84 terms

equal groups

there is some number of groups each containing the same number of items

comparison problems

some amount is some number of times greater than some other amount (e.g., my dog is twice as heavy as your dog) **THESE ARE ͞MEASUREMENT QUANTITIES IN LA STANDARDS

area and array

items are arranged in rows and columns (i.e., a rectangle or square) or you are talking about the area taken up by a shape (also often a rectangle)

Combination Problems

finding the number of ways you can combine different things (e.g., 2 different shirts with 3 different pants)

measurement problems

we know the size of the groups but not the number of groups

partitive problems

we know the number of groups but not the size of groups

disposed

Remainders:

the remainder needs to be disposed

the remainder needs to be disposed

exact

Remainders:

the problem requires an exact quotient in the form of a decimal

the problem requires an exact quotient in the form of a decimal

extra

Remainders:

the problem requires us to have an extra package

the problem requires us to have an extra package

leftover

Remainders:

the remainder itself is the appropriate response

the remainder itself is the appropriate response

understanding and enthusiasm

The challenge is to get all students to learn math with ___________ and ______________

future jobs require more than simple computation

What is one reason that students need to learn math?

NAEP (National Assessment of Education Progress)

tool to measure overall improvement of US students over time

National Council of Teaching Math (nctm)

is the world's largest education organization that emphasizes whats best for learners

focus on importance of integrating assessment with instruction

What is the purpose of standards?

equity, curriculum, teaching, learning, assessment, technology

What are the six principles of math?

learning trajectories

selection of topics from grade level of what came before and what comes next

array model

What is a powerful tool to use when teaching the commutative problem of multiplication

basic facts

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

number sense

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

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

partitioning strategy

strategy of multiplication that has students break numbers in a variety of ways that reflect understanding of place values

compensations strategy

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.

cluster problem

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)

lattice method

method of multiplication with a grid with squares by diagonal lines to organize their thinking along diagonal place value columns

frontal end

Estimation Strategy:

focuses on the leading or left-most digit

focuses on the leading or left-most digit

Rounding

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.

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.

compatible numbers

Estimation Strategy

that are close in value to the actual numbers, and which make it easy to perform mental arithmetic.

that are close in value to the actual numbers, and which make it easy to perform mental arithmetic.

functional thinking

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

represent generalizations

students use words, symbols, tables, graphs and pictures to represent generalizations

justify generalizations

begin with numeric examples and move toward making general arguments

reason with generalizations

students make use of generalizations to solve problems

algebraic thinking

begins in kindergarten as young students represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out, verbal explanation, expressions, or equations

functional thinking

students consider situations that co-vary, such as the relationship between number of t-shirts purchased and the cost of those shirts

algebraic thinking

__________ _____________ 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

(a+b)+c=a+(b+c)

identity property of addition

a+0=a

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

ab=ba

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 __________ _____ _____________

repeating patterns

patterns that have a core that repeats

growing patterns

Patterns that involve progression from step to step

recursive patterns

description that tells how a pattern changes from step to step

covariational thinking

noticing how two quantities vary in relation to each other and being explicit in making connections

correspondence relationships

correlation between two quantities expressed as a function rule

graphs

_________ encourage covariational thinking which can lead to identifying the function

discrete

when isolated or selected values are the only ones appropriate for a context

continuous

if all values along a line or curve are solutions to the function

domain

comprises the possible values for the individual variable

range

corresponding possible values for the dependent variable

rate

type of change associated with how fast something travels

slope

describes the rate of change for linear functions

y=mx+b

What is slope-intercept form?

y coordinate

when a slope is 0, it means that all points have the same ___ ______________

x coordinate

when there is no slope, then all points have the same ___ _____________

true false

_________/___________ questions are a great way to explore the equal sign

operational view

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

independent variable

step number, or the input, or whatever value is being used to find another value

dependent variable

the number of objects needed, the output, or whatever value you get from using the independent variable

experimentation

gathering real data gets students engages

the goal is to determine whether there is a relationship between the IV and the DV

the goal is to determine whether there is a relationship between the IV and the DV

scatter plots

phenomena are observed seem to suggest functional relationship, but they are not necessarily well defined

algorithms

sets of steps that always work to solve the problem

linear functions

growing patterns can develop into ___________ ____________

multiplication

involves counting groups of like size and determining how many in all

joint problems

Change being "added to" the initial.

separate problems

Change is being removed from initial.

part part whole

Either missing the whole or one of the parts must be found

compare problems

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

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?

number combinations

looking for generalizations begins early with decomposition of numbers.