Kategorier: Alla - vocabulary - sets - mathematics - fractions

av Yvonne Cabrera för 11 årar sedan

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MAT 156 Mindmap

Understanding mathematics for elementary teachers involves mastering various concepts such as problem-solving techniques, fractions, and set theory. Teachers need to devise a plan for solving problems, which might include looking for patterns, understanding the problem, implementing the plan, and checking their work.

MAT 156 Mindmap

Understading Mathematics for Elementary Teachers

Fractions!

Parts and Wholes
Simplified NOT Reduced

This means that as teachers, we need to make sure that we know what vocabulary we use. For fractions, when we are turning the fraction from something like 6/8 to 3/4 we are not reducing the number but we are making the fraction simplified.

Egyptian Fractions

All fractions must be unit fractions 1/12 which means that all unit fractions must be unique.

For example: 3/2 is not quivalent to 1/2 + 1/2+ 1/2

Relative Size of Fractions

With this we are finding whether or not the fractions are greater than, less than or equal to.

We can find the relative fractions by finding the common denominator, or by knowing the numerator, and distance parts of the whole.

Modular Arithmetic

Properties of Modular Arithmetic

Closure property of Addition

Commutative Property of Addition

Identity Property of Addition

Inverse property of Addition

Inverse property of Addition

Closure property of Multiplication

Commutative property of Multiplication

Identity property of multiplication

Inverse property of Multiplication

Arithmetic with Integers
Understanding Integers

Number Line Approach

Chip Method

Absolute Value with the Number Line Approach

Absolute Value with the Chip Method

Percents

We can use the 100 grid as a way to explore percents even further. This is a great way to apply this to real life situations.

Grids

Decimals

Rational Numbers- Any numbers that can be written in the form a/b where a and be are equvalent integers.

Operations With Decimals

Finding the:

Sum- make sure the decimals are lined up because it promotes oraganization and wont be confusing for students

Is a commputative property

Difference- We are lining up with what is given first as the top of our equation.

Is NOT a commutative property

Quotient

have to at least het to the decimal point AND get a 0

product- lining up the decimals are not necessary.

Understanding Divisibility and Algorithms

Divisibility- A whole number (a) is divisible by a whole number (b) if and only if there exists a third whole number (c) such that a=bc

Four Equivalent Statements

Let us assume that bla:

-then b is a factor of a

-a is a multiple of b

-b is adivisor of a

-a is divisible by b

Sieve of Erastothenes
Number Theory

Even- A number is even if the number is a multiple of two that is 2n.

Odd- if the number is one more than the even number that is, 2n+1

Alternate Division Algorithms

The idea of understanding divisibilty is that finding different ways of interpreting the problem.

3I27

This equation is true because 3 is a factor of 27.

OR

27 is a multiple of 3, which is also true.

Adding, Subtracting, Multiplying and Dividing

Dividing
Models of Division

Partition( equal sharing)

Distributing an even amount equally among the number of groups.

Measurement (repeated subtraction)

Using a given quantity to create groups of a specified size.

Four Fact Families in Division
Division Using Manipulatives
Multiplying

-Partial Product

-Expanded Method

- Lattice Method

Multiplication Tables
Closure Property of Multiplication

These are all of the properties that are needed to know to understand how multiplication works:

-Commutative Preoperty of Whole Numbers

-Associative Property of Whole Numbers

-Identity Property of Whole Numbers

-Zero Property of Whole Numbers

-Distributive Property of Whole Numbers

Models of Multiplication

-Multiplication as repeated addition

-Repeated addition continuous

-Area Model

- Cartesian Product

Subtracting

There are two alternative algorithms that can help students understand subraction.

Scratch Method:

In the Scratch Method we start from left to right and work with our bases we scratch out the number and continue on to find the answer.

European Method:

In the European Method we start from right to left. And just like the scratch method we will apply it to the European method.

Subtraction Models

Take Away:

When we start with an initial quantity and then we begin to take away a certain amount that is given.

EX: Sally has four apples. She eats two apples. How many apples does Sally have left?

Comparison:

We start with two quantities. Then we find out either how much larger or how much smaller one quantity is compared to the other.

EX: Johnny has three baseball cards. Tim has 5 baseball cards. How many more baseball cards does Tim have than Johnny?

Missing Addend:

This is used to determine what quantity must be added to a specified number to reach the targeted amount.

EX: Ted has one racecar. After Christmas, he has four racecars. How many racecars did he have for Christmas?

Subtraction is the inverse operation of addition.

We have the four fact families.

For example, we have something simple like:

7+3=10

which is the same as

3+7=10

When we subract we would use

10-7=3

and

10-3=7

Adding
Closure property of Addition
Addition Tables
Addition Models

Discrete set model:

This represents the quantities that are counted.

EX: Sarah has two blocks. Sally has two blocks. How many blocks do Sarah and Sally have altogether?

Continuous Set model:

This is used to measure quantities like time or distance.

EX: Rebecca ran three miles. John ran three miles . How many miles did they run altogether?

Alternative Algorithms

Any Column First:

First, we pick any column. Then we add the values in the column then add zeros according to the place values. The we add al oth the values from the column.

Lattice Method:

We first draw a box and then we divide according to the number of place values there are. We then put diagonals in each individual box. Then we will add diagonally and accordingly to the base that we are using.

Scratch Method:

We would start from the left to the right. The number regrouped would be added to the next value.

Low Stress:

We would start with something like five numbers that we would add together. We would take two at a time to add them.

Number Systems, Bases and Properties of Addition, Oh My!

Unit

A unit is a single unit, this can be used as b^0. Because b^0 equals one it will be the number of units times one.

Long

A long is the second smallest and they are written as the base to the first power or b^1

Flat

A flat is the second highest right after the cube. A flat is a base squared or b^2.

Cube

A cube is the base cubed. So for example we have a base. To find the cube of this base we will use b^3, or b cubed.

Properties of Addition
Identity Property of Addition

If all of the elements in a are whole numbers then:

a+0=a=0+a

Associative Property

If all the elements in a are whole numbers and all of the elements in b are whole and all of the elements in c are whole numbers then : (a+b)+c=a+(b+c)

Commutative Property Of Addition

Whole Numbers

If all the elements in a are a whole number and all of the elements in b are whole then: a+b=b+a

Closure Property

Of Whole Numbers

If all the elements in a are whole and all of the elements in b are whole then: a+b= a whole number

Of Addition

If all of the elements in a is x and all of the elements in b are x then: a+b is the elements of x

Ways of Recording Numbers

These are some of the different ways that numbers were recorded.

Tally System- This is one of the oldest method used to record numbers. These "numbers" consisted of lines.

Egyptians- The Egyptians drew pictures such as a heel bone that represented as the number ten. Drawing pictures was their way of recording numbers.

Mayans- The mayans used dots and lines that represented their numbers.

Babylonians- The number system for the babylonians consisted of upside down triangles. each number of triangle represented the number one.

Romans- the romans consisted of various X, V and I.

Hindu-Arabic- This consists of what the modern day uses such as 1,2,3...

Hindu-Arabic
Roman
Babylonians
Mayans
Egyptians
Tally System

Dividing with Fractions

Here is an example of dividing with fractions:

1/3 divided by 2/12

1/3 is our initial quantity

2/12 is the size of the group

We would use fractions bars to see how much of a difference we have the we would see that there are 2 groups of 2/12 in 1/3

Quotients of Fractions using Pattern Blocks

Modes of Multiplication

Area Model
Repeated Addition Continuous
Multiplication as repeated addition

here is an example to help understand multipilcation as repeated addition:

4 jars of pencils and each jar has 3 pencils. How many pencils do we have?

Ratio and Proportional Reasoning

here is an example of ratio and proportional reasoning:

Plant A Plant B

Week 3 5 in 8in

Week 4 7 in 10 in

Which plant grew more between weeks 3 and 4?

There are two types reasoning that will answer the question.

Absolutive Reasoning: Both plants grew a same 2 inches

Realative Reasoning: Plant A grew more because the growth of 2 inches Compared to the starting height is a bigger ratio than plant b.

Bases

Sets

Sequences
Recurrence Relationship Sequence

a sequence where the current term is dependent on a previous term.

Geometric Sequence

This is a sequence of numbers that have a common ratio.

Arithmetic Sequence

It is a sequence of numbers that have a common difference.

For example, the rule for an arithmetic sequence is that the for

Sequence

A sequence is an ordered list of objects, etc. For example, when the sequence of numbers increase the number of diamonds increase by two.

Equivalent

The equivalence is that the number of elements in A are the EQUAL to the number of elements in B

Set Complement

A complement is whatever IS NOT in the set.

Set Union

A set union is what ever is in set A OR set B.

Set intersection

A set intersection means what ever is in set A AND set B.

How to solve a problem

4. Check Work
3. Implement the Plan
2. Devise a plan
Ex: look for patterns in the problem
1. Understand the problem