copyright 2010 Dr. Carol J.V. Fisher and OpenVES.org
Revision date: April 25, 2010
Completes structure needed for National Core Standards through beginning of Grade 4.
(Additional structure is also included.)

# K12 Math Taxonomy

The content for this mathematics taxonomy derives from several places:

• EXPRESSIONS
• definition of expression
• expression concepts
• expressions can have measurable attributes; the attribute(s) depend on the expression
• measurable attributes can be compared, to see which expression has "more of" the attribute
• measurable attributes allow expressions to be separated into categories
• the Substitution Principle for Expressions: expressions have lots of different "names"; the name used depends on what is being done with the expression
• types of expressions
• numbers
• real numbers
• definition of real numbers
• representations of real numbers
• real number line
• representations of a real number line
• horizontal line
• arrows at both ends
• arrow at right end only
• vertical line
• arrows at both ends
• arrow at top only
• construction of a real number line
• standard construction: choose locations for 0 and 1
• use distance between 0 and 1 as the "unit length"
• use the unit length to determine locations of whole numbers
• uses of real number lines
• distance problems (interval from 0 to 1 represents a unit of length)
• elapsed time problems (interval from 0 to 1 represents a unit of time)
• money problems (interval from 0 to 1 represents a unit of currency)
• fractions
• definition of fraction
• fraction vocabulary
• numerator
• denominator
• representations of fractions
• diagonal slash: e.g., 1/3
• horizontal fraction bar: e.g., $\frac 13$
• fraction concept
• the word "fraction" refers to the name, not the number; e.g., even though $3=\frac 62\,$, we say that $\frac 62$ is a fraction, but $3$ is not a fraction
• types of fractions
• unit fraction
• definition of unit fraction: a fraction of the form $\frac1n$, for $n=2,3,4,...$
• representations of unit fractions
• point on a number line: e.g., $\frac13$ represents the point obtained by decomposing the interval from 0 to 1 into three equal parts and taking the right-hand endpoint of the first part
• arithmetic with unit fractions
• using unit fractions to "build" fractions: e.g., $\frac34 = \frac14+\frac14+\frac14$
• number line representation: e.g., locating $\frac34$ on a number line by marking off three lengths of $\frac 14$ to the right of 0
• related fractions: when one denominator is a factor of the other; e.g., $\frac 25$ and $\frac 3{10}$ are related, since 5 is a factor of 10
• operations with fractions
• addition/subtraction of fractions
• adding/subtracting fractions with the same denominator
• as adding/subtracting unit fractions; e.g., $\frac 23 + \frac 43 = (\frac 13 + \frac 13) + (\frac 13 + \frac 13 + \frac 13 + \frac 13) = \frac 63 = 2$
• multiplication/division of fractions
• equivalence of fractions
• definition of equivalence of fractions: two fractions are equivalent if and only if they correspond to the same point on a number line (i.e., represent hte same number)
• recognize equivalent fractions
• with denominator 2, 3, 4, 6 (e.g., $\frac12 = \frac 24$)
• explain reasoning
• generate equivalent fractions
• with denominator 2, 3, 4, 6 (e.g., $\frac12 = \frac 24$)
• explain reasoning
• express whole numbers as fractions
• express the number 1 as a fraction with a specified denominator: e.g., $1 = \frac 44$
• express a whole number as a fraction with a denominator of 1: e.g., $4 = \frac 41$
• express a whole number as a fraction with a specified denominator: e.g., $4 = 4\cdot\frac 33 = \frac{12}{3}$
• compare/order fractions
• with equal numerators and different denominators: e.g., $\frac 52$ and $\frac 53$
• using the fractions themselves: e.g., $\frac 52 > \frac 53$
• using tape diagrams (measure; report result)
• using number line representations (locate both on number line; report result)
• using area models (make area model; report result)
• with equal denominators and different numerators: e.g., $\frac 25$ and $\frac 35$
• using the fractions themselves: e.g., $\frac 25 < \frac 35$
• using tape diagrams (measure; report result)
• using number line representations (locate both on number line; report result)
• using area models (make area model; report result)
• uses of fractions
• to decompose a whole into equal parts
• describe parts of a whole: e.g., to show $\frac 13$ of a length, decompose the length into 3 equal parts and show one of the parts
• decimals
• base ten
• base ten concepts
• Understand that a digit in one place represents ten times what it represents in the place to its right. For example, 7 in the thousands place represents 10 times as much as 7 in the hundreds place.
• base ten numbers
• zero
• word name: zero
• numeral: 0
• ones
• word names: one, two, three, four, five, six, seven, eight, nine, ten
• numerals: 1, 2, 3, 4, 5, 6, 7, 8, 9
• decades
• decade word names: ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety
• decade symbols: 10, 20, 30, 40, 50, 60, 70, 80, 90
• a "ten" is a bundle of 10 ones
• decade concept: 20 is two tens, 30 is three tens, etc.
• teens
• teen word names: eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen
• teen symbols: 11, 12, 13, 14, 15, 16, 17, 18, 19
• compose teen numbers: e.g., $10+7=17$
• decompose teen numbers: e.g., $17=10+7$
• two-digit numbers
• word names: e.g., 23 is "twenty-three"
• a two-digit number represents tens plus ones; e.g., 23 is 2 tens plus 3 ones
• expanded form: e.g., $23 = 2\cdot 10 + 3\cdot 1$
• Given two different two-digit numbers: if the tens digits are different, then the number with more tens is greater; if the tens digits are the same, then the number with more ones is greater.
• three-digit numbers
• a "hundred" is a bundle of 10 tens
• word names: e.g., 234 is "two hundred thirty-four"
• a three-digit number represents hundreds plus tens plus ones; e.g., 234 is 2 hundreds plus 3 tens plus 4 ones
• expanded form: $234=2\cdot 100 + 3\cdot 10 + 4\cdot 1$
• Compare three-digit numbers by first comparing their hundreds digits. If the hundreds digits are different, then the number with more hundreds is greater. If the hundreds digits are the same, then the number with more tens is greater. If the hundreds digits and the tens digits are the same, then the number with more ones is greater.
• four-digit numbers
• a "thousand" is a bundle of 10 hundreds
• word names: e.g., $2{,}345$ is "two thousand three hundred forty-five"
• a four-digit number represents thousands plus hundreds plus tens plus ones; e.g., $2{,}345 is 2 thousands plus 3 hundreds plus 4 tens plus 5 ones • expanded form:$2345=2\cdot 1000 + 3\cdot 100 + 4\cdot 10 + 5\cdot 1$• Compare four-digit numbers by first comparing their thousands digits. If the thousands digits are different, then the number with more thousands is greater. If the thousands digits are the same, then the number with more hundreds is greater. If thousands/hundreds are the same, then the number with more tens is greater. If thousands/hundreds/tens are the same, then the number with more ones is greater. • some people use a comma between the thousands digit and hundreds digit; e.g.,$2{,}345$• five-digit numbers • "ten-thousand" is a bundle of 10 thousands • word names: e.g.,$23{,}456$is "twenty-three thousand four hundred fifty-six" • a five-digit number represents ten-thousands plus thousands plus hundreds plus tens plus ones; e.g.,$23{,}456$is 2 ten-thousands plus 3 thousands plus 4 hundreds plus 5 tens plus 6 ones • expanded form:$23{,}456=2\cdot 10000 + 3\cdot 1000 + 4\cdot 100 + 5\cdot 10 + 6\cdot 1$• Compare five-digit numbers using prior strategies. • use a comma between the thousands digit and hundreds digit; e.g.,$23{,}456$• six-digit numbers • "hundred-thousand" is a bundle of 10 ten-thousands • word names: e.g.,$234{,}567$is "two hundred thirty-four thousand, five hundred sixty-seven" • a six-digit number represents hundred-thousands plus ten-thousands plus thousands plus hundreds plus tens plus ones; e.g.,$234{,}567$is 2 hundred-thousands plus 3 ten-thousands plus 4 thousands plus 5 hundreds plus 6 tens + 7 ones • expanded form:$234{,}567=2\cdot 100{,}000 + 3\cdot 10{,}000 + 4\cdot 1000 + 5\cdot 100 + 6\cdot 10 + 7\cdot 1$• Compare six-digit numbers using prior strategies. • use a comma between the thousands digit and hundreds digit; e.g.,$234{,}567$• higher place values • million, billion • real number concepts • the Substitution Principle for real numbers: if$a=b$, then$a$and$b$can be substituted, one for the other, in any situation. That is,$a$and$b$are just different names for the same number. • attributes of real numbers • opposite: the opposite of$x$is$-x$• reciprocal (multiplicative inverse): for$x\ne 0$, the reciprocal of$x$is$\frac 1x$; the number$0$does not have a reciprocal • size (distance from zero) • sign • positive • negative • for all real numbers$a$and$b$, either$a=b$or$a > b$or$a < b$• arithmetic with real numbers • addition/subtraction of real numbers • addition/subtraction vocabulary • sum: the result of an addition problem is called a sum; e.g.,$2 + 3$is a sum • difference: the result of a subtraction problem is called a difference; e.g.,$3 - 2$is a difference • addend: in an addition problem, the numbers being added are called the addends; e.g., in the sum$2 + 3$,$2$and$3$are the addends • term: in an addition/subtraction problem, the numbers being added are called the terms; e.g., in the expression$3-2 = 3 + (-2)$, the terms are$3$and$-2$• commutative property of addition:$x+y=y+x$• changing the order of the addends in an addition problem does not change the sum • associative property of addition:$(x+y)+z=x+(y+z)$• changing the grouping of addends in an addition problem does not change the sum • because of the associative property, we can write$x+y+z$without ambiguity • zero is the additive identity:$0+x=x+0=x$• adding zero to any number does not change the number • a sum is unchanged when one addend is increased by 1 and another decreased by 1:$x+y=(x+1)+(y-1)$• subtracting one addend from a sum of two numbers results in the other addend; i.e.,$(a + b) - a = b$• a number, subtracted from itself, gives zero:$x-x=0$a number, when added to its opposite, gives zero:$x + (-x) = 0$• every subtraction problem is an addition problem in disguise:$a - b = a + (-b)$; subtracting a number is the same as adding its opposite • multiplication/division of real numbers • multiplication/division vocabulary • -- product: the result of a multiplication problem is called a product ; e.g.,$2\cdot3$is a product • -- quotient: the result of a division problem is called a quotient; e.g.,$\frac 52$is a quotient • -- factor: in a multiplication problem, the numbers being multiplied are called the factors; e.g., in the product$2\cdot3$,$2$and$3$are the factors • multiplication/division notation • multiplication notation • multiplication symbol:$2\times 3$denotes the product of$2$and$3$• centered dot:$a\cdot b$denotes the product of$a$and$b$; this notation should be used once students get into algebra, so the "times" symbol is not confused with the variable$x$• juxtaposition (putting next to each other):$ab$denotes the product of$a$and$b$• division notation • division symbol:$a\div b$denotes$a$divided by$b$• diagonal slash:$a/b$denotes$a$divided by$b$• horizontal fraction bar:$\frac ab$denotes$a$divided by$b$• commutative property of multiplication:$xy=yx$• changing the order of the factors in a multiplication problem does not change the product • associative property of multiplication:$(xy)z=x(yz)$• changing the grouping of the factors in a multiplication problem does not change the product • because of the associative property, we can write$xyz$without ambiguity • one is the multiplicative identity:$1\cdot x=x\cdot1=x$• multiplying a quantity by a nonzero number, then dividing by the same number, yields the original quantity:$(a\cdot b)/b = a$• when one factor in a product is multiplied by a nonzero number and another factor divided by the same number, the product is unchanged:$ab = (ac)\cdot(\frac bc)$• limit to multiplying/dividing by numbers that result in whole-number quotients • a nonzero number, divided by itself, gives one:$\frac xx=1$a nonzero number, multiplied by its reciprocal, gives one:$x\cdot\frac 1x = 1$• every division problem is a multiplication problem in disguise:$\frac ab = a\cdot \frac 1b$; dividing a number is the same as multiplying by its reciprocal • The expression$\frac ab$represents the number which, when multiplied by$b$, yields$a\,$;$\ \frac ab\cdot b = a\,$. For example,$\frac{35}5 = 7\,$, since$7\cdot 5 = 35\,$. • the distributive property:$a(b+c) = ab + ac$(multiplication distributes over addition) • word problems involving addition/subtraction/multiplication/division of real numbers • types of word problems • one-step • using whole numbers and fractions • using whole numbers and decimals • using whole numbers, fractions, and decimals • two-step • using whole numbers and fractions • using whole numbers and decimals • using whole numbers, fractions, and decimals • word problem concepts • Understand that while quantities in a problem might be described with whole numbers, fractions, or decimals, the operations used to solve the problem depend on the relationships between the quantities regardless of which number representations are involved. • operations with word problems • estimate the answer • estimation strategies • mental computation, rounding numbers to the nearest 10 or 100 • important subsets of the real numbers • counting (natural) numbers • definition of counting numbers • attributes of counting numbers • factors of a counting number • definition of factor: a counting number that goes into$n$evenly is called a factor of$n$. • 1 is a factor of every number, since 1 goes into every number evenly •$n$is a factor of$n$, since$n$goes into itself evenly • every number has itself and 1 as factors • definition of factor pair: a factor pair for$n$is a pair of counting numbers which, when multiplied together, give$n$• find factor pairs for$n$; e.g., the factor pairs of 20 are$\{1,20\}, \{2,10\}, \{4,5\}$•$n \le 100$• prime numbers • definition of prime number: if the only factors of a number are itself and 1, then it is prime. (By definition, the number 1 is NOT a prime number.) • determine if a number is prime • representations of counting numbers • roman numerals • arithmetic with counting numbers • addition/subtraction • addition • mental addition • addition facts • within ten (memorization) • adding to a number • adding 10 to a number • adding 100 to a number • within 10000 • adding 1000 to a number • within 10000 • sums of multiples • sums of multiples of ten (e.g.,$30+80$) • sums of multiples of one hundred (e.g.,$300+800$) • sums of multiples of one thousand (e.g.,$3000+8000$) • mixed sums ten/hundred (e.g.,$30+800$) • mixed sums hundred/thousand (e.g.,$300+8000$) • compose numbers in different ways: e.g.,$1+4 = 2 + 3 = 5$• addition concept as "putting together": i.e., finding the number of objects in a group formed by putting two groups together; addition concept as "adding to": i.e., finding the number of objects in a group formed by adding members to an existing group • represent addition with objects • represent addition with fingers • represent addition with mental images • represent addition with drawings • represent addition with sounds (e.g., claps) • represent addition with acting out situations • represent addition with verbal explanations • represent addition with expressions (e.g.,$1+2$) • represent addition with equations (e.g.,$1 + 2 = 3$) • strategies for addition • counting on: e.g.,$5+2 = 5 + 1 + 1$• making ten: e.g.,$7+6 = 7+3+3=10+3=13$• algorithms for addition • right-to-left • add like units; carry as needed • addition of one-digit numbers • two addends • three addends •$n$addends,$n > 3$• addition of two-digit numbers • two addends • 2-digit + 2-digit • mixed: 1-digit, 2-digit • three addends • 2-digit + 2-digit + 2-digit • mixed: 1-digit, 2-digit •$n$addends,$n > 3$• all 2-digit • mixed: 1-digit, 2-digit • addition of three-digit numbers • two addends • 3-digit + 3-digit • mixed: 1-digit, 2-digit, 3-digit • three addends • 3-digit + 3-digit + 3-digit • mixed: 1-digit, 2-digit, 3-digit •$n$addends,$n > 3$• all 3-digit • mixed: 1-digit, 2-digit, 3-digit • addition of$n$-digit numbers,$n > 3$• two addends •$n$-digit +$n$-digit • mixed: 1-digit, 2-digit, ...,$n$-digit • three addends •$n$-digit +$n$-digit +$n$-digit • mixed: 1-digit, 2-digit, ...,$n$-digit •$m$addends,$m > 3$• all$n$-digit • mixed: 1-digit, 2-digit, ...,$n$-digit • left-to-right • subtraction • mental subtraction • subtraction facts • within ten (memorization) • subtracting from a number • subtracting 10 from a number • subtracting 100 from a number • within 10000 • subtracting 1000 from a number • within 10000 • differences of multiples • differences of multiples of ten (e.g.,$80-30$) • differences of multiples of one hundred (e.g.,$800-300$) • differences of multiples of one thousand (e.g.,$8000-3000$) • mixed differences ten/hundred (e.g.,$800-30$) • mixed differences hundred/thousand (e.g.,$8000-300$) • decompose numbers in different ways: e.g.,$5 = 1+4 = 2+3$• subtraction as "taking apart" or "taking from": i.e., finding the number of objects left when one group is separated from another • represent subtraction with objects • represent subtraction with fingers • represent subtraction with mental images • represent subtraction with drawings • represent subtraction with sounds (e.g., claps) • represent subtraction with acting out situations • represent subtraction with verbal explanations • represent subtraction with expressions (e.g.,$3-1$) • represent subtraction with equations (e.g.,$3 - 1 = 2$) • algorithms for subtraction • right-to-left • subtract like units; borrow as needed • subtraction of one-digit numbers • subtraction of two-digit numbers • 2-digit minus 2-digit • 2-digit minus 1-digit • subtraction of three-digit numbers • 3-digit minus 3-digit • 3-digit minus 2-digit • 3-digit minus 1-digit • subtraction of$n$-digit numbers,$n > 3$•$n$-digit minus$n$-digit •$n$-digit minus$m$-digit,$m < n$• relationship between addition and subtraction; e.g., if$1+4=5$, then both$5-1=4$and$5-4=1$• addition/subtraction word problems [Glossary Table 1, Core National Math Standards] • "add to" word problems • add to/result unknown: Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? $$2+3 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • add to/change unknown: Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? $$2+\text{?} = 5$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • add to/start unknown: Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? $$\text{?} + 3 = 5$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • "take from" word problems • take from/result unknown: Five apples were on the table. I ate two apples. How many apples are on the table now? $$5-2=\text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • take from/change unknown: Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? $$5-\text{?}=3$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • take from/start unknown: Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? $$\text{?}-2=3$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • "put together/take apart" word problems • put together/take apart/total unknown: Three red apples and two green apples are on the table. How many apples are on the table? $$3+2=\text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • put together/take apart/addend unknown: Five apples are on the table. Three are red and the rest are green. How many apples are green? $$3+\text{?}=5,\quad 5-3=\text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • put together/take apart/both addends unknown: Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? $$\displaylines{ 5=0+5,\quad 5=5+0\cr 5=1+4,\quad 5=4+1 }$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • "compare" word problems • compare/difference unknown: • "How many more?" version: Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? $$2 + \text{?}=5,\quad 5-2=\text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • "How many fewer?" version: Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? $$2 + \text{?}=5,\quad 5-2=\text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • compare/bigger unknown: • version with "more": Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? $$2 + 3 = \text{?}, \quad 3 + 2 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • version with "fewer": Lucy has three fewer apples than Julie. Lucy has two apples. How any apples does Julie have? $$2 + 3 = \text{?}, \quad 3 + 2 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • compare/smaller unknown: • version with "more": Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? $$5 - 3 = \text{?}, \quad \text{?} + 3 = 5$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • version with "fewer": Lucy has three fewer apples than Julie. Julie has five apples. How many apples does Lucy have? $$5 - 3 = \text{?}, \quad \text{?} + 3 = 5$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • multiplication/division • multiplication • mental multiplication • multiplication facts (memorization) • one:$1\times 1$,$1\times 2$, ...,$1\times 10$• two:$2\times 1$,$2\times 2$, ...,$2\times 10$• three:$3\times 1$,$3\times 2$, ...,$3\times 10$• four:$4\times 1$,$4\times 2$, ...,$4\times 10$• five:$5\times 1$,$5\times 2$, ...,$5\times 10$• six:$6\times 1$,$6\times 2$, ...,$6\times 10$• seven:$7\times 1$,$7\times 2$, ...,$7\times 10$• eight:$8\times 1$,$8\times 2$, ...,$8\times 10$• nine:$9\times 1$,$9\times 2$, ...,$9\times 10$• multiply by tens • multiply one-digit numbers by 10: e.g.,$7\times 10$• multiply one-digit numbers by one-digit multiples of 10: e.g.,$7\times 20$, ...,$7\times 90$• multiply by hundreds • multiply one-digit numbers by 100: e.g.,$7\times 100$• multiply one-digit numbers by one-digit multiples of 100: e.g.,$7\times 200$, ...,$7\times 900$• multiply by thousands • multiply one-digit numbers by 1000: e.g.,$7\times 1000$• multiply one-digit numbers by one-digit multiples of 1000: e.g.,$7\times 2000$, ...,$7\times 9000$• compose numbers in different ways: e.g.,$2\cdot 6 = 3\cdot 4 = 12$• multiplication concept as "repeated addition": i.e.,$2\times 3 = 3 + 3$(two groups of 3) or$2\times 3 = 2 + 2 + 2$(three groups of 2) • represent multiplication with rectangular arrays: one factor is the number of rows, the other is the number of columns • represent multiplication with verbal explanations (e.g., two groups of three) • represent multiplication with expressions (e.g.,$2\cdot 3$or$2\times 3$) • represent multiplication with equations (e.g.,$2\cdot 3 = 6$) • multiplication concept as area • a rectangular region that is$a$lengths units by$b$length units (where$a$and$b$are counting numbers) and tiled with unit squares illustrates why the rectangle encloses an area of$a\times b$square units • strategies for multiplication • using the distributive law • one-digit number times one-digit number: e.g.,$2\times 7 = 2(6+1) = 12 + 2 = 14$• one-digit number times multi-digit number in expanded form: e.g.,$2\times 37 = 2(30 + 7) = 2\times 30 + 2\times 7 = 60 + 14 = 74$• -- illustrate numerically using equations • -- illustrate using rectangular arrays • -- illustrate using area models • -- illustrate using tape diagrams • algorithms for multiplication • one-digit multiplier • times two-digit; e.g.,$7\times 23$• right-to-left • times three-digit; e.g.,$7\times 234$• right-to-left • times four-digit; e.g.,$7\times 2345$• right-to-left • two-digit multiplier • times two-digit • times three-digit • times four-digit • estimation techniques for multiplication • division • mental division • division facts within ten, giving whole number results (memorization) • divide by one:$1\div 1$,$2\div 1$, ...,$10\div 1$• divide by two:$2\div 2$,$4\div 2$, ...,$10\div 2$• divide by three:$3\div 3$,$6\div 3$, ...,$9\div 3$• divide by four:$4\div 4$,$8\div 4$• divide by five:$5\div 5$,$10\div 5$• divide by six:$6\div 6$• divide by seven:$7\div 7$• divide by eight:$8\div 8$• divide by nine:$9\div 9$• divide by tens • divide one-digit multiples of 10 by 10: e.g.,$70\div 10$• divide by hundreds • divide one-digit multiples of 100 by 100: e.g.,$700\div 100$• divide by thousands • divide one-digit multiples of 1000 by 1000: e.g.,$7000\div 1000$• division as... • algorithms for division • one-digit divisor (with whole number results) • two-digit divided by one-digit; e.g.,$28\div 4$• long division • short division • three-digit divided by one-digit; e.g.,$284\div 4$• long division • short division • four-digit divided by one-digit; e.g.,$2828\div 4$• long division • short division • estimation techniques for division • applications of division • Given two whole numbers, find an equation displaying the largest multiple of one which is less than or equal to the other. For example, given 325 and 7:$325 = 46\times 7 + 3\,$. • relationship between multiplication and division; e.g., if$2\cdot 3=6$, then both$\frac62=3$and$\frac63=2$• multiplication/division word problems [Glossary Table 2, Core National Math Standards] • "equal groups" word problems • equal groups/unknown product: There are 3 bags with 6 plums in each bag. How many plums are there in all? $$3\times 6 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • measurement example: You need 3 lengths of string, each 6 inches long. How much string will you need altogether? • equal groups/group size unknown: If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? $$3\times\text{?}=18\ \text{ and } \frac{18}3 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • measurement example: You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? • equal groups/number of groups unknown: If 18 plums are to be packed 6 to a bag, then how many bags are needed? $$\text{?}\times 6 = 18 \text{ and } \frac{18}6 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • measurement example: You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? • "arrays" word problems • arrays/unknown product: There are 3 rows of apples with 6 apples in each row. How many apples are there? $$3\times 6 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • area example: What is the area of a$3\text{ cm}$by$6\text{ cm}$rectangle? • arrays/group size unknown ("How many in each group?"): If 18 apples are arranged into 3 equal rows, how many apples will be in each row? $$3\times \text{?} = 18 \text{ and } \frac{18}3 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • area example: A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it? • arrays/number of groups unknown ("How many groups?"): If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? $$\text{?}\times 6 = 18 \text{ and } \frac{18}6 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use an equation with a symbol for the unknown to represent the problem • use a data set to supply info for word problem • area example: A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it? • "compare" word problems • compare/unknown product: A blue hat costs \$6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? $$3\times 6 = \text{?}$$
• use numbers within 10
• use numbers within 100
• use an equation with a symbol for the unknown to represent the problem
• use a data set to supply info for word problem
• measurement example: A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?
• compare/item being compared unknown: A red hat costs \$18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? $$3\times\text{?} = 18 \text{ and } \frac{18}3 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • measurement example: A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? • compare/scaling factor unknown: A red hat costs \$18 and a blue hat costs \$6. How many times as much does the red hat cost as the blue hat? $$\text{?}\times 6 = 18 \text{ and } \frac{18}6 = \text{?}$$ • use numbers within 10 • use numbers within 100 • use a data set to supply info for word problem • measurement example: A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? • addition/subtraction/multiplication/division word problems (whole-number quantities and quotients) • one-step word problems • two-step word problems • uses for counting numbers • counting • counting forward • by ones • starting at 1 • within 100 • within 1000 • starting at$n$, for$n = 2,3,4,...$• within 100 • within 1000 • by twos • starting at 2 • within 100 • within 1000 • starting at$n$, for$n = 4,6,8,...$• within 100 • within 1000 • by fives • starting at 5 • within 100 • within 1000 • starting at$n$, for$n = 10,15,20,...$• within 100 • within 1000 • by tens • starting at 10 • within 100 • within 1000 • within 10000 • starting at$n$, for$n = 20,30,40,...$• within 100 • within 1000 • by hundreds • starting at 100 • within 1000 • within 10000 • starting at$n$, for$n = 200,300,400,...$• within 1000 • by thousands • starting at 1000 • within 10000 • starting at$n$, for$n = 2000,3000,4000,...$• within 10000 • counting backward • by ones • starting at$n$, for$n = 2,3,4,...$• within 100 • within 1000 • by twos • starting at$n$, for$n = 4,6,8,...$• within 100 • within 1000 • by fives • starting at$n$, for$n = 10,15,20,...$• within 100 • within 1000 • by tens • starting at$n$, for$n = 20,30,40,...$• within 100 • within 1000 • by hundreds • starting at$n$, for$n = 200,300,400,...$• within 1000 • ordering • given a set of counting numbers, put them in increasing order • set presented as pictures • set presented as numerals • set presented as written words • given a set of counting numbers, put them in decreasing order • set presented as pictures • set presented as numerals • set presented as written words • identify positions of objects in sequences • first, second, third, fourth, fifth • sixth, seventh, eight, ninth, tenth • comparing sizes of groups • matching strategy • match pictures, numerals (1,2,3,...,10), and written/spoken words (one,two,three,...,ten) • pictures to pictures (e.g., 3 cats to 3 dogs) • pictures to numerals • pictures to written words • pictures to spoken words • numerals to pictures • numerals to numerals (e.g., different fonts for the numerals) • numerals to written words • numerals to spoken words • written words to pictures • written words to numerals • written words to written words (e.g., different fonts for the written words) • written words to spoken words • spoken words to pictures • spoken words to numerals • spoken words to written words • spoken words to spoken words (e.g., male voice versus female voice) • Given two sets of concrete objects or pictures, compare the number of objects in each using appropriate language • more than, greater than • fewer than, less than • same number of • one more than, one less than • counting strategy • whole numbers • integers • rational numbers • definition of rational numbers • irrational numbers • definition of irrational numbers • special irrational numbers • pi • e • types of sentences involving real numbers • equations • definition of equation • "$a = b$" is read as "$a$is equal to$b$" or "$a$equals$b$" • "$a = b$" is equivalent to "$a$and$b$are at the same position on a number line" (i.e.,$a$and$b$are different names for the same number) • solving equations • the Addition Property of Equality: for all real numbers$a$,$b$, and$c$,$a=b$is equivalent to$a+c=b+c$• you can add/subtract the same number to/from both sides of an equation, and it won't change the truth of the equation • addition and subtraction have an inverse relationship: if$x+y=z$, then both$z-x=y$and$z-y=x$• in a sentence of the form$x+y=z$, when two of the three numbers are known, then the unknown number can be found • the Multiplication Property of Equality: for all real numbers$a$and$b$, and for$c\ne 0$,$a=b$is equivalent to$ac=bc$• you can multiply/divide both sides of an equation by the same nonzero number, and it won't change the truth of the equation • multiplying by zero can change the truth of an equation: "$1=2$" is false, but "$1\cdot0 = 2\cdot 0$" is true • multiplication and division have an inverse relationship: if$x$and$y$are nonzero, and$xy=z$, then both$\frac zx=y$and$\frac zy=x$• in a sentence of the form$xy=z$, when two of the three nonzero numbers are known, then the unknown number can be found • limit to cases where unknown number is a whole number • in a sentence of the form$\frac{x}{y}=z$, when two of the three nonzero numbers are known, then the unknown number can be found • limit to cases where unknown number is a whole number • inequalities • definition of inequality • order • greater than • "$a > b$" is read as "$a$is greater than$b$" • "$a > b$" is equivalent to "$a$lies to the right of$b$on a number line" • less than • "$a < b$" is read as "$a$is less than$b$" • "$a < b$" is equivalent to "$a$lies to the left of$b$on a number line" • greater than or equal to • "$a \ge b$" is read as "$a$is greater than or equal to$b$" • "$a \ge b$" is equivalent to "$a > b$or$a = b$" • less than or equal to • "$a \le b$" is read as "$a$is less than or equal to$b$" • "$a \le b$" is equivalent to "$a < b$or$a = b$" • solving inequalities • the Addition Property for Inequalities: For all real numbers$a$,$b$, and$c$, $$a < b \iff a + c < b + c\ .$$ The inequality symbol can be any of the following:$<$,$\le$,$>$,$\ge$. • translation: You can add/subtract the same number to/from both sides of an inequality, and it won't change the truth of the inequality. • consequences: • If more is subtracted from a number, then the difference is decreased: e.g.,$n - 5 < n - 3$. (Note:$-5 < -3 \iff -5+n < -3+n$) If less is subtracted from a number, then the difference is increased: e.g.,$n - 3 > n - 5$. (Note:$-3 > -5 \iff -3+n > -5+n$) • the Multiplication Property for Inequalities (negative multiplier): For all real numbers$a$and$b$, and for$c < 0$, $$a < b \iff ac > bc\ .$$ The inequality symbol can be any of the following:$<$,$\le$,$>$,$\ge$, with appropriate changes made to the equivalence statement. • translation: If you multiply/divide both sides of an inequality by the same negative number, then the direction of the inequality symbol must be changed in order to preserve the truth of the inequality. • systems • complex numbers • geometric figures • definition of space • definition of geometric figure (subset of space) • geometric figure concepts • different categories of geometric figures (e.g., rhombuses, trapezoids, rectangles, and others) can be united into a larger category (e.g., quadrilaterals) on the basis of shared attributes • a given category of geometric figures (e.g., triangles) can be divided into subcategories defined by special properties (e.g., equilateral and non-equilateral) • attributes of geometric figures • types of attributes • defining attributes • non-defining attributes • operations with attributes • classify attribute(s) as defining or non-defining • spatial reasoning of geometric figures • orientation • location in space • relative position • above, below • beside, next to • in front of, behind • types of geometric figures • zero-dimensional (point) • one-dimensional • definition of one-dimensional geometric figures • measurable attributes of one-dimensional geometric figures • length • definition of length • types of one-dimensional geometric figures • line • line segment • definition of line segment • operations with line segments • measure the length of a line segment • using a ruler • using another object as a length unit • compare, based on length • compare the length of two objects by measuring each with a ruler • compare the length of two objects by using a third object • order, based on length • order three objects by length • curve • two-dimensional • definition of two-dimensional geometric figure • measurable attributes of two-dimensional geometric figures • area (finite or infinite) • types of two-dimensional geometric figures • a plane • definition of plane • operations with planes • tilings ("tile a plane") • two-dimensional geometric figures lying in a plane • concepts for two-dimensional figures lying in a plane • different categories of geometric figures (e.g., rhombuses, trapezoids, rectangles, and others) can be united into a larger category (e.g., quadrilaterals) on the basis of shared attributes • a given category of geometric figures (e.g., triangles) can be divided into subcategories defined by special properties (e.g., equilateral and non-equilateral) • types of two-dimensional geometric figures lying in a plane • circles • definition of circle • attributes of circles • center • radius • diameter • half-circle • quarter-circle • operations with circles • draw a circle, given a center and radius • decompose a circle into two equal parts • describe each part as a "half" • describe each part as a "half-circle" • describe the whole as "two half-circles" • decompose a circle into four equal parts (quarter-circles) • describe each part as a "fourth" or a "quarter" • describe each part as a "quarter-circle" • describe the whole as "four quarter-circles" • decompose a circle into$n$equal parts,$n > 1$• the greater$n$is, the smaller the parts are • polygons • definition of polygon • attributes of polygons • area • perimeter • sides • number of sides • adjacent sides • angles • interior angles • number of angles • adjacent angles • exterior angles • vertex/vertices/corner • concave/convex • regular • interior • boundary • types of polygons • triangles • definition of triangle: 3-sided polygon • types of triangles • equilateral • equiangular • isosceles • scalene • acute • right • obtuse • attributes of triangles • operations with triangles • decide if two triangles are similar • decide if two triangles are congruent • quadrilaterals (4-sided polygon) • attributes of quadrilaterals • opposite sides • number of parallel sides • types of quadrilaterals • rectangle • definition of rectangle • operations with rectangles • decompose a rectangle into parts • decompose a rectangle into two equal parts • describe each part as a "half" • describe each part as a "half of the rectangle" • describe the whole as "two halves" • decompose a rectangle into four equal parts • describe each part as a "fourth" or a "quarter" • describe each part as a "fourth of the rectangle" or a "quarter of the rectangle" • describe the whole as "four fourths" or "four quarters" • decompose a rectangle into$n$equal parts,$n > 1$• the greater$n$is, the smaller the parts are • parts have the same area, regardless of their shape (e.g., dividing horizontally versus vertically) • into rows • determine the number of squares by multiplication; e.g., 3 squares in each row, 4 rows, hence$3\times 4 = 12$squares • determine the number of squares by skip-counting; e.g., 3 squares in each row, 4 rows, so skip-count: 3, 6, 9, 12 • into columns • determine the number of squares by multiplication; e.g., 3 squares in each column, 4 columns, hence$3\times 4 = 12$squares • determine the number of squares by skip-counting; e.g., 3 squares in each column, 4 columns, so skip-count: 3, 6, 9, 12 • tile a rectangle with squares • in rows (perhaps alternate color, to distinguish rows) • in columns (perhaps alternate color, to distinguish columns) • operations involving area/perimeter of rectangles • exhibit rectangles with the same perimeter and different area • exhibit rectangles with the same area and different perimeter • types of rectangles • square • rhombus • parallelogram • kite • trapezoid •$n$-gons ($n > 4$) • pentagons • hexagons • heptagons • octagons • nonagons • decagons • operations with polygons • name polygons • from a word description (e.g., "a 3-sided polygon") • from pictures • recognize • members of a given class • triangles: equilateral, equiangular, isosceles, right, acute, obtuse, scalene • quadrilaterals: rectangle, trapezoid, square, rhombus, parallelogram, kite, other (e.g., "Circle all the quadrilaterals") • give example(s) of (using a sketch): • polygons with specific attributes: • one attribute (e.g., a triangle) • two attributes (e.g., an isosceles triangle) • three attributes (e.g., a quadrilateral with a pair of parallel sides and a 90-degree angle) • operations involving perimeter • find the perimeter • add side lengths • if equal side lengths: find the length of one side, multiply by the number of sides • known perimeter, unknown side length • known perimeter, all sides known except one; determine unknown side • represent problem with an equation involving a letter for the unknown quantity • known perimeter, known number of sides, find length of side • represent problem with an equation involving a letter for the unknown quantity • operations with two-dimensional geometric figures lying in a plane • -- compose shapes to create a unit • decompose shapes into smaller pieces • understand that shapes can be decomposed into parts with equal areas; the area of each part is a unit fraction of the whole • three-dimensional • definition of three-dimensional geometric figure • measurable attributes of three-dimensional geometric figures • volume • types of three-dimensional geometric figures • sphere • prisms • definition of prism • types of prisms • rectangular • right rectangular prism • cube (right-rectangular prism, base is a square) • cylinder (base is a circle) • attributes of prisms • base • base shape • base area • base perimeter • height • right (bases directly above each other) • cones • definition of cone • types of cones • pyramid (base is a polygon) • circular cone (base is a circle) • right circular cone • attributes of cones • base • base shape • base area • base perimeter • height • right (base has a "center"; point above center of base) • operations with three-dimensional geometric figures • compose objects to create a unit • decompose objects into smaller pieces • recognize • objects as resembling: • spheres • right circular cylinders • right rectangular prisms ("boxes") • time • definition of time • representations of time • digital • operations with digital time • telling time • find time intervals • analog • operations with analog time • telling time • tell time in hours • tell time in half-hours • tell time in quarter-hours • find time intervals • between hours in a day (e.g., how many hours between 8AM and 9PM) • attributes of time • AM • PM • money • types of money • dollar (dollars) • unit abbreviation: \$
• alternate name: dollar bill (dollar bills)
• conversion information
• to smaller units
• dollar to quarter
• 1 dollar is 4 quarters
• dollar to dime
• 1 dollar is 10 dimes
• dollar to nickel
• 1 dollar is 20 nickels
• dollar to penny
• 1 dollar is 100 pennies
• quarter (quarters)
• conversion information
• to smaller units
• quarter to nickel
• 1 quarter is 5 nickels
• quarter to penny
• 1 quarter is 25 pennies
• to bigger units
• 1 quarter is one-fourth of a dollar
• dime (dimes)
• conversion information
• to smaller units
• dime to nickel
• 1 dime is 2 nickels
• dime to penny
• 1 dime is 10 pennies
• to bigger units
• 1 dime is one-tenth of a dollar
• nickel (nickels)
• conversion information
• to smaller units
• nickel to penny
• 1 nickel is 5 pennies
• to bigger units
• 1 nickel is one-twentieth of a dollar
• 1 nickel is one-fifth of a quarter
• 1 nickel is one-half of a dime
• penny (pennies)
• conversion information
• to bigger units
• 1 penny is one-hundredth of a dollar
• 1 penny is one-twenty-fifth of a quarter
• 1 penny is one-tenth of a dime
• 1 penny is one-fifth of a nickel
• arithmetic with money
• word problems
• single-unit word problems (e.g., dollars only)
• two-unit word problems (e.g., dollars and quarters)
• sets
• definition of set
• important types of sets
• intervals
• representations of sets
• finite sets
• infinite sets
• attributes of sets
• operations with sets
• data sets
• definition of data set
• types of data sets
• representations of data sets
• dot plots
• picture graphs
• scaled picture graphs
• representations of scaled picture graphs
• single-unit scale; e.g., a picture of a cat represents 1 cat
• multiple-unit scales; e.g., a picture of a cat represents 5 cats
• operations with scaled picture graphs
• "how many more/less" problems
• one-step problems
• two-step problems
• bar graphs
• scaled bar graphs
• representations of scaled bar graphs
• single-unit scale; e.g., a square represents 1 cat
• multiple-unit scales; e.g., a square represents 5 cats
• operations with scaled bar graphs
• "how many more/less" problems
• one-step problems
• two-step problems
• attributes of data sets
• data points
• number of data points
• number of data points in a given category
• categories
• measurable attributes of categories
• number of categories
• names of categories
• operations with categories
• compare categories
• compare number of data points in categories
• how many more/less in one category than another
• operations with data sets
• units
• attributes of units
• system of measurement (e.g., metric/English)
• what it measures (e.g., length, time)
• representations of units
• unit names
• unit abbreviations
• types of units
• length
• tools for measuring length
• rulers, yardsticks, measuring tapes
• to measure an object's length: find out how many standard length units span the object with no gaps or overlaps, when the 0 mark of the tool is aligned with an end of the object
• operations with length
• comparing lengths
• Lengths can be compared by placing objects side by side, with one end lined up. The difference in lengths is how far the longer extends beyond the end of the shorter.
• use addition to find a sum of lengths
• use subtraction to find a difference of lengths
• types of length units
• English
• inch
• abbreviation: in
• foot
• abbreviation: ft
• metric
• centimeter
• abbreviation: cm
• meter
• abbreviation: m
• area
• ways to measure area
• unit square: a square with side length 1 unit (e.g., 1 cm by 1 cm); this has "one square unit" of area
• to measure the area of a plane figure using unit squares: cover the plane figure with unit squares or fractions of unit squares; count the number of unit squares used
• use only unit squares; e.g., 2 unit squares
• use unit squares AND fractions of unit squares; e.g., 2.5 unit squares
• decompose into known areas: decompose a plane figure into pieces, each of which has known area (e.g., triangles, circles, rectangles)
• operations with area
• compare areas
• compare areas by counting square units
• use addition to find a sum of areas
• use subtraction to find a difference of areas
• types of area units
• English
• square inch
• abbreviation: ${\text{in}}^2$
• square foot
• abbreviation: ${\text{ft}}^2$
• metric
• square centimeter
• abbreviation: ${\text{cm}}^2$
• square meter
• abbreviation: ${\text{m}}^2$
• improvised units (e.g., ${\text{blah}}^2$)
• mass/weight
• English
• metric
• time
• English
• metric
• volume
• English
• metric
• unit concepts
• when measuring an object with an appropriate unit, if a smaller unit is used, then more copies of that unit are needed than would be necessary if a larger unit were used
• units can be decomposed into smaller units (e.g., 1 foot is 12 inches)
• arithmetic with units
• functions
• matrices
• vectors
• operations with expressions
• compare two expressions, based on measurable attributes
• order two or more expressions, based on measurable attributes
• operation vocabulary
• algorithm: predefined steps that give the correct result in every case
• strategy: a purposeful manipulation that may be chosen for a specific problem, may not have a fixed order, and may be aimed at converting one problem into another

• MATHEMATICAL SENTENCES
• definition of mathematical sentence
• operations on mathematical sentences
• negation
• the mathematical word "and"
• the mathematical word "or"
• equivalence of sentences