Understading Mathematics for Elementary Teachers

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

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

A complement is whatever IS NOT in the set.

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

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

It is a sequence of numbers that have a common difference.For example, the rule for an arithmetic sequence is that the for

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

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

here is an example of ratio and proportional reasoning: Plant A Plant BWeek 3 5 in 8inWeek 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.

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?

Here is an example of dividing with fractions:1/3 divided by 2/121/3 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

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...

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

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

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

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)

If all of the elements in a are whole numbers then: a+0=a=0+a

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.

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

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

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.

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.

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?

We have the four fact families. For example, we have something simple like:7+3=10which is the same as 3+7=10When we subract we would use 10-7=3and 10-3=7

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?

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.

-Multiplication as repeated addition-Repeated addition continuous-Area Model - Cartesian Product

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

-Partial Product-Expanded Method- Lattice Method

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.

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

The idea of understanding divisibilty is that finding different ways of interpreting the problem. 3I27This equation is true because 3 is a factor of 27.OR27 is a multiple of 3, which is also true.

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

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

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

Finding the:Sum- make sure the decimals are lined up because it promotes oraganization and wont be confusing for students Is a commputative propertyDifference- We are lining up with what is given first as the top of our equation. Is NOT a commutative propertyQuotient have to at least het to the decimal point AND get a 0 product- lining up the decimals are not necessary.

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.

Number Line Approach Chip Method Absolute Value with the Number Line Approach Absolute Value with the Chip Method

Closure property of AdditionCommutative Property of AdditionIdentity Property of Addition Inverse property of AdditionInverse property of Addition Closure property of MultiplicationCommutative property of MultiplicationIdentity property of multiplicationInverse property of Multiplication

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.

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

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.