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 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
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)
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
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
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
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
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
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., $8000-3000$)
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? $$5 - 3 = \text{?}, \quad \text{?} + 3 = 5$$
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")
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