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# Child Development Tracker: Mathematics From Age 5 to 6

Five-year-olds know the characteristics of various shapes, have improved number sense, and can think more abstractly. They can count out a collection of up to "20" items, conduct simple addition and subtraction, and identify which number in a set is larger. Five-year-olds understand and use words related to position, such as "under" or "behind." They sequence events chronologically and are learning to tell time. They can also sort objects based on more than one characteristic.

### Numbers

• At age five, some children may still be gaining an understanding of the number words up to "four" (e.g., distinguishes one-four items from "many"; can identify collections of up to four items with a corresponding number; asks for up to "four" of something; knows age; can put out "one," "two," "three," or "four" items upon request).
• Some children at the beginning of this year are still learning how to verbally count by ones up to "ten." The average five-year-old, however, will be able to use the "teen" pattern to accurately count to "20." (Some children may not be able to count up to "20" until age six.) Other children will be able to use repeating patterns to accurately count up to "42," with the average child able to do this by the second half of this year. (Some children may not be able to count up to "42" until age six.) In the second half of this year, a few five-year-olds will be able to use repeating patterns to accurately count up to "200." (The average child can count to "200" at age six.)
• At the beginning of this year, a small number of children may still be figuring out how to accurately determine the number of items in a collection of up to five items using one-to-one counting, or "enumeration" (i.e., the child labels each item in a collection with one and only one number word from the counting sequence to determine the total number of items in the collection). The average five-year-old, however, can accurately enumerate a collection of up to "ten" items. In the second half of this year, a few children may even be able to enumerate a collection of up to "20" items. (The average child can do this at age six.) Also, a few children may still be learning to recognize that the last number word used to count (enumerate) a collection has special significance because it represents the total number of items in the collection.
• In response to a verbal request, some children will still be learning how to accurately count out a collection of up to "five" items, while the average child can count out a collection of up to "ten" items. (Some children may not achieve this skill until age six.) Others may be able to count out a collection of up to "20" items, and the average five-year-old can do this during the second half of this year. (Some children learn how to do this at age six.)
• At the beginning of this year, some children may still be learning how to verbally count one-by-one from a starting point other than "one."
• In the first half of this year, the average five-year-old can name the number after a specified count term between "one" and "nine" (e.g., "What number comes after five?") without being prompted with the number's preceding sequence. Some will be able to name the number after a specified count term between "ten" and "40," which the average five-year-old can do in the second half of this year. (Some children may not be able to do this until age six.) A few children may be able to name the number after a specified count term between "29" and "99," but the average child can do this at age six, and some not until age seven.
• The average child in the first half of this year will be able to name the number that comes before a specified count term between "two" and "ten" (e.g., What number comes before "seven"?). Some children may be able to do this working with numbers up to "29," and the average five-year-old can do this in the second half of this year. (Some may not learn how to do this until age six.)
• In the first half of this year, some children may be able to verbally count backward from "five" or "ten," but the average child can count this way during the second half of the year. A few may even be able to count backwards from "20," but the average child learns how to do this at age six, and others not until age eight.
• Some five-year-olds can count to "100" by tens at the beginning of the year, but the average five-year-old is able to do this in the second half of this year. (Some children understand how to do this at age six.)
• A very small number of five-year-olds will understand terms related to estimation (e.g., "about," "near," "closer to," "between," "a little less than"). The average child, however, understands these terms at age seven.
• In the first half of this year, the average child will be able to make a reasonable estimate of the number of items in a collection involving up to "five" items (others will master this skill later in the year), and some may even be able to make such estimates with collections involving up to "20" items. The average five-year-old, however, can make reasonable estimates of the number of items in a collection involving up to "20" items during the second half of this year. Some children may even be able to make such estimates in a collection of "100" items, but most children develop this skill at age six.
• At the beginning of this year, some children will be able to correctly use formal relational terms (e.g., "greater than," "less than," and "equal to"). The average child can effectively apply these terms in the second half of this year, while others may not until age seven.
• In the first half of this year, some children may still be learning how to determine which of two numbers less than "ten" and widely separated in the counting sequence (e.g., "nine" and "three") is "more" by understanding the larger-number principle (the later a number appears in the counting sequence, the larger the quantity represented). Other children will be able to make such comparisons working with numbers up to "100," with the average child able to do so during the second half of the year, and others at age six. Also in the first half of this year, the average child will be able to determine which of two numbers less than "ten" and widely separated in the counting sequence (e.g., "nine" and "three") is "less." Other children will understand how to do this in the second half of this year.
• At the beginning of this year, the average child will use the "larger-number principle" (i.e., the later a number appears in the counting sequence, the larger the quantity represented) and number-after knowledge to determine which of two "neighboring" numbers (e.g. "three" and "four") in the counting sequence is "more," working with numbers from "one" to "five." Some children will not be able to make such determinations with numbers from "one" to "five" until the second half of the year, when the average child is able to effectively make such comparisons with numbers up to "ten" (e.g. "Which number is more, 'seven' or 'eight'?"). Also in the second half of this year, some children will be able to determine which of two "neighboring" numbers is "more," working with numbers up to "100." (The average child understand how to make such determinations with numbers up to "100" at age six.)
• In the second half of this year, the average child will be able to determine which of two adjacent numbers in the counting sequence from "one" to "ten" is "less" (e.g., "Which number is less, 'seven' or 'eight'?"), with some children understanding how to do this earlier in the year, and some not until age six. At the same time, some children can determine which of two adjacent numbers in the counting sequence from "one" to "100" is "less," with the average child able to do this at age six.
• In the first half of this year, some children will be able to use a mental number line to determine the relative proximity of one-digit numbers (e.g., recognizes that "five" is closer to "three" than to "nine"). The average child develops such number sense during the second half of this year, and some not until age six. During the second half of this year, some children may also be able to gauge the relative proximity of two-digit numbers (e.g., recognizes that "63" is closer to "77" than to "32"). The average child has such two-digit number sense at age six, and others at age seven.
• Finally, at the beginning of this year, a few children will still be learning how to understand and effectively apply the ordinal terms "first" and "last."
• At the beginning of this year, the average child can draw objects, make a tally, or use some other informal symbol to represent a spoken number. (Some children may not understand how to do this until age six.)
• A very few children may begin to use informal and symbolic representations (e.g., drawings of objects, a tally, etc.) to represent the number of items in a collection up to "nine." (The average child can do this at age six, but other children may not be able to until age seven.)
• Some children may still be learning how to recognize or read numerals "0" to "9" (e.g., is able to point out a "three" given a choice of five numerals, or identifies the numeral "3" as "three"). The average child is able to copy or write numerals "0" to "9," but some children may not learn to do this until age six. During the first half of this year, the average child can also connect at least some numerals to both number words and the quantities they represent (e.g., uses one-digit written numerals to represent the value of a collection, identifies the larger of two written numerals, recognizes that "0" can mean "none"). Some children make these connections in the second half of this year.
• Throughout this year, some children will be able to understand that numbers can be represented on a number line, although the average child learns this at age six and some not until age eight.
• During the second half of this year, some children may be able to informally show if two collections are equal or not. The average child understands how to do this at age six.
• In the later part of this year, some children will be able to identify the written number words "one" through "nine" with their corresponding pronunciations and written numerals "1" through "9," as well as use them to represent the number of items in a collection. (The average child achieves this skill at age six.)

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