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## Mathematics

### Sagwa

How many fish can you catch? Count and see. Couting Koi

### Sagwa

Click and drag shapes to solve these puzzles. Matching Shapes

### Arthur

Scooping up floating balls sharpens counting skills. Water Fun

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.)

### 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.

### Geometry and Spatial Sense

• During the first half of this year, some children may still be learning to recognize and name some variations of a circle, square, triangle, and rectangle. Throughout this year, a small number of five-year-olds will recognize and name circles, squares, triangles and rectangles in any size or orientation, including varying shapes for triangles and rectangles. The average child can do this at age seven, and others at age eight. A very small number of children may even be able to recognize and name a variety of shapes in any orientation, such as semi-circles, quadrilaterals, trapezoids, rhombi, hexagons, etc. The average child can recognize such shapes at age eight. Finally, some five-year-olds can use shape class names to classify and sort (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). The average child can do this at age six, but some not until age seven.

• During the first half of this year, some children will still be learning how to build, copy and informally describe two-dimensional shapes. The average child will be able to copy a shape from memory after seeing a model for several seconds. Other children will develop this skill at age six. Throughout this year, a small number of five-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). The average child can do this at age seven, and others at age eight. A small number of five-year-olds will also be able to create shapes from verbal directions. The average child learns how to do this at age seven, and others at age eight.

• During the first half of this year, some children will still be learning how to recognize congruence by matching shapes with other objects that have the same shape and size. Throughout the year, some five-year-olds will be able to explicitly define the term "congruent" as two shapes with the same size and shape. The average child will learn this definition at age six, and others not until age seven. Some five-year-olds will also be able to match shapes and parts of shapes to justify congruency. The average child understands how to do this at age six, and others at age seven.

• Throughout this year, some children will be able to identify and count the sides of shapes. The average child develops this skill at age six, and others at age seven.

• Throughout the year, children can complete increasingly complex puzzles (e.g., puzzles with smaller and more numerous pieces) and progress in their abilities to put together and take apart complex shapes. Children also build three-dimensional structures using multiple types of items (e.g., a rectangular prism, cube, and arches), and create drawings that involve more than two geometric forms.

• In the first half of the year, some children may still be learning how to make a picture by combining shapes. The average child can cover an outline of a shape with other shapes without leaving gaps, first with trial-and-error, and then with foresight. Some children will learn how to do this at age six. In the second half of this year, some children may be able to combine shapes to create a new shape. The average child can do this at age six, and others at age seven.

• In the first half of this year, the average five-year-old can break apart simple two-dimensional shapes that have obvious clues for breaking them apart. Some children won't understand how to do this until age six.

• In the first half of this year, some children can create tilings (i.e. covering a flat surface with small shapes, allowing no gaps between shapes or overlaps) with single shapes. The average child tiles with single shapes during the second half of this year, and some children at age six. In the second half of this year, some five-year-olds can tile with both single shapes and combinations. The average child can do this at age six, and others at age seven.

• In the first half of this year, the average child can find some shapes "hidden" in arrangements in which the shapes overlap each other, but are not embedded inside one another. Some children are able to do this at age six. During the second half of this year, some children will be able to find shapes "hidden" inside of other shapes. The average child can do this at age six, but others at age seven.

• During the first half of this year, some five-year-olds will understand and use words representing 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"). The average child understands and uses these words during the second half of this year, and others at age seven.

• During this year, the average child will be able to place toy objects in correct relative position to make a map of a room, and will also be able to follow simple route maps (e.g., uses pictures of desks, tables, windows, and doors to create a map of a classroom, and then uses it to follow directions).

• During the first half of this year, some five-year-olds will still be learning how to orient objects vertically or horizontally. Throughout the year, some children will understand how to use coordinate labels to locate objects or pictures in simple situations (e.g., uses a grid to locate ships in the game, "Battleship"). The average child will be able to do this at age six, and others at age seven.

• During the first half of this year, the average child can informally recognize when a rigid two-dimensional shape has been turned, flipped or otherwise moved, and will also move such shapes in this way. Some children will not informally identify and move two-dimensional shapes in this way until age six.

• During the first half of this year, some children will still be learning how to informally create two-dimensional shapes and three-dimensional buildings that have symmetry. At the same time, some children will be able to specifically identify and create shapes that have symmetry.

• Throughout this year, a few five-year-olds will be able to recognize that with elastic objects or surfaces, certain characteristics (e.g., closed versus open figures, inside or outside a figure, intersecting or nonintersecting lines) do not change; however, other characteristics (e.g., length and straightness) do change when a flexible object or surface is bent, twisted, enlarged or shrunk. In addition, a few children will informally recognize that shapes with no holes, one hole, or two holes ("genus 0, 1, and 2" objects respectively) can maintain the same absence or number of holes despite being bent, twisted, enlarged or shrunk (e.g., a cube of clay molded into a ball shape and then flattened into a pie shape are all "genus 0;" a donut-shaped piece of clay that is formed into a cup still has one hole through and through, so both shapes are "genus 1"). Children will continue to progress in their understanding of the geometry of elastic objects or surfaces through age eight.

• Throughout this year, a few five-year-olds may recognize that with the shadows of shapes, some characteristics (e.g., straightness) do not change, but other characteristics (e.g., length) do. Children will continue to progress in their understanding of this aspect of projective geometry through age eight.

### Measurement

• During the first half of this year, some five-year-olds will still be learning how to recognize, informally discuss, and develop language to describe attributes such as "big" or "small" (height/area/volume), "long" and "tall" or "short" (length/height), "heavy" or "light" (weight), and "fast" or "slow" (speed).

• Throughout this year, some children will still be learning how to compare a single attribute of several objects (e.g., says, "She has a bigger piece of cake than I do."). Some will also still be learning how to order objects from smallest to largest (e.g., lines up from shortest to tallest, nests cups, etc.) and describe relationships among objects (e.g., "big," "bigger," "biggest").

• Throughout the year, children continue to develop their sense of time. Some five-year-olds may still be learning how to recite the days of the week and seasons, and to recognize that a specific time is associated with certain events (e.g., favorite TV show comes on at 4:00). During the second half of the year, typical five-year-olds will have developed a strong sense of time and will know when events close to them take place. They will know the days of the week, the months, and the seasons, but will still be learning how to tell time. Some children do not master these concepts until age six.

• Throughout this year, some five-year-olds will begin to 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. The average child understands this "law of conservation" at age six, and other children grasp this idea at age seven.

• During the first half of this year, some children may still be learning how to solve a problem by comparing lengths directly (e.g. placing two sticks side by side to see which is longer). Also, some children may 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. The average child can make such transitive comparisons during the second half of the year, but others will not understand how to do this until age six.

• During the first half of this year, some children may still be learning how to compare the areas of two objects by placing one object on another. Also, some children during play may intuitively compare angles and how much "turn" angles have. The average child does this during the second half of the year, and others at age six.

• During this year, some five-year-olds will measure the length of an object by laying end-to-end an informal and same-size unit of length (e.g., paper clips). The average child measures this way at age six, and others at age seven. During the second half of this year, some children may be able to use a simple ruler to measure units. The average child can use a ruler at age seven, and others at age eight.

• During the first half of this year, some children will measure area by covering an area with informal units (e.g., 1" x 1" squares) and counting the individual squares (not necessarily in an organized way). The average child can do this during the second half of this year, and others will learn this concept at age six.

• During the first half of this year, the average child makes informal comparisons and estimates (e.g., says, "I'm as tall as the yellow bookshelf."). Some children will make such comparisons and estimates at age six. Throughout the year, a small number of five-year-olds will identify common objects to use as referents when estimating standard measures of length (e.g., the top of the door knobs are about a meter from the floor). The average child develops such referents at age seven, and others do this at age eight.

### Patterns, Reasoning, and Algebra

• Throughout the year, some children are just beginning to understand a sequence of events when it is clearly explained (e.g., parent says, "First we plug the drain, then we run the water, and finally we take the bath."). In addition, some children recognize regularities in a variety of contexts (e.g., events, designs, shapes, sets of numbers). The average child easily recognizes these regularities at age six, and others at age eight.

• Throughout this year, some children can 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"). Children show varying levels of progress with this skill through age six. This development is also true for when children imitate pattern sounds and physical movements (e.g., clap, stomp, clap, stomp...).

• Throughout this year, some five-year-olds recognize the growing pattern involved with counting, where "one" is added each time to get to the next number in a basic arithmetic progression. Since this process, in principle, could go on forever, this understanding is the basis of the concept of "infinity." The average child understands these concepts at age six, and others discover them at age eight. During the second half of this year, a few five-year-olds 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.). The average child understands such progressions at age seven, but others not until age eight.

• Throughout this year, some children 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). The average child understands about "even" and "odd" numbers at age six, and others will at age seven. During the second half of this year, a few five-year-olds will discover "rectangular" numbers, or the number of square tiles that can be used to form a rectangle composed of at least two rows. The average child understands such numbers at age eight. Also during the second half of this year, a few five-year-olds will grasp the concept of integers ("positive integers," which indicate credits, positive charges, numbers to the right of "zero" on a number line; and "negative integers," which indicate debits, negative charges, numbers to the left of "zero" on a number line). The average child understands the concept of integers at age eight.

• Throughout this year, a small number of five-year-olds may be able to 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..."). The average child develops this skill at age eight. During the second half of this year, a small number of 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). The average child understands this concept at age eight.

• Throughout this year, a few five-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."). The average child understands these concepts at age eight. Also by this age, some children can summarize these principles using algebraic shorthand (e.g., "n + 0 = n" for additive identity, "n - 0 = 0" for subtractive identity, and "n - n = 0" for subtractive negation). During the second half of this year, a small number of children may also verbally summarize "additive commutativity" (e.g., says, "You can add numbers in any order."). The average child understands this concept at age eight. Finally, a small number of children will be able to verbally summarize the concept of "inverse principle" (e.g., says, "You added and took away the same, so it is the same."). Again, the average child recognizes this concept at age eight.

• Throughout the year, a small number of five-year-olds 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") or artificial ones (e.g., "in-out" machines where a rule can be determined based on the input and output values). The average child understands these concepts at age eight, when he or she may also be able to represent functional relations using the shorthand of algebra.

• Throughout this year, a small number of five-year-olds may start to 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. The average child will develop this thought process at age seven, and others at age eight.

• During the second half of this year, a small number of children may be able to use estimation procedures such as rounding up, rounding to the nearest decade, and so forth. The average child understands how to use these procedures at age seven, and other children will at age eight.

• Throughout this year, some five-year-olds will still be learning how to use deductive reasoning (using what we know to logically reason out a conclusion about what we do not know) to solve everyday problems (e.g., figures out which child is missing by looking at children who are present).

• 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. Some children won't be able to do this until age six. During the first half of this year, some children will be able to sort and classify on the basis of one or more characteristics (e.g., color, size, etc.), and can articulate why items are grouped together. The average child is able to classify this way during the second half of this year, and other children develop these skills at age seven.

• Throughout the year, some children will also still be learning how to reason "transitively" (e.g., if Abby is older than Betsy, and Betsy is older than Charlene, then Abby is also older than Charlene). During the first half of the year, some children will be able to sequence events chronologically. The average child can do such sequencing of events during the second half of this year, but others won't understand how to do this until age seven.

• During the second half of the year, a small number of five-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). The average child understands such concepts at age seven, and others will at age eight. At the same time, a small number of five-year-olds will also be able to use patterns within different rows of data and additive reasoning to logically solve problems (e.g., for the input, "1, 2, 3, 4", the next value in the output, "3, 4, 5, ?" would be "6" since in the first three cases, "2" was added to the input to make the output). The average child understands how to reason through these types of problems at age eight.

• During the second half of this 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." Children will progress in their abilities to make such estimates through age seven.

• During the second half of this 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. Children continue to develop this "sense" through age eight.

• Throughout this year, 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.). Children continue to develop their understanding of this idea through age eight.

• During the second part of this year, some five-year-olds will understand the "other-name-for-a-number" concept (e.g., "12 = 12 + 0, 11 + 1, 10 + 2, 12 - 0, 13 - 1, 14 - 2,..."). Children continue to develop their knowledge about this idea through age eight. At the same time, some five-year-olds understand the "balance beam" analogy of equals, where the fulcrum of a level balance beam visually represents "=". These children can use this analogy to simplify a variety of mathematical expressions. Children continue to develop their understanding of this analogy through age eight.

### Statistics and Probability

• 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, children will be able to collect relevant data for addressing a question or making a decision of personal importance.

• Throughout this year, children will learn to organize and describe data (e.g., by constructing real or picture 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?).

• Children will develop skills to read and interpret real graphs or picture graphs that summarize information needed to address a question, make a prediction, communicate to others, or make a decision of personal importance.

• Children will have some understanding 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").

• 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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