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## Mathematics

### Early Math

Use addition to fill up the seats at the Fun Park and set the rides in motion. Fun Park Math

### Early Math

Keep kids entertained at the store while you nurture early math skills. Grocery Store Bingo

Seven-year-olds have strong number sense and estimation skills. Children this age can do simple addition and subtraction and can apply strategies necessary to solve related word problems. They can also effectively work in many ways with three-digit numbers, and have improved abilities for solving arithmetic problems mentally. Seven-year-olds use rulers to measure units. They also understand how to measure angles, and can apply their knowledge of shapes to three-dimensional objects and structures in the environment.

### Numbers

• During the first half of this year, the average seven-year-old will be able to use repeating patterns to count to "100." Others will be able to count to "200," and some may be able to count to "1,000."

• Throughout the year, 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).

• At the same time, some children will still be figuring out how to name the number after a specified number between "29" and "99." During the first half of the year, some children can name the number after a specified number in the hundreds (e.g., "What number comes after 188?"), but the average child learns how to do this during the second half of the year.

• Throughout the year, some children may still be learning how to verbally count backwards from "20."

• Some children may 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, the average child can verbally count by fives to "100" as well as count objects by fives. In addition, the average child can count by twos to "20" as well as count objects by twos. During the second half of the year, some children can also count to "19" using odd numbers. Throughout the year, a few seven-year-olds will be able to verbally count by fours up to "24."

• During the first half of the year, the average child understands terms related to estimation (e.g., "about," "near," "closer to," "between," "a little less than"). At the same time, the average seven-year-old knows 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, some children will still be learning to correctly use formal relational terms (e.g., "greater than," "less than," "equal to").

• Throughout the year, some children will still be learning how to use a mental number line to determine the relative proximity of two-digit numbers (e.g., recognizes that "63" is closer to "77" than to "32"). Also, some seven-year-olds may be able to use a mental number line to determine the relative proximity of three- and four-digit numbers (e.g., "5,000" is closer to "3,000" than "8,000").

• During the first half of the year, some children can recite the ordinal terms (e.g., "first," "second," "third," "fourth") up to "tenth." The average child can do this by the end of the year. Also during the first half of the year, some children can describe the similarities and differences between the ordinal and cardinal (e.g., "one," "two," "three") counting sequence. The average child can do this by the end of the year. Some children will also recognize in the first half of the year that ordinal terms are only meaningful if a point of reference is specified. The average child understands this by the end of the year. During the second half of the year, some seven-year-olds can recite and effectively apply ordinal terms up to "29th."

• Throughout the year, some children will still be learning how to use informal and symbolic representations (e.g., drawings of objects, a tally) to represent the number of items in a collection up to "nine." Some children will also still be gaining an understanding that numbers can be represented on a number line. During the first half of the year, some children will still be learning how to describe the parallels between abbreviated ordinal terms (e.g., "1st," "2nd," "3rd"..."9th") and cardinal terms (e.g., "1," "2," "3"..."9"). The average child can identify written ordinal terms (e.g., "first," "second"..."ninth") with their corresponding verbal words and use them to represent ordinal relationships. At the same time, some children recognize the written terms, "equals," "unequal," "greater than" and "less than," along with their corresponding verbal words and written symbols. The average child will understand these concepts by the end of the year.

### Operations on Numbers

• During the first half of the year, the average seven-year-old can 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." At the same time, average seven-year-olds 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 "18" and their corresponding differences.

• Throughout the year, some children will still be figuring out 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 children might still be learning for how 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.).

• During the first half of the year, the average child can use a "counting down" strategy to solve subtraction word problems (e.g., to solve, "Five take away three," counts, "Five, four is one less, three is two less, two is three less, so two are left."), perhaps using fingers or objects to keep track of "how many less" a number is. Some children will also be able 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."). The average child can count up to solve such problems by the end of the year. Some seven-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.").

• During the first half of this year, the average child can translate addition and subtraction word problems and their solutions into a number sentence and vice versa, thereby making connections between formal addition/subtraction and concrete or informal knowledge.

• Throughout the year, some seven-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 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 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"), the "number-before" rule for "n - 1" facts (e.g., "6 - 1 = 5" and "5 - 1 = 4"), the "additive commutativity" rule (e.g., if "5 + 3 = 8" and "5 + 3 = 3 + 5," then "3 + 5 = 8" also) and the "negation" rule for "n - n" facts (e.g., "6 - 6 = 0" and "5 - 5 = 0").

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

• Throughout the year, some children will also be able to effectively solve "10 - n" subtraction problems, and during the second half of the year, a few can easily solve addition problems with solutions in the "teens" (e.g., "9 + 4," "8 + 7").

• During the first half of the year, some children will still be learning 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"). At the same time, the average child can add number partners involving decades up to "100" (e.g., "50 = 10 + 40, 20 + 30, 40 + 10").

• Throughout the year, some seven-year-olds will still 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"). During the first part of the year, the average child can informally solve subtraction problems that "compare" (e.g., Ann has five pennies and Barb has three. How many more pennies does Ann have?) or "equalize" (e.g., Ann has five pennies and Barb has three. How many more pennies does Barb need to have the same number of pennies as Ann?).

• During the first half of the year, the average seven-year-old can accurately read multi-digit numerals up to "999." At the same time, some children will be able to read numbers up to "5,000." Some children will also be able to accurately write multi-digit numerals up to "999." The average child can do so by the end of the year.

• During the first half of the year, the average child recognizes that "1 'ten' = 10 'ones'." By the end of the year, the average child understands that "1 'hundred' = 10 'tens' or 100 'ones'," and throughout the year, some children will recognize that "1 'thousand' = 10 'hundreds' or 1,000 'ones'."

• During the first half of the year, the average child can 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'). In addition, some children can meaningfully represent multi-digit numerals up to "1,000" in these different forms.

• During the second half of the year, some children recognize the largest and smallest one-digit number, two-digit number and three-digit number. During the first half of the year, the average seven-year-old can invent mental procedures for adding and subtracting multi-digit numbers, view sums and differences as a composite of "tens" and "ones" and create shortcuts involving "tens" for sums up to "20" (e.g., recognizes that "10 + n" = "n + 'teen'" such as "10 + 7 = 17"; also "10 + 10 = 20" and "20 - 10 = 10").

• Some children also understand 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"). The average child recognizes this principle by the end of the year. Other shortcuts involving "tens" that some seven-year-olds may understand throughout the year include adding multiples of ten (e.g., "5 + 20"), adding decades (e.g., "50 + 40 = 5 'tens' + 4 'tens' = 9 'tens'"), subtracting multiples of ten (e.g., "45 - 20"), subtracting single-digit numbers from teen numbers (e.g., "17 - 9," "18 - 5"), adding teen numbers (e.g., "15 + 13") and subtracting two-digit numbers (e.g., "18 - 13," "22 - 15").

• Throughout the year, a few seven-year-olds will be able to invent concrete procedures for adding and subtracting two-digit and three-digit numbers. At the same time, some children can invent or accurately apply written addition procedures for problems with two- and three-digit numbers, and in the second half of the year, a few children may be able to do so with four-digit numbers. During the second half of 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.

• Throughout the year, some children can use grouping/place-value knowledge and a front-end strategy to make reasonable estimates with two-digit numbers (e.g., "51 + 36 + 7" is at least "5 'tens' + 3 'tens', or 80") as well as 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 seven-year-olds will still be learning how 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?) as well as problems where up to "20" items are divided evenly among three to five people. The average child will be able to solve such problems with "100" items (grouped by tens and ones) divided evenly among up to "10" people, and in the second half of the year, a few children may 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 "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?). In the first half of the year, some children can solve such problems with "100" items (grouped by tens and ones) and shares of up to "10" items. The average child can do this by the end of the year.

• Throughout the year, some seven-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?). During the first half of the year, the average child can solve such problems with one to ten wholes and six to ten people.

• At the same time, the average child can verbally label one of two as "half" or "one-half," and by the end of the year, the average child can similarly label one of three as "one-third," one of four as "one-fourth" and one of five as "one-fifth." Some seven-year-olds may also be able to label non-unit fractions (e.g., labels three of eight equal pieces as "three-eighths"). During the first half of the year, some children can compare unit fractions (e.g., knows that "one-half" is larger than "one-third"). Some children throughout the year 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).

• Throughout the year, some seven-year-olds are still learning how to use concrete objects to solve repeated-addition problems, or those involving groups of items (e.g., uses blocks to solve the problem, "If four boxes each have three toys, how many toys are there altogether?"). Other children may be able to mentally solve such problems using strategies like verbal counting (e.g., says, "1, 2, 3 in one box, 4, 5, 6 in two boxes, 7, 8, 9 in three boxes, 10, 11, 12 in four boxes"), addition (e.g., says, "3 and 3 is 6, and 6 and 3 is 9, and 9 and 3 is 12"), skip counting (e.g., says, "3, 6, 9, 12") or some combination of these strategies.

• During the first half of this year, the average child can represent repeated-addition problems symbolically as addition (e.g., writes, "3 + 3 + 3 + 3"). In the second half of this year, some children may also be able to symbolically represent repeated-addition problems as multiplication (e.g., writes, "4 x 3"). Throughout the year, some seven-year-olds can effectively solve problems that multiply numbers by "zero" and "one." During the second half of the year, some children can also solve problems that multiply numbers by "two."

### Geometry and Spatial Sense

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

• During the first half of the year, the average seven-year-old 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 also create shapes from verbal directions.

• Throughout the year, some children will be learning how to define the term "congruent" as two shapes with the same size and shape. In addition, some children during the first half of the year will still be learning to match shapes and parts of shapes to show that they are congruent. By the end of the year, the average seven-year-old can manipulate shapes (e.g., slide, flip, turn, superimpose) to show congruency.

• During the first half of the year, some seven-year-olds are still learning to identify and count the sides of two-dimensional shapes. In addition, the average child will be able to identify and count both sides and angles. By the end of the year, the average seven-year-old can independently identify shapes in terms of their defining attributes (e.g., says, "It has one, two, three sides...it is a triangle."), but won't see relations between or among shapes (e.g., does not recognize a square as a rectangle). A few, 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, the average seven-year-old will be able to informally name, describe, compare and sort solids. In the second half of the year, the average child will also be able to identify and describe the faces of three-dimensional shapes as specific two-dimensional shapes (e.g., a face of a cube is a square).

• Throughout the year, some children will still be learning to combine shapes to create a new shape. During the first half of the year, some will also be able to create new shapes by combining other shapes and substituting a combination of smaller shapes for a larger shape. The average child understands how to do this by the end of the year.

• In the first half of the year, the average seven-year-old understands how to break apart two-dimensional shapes to make new shapes by picturing images suggested by the task or an adult (e.g., break a square into two triangles). At the same time, some children will be able to break apart shapes to form new shapes by using their own imagery. The average child is able to do this by the end of the year. In the first half of the year, the average child can also understand and 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 create tilings (i.e., covering a flat surface with small shapes, allowing no gaps between shapes or overlaps) with combinations of shapes.

• Throughout the year, some seven-year-olds will still be figuring out how to find shapes "hidden" inside of other shapes. In the first half of the year, some children can find more complicated shapes that are "hiding" inside of other shapes. The average child can find such shapes by the end of the year.

• Throughout the year, some seven-year-olds will still be learning how to understand and use words that represent physical relations or positions (e.g., "over," "under," "above," "on," "beside," "next to," "in front," "behind," "in," "inside," "outside," "between," "up," "down," top," "bottom," "front," "back," "near," "far," "left," "right").

• During the first half of the year, the average child can recognize and point out geometric shapes and structures in the environment. At the same time, some children are still figuring out how to give and follow directions for moving in physical space and on a map. Some children may be able to make and follow maps of familiar areas. The average child can do this by the end of the year. During the first half of the year, some children will understand that maps answer questions about direction, distance and location. The average seven-year-old understands these things by the end of the year.

• During the first half of the year, some seven-year-olds understand that objects can be represented from different points of view and can show shapes from different perspectives. The average child makes these connections by the end of the year. In the first half of the year, some seven-year-olds will still be learning how to use coordinates to locate objects or pictures in simple situations. Some children will also be able to use coordinates to locate positions. The average child can do this by the end of the year.

### 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.

• In the first part of this year, some seven-year-olds will still be learning how to 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 will be able to use a simple ruler to measure units. During the first half of this year, some seven-year-olds can compare the effects of measuring length using units of different size and determine the need for using a standard unit of measurement. Some will also be able to measure the perimeter of an object, and can use a single unit over and over to measure something larger (e.g., measures the length of a room with a single meter stick). The average child will understand these concepts of measurement during the second half of this year. In terms of measuring area, some seven-year-olds will begin to understand 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!"). The average child will understand this during the second half of the year.

• During the first half of the year, the average child will be able to identify common objects to use as referents when estimating standard measures of length (e.g., says, "The top of the door knobs are about a meter from the floor."). In the first half of the year some seven-year-olds will also be able to recognize that a unit can be sub-divided into equal size sub-units, and that both types of units can be used to make measurements (e.g., uses units and fractional units such as 1 1/2 inches). The average child will be able to do this by the end of the year. A few seven-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).

### 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). Also, 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, the average seven-year-old recognizes 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 can identify other obvious growing patterns (e.g., 121121112...). The average child recognizes such growing patterns by the end of the year.

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

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

• Throughout this year, some children will be able to translate simple addition and subtraction word problems or real-world situations into number sentences with a self-chosen symbol to represent the unknown (e.g., "5 + ? = 8"). 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").

• Throughout this year, some seven-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."). Some children may also verbally summarize "additive commutativity" (e.g., says, "You can add numbers in any order.") and the concept of "inverse principle" (e.g., says, "You added and took away the same, so it is the same."). Throughout the year, some children may begin to summarize with natural language, and then later with algebraic expressions or equations, real functional relations (e.g., "12 inches equals a foot").

• During the first half of the year, some seven-year-olds may recognize that the act of looking for patterns can be a useful problem-solving method. They may also use a pattern to justify a solution. These children will likely assume, however, that the first pattern identified must be the correct solution. At the same time, there will be some children this age that recognize that finding a pattern does not automatically mean it is the correct solution. They will understand that evidence (e.g., patterns, examples) is needed to support their ideas, and they may use multiple patterns or examples to justify a solution.

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

• During the first half of this year, some children will still be learning to sort and classify on the basis of one or more characteristics (e.g., color, size, etc.), as well as how to articulate why items are grouped together. At the same time, some children will still be learning how to sequence events chronologically.

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

• During the first half of this year, children will be able to use known quantities (mental numerical benchmarks or mental images of "5," "10," or "100") to make reasonable estimates of collections with quantities such as "17," "24," "78," or "125." At the same time, some children use known basic number combinations to make reasonable estimates of the answers to large, difficult computations (e.g., "45 + 37").

• Throughout the year, some children may also learn to use their knowledge of the counting sequence to round numbers to the nearest tens, hundreds or thousands and estimate an answer to a problem.

• 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. Finally, some children will be learning how to distinguish among types of data (e.g., "names" versus "numbers," "measures" versus "number of items").

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

### Probability

• Some children will begin to understand that some events are more likely to occur than others (e.g., snow is more likely in winter than in summer). They will also have some understanding and use the language of probability (e.g., "certain" or "sure," "uncertain" or "unsure," "likely" or "probable," "unlikely" or "improbable," "maybe" or "possible," and "impossible"). In addition, some children will understand that uncertainty and chance play a role in many everyday situations. A few children will recognize that the outcomes of an event are not necessarily equally likely (e.g., the outcome of two "heads" in a coin toss is not as likely as a "heads" and a "tails").

• 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).

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