advanced perspective, most of these situations can be formulated in a natural way with an equation of the form

in which two of the three numbers in the equation are known and the problem is to determine the other number that makes the equation true. The types of situations that are naturally formulated with these equations are change plus and change minus situations, put together situations, and comparison situations.

In change plus and change minus situations, there is a starting quantity (A), an amount by which this quantity changes (B), and the resulting quantity (C). Problems in which A and B are the known amounts and C is to be determined are the classic, most readily recognized addition and

ent numbers helps them learn number triads related by this total-addend-addend relationship, which they can use when adding and subtracting. Eventually with much experience, children move to thinking of embedded number situations in which one considers the total and the two addends (partners) that are “hiding inside” the total simultaneously instead of needing to shift back and forth.

Equations with the total alone on the left describe take apart situations: 3 = 2 + 1. Such equations help children understand that the = sign does not always mean makes or results in but can also mean is the same number as. This helps with algebra later.

Comparison Situations

Children first learn the comparing relations equal to, more than, and less than for two groups of things or two numbers. They find out which one is bigger and which one is smaller (or if they are equal) by matching and by counting.

Eventually first grade children come to see the third quantity involved in a more than/less than situation: the amount more or less (the difference). Children then can solve additive comparison problems in which a larger quantity is compared to a smaller quantity to find the difference. Children may write different equations to show such comparisons and may also still solve by matching or counting. As with the other addition and subtraction situations, any of the three quantities can be unknown. The language involved in such situations is complex because the comparing sentence gives two kinds of information. “Julie has six more than Lucy” says both that “Julie has more than Lucy” and that the amount more is six. This is a difficult linguistic structure for children to understand and to say.

NOTE: Researchers use different names for these types of addition and subtraction situations, and some finer distinctions can be made within the categories. However, there is widespread agreement about the basic types of problem situations despite the use of different terminology.

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