Mathematics

Mathematics is the science of quantity.

Everyday Mathematics
Everyday mathematics usually takes on the form of conversations, using numbers to figure out something the child wants to know, games, allowance, weighing things in the grocery store, finding the best deal among several choices - which is what you would normally do, board games, figuring out “how long until?” when she asks, budgets, doing a rough estimation of the items in the grocery cart to see if you have enough money, baseball statistics, crafts, origami, wrapping presents, etc.

Use whatever naturally come up in your life. There is more math in life than school math lets you see.

Numerals, numbers
When we write, we use different symbols to mean certain things. Some of these symbols are called “numerals” because they are symbols that represent numbers. The most common numerals used to represent numerals are: 1 2 3 4 5 6 7 8 9 0

These numerals are arranged in a specific way to represent a certain number. We use the numerals 1 and 2 to represent the number “twelve” if they are arranged as 12. We use the numerals 1 and 2 to represent the number “twenty-one” if they are arranged as 21. We use the numerals 1 and 0 to represent the number “ten” if they are arranged as 10.

Values of individual common numerals
We use each of the individual numerals to represent a unique number:

We use the numeral 1 by itself to represent the number “one.”

We use the numeral 2 by itself to represent the number “two.”

We use the numeral 3 by itself to represent the number “three.”

We use the numeral 4 by itself to represent the number “four.”

We use the numeral 5 by itself to represent the number “five.”

We use the numeral 6 by itself to represent the number “six.”

We use the numeral 7 by itself to represent the number “seven.”

We use the numeral 8 by itself to represent the number “eight.”

We use the numeral 9 by itself to represent the number “nine.”

We use the numeral 0 by itself to represent the number “zero.”

Here is a chart showing each numeral with the number it represents and a visual picture of the number's value:

Organizing numerals
Rather than use an individual numeral for each number we want to represent, we use a combination of numerals to represent most numbers. We organize numbers using a “base-ten” system, which means that a numeral represents ten times the value the same numeral to the right would represent.

Adding two numbers; addends; sum
Three people went into the chapel to pray. Two more went in to pray with them. How any people were in the chapel now?

We can also write this faster as


 * 3 people
 * + 2 people


 * or 3
 * +2

or 3 + 2 = _____

If we count the people we find there are five of them. We can answer the questions we wrote more quickly as


 * 3 people
 * + 2 people
 * 5 people


 * or 3
 * +2
 * 5

If we write our question in any of these three ways they will all have the same answer: 5

We call the numbers we add together “addends” (in this case 2 and 3 were the addends) and we call the total of the addends the “sum.”


 * 3 people <--addend

+ 2 people <--addend
 * 5 people <--sum

or 3 <--addend
 * +2 <--addend
 * 5 <--sum

Adding with zero
If we have a full pie and an empty pie pan, how many pie do we have?

We can also write this as 1 + 0 = ____

Since we know that there is only one pie, we write 1 on the line so it looks like this= 1 + 0 = 1

We can add 0 to any number and the sum (total) will be the same. So if we have 12 eggs and get 0 more we will still have 12 eggs. This is called the Identity Property of Addition.

Adding more than two numbers/addends

When we need to add more than two groups of numbers at a time, it may be easier to add just two numbers at a time rather than trying to answer a question all at once.

If are three clean plates in the cupboard, two clean plates in the dish drainer, and four clean plates on the table, how many dishes do we have that we can use for the meal?

We can also write this as


 * 3 plates
 * 2 plates
 * +4 plates

or


 * 3
 * 2
 * +4

or 3+2+4=___

If we want to add two of the numbers together before we add the other number or numbers to it, we can use parentheses to enclose what should be added together first. If we wanted to add 2 and 4 together first, we can write it as 3+ (2 + 4)=_______ When we add numbers together that are inside the parentheses, we replace the numbers and signs inside the parentheses with the total of the numbers. So since 2 + 4 = 6, we replace the “2 + 4” with “6” and our question will now look like this: 3 + (6) = ___

We can remove the parentheses from around the 6 because we no longer need them. It will now look like 3 + 6 = ____ We then add the 3 and 6 and write the answer on the line and it looks like this:

3 + 6 = 9

If we write it as


 * 3
 * 2
 * +4

we can first write it on the scratch paper like this


 * 3
 * 2
 * +4

then if we want to add the 2 and 4 four together first we can make a line come from each number and join together, then write the sum of the two numbers where they join so that it looks like this:


 * 3
 * 2   \ 6 (show better picture of this)
 * +4   /

then write a new question adding the 3 and 6


 * 3
 * +6

and since we know that the sum of 3 and 6 is 9, we write the sum below the line like this:


 * 3
 * +6
 * 9

so we know there are nine clean dishes we can use for the meal.

Subtraction
If we had four clean large bowls and used two for a meal, how many are left? We can write this as


 * 4 bowls
 * - 2 bowls


 * or 4
 * - 2

or 4 – 2 = _____

If we count the bowls we see there are two clean large bowls left, so we write 2 under the line for the first two ways we wrote the question, and on the line for the third way we wrote the question:


 * 4 bowls
 * - 2 bowls
 * 2 bowls


 * or 4
 * - 2
 * 2

or 4 – 2 = 2

So we have two large clean bowls left.

We call the original number the minuend and the amount we take away the subtrahend. We call the total the difference.


 * 4 bowls <--minuend
 * - 2 bowls <--subtrahend
 * 2 bowls <--difference


 * or 4 <--minuend
 * -2 <--subtrahend
 * 2 <--difference

Subtracting with zero
If we have a full pie and do not take any away, how many pies do we have?

We can also write this as 1 - 0 = ____

Since we know that we still have the one pie, we write 1 on the line so it looks like this= 1 - 0 = 1

We can subtract 0 from any number and the difference (total) will be the same. So if we have 12 eggs and take away 0 eggs we will still have 12 eggs. This is called the Identity Property of Subtraction.

Checking subtraction with addition; Checking addition with subtraction If we add the original difference and the original subtrahend and it equals the original minuend, we can see if we subtracted it correctly. So if we have


 * 4 bowls <--original minuend
 * - 2 bowls <--original subtrahend
 * 2 bowls <--original difference

we can setup to add the original minuend and the original difference


 * 2 bowls <--original subtrahend
 * + 2 bowls <--original difference

and add them


 * 2 bowls <--original subtrahend
 * + 2 bowls <--original difference
 * 4 bowls

As we see, our answer (4 bowls) is the same as the original minuend (also 4 bowls).

Also, we can check our addition by subtracting one of the original addends from the sum, and it will equal the other original addend. So if we have


 * 3 people <--original addend
 * + 2 people <--original addend
 * 5 people <--original sum

we can setup to subtract one of the original addends from the sum


 * 5 people <--original sum
 * - 2 people <--original addend

then subtract
 * 5 people <--original sum
 * - 2 people <--original addend
 * 3 people

As we see, our answer (3 people) is the same as the other original addend (also three people).

As with anything else in life, addition undoes subtraction, and subtraction undoes addition.

Outline

 * circles, radius, diameter--already covered partially?
 * circle graphs
 * decimal number lines--merge?
 * fractions equal to one--merge
 * improper fractions to whole and mixed numbers--merge
 * dividing by 10--merge?
 * adding decimal numbers--teach earlier--merge?
 * subtracting decimal numbers--teach earlier--merge?
 * alligning decimal points in addition and subtraction--merge
 * dividing by multiples of 10
 * adding fractions with like denomiators--teach earlier--already covered?
 * subtracting fractions with like denominatrs--each earlier--already covered?
 * equivalent fractions--merge?
 * square units (like square inches)
 * find area of a rectangle--merge?
 * multplying a three digit number by a two digit number
 * reducing fractions--like making 4/6 into 2/3 --merge?
 * dividing by two digit numbers (like 95 divided by 21)
 * adding and subtracting mixed numbers--merge?
 * simplifying fraction problem answers--like 4 5/3 into 5 2/3--merge
 * renaming fractions--merge