(decomposed/composed numerical triads) to permit flexible breaking apart and combining of numbers to turn them into teen addition or subtraction problems. For example, all of the following addition problems—9 + 2, 9 + 3, 9 + 4, … , 9 + 9—require the same first step: 9 needs 1 more to make ten, so separate the second number into 1 + ?. This triad then becomes 9 + 1 + ? = 10 + ?, which is an easier problem to solve if you know the tens in teen numbers. However, each problem requires a different second step: decomposing the second number to identify the rest of the second addend that will be added to ten (prerequisite 3 for derived facts methods in Box 5-11). For example, 9 + 4 = 9 + 1 + 3 + 10 = 3 13, but 9 + 6 = 9 + 1 + 5 = 10 + 5 + 15. So kindergarten children need experiences with finding and learning the partners of various numbers under 10.

Children’s counting and matching knowledge is now sufficient to extend to relations on sets up through 10 and to more abstract ways of presenting such relational situations as two rows of drawings that can be matched by drawing lines connecting them. As discussed above for Step 2, children will be more accurate when these objects are already matched instead of being visually misleading (for example, the longer row has less). They therefore can start with the simpler nonmisleading situations and extend to the visually misleading situations when they have mastered such matched situations. Again, differentiating length and number meanings of more will be helpful (which looks like more and which really is more). Children who have not had sufficient experiences matching objects at Step 2 will need such experiences to support the more advanced activities in which matching is done by drawing lines.

Working with the terms more and less can also be an opportunity to discuss and emphasize that length units used in measuring a length must touch each other and cover the whole length from beginning to end to get an accurate length measurement. But things children are counting can be spread apart or moved around and they will still have the same number of things. Comparing objects spaced evenly in two rows can also be related to picture graphs, which record numbers of different kinds of data as a row of the same pictures (see the Chapter 2 discussion in the Mathematical Connections section). Activities in which children compare two rows of drawings by counting or matching them can be considered as using picture graphs if each drawing in one row is the same. What is important about such activities is that children talk about them using comparison language (There are more suns than clouds or There are fewer clouds than suns) and describe how they found their answer.

Children at this level can also prepare for the comparison problems at Grade 1 by beginning to equalize two related sets. For example, for a row of 5 above a row of 7, they can be asked to add more to the row of 5 to



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