— PBS Parents

Updated on May 5, 2014

**Operations on Numbers**

- During the first half of this year, the average child will be able to nonverbally and mentally determine sums up to "four" and their subtraction counterparts (e.g., "3 + 1," "4 - 1," "2 + 1," "3 - 2").

- Some children will still be learning how to use informal knowledge gained from everyday experiences to nonverbally estimate sums up to "five" (e.g., for "3 + 2," puts out four to six items to estimate the answer) and their subtraction complements (e.g., for "5 - 2," puts out around three items to estimate the answer). In the second half of this year, some children will be able to use informal knowledge to estimate the sums of addition word problems or their subtraction complements up to "ten."

- Throughout this year, some children will be able 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 corresponding differences.

- In the second half of this year, some children can use various addition strategies to mentally determine sums up to "nine."

- In the second 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"...).

- During the first part of this year, some children may still be learning to understand that if you change the size of a part of a collection, then you also change the size of the whole collection. Throughout this year, some children may intuitively recognize that adding to a collection creates a sum greater than the starting amount.

- During the second half of this year, a few children will 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?). A few children will also 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?).

- During the second half of this year, a few four-year-olds 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").

- During the second half of the year, a few 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?).

- During the first half of this year, the average child will trade several small items for a larger one (e.g., trades four small candies for a candy bar). Throughout this year, some children will 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). During the second half of this year, some children can break down a larger unit (especially "10" and "100") into smaller units, and can combine smaller units into a larger unit.

- During the second half of this year, a few four-year-olds will be able to accurately read multidigit numerals up to "19." At the same time, a very few children may even be able to accurately read multidigit numerals up to "99."

- During the second half of this year, a few four-year-olds may be able to write multidigit numerals up to "99" (e.g., writes "twenty-four" as "24" and not "204").

- During the second half of this year, there may be a small number of four-year-olds 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").

- During the second half of this year, a few four-year-olds 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?).

**Geometry and Spatial Sense**

- During the first half of this year, some children may still be learning to match shapes, first with same size and orientation, then with different sizes and orientation (e.g., matches simple shapes in form boards and puzzles, sorts simple shapes in a sorter box, etc.). At the same time, some children may be able to recognize and name some variations of a circle, square, triangle and rectangle. The average child can recognize and name these shapes during the second half of this year.

- During the first half of the year, some four-year-olds can build, copy and informally describe two-dimensional shapes. The average child can do this during the second half of the year. Throughout the year, some children will also be able to copy a shape from memory after seeing a model for several seconds.

- In the first half of this year, some four-year-olds will intuitively recognize congruence by matching shapes with other objects that have the same shape and size. The average child will be able to do this during the second half of this year. During the second half of this year, a few four-year-olds can explicitly define the term "congruent" as two shapes with the same size and shape.

- Throughout the year, children can complete increasingly complex puzzles (e.g., four-piece interlocking to eight- or ten-piece puzzles, to puzzles with smaller and up to 15 pieces) and progress in their abilities to put together and take apart shapes (e.g., understands that a whole object such as a pizza can be separated into parts). Children also build three-dimensional structures using multiple types of items (e.g., a rectangular prism, cube and arches). In the second half of the year, some children may create drawings that involve more than two geometric forms.

- In the first half of the year, some children may still create pictures using one shape, but won't yet use shapes in combination. Others may make a picture by combining shapes. By the second half of this year, the average four-year-old will be able to combine shapes to form a picture. Throughout this year, some children may also be able to cover an outline of a shape with other shapes without leaving gaps, first with trial-and-error, and then with foresight.

- Throughout the year, some four-year-olds can break apart simple two-dimensional shapes that have obvious clues for breaking them apart.

- Throughout the year, some children can find some shapes "hidden" in arrangements in which the shapes overlap each other, but are not embedded inside one another.

- Throughout this year, some four-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").

- During this year, the average child can use a model of a room or simple picture maps to locate where an object is hidden in a real room.

- During the first half of this year, some four-year-olds will be able to orient objects vertically or horizontally. The average child will be able to do this during the second half of the year.

- Throughout this year, some four-year-olds will informally recognize when a two-dimensional shape has been turned, flipped or otherwise moved, and will also move such shapes in this way.

- During the first half of this year, the average child can informally create two-dimensional shapes and three-dimensional buildings that have symmetry.

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