selection of topics from grade level of what came before and what comes next
What is a powerful tool to use when teaching the commutative problem of multiplication
number combinations where both addends or both factors are less than 10
flexibility, accuracy, efficiency, and appropriately solved
What are the four factors of fluency
research shows that early ___________ _____________ predicts school success more than verbal, spatial, or memory skills
do not assess fluency negatively affect students number sense and recall not needed/ waste of time
Why are time tests ineffective?
partitive and measurement
What are two concepts of division
complete number strategy
strategy of multiplication that encourages students to recognize if they add 2 numbers the next 2 numbers will have the same sum
strategy of multiplication that has students break numbers in a variety of ways that reflect understanding of place values
strategy of multiplication that is the process of reformulating an addition, subtraction, multiplication, or division problem to one that can be computed more easily mentally.
strategy of multiplication that encourages students to use facts and combinations that they already know in order to figure out more complex problems (allow students to brainstorm)
method of multiplication with a grid with squares by diagonal lines to organize their thinking along diagonal place value columns
Estimation Strategy: focuses on the leading or left-most digit
Estimation Strategy: changing numbers in the problem to others that are easier to compute mentally; can be used when an exact number isn't needed, and an approximate answer will do.
Estimation Strategy that are close in value to the actual numbers, and which make it easy to perform mental arithmetic.
involves reasoning about and expressing how two quantities vary in relation to each other case
generalize math relationships and structure
students notice relationships and structure in arithmetic operations, expressions, equations, or function data that can be generalized beyond the given
students use words, symbols, tables, graphs and pictures to represent generalizations
begin with numeric examples and move toward making general arguments
reason with generalizations
students make use of generalizations to solve problems
begins in kindergarten as young students represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out, verbal explanation, expressions, or equations
students consider situations that co-vary, such as the relationship between number of t-shirts purchased and the cost of those shirts
__________ _____________ is used to generalize arithmetic, to notice patterns that hold true in algebra and with properties and to reason quantitatively about such things as whether expressions are equal or not.
properties of operations
focus on helping students recognize and understand important generalizations and also use them to generate equivalent expressions to solve problems efficiently and flexibly
commutative property of addition
a+b = b+a
associative property of addition
identity property of addition
inverse property of addition
a + (-a) = 0
inverse of addition and subtraction
if a+b=c, then c-b=a, and c-a=b
commutative property of multiplication
associative property of multiplication
(ab)c = a(bc)
identity property of multiplication
a x 1 = a
inverse property of multiplication
a x 1/a = 1
multiplication and division inverse
if axb=c, then c/b=a and c/a=b
distributive property of multiplication
a(b + c) = ab + ac
heart of math
noticing generalizations of properties and attempting to prove they are true is a significant form of algebraic reasoning and is the __________ _____ _____________
patterns that have a core that repeats
Patterns that involve progression from step to step
description that tells how a pattern changes from step to step
noticing how two quantities vary in relation to each other and being explicit in making connections
correlation between two quantities expressed as a function rule
_________ encourage covariational thinking which can lead to identifying the function
when isolated or selected values are the only ones appropriate for a context
if all values along a line or curve are solutions to the function
comprises the possible values for the individual variable
corresponding possible values for the dependent variable
type of change associated with how fast something travels
describes the rate of change for linear functions
What is slope-intercept form?
when a slope is 0, it means that all points have the same ___ ______________
when there is no slope, then all points have the same ___ _____________
_________/___________ questions are a great way to explore the equal sign
equal sign means to do something
relational computational view
students understand the equal sign represents a relationship between answers and two calculations, but they see computation to determine if both sides are equal
relational structural view
student uses numeric relationship between two sides of equal signs rather than computing the amounts
step number, or the input, or whatever value is being used to find another value
the number of objects needed, the output, or whatever value you get from using the independent variable
gathering real data gets students engages the goal is to determine whether there is a relationship between the IV and the DV
phenomena are observed seem to suggest functional relationship, but they are not necessarily well defined
sets of steps that always work to solve the problem
growing patterns can develop into ___________ ____________
involves counting groups of like size and determining how many in all
Change being "added to" the initial.
Change is being removed from initial.
part part whole
Either missing the whole or one of the parts must be found
There are three ways to present compare problems, corresponding to which quantity is unknown (smaller, larger, or difference).
1. focus on problem and meaning of answer 2.identify the important numbers as well as the unimportant ones 3. this leads to a rough estimate and unit of the answer
What are the three steps to analyzing a contextual problem?
looking for generalizations begins early with decomposition of numbers.