Math 210

how

how

describes process or steps

why

gives reasons for doing that thing

List Polya's Four Steps, in order. Each step is a 3-word phrase/sentence, except the last, which is only 2 words.

1. Understand the problem

2. devise a plan

3. carry it out

4. look back

What are the deeper meanings behind Polya’s first step: Understand the Problem

context (setting)

terminology, notation

What are the deeper meanings behind Polya’s second step: Devise a plan

need a good toolbox (list of options)

no guarantees that it will work

What are the deeper meanings behind Polya’s third step: Carry it out

actual work happens

need good labels

if explaining don’t make big leaps that are hard to follow

explain source of computation

What are the deeper meanings behind Polya’s fourth step: Look Back

check computation

is answer reasonable

answer what was asked

Look for a pattern

The problem's information, predicted work, or possible answers feature some REPETITION.

Make a table or list

There are several options/lots of information to keep ORGANIZED

Examine a simpler problem

The problem's numbers are TOO BIG or the situation TOO COMPLICATED.

Identify a sub-goal

The problem has some issue or step that must be addressed BEFORE ANYTHING ELSE can be done.

Write an equation

The answer is an UNKNOWN NUMBER AND the problem gives enough information to create a RELATIONSHIP about it.

Draw a diagram or picture

The problem contains information that you need/want to VISUALIZE.

Guess and check

There are a REASONABLE number of possible options AND the problem gives a value or condition to CHECK THEM AGAINST.

Work backward

The problem gives a clear CHAIN OF EVENTS or STORY in which you know about the "end" and need to find out about the "beginning."

Use elimination

The problem or your work involves some possibilities that can be RULED OUT.

Use direct arithmetic

The problem requires STRAIGHT-FORWARD adding, subtract- ing, multiplying, or dividing numbers with NO ADDITIONAL INTERPRETATION.

Break into cases

The problem features TOTALLY SEPARATE QUANTITIES or SITUATIONS, like positive/negative, small/medium/large, having different fixed amounts of something (like 1 dime vs. 2 dimes), etc.

Geometric sequence

a sequence where we always multiplying (or dividing) each term by the same amount to create the next term

Fibonacci-type sequence

sequence where we always add two terms to get the next

Arithmetic Sequence

sequence that always adds (or subtracts) the same amount to change each term into the next

Difference sequence

new sequence that shows how each term of an original sequence changes to make next term

2+3=5 What are the highlighted numbers called in an addition sentence?

Addends

2+3=5 What is the highlighted number called in an addition sentence?

Sum

7-4=3 What is the highlighted number called in an subtraction sentence?

minuend

7-4=3 What is the highlighted number called in an subtraction sentence?

subtrahend

7-4=3 What is the highlighted number called in an subtraction sentence?

difference

7 x 8 = 56 What are the highlighted numbers called in an multiplication sentence?

factors

7 x 8 = 56 What is the highlighted number called in an multiplication sentence?

product

18/6=3 What is the highlighted number called in an division sentence?

dividend

18/6=3 What is the highlighted number called in an division sentence?

divisor

18/6=3 What is the highlighted number called in an division sentence?

quotient

Fact Family

set of related number sentences that all 1. use the name #’s and 2. show addition and subtraction or multiplication and division

(a) changing the order of addends does not change the sum

Commutative Property of Addition

(b) a+b = b+a

Commutative Property of Addition

(a) changing the order of the factors does not change the products

Commutative Property of Multiplication

(b) ab = ba

Commutative Property of Multiplication

(a) Changing the grouping of the addends does not change the sum

Associative Property of Addition

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

Associative Property of Addition

(a) Changing the grouping of the factors does not change the product

Associative Property of Multiplication

(b) (ab)c =a(bc)

Associative Property of Multiplication

(a) Adding zero to any number leaves that number unchanged (We call the 0 the additive identity)

Identity Property of Addition

(b) a+0 = 0+a=a

Identity Property of Addition

(a) Multiplying any number by 1 leaves that number unchanged. (1 is the multiplicative identity)

Identity Property of Multiplication

(b) a x 1 = 1 x a = a

Identity Property of Multiplication

(a) mulitplying any number by 0 gives 0

Zero Property of Multiplication

(b) a x 0 = 0 x a = 0

Zero Property of Multiplication

a(b+c)= ab +ac or (b+c)a = ba+ca

Distributive property of multiplication over addition

a (b-c)= ab - ac or (b-c)a = ba-ca

Distributive Property of Multiplication over subtraction

number

idea of how many

system of writing numerals

hindu arabic

place value

means where a digit occurs in an numeral-offsets how much it is worth

numeral

how we write a number

Digit

0-9

Common Difference (in Arithmetic)

same amount to get to each next term

Common Ratio (in Geometric)

same amount to get to each next term (will look like fraction in division

sequence

an ordered list of items called terms

term

A term is an individual item that occurs within a sequence