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Mathematics

Early Math

String together colorful pasta necklaces and practice pattern-making at the same time! Pasta Patterns

Use counting and matching skills to help owners find their lost pets. Animal Lost and Found

Clifford

Help Clifford compare sizes and make the right choice in this measuring game. Find the Right One!

Six-year-olds can typically count up to "200" and count backwards from "20." They understand the concept of "odd" and "even" numbers, and can represent numbers on a number line or with written words. They use increasingly more sophisticated strategies to solve addition and subtraction problems. They also count the sides of shapes to identify them, and can combine shapes to create a new one. Six-year-olds can also give and follow directions for moving around a room or on a map.

Numbers

• During the first half of this year, some children will still be learning how to use the "teen" pattern to accurately count to "20." Some children will also still be figuring out how to use repeating patterns (e.g., count by "twos" or "tens") to accurately count up to "42" while others use such repeating patterns to accurately count up to "100" and possibly "200." The average child can count to "200" by the end of the year.

• At the beginning of this year, some six-year-olds can accurately determine the number of items in a collection of up to "20" items using one-to-one counting, or "enumeration" (e.g., 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 child can enumerate a collection of "20" items by the end of the year. In response to a verbal request, some children will still be learning how to accurately count out a collection of up to "ten" items, while others will still be figuring out how to count out a collection of up to "20" items.

• Throughout the year, some six-year-olds will still be learning how to name the number after a specified number between "ten" and "40" (e.g., "What number comes after 25?") without being prompted with the number's preceding sequence. The average child, however, can name the number after a specified number between "29" and "99" during the first half of the year. A few children may even be able to name the number after a number in the hundreds (e.g., "What number comes after 188?"). During the first half of the year, some children will still be learning how to name the number that comes before a specified number between "11" and "29" (e.g., What number comes before "17"?).

• Throughout the year, some children may still be learning how to verbally count backwards from "five" or "ten," but the average child can count backwards from "20" during the second half of the year. At the same time, some six-year-olds may still be learning to count up to "100" by tens. Others will be able to name the decade after "10" and up to "90" without the preceding decade counting sequence. Some six-year-olds can verbally count by fives to "100" as well as count objects by fives. In addition, some children can count by twos to "20" as well as count objects by twos.

• Throughout the year, some children understand terms related to estimation (e.g., "about," "near," "closer to," "between," "a little less than"). During the first half of the year, some six-year-olds will still be learning how to make a reasonable estimate of the number of items in a collection involving up to "20" items. The average six-year-old, however, can make reasonable estimates of the number of items in a collection involving up to "100" items, and some can do this with collections of up to "1,000" items.

• Throughout the year, some children will still be learning to correctly use formal relational terms (e.g., "greater than," "less than," "equal to").

• During the first half of the year, some children will still be learning how to determine which of two numbers up to "100" and widely separated in the counting sequence (e.g., "30" and "63") is "more" by understanding the "larger-number principle" (e.g., the later a number appears in the counting sequence, the larger the quantity represented). Throughout the year, some children will still be figuring out how to use the larger-number principle 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 "ten" (e.g., "Which number is more, 'seven' or 'eight'?"). During the first half of the year, the average six-year-old can determine which of two "neighboring" numbers is "more," working with numbers up to "100." At the same time, some children will still be learning how 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'?"). The average child can do this with numbers from "one" to "100."

• Throughout the year, some children will still be learning how 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"). Some children will be able to make such determinations with two-digit numbers (e.g., recognizes that "63" is closer to "77" than to "32"), and the average child can do this by the end of the year. During the second half of the year, a small number of six-year-olds may be able to use a mental number line to determine the relative proximity of three- and four-digit numbers (e.g., "5,000" is closer to "3,000" than "8,000").

• Throughout the year, some children will still be learning how to draw objects, make a tally or use some other informal symbol to represent a spoken number. During the first half of the year, the average six-year-old can use informal and symbolic representations (e.g., drawings of objects, a tally) to represent the number of items in a collection up to "nine." At the same time, some children will still be learning how to copy or write one-digit numerals. Some children also understand that numbers can be represented on a number line, and the average child will recognize this by the end of the year.

• The average child can 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.

• Some children may be able to describe the parallels between abbreviated ordinal terms (e.g., "1st," "2nd," "3rd"..."9th") and cardinal terms (e.g., "1," "2," "3"..."9"). The average six-year-old can do this by the end of the year. In the second half of the year, some children can identify written ordinal terms (e.g., "first," "second"..."ninth") with their corresponding verbal words and use them to represent ordinal relationships.

Operations on Numbers

• During the first half of the year, some children will still be learning how to nonverbally and mentally determine sums up to "five" (e.g., "2 + 3") and their subtraction counterparts (e.g., "5 - 3"). Throughout the year, some six-year-olds will still be finding out how 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." At the same time, some children will be able to make such estimates up to "twenty."

• Throughout the year, some children will still be learning how to 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 their corresponding differences. At the same time, some children will be able to use such strategies with sums to "18" and their corresponding differences.

• Throughout the year, some six-year-olds are still learning how to use various addition strategies to mentally determine sums up to "nine." Some children can also 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 six-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.). The average six-year-old understands how to use both strategies by the end of the year.

• In the second half of the year, some children can use a "counting down" strategy to solve subtraction word problems (e.g., to solve, "Five take away three," counts, "Five, four is one less, three is two less, two is three less, so two are left."), perhaps using fingers or objects to keep track of "how many less" a number is.

• Throughout this year, some children can translate addition and subtraction word problems and their solutions into a number sentence and vice versa, thereby making connections between formal addition/subtraction and concrete or informal knowledge.

• Throughout the year, some six-year-olds will still be learning how to apply 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"...). During the first half of this year, the average child can solve addition and subtraction problems up to "18" 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 six-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"), the "additive commutativity" rule (e.g., if "5 + 3 = 8" and "5 + 3 = 3 + 5," then "3 + 5 = 8" also) and the "negation" rule for "n - n" facts (e.g., "6 - 6 = 0" and "5 - 5 = 0"). The average child can apply such strategies by the end of the year.

• Throughout the year, some six-year-olds will also understand how to solve addition and subtraction problems up to "18" by using the "difference between number neighbors is one" rule (e.g., "7 - 6 = 1", "8 - 7 = 1", "9 - 8 = 1"), the "make-a-ten" addition strategy (e.g., "8 + 5" can be solved "8 + 2 + 3 = 10 + 3 = 13) and the "related-addition-fact" strategy (e.g., "8 - 5 = ?" can be thought of as "5 + ? = 8). Throughout the year, some six-year-olds will be able to efficiently solve addition problems up to "nine" regardless of the strategy used. During the second half of the year, some children will also be highly effective with sums of "ten" and small doubles (e.g., "2 + 2," "5 + 5"), sums of large doubles (e.g., "8 + 8," "9 + 9") and subtraction problems related to the addition doubles (e.g., "14 - 7").

• During the first half of this year, some six-year-olds will still be learning to recognize that adding to a collection creates a sum greater than the starting amount.

• Throughout the year, some children will also still be understanding 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 still be learning to recognize 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?).

• Throughout the year, some children will also be learning how to 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"). During the first half of the year, the average six-year-old can add 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"). Throughout the year, some children can add number partners involving decades up to "100" (e.g., "50 = 10 + 40, 20 + 30, 40 + 10").

• Throughout the year, some children will still be figuring out 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?).

• During the first half of this year, some seven-year-olds can 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 understands these concepts during the second half of the year.

• Throughout the year, some children can informally solve subtraction problems that "compare" (e.g., Ann has five pennies and Barb has three. How many more pennies does Ann have?) or "equalize" (e.g., Ann has five pennies and Barb has three. How many more pennies does Barb need to have the same number of pennies as Ann?).

• During the first half of this year, some children will still be learning how to group objects into fives or tens, and recognize that the position of a digit in a number affects its value (e.g., recognizes that "23" and "32" are different). Throughout the year, some six-year-olds will still be figuring out how to break down a larger unit (especially "10" and "100") into smaller units, and can combine smaller units into a larger unit.

• Some children will still be learning how to accurately read multi-digit numerals up to "19." In the first half of the year, the average child can accurately read multi-digit numerals up to "99," and throughout the year, some children can read numbers up to "999." During the first half of the year, the average six-year-old can accurately write multi-digit numerals up to "99" (e.g., writes "twenty-four" as "24" and not "204"). During the second half of the year, some children will be able to write numbers up to "999."

• Throughout the year, some children recognize that "1 'ten' = 10 'ones'," that "1 'hundred' = 10 'tens' or 100 'ones'," and in the second half of the year, some children will understand that "1 'thousand' = 10 'hundreds' or 1,000 'ones'."

• Some children may be able to 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'). In addition, some children can meaningfully represent multi-digit numerals up to "1,000" in these different forms.

• Throughout the year, some children are able to invent mental procedures for adding and subtracting multi-digit numbers, view sums and differences as a composite of "tens" and "ones" and create shortcuts involving "tens" 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"). Another shortcut involving "tens" that some children may understand is that a decade + ten = the next decade (e.g., "60 + 10 = 70") and a decade - ten is the previous decade (e.g., "60 - 10 = 50"). In the second half of the year, a few six-year-olds can add multiples of ten (e.g., "5 + 20"). A few children may also be able to create shortcuts for "10's" when adding decades (e.g., "50 + 40 = 5 tens + 4 tens = 9 tens").

• Throughout the year, a few six-year-olds will be able to invent concrete procedures for adding and subtracting two-digit numbers, and in the second half of the year, a few might even be able to do this with three-digit numbers. Throughout the year, a few six-year-olds will be able to invent or accurately apply written addition procedures for problems with two-digit numbers. In addition, a few children can 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"), and in the second half of the year, a few might be able to do so with three- and four-digit numbers (e.g., "563 + 222 + 87" is at least "5 'hundreds' + 2 'hundreds', or 700").

• During the first half of the year, the average six-year-old can 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?) as well as problems where up to "20" items are divided evenly among three to five people. Some children may also be able to solve such problems with "100" items (grouped by tens and ones) divided evenly among up to "10" people.

• In the first 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 by the end of the year. In the second half of the year, some children can also solve such problems with "100" items (grouped by tens and ones) and shares of up to "10" items.

• In the first half of this year, some children can use informal strategies to solve "divvy-up/fair-sharing" problems with continuous quantities (i.e., when a whole can be divided into whatever number of parts are needed) 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 by the end of the year. Some six-year-olds may also be able to solve such problems with one to ten wholes and six to ten people.

• Throughout the year, some children can verbally label one of two as "half" or "one-half," one of three as "one-third," one of four as "one-fourth" and one of five as "one-fifth." During the second half of the year, a few children may similarly be able to label non-unit fractions (e.g., labels three of eight equal pieces as "three-eighths"). At the same time, some children can compare unit fractions (e.g., knows that "one-half" is larger than "one-third").

• During the first half of the year, some six-year-olds can use concrete objects to solve repeated-addition problems, or those involving groups of items (e.g., uses blocks to solve the problem, "If four boxes each have three toys, how many toys are there altogether?"). By the end of the year, the average child can solve such problems using concrete objects. During the second half of the year, a few children can mentally solve such problems using strategies like verbal counting (e.g., says, "1, 2, 3 in one box, 4, 5, 6 in two boxes, 7, 8, 9 in three boxes, 10, 11, 12 in four boxes"), addition (e.g., says, "3 and 3 is 6, and 6 and 3 is 9, and 9 and 3 is 12"), skip counting (e.g., says, "3, 6, 9, 12") or some combination of these strategies. During the second half of the year, some children can represent repeated-addition problems symbolically as addition (e.g., writes, "3 + 3 + 3 + 3").

Geometry and Spatial Sense

• Throughout this year, some six-year-olds will recognize and name circles, squares, triangles and rectangles in any size or orientation, including varying shapes for triangles and rectangles. In addition, a small number of children may be able to recognize and name a variety of other shapes in any orientation (e.g., semi-circles, quadrilaterals, trapezoids, rhombi, hexagons). In the first half of the year, the average six-year-old can use class names to classify and sort shapes (e.g., when asked to identify "circles," places different examples of a circle together on a mat, but does not put down other shapes such as squares and triangles).

• During the first half of the year, some children will still be learning how to copy a shape from memory after seeing a model for several seconds. Throughout this year, some six-year-olds will be able to accurately visualize two-dimensional shapes and draw them from memory, including geometric paths that represent "route maps" (e.g., can mentally represent and then draw a “train” or a line of shapes composed of a square, circle and triangle). Some children will also be able to create shapes from verbal directions.

• During the first half of the year, the average six-year-old can define the term "congruent" as two shapes with the same size and shape. In addition, the average child will be able to match shapes and parts of shapes to show that they are congruent.

• During the first half of the year, the average child can identify and count the sides of two-dimensional shapes. Some children will also be able to identify and count both sides and angles. By the end of the year, some six-year-olds can independently identify shapes in terms of their defining attributes (e.g., says, "It has one, two, three sides...it is a triangle."), but won't see relations between or among shapes (e.g., does not recognize a square as a rectangle).

• Some six-year-olds will be able to informally name, describe, compare and sort solids. In the second half of the year, some children will also be able to identify and describe the faces of three-dimensional shapes as specific two-dimensional shapes (e.g., a face of a cube is a square).

• Throughout the year, children develop the ability to create drawings that involve more than two geometric forms. During the first half of the year, some children will still be learning how to cover an outline of a shape with other shapes without leaving gaps, first with trial-and-error, and then with foresight. In addition, some children may be able to combine shapes to create a new shape. The average child will be able to do so by the end of the year. At the same time, some six-year-olds may be able to create new shapes by combining other shapes and substituting a combination of smaller shapes for a larger shape.

• In the first half of this year, some six-year-olds will still be learning how to break apart simple two-dimensional shapes that have obvious clues for breaking them apart to make new shapes (e.g., break a square into two triangles). In the second half of the year, some children will also understand how to break apart two-dimensional shapes to make new shapes by picturing images suggested by the task or an adult. Throughout the year, some children will understand and be able to predict how shapes will change when they are put together and broken apart in different sequences.

• During the first half of the year, some children will still be learning how to create tilings (i.e., covering a flat surface with small shapes, allowing no gaps between shapes or overlaps) with single shapes. By the end of the year, the average child will also be able to create tilings with combinations of shapes.

• In the first half of this year, some children will still be figuring out how to find some shapes "hidden" in arrangements in which the shapes overlap each other, but are not embedded inside one another. Some children may also be able to find shapes "hidden" inside of other shapes. The average child will be able to do this by the end of the year. At the same time, a few children will be able to find more complicated shapes that are "hiding" inside of other shapes.

• Throughout the year, some six-year-olds will still be learning how to understand and use words that represent physical relations or positions (e.g., "over," "under," "above," "on," "beside," "next to," "in front," "behind," "in," "inside," "outside," "between," "up," "down," top," "bottom," "front," "back," "near," "far," "left," "right"). In addition, some children recognize and can point out geometric shapes and structures in the environment. During the first half of the year, some children can give and follow directions for moving in physical space and on a map. The average child can do such things by the end of the year. Some children can also make and follow maps of familiar areas, such as their room. A few children will understand that maps answer questions about direction, distance and location. During the second half of the year, some six-year-olds understand that objects can be represented from different points of view and can show shapes from different perspectives. In the first half of the year, the average child can use coordinates to locate objects or pictures in simple situations. By the end of the year, some children will even be able to use coordinates to locate positions.

Measurement

• Throughout the year, some six-year-olds will still be developing a sense of time and will primarily know when events close to them take place. Some will also still be mastering the days of the week, the months, the seasons, and how to tell time. During the first half of this year, the average six-year-old will understand that unless more is added or removed, the number of objects in a collection remains the same (is conserved), even if the appearance (e.g., shape) of those objects changes. Some children may not grasp this idea until later in the year. Similarly, some six-year-olds may begin to recognize that the length of something remains the same (is conserved) regardless of appearances, unless more is added or removed.

• During the first half of this year, some six-year-olds will still be learning how to compare the lengths of two objects by representing the lengths with strings or strips of paper and then using these representations to determine which is longer. Some children during play may also be just learning how to intuitively compare angles and how much "turn" they have.

• In the first part of this year, the average six-year-old will measure the length of an object by laying end-to-end an informal and same-size unit of length (e.g., paper clips). Some children may also be able to use a simple ruler to measure objects. During the second half of this year, some six-year-olds can compare the effects of measuring length using units of different size and determine the need for using a standard unit of measurement such as a ruler. Some will also be able to measure the perimeter of an object, and can use a single unit over and over to measure something larger (e.g., measures the length of a room with a single meter stick). In terms of measuring area, some six-year-olds will still be learning how to measure area by covering an area with informal units (e.g., 1" x 1" squares) and counting the individual units (not necessarily in an organized way). However, the average child, during the first half of the year, will partially count such units by using row or column structuring (e.g., says, "Three in this row and three in this one make six. Ummm..." and then continues counting by ones, "...seven, eight, nine..."). In the second half of this year, some children will begin to understand area as an array with both rows and columns (e.g., says, "Four rows with three in each row... so three, six, nine, twelve!").

• During the first half of this year, some six-year-olds will still be learning how to make informal comparisons and estimates (e.g., says, "I'm as tall as the yellow bookshelf."). Throughout the year, some children will be able to identify common objects to use as referents when estimating standard measures of length (e.g., says, "The top of the door knobs are about a meter from the floor."). Some six-year-olds will also be able to recognize that a unit can be sub-divided into equal size sub-units, and that both types of units can be used to make measurements (e.g., uses units and fractional units such as 1 1/2 inches).

Patterns, Reasoning, and Algebra

• During the first half of this year, some children can recognize regularities in a variety of contexts (e.g., events, designs, shapes, sets of numbers). The average child recognizes such regularities by the end of the year. During the first half of the year, some children are still learning to identify the "core" of simple repeating patterns (i.e., the basic sequence or building block that is repeated) and extend the pattern by replicating the core (e.g., for the pattern "red/blue/red/blue/red/blue," the child will add "red/blue"). This development is also true for when children imitate pattern sounds and physical movements (e.g., clap, stomp, clap, stomp...). At the same time, some children recognize the growing pattern involved with counting, where "one" is added each time to get to the next number in a basic arithmetic progression. The average child can recognize such patterns by the end of the year. Throughout the year, some children will also recognize arithmetic progressions where numbers other than "one" are added (e.g., "2, 4, 6, 8,..." involves adding "two" each time; "5, 10, 15, 20..." involves adding "five" each time, etc.). In addition, some children can identify other obvious growing patterns (e.g., 121121112...).

• During the first half of the year, the average child will discover the concepts of "even" numbers (i.e., a number of items that can be shared fairly between two people), and "odd" numbers (i.e., sharing between two people results in a leftover item). Also, some children will grasp the concept of integers (i.e., "positive integers," numbers to the right of "zero" on a number line; and "negative integers," numbers to the left of "zero" on a number line). Finally, a few six-year-olds will discover odd-even rules for addition and subtraction (e.g., the sum of two odd numbers is an even number).

• Throughout this year, some children can use letters to represent the "core" of a repeating pattern (i.e., the basic sequence or building block that is repeated) of up to "three" elements (e.g., "ABC" for "123123123..."). Some children may also explicitly recognize that the same pattern can be manifested in many different ways (e.g., recognizes that "123123123...", "do re mi do re mi do re mi...", and "triangle/square/circle/triangle/square/circle..." are all examples of an "ABC" repeating pattern).

• Throughout this year, a few children will be able to translate simple addition and subtraction word problems or real-world situations into number sentences with a self-chosen symbol to represent the unknown (e.g., "5 + ? = 8"). During the second half of this year, a few children will also be able to translate number sentences with a variable into realistic word problems. In addition, a few will be able to determine the specific unknown of number sentences, including those that represent arithmetic principles, properties and relations (e.g., "5 + ? = 5," "5 - ? = 5," "5 + 3 = 5 + ?," "5 + 3 - ? = 5").

• Throughout this year, a few six-year-olds may begin to summarize with natural language the ideas of "additive identity" (e.g., says, "You did not add anything, so it is still the same"), "subtractive identity" (e.g., says, "You did not take anything away, so it is still the same"), and "subtractive negation" (e.g., says, "You took it all; there is nothing left."). A few children may also verbally summarize "additive commutativity" (e.g., says, "You can add numbers in any order.") and the concept of "inverse principle" (e.g., says, "You added and took away the same, so it is the same."). Throughout the year, some children may begin to summarize with natural language, and then later with algebraic expressions or equations, real functional relations (e.g., "12 inches equals a foot").

• Throughout this year, some six-year-olds may recognize that the act of looking for patterns can be a useful problem-solving method. They may also use a pattern to justify a solution. These children will likely assume, however, that the first pattern identified must be the correct solution. At the same time, there will be some children this age that recognize that finding a pattern does not automatically mean it is the correct solution. They will understand that evidence (e.g., patterns, examples) is needed to support their ideas, and they may use multiple patterns or examples to justify a solution. Throughout the year, some six-year-olds can use estimation procedures such as rounding up, rounding to the nearest decade, and so forth. At the same time, some children can use the renaming procedure (i.e., "carrying over" or "borrowing" from neighboring numbers) for multi-digit addition and subtraction with whole numbers.

• Throughout this year, some children will still be learning how to move beyond using arbitrary rules (e.g., creating a category for "because I like it") to complete an adult-imposed classification task. As they develop this ability, these children can stick with one feature (e.g., color, shape, size) in sorting objects into a class. Also, some children will still be learning to sort and classify on the basis of one or more characteristics (e.g., color, size, etc.), as well as how to articulate why items are grouped together.

• Throughout this year, some children will still be learning how to sequence events chronologically. At the same time, a few six-year-olds will be able to use patterns within the same row of data and additive reasoning to logically solve problems (e.g., in the sequence, "3, 4, 5", the next value would be "6" since each preceding value increased by "one" for each step in the sequence). Throughout the year, some children will be able to use known quantities (mental numerical benchmarks or mental images of "5," "10," or "100") to make reasonable estimates of collections with quantities such as "17," "24," "78," or "125." At the same time, some children use known basic number combinations to make reasonable estimates of the answers to large, difficult computations (e.g., "45 + 37"). Some may also learn to use their knowledge of the counting sequence to round numbers to the nearest tens, hundreds or thousands and estimate an answer to a problem.

• Throughout the year, some children will be constructing "algebra sense," where they can use a variety of informal problem-solving strategies (e.g., drawing a picture, try-and-adjust, and working backward) to solve algebra problems. At the same time, some children will understand that the "equal" sign can be interpreted as "the same number as" or "the same as" in a variety of contexts and comparisons (e.g., "12 inches = 1 foot," "10 pennies = 1 dime," "3 = 1 + 2," "1 + 2 = 4 - 1," etc.). In addition, some children will understand the "other-name-for-a-number" concept (e.g., "12 = 12 + 0, 11 + 1, 10 + 2, 12 - 0, 13 - 1, 14 - 2,...").

Statistics

• Throughout this year, some children will recognize that some questions, issues, or areas of disagreement are "empirical questions" that cannot be answered without first collecting data. Also, some children will be able to collect relevant data for addressing a question or making a decision of personal importance. Finally, some children will be learning how to distinguish among types of data (e.g., "names" versus "numbers," "measures" versus "number of items").

• Children continue to learn how to organize, describe and interpret data (e.g., by constructing picture or bar graphs) to address a question (e.g., What eye color is most common in the family?) or make a decision of personal importance (e.g., Which ice cream shop has the most flavors?).

Probability

• Some children will begin to understand that some events are more likely to occur than others (e.g., snow is more likely in winter than in summer). They will also have some understanding and use the language of probability (e.g., "certain" or "sure," "uncertain" or "unsure," "likely" or "probable," "unlikely" or "improbable," "maybe" or "possible," and "impossible"). In addition, some children will understand that uncertainty and chance play a role in many everyday situations.

• Some children can conduct a simple experiment to see if all players have the same chance of winning a game, or to solve other simple probability problems.

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