = C when C is B less than A) to distinguish them from multiplicative comparison situations, which can be formulated in terms of multiplication or division. Students solve multiplicative comparison problems in the middle and later elementary grades.

In take apart situations, a total amount, C, is known and the problem is to find the ways to break the amount into two parts (which do not have



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