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# Child Development Tracker: Mathematics From Age 5 to 6 (page 2)

PBS Parents
Updated on Jul 9, 2010

### Operations on Numbers

• Throughout this year, some children will still be learning how to nonverbally and mentally determine sums up to "four" and their subtraction counterparts (e.g., "3 + 1," "4 - 1," "2 + 1," "3 - 2"). The average child will be able to nonverbally and mentally determine sums up to "five" (e.g., "2 + 3") and their subtraction counterparts (e.g., "5 - 3"), although some children will understand how to do this later, at age six.
• The average child will be able to use informal knowledge to estimate the sums of addition word problems (e.g., for "3 + 2," puts out four to six items to estimate the answer) or their subtraction complements (e.g., for "5 - 2," puts out around three items to estimate the answer) up to "ten." Other children will develop such estimation skills at age six.
• During the first half of this year, the average child can use concrete counting strategies to solve addition word problems (e.g., for a problem involving three and two more, the child counts out three items, puts out two more items, and then counts all the items to determine the answer) and concrete take away strategies to solve subtraction word problems (e.g., for a problem involving five take away two, counts out five items, removes two, and counts the remaining three items to determine the answer) with sums up to "ten" and corresponding differences. Other children learn how to use counting strategies this way at age six.
• In the first half of this year, some children can use various addition strategies to mentally determine sums up to "nine." The average child, however, understands how to apply such strategies in the second half of the year, and other children acquire these skills at age six.
• In the second half of this year, some five-year-olds can apply more advanced and abstract counting strategies to solve addition word problems with sums to "18" (e.g., solves "3 + 2," by verbally counting, "One, two, three, four is one more, five is two more," perhaps using fingers or other objects to keep track of the "one more," "two more" count). Another advanced strategy that some five-year-olds might use to find a sum is to begin counting from the number being added, rather than starting with "one" (e.g., for solving "3 + 2," starts counting from "three" instead of "one" by saying, "Three, four is one more, five is two more," perhaps by using fingers or other objects to keep track of the "one more," "two more" count.)
• In the first half of this year, some children can use existing knowledge and reasoning strategies to logically determine unknown sums up to "18" and their subtraction counterparts, including the "additive and subtractive identity" rule (e.g., "n + 0 = n" and "n - 0 = n"), the "number-after" rule (e.g., "7 + 1" equals the number eight when we count), and the notion that addition doubles have an even sum or form part of the skip count by two's sequence (e.g., "3 + 3 = 6," "4 + 4 = 8," "5 + 5 = 10"...). The average child can use these reasoning strategies in the second half of this year.
• Throughout this year, some five-year-olds may be able to solve addition and subtraction problems using the idea that near doubles are one more or less than doubles are, or in other words, their sums are in-between doubles and are "odd" (e.g., "7 + 6" is one more than "6 + 6," or "13"). Other strategies that some five-year-olds might use to solve addition and subtraction problems are the "number-before" rule for "n - 1" facts (e.g., "6 - 1 = 5" and "5 - 1 = 4") and the "negation" rule for "n - n" facts (e.g., "6 - 6 = 0" and "5 - 5 = 0"). The average child learns how to apply these strategies at age six, and some not until age seven.
• Finally, in the second half of this year, some five-year-olds may be able to solve addition and subtraction problems by applying the "additive commutativity" rule (e.g., if "5 + 3 = 8" and "5 + 3 = 3 + 5," then "3 + 5 = 8" also). The average child can apply such reasoning strategies at age six, but other children understand such concepts at age seven.
• During the first half of this year, the average child recognizes that adding to a collection creates a sum greater than the starting amount. Some children may not understand this concept until age six. Some children will also see that a part is less than the whole as they solve addition word problems (e.g. Bret had three cookies. His mother gave him some more, and now he has five cookies. How many cookies did Bret's mother give him?). In addition, some children will see that the whole is larger than its composite parts as they solve subtraction word problems (e.g., Chico had five cookies. He ate some, and now he has three left. How many cookies did Chico eat?). The average child will understand these concepts during the second half of this year, but other children will learn them at age six.
• During the first half of this year, some children can use up to ten objects to construct number partners up to "5" (e.g., 5 = "1 + 4," "2 + 3," "3 + 2," "4 + 1"), and doubles partners up to "10" (e.g., "3 + 3 = 6"). The average child understands these concepts during the second half of this year, but some children will learn them at age six. Throughout this year, some five-year-olds may also know number partners up to "10" (e.g., "1 + 9"), especially with "5" as a partner (e.g., "6 = 5 + 1"), and doubles to "20" (e.g., "12 = 6 + 6"). The average child will have this number sense at age six, and other children will develop this knowledge at age seven.
• During the first half of the year, some children will understand the "part-whole" relationship of addition and will be able to informally solve "part-part-whole" word problems that have a missing whole and sums up to "10" (e.g., Deborah had five chocolate chip cookies and three ginger snap cookies. How many cookies did she have altogether?). The average child understands this concept during the second half of this year, but others not until age six. During the second half of this year, a few five-year-olds will also recognize the "additive-commutativity" principle (e.g., "3 + 6 = 6 + 3"), the "addition-subtraction complement" principle (e.g., "5 - 3 = ?" can be thought of as "3 + ? = 5"), and the "inverse" principle (e.g., "5 + 3 - 3 = 5"). The average child recognizes these principles at age six, but others not until age eight.
• During the first half of this year, some children will still be learning how to trade several small items for a larger one (e.g., trades four small candies for a candy bar). Also, the average child can group objects into 5's or 10's, and recognize that the position of a digit in a number affects its value (e.g., recognizes that "23" and "32" are different). Some children will learn these ideas at age six. Finally, some children can break down a larger unit (especially "10" and "100") into smaller units, and can combine smaller units into a larger unit. The average child will be able to do this during the second half of this year, but other children will learn how to do this at age six.
• During the first half of this year, some children will be able to accurately read multidigit numerals up to "19." The average child will be able to do this during the second half of this year, and others at age six. At the same time, some children may even be able to accurately read multidigit numerals up to "99." The average child is able to do this at age six. Finally, during the second half of this year, a few five-year-olds will be able to accurately read multidigit numerals up to "999," but the average child develops this skill at age seven, and some not until age eight.
• Throughout this year, some children may be able to write multidigit numerals up to "99" (e.g., writes "twenty-four" as "24" and not "204"). The average child is able to do this at age six.
• Throughout this year, there may be a few five-year-olds who recognize that "1 ten = 10 ones." The average child understands this concept at age seven, but others not until age eight.
• During the second half of this year, there may be a few children who can meaningfully represent multidigit numerals up to "100" in different forms, such as with numerals and grouping/place-value models (e.g., recognizes that "2" in "27" represents two "tens" and "7" indicates seven "ones"). The average child understands these concepts at age seven, and others will learn them at age eight. During the second half of this year, a few five-year-olds may be able to meaningfully represent multidigit numerals up to "1000" in these different forms. The average child can do this with numbers up to "1000" at age eight.
• Throughout this year, a few five-year-olds will be able to invent mental procedures for adding and subtracting multidigit numbers, views sums and differences as a composite of "tens" and "ones," and creates shortcuts involving "10's" for sums up to "20" (e.g., recognizes that "10 + n" = "n + 'teen'" such as "10 + 7 = 17"; also "10 + 10 = 20" and "20 - 10 = 10"). The average child understands how to do this at age seven, and others at age eight.
• During the second half of this year, a few five-year-olds may be able to invent or accurately apply written addition procedures for problems with two-digit numbers. The average child can do this at age eight.
• During the second half of this year, a few five-year-olds may be able to use grouping/place-value knowledge and a front-end strategy to make reasonable estimates with two-digit numbers (e.g., "51 + 36 + 7" is at least "5 'tens' + 3 'tens', or 80"). The average child can make such estimates at age eight.
• Throughout this year, some children will be able to use informal strategies to solve "divvy-up/fair-sharing" problems where up to "10" items are distributed evenly to two or three people (e.g., if Este and Freeha share fairly the "12" cookies they baked, how many cookies would each get?). The average child can solve such problems at age six, and some children learn how to do this at age seven. Some five-year-olds may also be able to solve "divvy-up/fair-sharing" problems where up to "20" items are divided evenly among three to five people. The average child understands how to do this at age six, and others at age seven.
• In the second half of this year, some children can use informal strategies to solve "measure-out/fair-sharing" problems that divide up to "20" items into shares of two to five items each (e.g., If Este and Freeha baked 12 cookies and put three cookies in a bag, how many bags of cookies can they make?). The average child can solve such problems at age six, and others at age seven.
• In the second half of this year, some children can use informal strategies to solve "divvy-up/fair-sharing" problems with continuous quantities of one to ten wholes and two to five people (e.g., if four friends shared two pizzas fairly among them, how much pizza would each friend get?). The average child can solve such problems at age six, with other children learning how to solve these problems at age eight.
• In the second half of this year, a small number of five-year-olds can verbally label one of two as "half" or "one-half." The average child can apply this label at age seven, while some children learn how to do this at age eight.