— PBS Parents

Updated on Jul 9, 2010

**Operations on Numbers**

- Throughout this year, some children will still be learning how to nonverbally and mentally determine sums up to "four" and their subtraction counterparts (e.g., "3 + 1," "4 - 1," "2 + 1," "3 - 2"). The average child will be able to nonverbally and mentally determine sums up to "five" (e.g., "2 + 3") and their subtraction counterparts (e.g., "5 - 3"), although some children will understand how to do this later, at age six.

- The average child will be able to use informal knowledge to estimate the sums of addition word problems (e.g., for "3 + 2," puts out four to six items to estimate the answer) or their subtraction complements (e.g., for "5 - 2," puts out around three items to estimate the answer) up to "ten." Other children will develop such estimation skills at age six.

- During the first half of this year, the average child can use concrete counting strategies to solve addition word problems (e.g., for a problem involving three and two more, the child counts out three items, puts out two more items, and then counts all the items to determine the answer) and concrete take away strategies to solve subtraction word problems (e.g., for a problem involving five take away two, counts out five items, removes two, and counts the remaining three items to determine the answer) with sums up to "ten" and corresponding differences. Other children learn how to use counting strategies this way at age six.

- In the first half of this year, some children can use various addition strategies to mentally determine sums up to "nine." The average child, however, understands how to apply such strategies in the second half of the year, and other children acquire these skills at age six.

- In the second half of this year, some five-year-olds can apply more advanced and abstract counting strategies to solve addition word problems with sums to "18" (e.g., solves "3 + 2," by verbally counting, "One, two, three, four is one more, five is two more," perhaps using fingers or other objects to keep track of the "one more," "two more" count). Another advanced strategy that some five-year-olds might use to find a sum is to begin counting from the number being added, rather than starting with "one" (e.g., for solving "3 + 2," starts counting from "three" instead of "one" by saying, "Three, four is one more, five is two more," perhaps by using fingers or other objects to keep track of the "one more," "two more" count.)

- In the first half of this year, some children can use existing knowledge and reasoning strategies to logically determine unknown sums up to "18" and their subtraction counterparts, including the "additive and subtractive identity" rule (e.g., "n + 0 = n" and "n - 0 = n"), the "number-after" rule (e.g., "7 + 1" equals the number eight when we count), and the notion that addition doubles have an even sum or form part of the skip count by two's sequence (e.g., "3 + 3 = 6," "4 + 4 = 8," "5 + 5 = 10"...). The average child can use these reasoning strategies in the second half of this year.

- Throughout this year, some five-year-olds may be able to solve addition and subtraction problems using the idea that near doubles are one more or less than doubles are, or in other words, their sums are in-between doubles and are "odd" (e.g., "7 + 6" is one more than "6 + 6," or "13"). Other strategies that some five-year-olds might use to solve addition and subtraction problems are the "number-before" rule for "n - 1" facts (e.g., "6 - 1 = 5" and "5 - 1 = 4") and the "negation" rule for "n - n" facts (e.g., "6 - 6 = 0" and "5 - 5 = 0"). The average child learns how to apply these strategies at age six, and some not until age seven.

- Finally, in the second half of this year, some five-year-olds may be able to solve addition and subtraction problems by applying the "additive commutativity" rule (e.g., if "5 + 3 = 8" and "5 + 3 = 3 + 5," then "3 + 5 = 8" also). The average child can apply such reasoning strategies at age six, but other children understand such concepts at age seven.

- During the first half of this year, the average child recognizes that adding to a collection creates a sum greater than the starting amount. Some children may not understand this concept until age six. Some children will also see that a part is less than the whole as they solve addition word problems (e.g. Bret had three cookies. His mother gave him some more, and now he has five cookies. How many cookies did Bret's mother give him?). In addition, some children will see that the whole is larger than its composite parts as they solve subtraction word problems (e.g., Chico had five cookies. He ate some, and now he has three left. How many cookies did Chico eat?). The average child will understand these concepts during the second half of this year, but other children will learn them at age six.

- During the first half of this year, some children can use up to ten objects to construct number partners up to "5" (e.g., 5 = "1 + 4," "2 + 3," "3 + 2," "4 + 1"), and doubles partners up to "10" (e.g., "3 + 3 = 6"). The average child understands these concepts during the second half of this year, but some children will learn them at age six. Throughout this year, some five-year-olds may also know number partners up to "10" (e.g., "1 + 9"), especially with "5" as a partner (e.g., "6 = 5 + 1"), and doubles to "20" (e.g., "12 = 6 + 6"). The average child will have this number sense at age six, and other children will develop this knowledge at age seven.

- During the first half of the year, some children will understand the "part-whole" relationship of addition and will be able to informally solve "part-part-whole" word problems that have a missing whole and sums up to "10" (e.g., Deborah had five chocolate chip cookies and three ginger snap cookies. How many cookies did she have altogether?). The average child understands this concept during the second half of this year, but others not until age six. During the second half of this year, a few five-year-olds will also recognize the "additive-commutativity" principle (e.g., "3 + 6 = 6 + 3"), the "addition-subtraction complement" principle (e.g., "5 - 3 = ?" can be thought of as "3 + ? = 5"), and the "inverse" principle (e.g., "5 + 3 - 3 = 5"). The average child recognizes these principles at age six, but others not until age eight.

- During the first half of this year, some children will still be learning how to trade several small items for a larger one (e.g., trades four small candies for a candy bar). Also, the average child can group objects into 5's or 10's, and recognize that the position of a digit in a number affects its value (e.g., recognizes that "23" and "32" are different). Some children will learn these ideas at age six. Finally, some children can break down a larger unit (especially "10" and "100") into smaller units, and can combine smaller units into a larger unit. The average child will be able to do this during the second half of this year, but other children will learn how to do this at age six.

- During the first half of this year, some children will be able to accurately read multidigit numerals up to "19." The average child will be able to do this during the second half of this year, and others at age six. At the same time, some children may even be able to accurately read multidigit numerals up to "99." The average child is able to do this at age six. Finally, during the second half of this year, a few five-year-olds will be able to accurately read multidigit numerals up to "999," but the average child develops this skill at age seven, and some not until age eight.

- Throughout this year, some children may be able to write multidigit numerals up to "99" (e.g., writes "twenty-four" as "24" and not "204"). The average child is able to do this at age six.

- Throughout this year, there may be a few five-year-olds who recognize that "1 ten = 10 ones." The average child understands this concept at age seven, but others not until age eight.

- During the second half of this year, there may be a few children who can meaningfully represent multidigit numerals up to "100" in different forms, such as with numerals and grouping/place-value models (e.g., recognizes that "2" in "27" represents two "tens" and "7" indicates seven "ones"). The average child understands these concepts at age seven, and others will learn them at age eight. During the second half of this year, a few five-year-olds may be able to meaningfully represent multidigit numerals up to "1000" in these different forms. The average child can do this with numbers up to "1000" at age eight.

- Throughout this year, a few five-year-olds will be able to invent mental procedures for adding and subtracting multidigit numbers, views sums and differences as a composite of "tens" and "ones," and creates shortcuts involving "10's" for sums up to "20" (e.g., recognizes that "10 + n" = "n + 'teen'" such as "10 + 7 = 17"; also "10 + 10 = 20" and "20 - 10 = 10"). The average child understands how to do this at age seven, and others at age eight.

- During the second half of this year, a few five-year-olds may be able to invent or accurately apply written addition procedures for problems with two-digit numbers. The average child can do this at age eight.

- During the second half of this year, a few five-year-olds may be able to use grouping/place-value knowledge and a front-end strategy to make reasonable estimates with two-digit numbers (e.g., "51 + 36 + 7" is at least "5 'tens' + 3 'tens', or 80"). The average child can make such estimates at age eight.

- Throughout this year, some children will be able to use informal strategies to solve "divvy-up/fair-sharing" problems where up to "10" items are distributed evenly to two or three people (e.g., if Este and Freeha share fairly the "12" cookies they baked, how many cookies would each get?). The average child can solve such problems at age six, and some children learn how to do this at age seven. Some five-year-olds may also be able to solve "divvy-up/fair-sharing" problems where up to "20" items are divided evenly among three to five people. The average child understands how to do this at age six, and others at age seven.

- In the second half of this year, some children can use informal strategies to solve "measure-out/fair-sharing" problems that divide up to "20" items into shares of two to five items each (e.g., If Este and Freeha baked 12 cookies and put three cookies in a bag, how many bags of cookies can they make?). The average child can solve such problems at age six, and others at age seven.

- In the second half of this year, some children can use informal strategies to solve "divvy-up/fair-sharing" problems with continuous quantities of one to ten wholes and two to five people (e.g., if four friends shared two pizzas fairly among them, how much pizza would each friend get?). The average child can solve such problems at age six, with other children learning how to solve these problems at age eight.

- In the second half of this year, a small number of five-year-olds can verbally label one of two as "half" or "one-half." The average child can apply this label at age seven, while some children learn how to do this at age eight.

**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.

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Reprinted with the permission of PBS. © PBS 2003 - 2008, all rights reserved.

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