Mathematics for Elementary Teachers

George Polya (1887-1985)

Mathematician and Teacher of 20th Century

"..'solving a problem means finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable'.." (Billstein 2)

Major contribution was his work in Problem Solving

Steps

1. Identify the problem

Understand the problem

2. Devise a plan

Guess and check

Think of solutions and plug into the problem to check answer

Solve a simpler problem

Use an easier problem based on your current problem

Work backwards

Use a model

Create charts, lists, and use objects

3. Carry out the plan

Be persistant with your current plan

Revise the plan if necessary

If the solution is not visible, rethink the plan

4. Look back (reflect)

Does the answer make sense?

Subtopic

Does it need to be revised?

Is there an easier way to solve it?

Universal Set

Collection of all objects in problem

Subset

Set of objects in which all objects in set are contained within another set

Proper Set

A subset of another set in which there is at least one element of the other that is not in the subset

Equal Set

Identical elements found in both sets

ex. {1, 2, 3} = {2, 3, 1}

Although different order both sets contain same elemets

Equivalent Set

Same number of elements but do not contain the same elements

ex. {1, 2, 3} ~ {a, b, c}

Both sets contain 3 elements

Complement of a set

Elemets not contained within specific set

Empty Set

Nothing contained in the set

symbol: { }

Addition

2 Base Models/ Context for Addition

Addition- Discrete

Characterized by combining two sets of discrete objects

ex. Bob has 3 apples and 2 oranges. How many pieces of fruit does he have?

Addition- Continuous

Characterized by the combining two continuous properties

Ex. Paul goes to the gym spends 10 min jogging and 20 min walking. How long are his workouts?

Properties of Addition
(W= set of all whole numbers)

Closure Property of Addition

If a is an element of W and b is an element of W is (a +b) an element of W

Sum of two whole numbers is another whole number.

Associative Property of Addition

If a is an element of W, b is an element of W and c is an element of W then

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

The order of the addition will not matter because both sets will bring the same result

Identity Property of Addition

If a is an element of W then
a+ 0 = a = 0+ a

Two sets , one contains a of jects and the other contains zero objects then when combined the new set will contain a objects

Communative Property of Addition

If a is an element of W then a +b = a+b

2 sets, one contains a objects and other contains b objects the two sets will contain the same a+b number of objects

Methods

Low Stress

Pyramid style. The sum of the first two numbers will begin the pyramid. The following number is added to the ones place but not to the tenths.

Left to Right

Start with the begining place value and work backwards. Make sure to place zeros as "place holders"

Any Column First

Addition is done using place values. Zeros are inserted after the number as "place holders"

Scratch

Addition done as regular but the "carry over number" is added to the following place value number

Subtraction

Context

Take Away:
starting with an initial quantity and removing (take away) a specific amount

Ex. Vince came to class with 5 pieces of candy. He gave 3 pieces away. How many pieces of candy does Vince have?

Comparison:
comparing relative sizes of two quantities to determine how much smaller one of the quantities is compared to the other quantity

Ex. Emily read 5 books. Jim read 3 books. How many more books did Emily read than Jim?

Missing-addend:
the need to determine what quantitiy must be added to a specified amount to reach some target quantity

Ex. Kelsie has 6 blocks. She wants 10 blocks. How many blocks does Kelsie need?

Properties of Subtraction

Closure Property of Subtraction

If a is an element of W and b is an element of W can we say (a-b) an element of W? NO

Ex. If a=7 and b=5
a-b= -2
-2 is not a whole number

Communative Property of Subtraction

If a is an element of W and c is an element of W can we say

a-b= b-a? NO

Ex. If a=5 and b=7
a-b=-2 and b-a=2
-2 is not equal to 2

Associative Property of Subtraction

If a is an element of W, b is an element of W and c is an element of W can we say

(a-b) -c = a- (b-c)? NO

Ex. If a=5 and b=4 and c=3
(a-b)-c=-2 and a-(b-c)=4
-2 is not equal to 4

Identity Property of Subtraction

If a is an element of W can we say a-0 = a and 0-a=a? NO

Ex. If a=4 then
a-0= 4 but 0-a= -4

Methods

European Algorithm

Adding method in order to subtract from numbers

Four Fact Families

relates the sum of 3 &4
3+4=7
4+3=7
7-4=3
7-3=4

Even Number

Ex. 12 even
2*6

Odd Number

Ex. 9 odd
(2*4)+1

Prime Number

Ex. 23 is prime
can only be divided by 1 and 23

Composite Number

Ex. 8 is a composite
composed of 1,2,4, and 8

Divisibility

2 distinct wats
a/b: a divided by b
a|b: a divides ("divides into") b

4 statements

a is a factor of b
b is a multiple of a
a is a divisor of b
b is divisible by a

Divisibility Tests
2, 3, 4, 5, 6, 8, 9, 10

GCF (greatest common factor)

Ex. GCF (4,6): 2
Factors of 4: 1, 2, 4
Factors of 6: 1, 2, 3, 6

LCM (least common multiple)

Ex. LCM (4, 6): 12
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24, 30...

Composed of elements (objects found within)

Types of Sequences

Arithmetic Sequences

Sequences or numbers with a common difference

ex. 2, 4, 6, 8, 10

Subtopic

common difference of 2, 4, 6, 8 is 2

Adding 2 to each time will create the next number

Geometric Sequences

Sequence of numbers with a common ratio

ex. 8, -4, 2,-1,..

common ratio -1/2

Recurrence relationship Sequences

Defines a sequence in which the current term is dependent on previous term(s)

ex. 6, 13, 27, 55..

Difference: multiply by 2 then add 1

Product of 1st term 13

13 multiplied by 2 equals 26 then add 1

Tally System

Using sticks and grouping in fives

Egyptians

Used symbols and position of the symbol determined the number

Mayan/ Babylonians

Mayans used base 20

In base 20, 0-19 units are grouped and when 20 is reached a new placement value is created

Babylonians used base 60

In base 60, 0-59 units are grouped and when 60 is reached a new placement value is created

Roman

Letter represent a number and depending on the position a number is created

Hindu-Arabic

Used base 10

0 -9 is used when 10 is reached a new palcement value is formed

Multiplication

2 Base Models/ Context for Multiplication

Repeated Addition- Discrete:

Characterized by repeatedly adding a quantity of discrete objects a specified number of times

Ex. Sammy's brother and sister both gave him 2 cars for his birthday. How many cars did he get? 2+2... 2*2

Repeated Addition- Continuous:

Characterized by repeatedly adding continuous quantities a specified number of times

Ex. During the week, Monday thru Friday Sandra practice the piano 30 minutes a day. How long did she practice this week
30+30+30+30+30... 30*5

Properties of Multiplication

Closure Property:
if a is an element of a whole number and b is an element of a whole number then (a*b) is an element of a whole number

Ex. a=5 and b= 7, 5*7 is 35, 35 is an element of a whole number

Communative Property:
If a is an element of a whole number, b is an element of a whole number, then a*b=b*a
a= 5, b=7
a*b is 5*7 and b*a is 7*5 not the same but produce the same value

Methods

Area (Arran) Model:
Characterized as a product of two numbers representing the sides of a rectangular region such that the product produces unit size squares

Ex. Tom is tiling his bathroom that measures 10ft by 15ft. To purschase the tile he needs to find area of bathroom floor. What is the are of the bathroom floor? 10*15

Cartesian Product:
Characterized by finding all the possible pairings between two or more sets of objects

Ex. Sara has 4 Jackets and 3 scarves. How many jacet and scarve outfits can she wear?
J1-s1, s2, s3, (J1,s1)
J2- s1, s2, s3 (J2, s2)
J3- s1, s2, s3
J4- s1, s2, s3

Division

Context

Partition (sharing):
Characterized by distributing a specified number of partisions( groups) and determining the size ( amount) in each partition (group)

Ex. Billy comes to school with ten pencils. He Decides to share the pencils with 5 friends. How many pencils does each friend get?

KNOW: 5 groups
FIND: amount in each group 2

Measurement (Repeated Subtraction Model):
Characterized by using a specified quantity to create groups (or partitions) of a specified size (amount) and determining the number of partitions (groups) that can be formed

Ex. Sally has 8 eggs and a recepie for brownies requires 2 eggs. How many batches fo brownies can Sally make?

KNOW: Size of group
FIND: Number of partitions/groups

Fractions

Part-to-whole

Ex. 2 Total of 2 of those 3 parts are used
3 Whole is divided into three equal parts

Quotient

interpreting the fraction as a division problem
2 2 divided by 3 equal 0.6666...
3

Ratio

Used to seperate two sprate things
3 oranges for $1