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## Mathematics

### Cyberchase

Play with geometric shapes to create a world of animal and object puzzles. Tangrams

How many ways can your child make "1/2" in this interactive game? Fraction Action!

Weigh the poddles and make the scale balance in this first "algebra" game. Weigh the Poddles

Get tips for weaving math into your everyday family life. Everyday Math

### DragonFly TV

Your child will make quick mathematical observations and decisions to keep the Weebits prospering! Weebit World

### Fetch!

Turn a small picture into a gigantic one. Seeing is believing! Size It Up!

### Struggling to Learn

Understanding and supporting children with math disabilities. Signs of Struggle

Eight-year-olds can count to "1,000" and gauge the relative proximity of three- and four-digit numbers to one another. They are able to apply a host of strategies when solving problems with three-digit numbers or less. They are also building early multiplication skills. Children this age recognize a wide variety of shapes and can readily identify patterns. They can also translate simple word problems into number sentences, and begin to apply more algebraic thinking and logic to solving problems with addition and subtraction.

### Numbers

• During the first half of this year, some eight-year-olds will still be learning how to use repeating patterns to count to "100." Others will still be trying count to "200," but the average child will be able to count to "1,000."

• Some children will still be learning how to 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).

• Throughout the year, some children will still be learning how to name the number after a specified number in the hundreds (e.g., "What number comes after 188?").

• During the first half of the year, some children may still be learning how to verbally count backwards from "20." At the same time, some children may also still be learning how to name the decade after "10" and up to "90" without the preceding decade counting sequence. During the first half of the year, some eight-year-olds may still be learning how to verbally count by fives to "100" as well as count objects by fives. Some children will also still be figuring out how to count by twos to "20" as well as count objects by twos. The average child can count to "19" using odd numbers. Throughout the year, a few eight-year-olds will be able to verbally count by fours up to "24."

• Some children will still be learning terms related to estimation (e.g., "about," "near," "closer to," "between," "a little less than"). During the first half of the year, some eight-year-olds will still be learning how to make a reasonable estimate of the number of items in a collection of up to "1,000" items, and a few can do so with collections of up to "10,000" items.

• During the first half of the year, the average eight-year-old can 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 recite the ordinal terms (e.g., "first," "second," "third," "fourth") up to "tenth." Also, some children will be learning how to describe the similarities and differences between the ordinal and cardinal (e.g., "one," "two," "three") counting sequence. In addition, some children will still be figuring out that ordinal terms are only meaningful if a point of reference is specified. Throughout the year, some eight-year-olds can recite and effectively apply ordinal terms up to "29th."

• During the first half of the year, some eight-year-olds will still be gaining an understanding that numbers can be represented on a number line. At the same time, some children will still be learning to recognize the written terms, "equals," "unequal," "greater than" and "less than," along with their corresponding verbal words and written symbols.

### Operations on Numbers

• During the first half of the year, some eight-year-olds will still be learning 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 "twenty."

• Some children will still be finding out 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 "18" and their corresponding differences.

• During the first half of the year, some children are still figuring out how to 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 the year, some children will be learning how to "count up" to solve difference problems (e.g., to solve, "How much more is five than three?", counts, "Three, four is one more, five is two more, so the answer is two more."). During the first half of the year, some eight-year-olds are able to minimize their effort by flexibly choosing between counting-up and counting-down strategies, regardless of the problem (e.g., to solve, "Five take away three," counts up instead of down because it is easier: "Three, four is one, five is two, so two are left."). The average child can do this by the end of the year.

• During the first half of this year, some eight-year-olds will still be learning how to 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.

• During the first half of the year, some eight-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 "number-before" rule for "n - 1" facts (e.g., "6 - 1 = 5" and "5 - 1 = 4"), 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). At the same time, some children will be learning how to efficiently solve addition problems up to "nine" regardless of the strategy used. They will also be figuring out how to 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 the year, some children will also be learning how to effectively solve "10 - n" subtraction problems. The average child understands such problems by the end of the year. And throughout the year, some eight-year-olds can easily solve addition problems with solutions in the "teens" (e.g., "9 + 4," "8 + 7").

• During the first half of the year, some eight-year-olds will still be learning how to add number partners involving decades up to "100" (e.g., "50 = 10 + 40, 20 + 30, 40 + 10"). Some children will also be learning to 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"). Finally, some eight-year-olds will still be trying to 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?).

• Throughout the year, some eight-year-olds are still learning how to accurately read multi-digit numerals up to "999." By the end of the year, however, the average child can accurately read numbers up to "5,000." Throughout the year, some children will still be learning how to accurately write multi-digit numerals up to "999."

• * Throughout the year, some children will still be figuring out that "1 'ten' = 10 'ones'" and that "1 'hundred' = 10 'tens' or 100 'ones'." During the first half of the year, however, the average child understands that "1 'thousand' = 10 'hundreds' or 1,000 'ones'."

• Some children will still be learning how to meaningfully represent multi-digit 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, however, can meaningfully represent multi-digit numerals up to "1,000" in these different forms. Some children can also recognize the largest and smallest one-digit number, two-digit number and three-digit number. The average child can do this by the end of the year.

• During the first half of the year, some eight-year-olds can invent mental procedures for adding and subtracting multidigit 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").

• Throughout the year, some children will still be gaining an understanding 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 first half of the year, the average child can add multiples of ten (e.g., "5 + 20"), and some children can add decades (e.g., "50 + 40 = 5 'tens' + 4 'tens' = 9 'tens'"). The average child can easily add decades by the end of the year. In the first half of the year, the average eight-year-old can add multiples of ten. The average child can subtract multiples of ten by the end of the year.

• Some children will be able to subtract single-digit numbers from teen numbers (e.g., "17 - 9," "18 - 5") and add teen numbers (e.g., "15 + 13"). The average child can add teen numbers by the end of the year. Some children will also be able to easily subtract two-digit numbers (e.g., "18 - 13," "22 - 15").

• Throughout the year, some eight-year-olds will be able to invent concrete procedures for adding and subtracting two-digit and three-digit numbers up to "1,000." During the first half of the year, the average child can invent or accurately apply written addition procedures for problems with two-digit numbers. By the second half of the year, the average child can do so with three-digit numbers, and throughout the year, a few children will be able to do so with four-digit numbers. Throughout the year, a few children may also be able to invent or accurately apply written subtraction procedures for problems with two- and three-digit numbers.

• During the first half of the year, the average child 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 some eight-year-olds may 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, some eight-year-olds will still be learning how to solve "divvy-up/fair-sharing" problems where "100" items (grouped by tens and ones) are divided evenly among up to "10" people. Throughout the year, some children may also still be able to evenly divide "1,000" items (grouped by hundreds, tens and ones) among up to "20" people.

• Throughout the year, some children may still be learning how to use informal strategies to solve "measure-out/fair-sharing" problems that divide up to "100" items (grouped by tens and ones) into shares of up to "10" items. Some children may also be able to solve such problems with up to "1,000" items (grouped by hundreds, tens and ones) and shares of up to "20" items.

• Throughout the year, some eight-year-olds will still be learning how to 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?), as well as such problems with one to ten wholes and six to ten people.

• Some children will still be figuring out how to 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 first half of the year, the average eight-year-old can label non-unit fractions (e.g., labels three of eight equal pieces as "three-eighths"). Throughout the year, some children will still be learning how to compare unit fractions (e.g., knows that "one-half" is larger than "one-third"). During the first half of the year, the average eight-year-old can also compare non-unit fractions (e.g., knows "two-thirds" is larger than "two-fifths" because the pieces of the former are larger or because it is more than "one-half" and "two-fifths" is not).

• The average child can mentally solve repeated-addition problems, or those involving groups of items, by 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 first half of this year, some children will still be learning how to represent repeated-addition problems symbolically as addition (e.g., writes, "3 + 3 + 3 + 3"). The average child, however, can symbolically represent repeated-addition problems as multiplication (e.g., writes, "4 x 3") during the first half of the year. At the same time, some children can effectively solve problems that multiply numbers by "zero" and "one." The average child can do this by the end of the year. Some eight-year-olds can also solve problems that multiply numbers by "two."

### Geometry and Spatial Sense

• During this year, some eight-year-old will still be learning to recognize and name circles, squares, triangles and rectangles in any size or orientation, including varying shapes for triangles and rectangles. In the first half of the year, the average eight-year-old can recognize and name a variety of other shapes in any orientation (e.g., semi-circles, quadrilaterals, trapezoids, rhombi, hexagons).

• During the first half of the year, some eight-year-olds will still be learning how 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). Throughout the year, some eight-year-olds will just be figuring out how to create shapes from verbal directions.

• During the first half of the year, some eight-year-olds are still learning to manipulate shapes (e.g., slide, flip, turn, superimpose) to show congruency.

• During the first half of this year, some children will still be learning how to identify and count both the sides and angles of two-dimensional shapes. Throughout the year, some children will still be learning to 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 be able to see relations between or among shapes (e.g., does not recognize a square as a rectangle). Some, however, will be able to reason logically using the key attributes of shapes to see relations between shapes that have different appearances (e.g., sees a square as both a "special rectangle" -- a four-sided figure containing a right angle with opposite sides equal and parallel -- and a "special rhombus" -- a parallelogram with all sides equal).

• In the first half of the year, some eight-year-olds will still be learning how to informally name, describe, compare and sort solids. In addition, some children will still be learning 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, some eight-year-olds will still be learning how to create new shapes by combining other shapes and substituting a combination of smaller shapes for a larger shape.

• Throughout the year, some children will still be trying to understand how to break apart two-dimensional shapes to form new shapes by independently picturing such shapes in their minds (e.g., break a square into two triangles). In addition, some children will still be figuring out and learning to predict how shapes will change when they are put together and broken apart in different sequences.

• Throughout the year, some children will still be learning how to symbolically record the pattern of their tilings (i.e., the covering of a flat surface with small shapes, allowing no gaps between shapes or overlaps).

• Throughout the year, some eight-year-olds will still be figuring out how to find complicated shapes that are "hiding" inside of other shapes.

• During the first half of the year, some children are still understanding how to recognize and point out geometric shapes and structures in the environment. Throughout the year, some children may still be learning how to make and follow maps of familiar areas. At the same time, some children will still be gaining an understanding that maps answer questions about direction, distance and location. Throughout the year, some eight-year-olds will still be learning that objects can be represented from different points of view and can show shapes from different perspectives. Also, some children will still be figuring out how to use coordinates to locate positions.

### Measurement

• During the first half of this year, some seven-year-olds will still be learning 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. Similarly, the average child will recognize that the length of something remains the same (is conserved) regardless of appearances, unless more is added or removed. Some seven-year-olds may understand that the same is true for the area of something.

• Not applicable at this age.

• Throughout this year, some eight-year-olds will still be learning how to use a simple ruler to measure units. Also, some children will still be figuring out the need for using a standard unit of measurement. In addition, some will be learning to measure the perimeter of an object, and how to 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 eight-year-olds will still be understanding area as an array with both rows and columns (e.g., covers the area of an object with 1" x 1" squares and says, "Four rows with three in each row... so three, six, nine, twelve!").

• Some eight-year-olds will still be learning how 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 children will also be figuring out 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). A few eight-year-olds may recognize that, in theory, the process of sub-dividing a unit into smaller units could go on forever. A few may also notice limitations of measurements (e.g., measurement tools are not perfect, human efforts to read measurement tools are imperfect). Finally, a few children may recognize the parallel between measurement and rounding.

### Patterns, Reasoning, and Algebra

• Throughout the year some children are still learning to recognize regularities in a variety of contexts (e.g., events, designs, shapes, sets of numbers). During the first half of the year, some children are still learning to recognize the growing pattern involved with counting, where "one" is added each time to get to the next number in a basic arithmetic progression. During the first half of the year, some children are still learning to 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.). At the same time, some children are still learning to identify other obvious growing patterns (e.g., 121121112...).

• 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). The average child understand the concept of integers by the end of the year.

• Some children will discover odd-even rules for addition and subtraction (e.g., the sum of two odd numbers is an even number).

• Throughout the year, some children will still be learning how 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..."). During the first half of the year, the average eight-year-old can 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 can 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"). Also, a few children will 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").

• During the first half of the year, some eight-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 can do this by the end of this year.

• During the first half of the year, some children may 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."). The average child understands these concepts by the end of the year.

• During the first half of 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"). The average child understands these concepts by the end of the year, when he or she may also be able to represent functional relations using the shorthand of algebra.

• Throughout this year, some eight-year-olds may still be learning 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. 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 eight-year-olds will still be learning to use estimation procedures such as rounding up, rounding to the nearest decade, and so forth. During the first half of the year, the average child can use the renaming procedure (i.e., "carrying over" or "borrowing" from neighboring numbers) for multi-digit addition and subtraction with whole numbers.

• Throughout the year, some eight-year-olds will still be learning 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). During the first half of the year, the average child can 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).

• Some eight-year-olds will be able to expand their between-row problem-solving abilities to include simple multiplicative reasoning (e.g., for the input, "1, 2, 3, 4", the next value of the output, "3, 5, 7, ?" would be "9" since in the first three cases, the input was multiplied by "2" and then "1" was added to get the output). The average child can use such multiplicative reasoning by the end of the year. Finally, some children will be able to determine whether "a" or "b" is larger in each of the following equations: "a + 2 = b", "a - 2 = b", "a = b + 2", "a = b - 2", "a + 3 - 2 = b", "a + 3 - 3 = b".

• 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, children will be able to collect relevant data for addressing a question or making a decision of personal importance. In addition, some children will be learning how to distinguish among types of data (e.g., "names" versus "numbers," "measures" versus "number of items"). Finally, some eight-year-olds may recognize that in some situations, it isn't practical to check the opinion of everyone in a population, and that a representative sample might work better. By the end of the year, some children can use a simple sampling technique to answer a simple question and solve simple problems.

• Children continue to learn how to organize, describe and interpret data (e.g., by constructing picture, bar, or line 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 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. A few children will recognize that the outcomes of an event are not necessarily equally likely (e.g., when rolling two dice, a sum of seven is more likely than a sum of four).

• 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. In addition, some may begin to use experiences or the results of a simple experiment to make predictions about the likelihood of an event (e.g., when randomly choosing among two blue marbles and a red marble, sees that drawing a blue marble is more likely). Eight-year-olds also start to use qualitative reasoning to determine how likely something is to happen (e.g., sees that it is more likely to get at least one "heads" if three coins are flipped instead of one since there are more opportunities to get what you want). Finally, children this age are also capable of systematically listing all the possible outcomes of a simple event, and then using this information to logically make predictions.

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