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# Child Development Tracker: Mathematics From Age 5 to 6 (page 4)

— PBS Parents
Updated on Jul 9, 2010

### 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.