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Lesson 1A : Positive and Negative Numbers

Lesson Outline:

1. Adding

2. Subtracting

• Second number is larger
•    Method 1
•    Method 2
• Second number is a negative number

3. Multiplication/Division

• Guide chart
• Helpful tip
• Decimals, fractions, proportions and percent

4. Converting

5. Mixed numbers and improper fractions

6. Repeating decimals

• Helpful tip

ADDING

When talking about addition, we often refer to one number as being greater than or less than another number. This is called an inequality. An inequality means that two amounts are not equal. A number that is greater than another is represented by a symbol that looks like a right-pointing arrow, while less than is represented by the opposite symbol.

6 is greater than 4: 6 > 4               2 is less than 9: 2 < 9             3 is equal to 3: 3 = 3 

When adding two numbers:

A positive plus a positive equals a positive , just as a negative plus a negative equals a negative .

When you add a negative to a positive , the answer has the sign of the larger number

-4 + 10 = ? -5 + 3 = -?    Add the two together:    -4 + 10 -5 + 3 = ?

Picture this: You have thirteen plus symbols and 9 minus symbols. How many more plus symbols do we have than minus symbols?

+ + + + + + + + + + + + +

- - - - - - - - -

Or: 13 – 9 = ?

Each plus and minus symbol represents the numbers we are considering (i.e. the nine minus symbols represent the negative four plus the negative five, alongside the thirteen pluses, which represent the positive ten plus the positive three.)

When we add them, we combine all the positives and negatives. Then, each unit cancels out another unit with the opposite sign on a one-to-one basis. We have thirteen plus signs and nine minus signs. We subtract nine from thirteen and that leaves us with 4.

13 – 9 = 4 

SUBTRACTING

Subtraction is the opposite of addition. When you subtract a number from another number, you are taking away a number, as opposed to adding or joining one number to another. The language of subtraction involves different words than that of addition.

• To subtract means to take away.
• The difference is the part that is left over after another part has been taken away.
• The minus sign signifies subtraction, or taking away.

Subtraction is the process of removing some objects from a group. When we say “7 minus 4 equals 3”, we mean that four objects were taken away from a group of seven objects, leaving three objects.

For example:

Lila has 7 peaches and gives 3 peaches to Roscoe. How many peaches does Lila have left?

+ + + + + + +

-  -  -

= + + + +

Lila has 4 peaches left.

As you know, in simple subtraction, the first number is larger than the second number that is subtracted from it. As in 2 – 1 = 1

So what happens when the second number is larger, as in 2 – 5 = ?

There are two methods you can choose from:

1. Treat it as a sum

Try 2 + (-5), or set up the equation with symbols to cancel out the negatives and positives. Below, you see that two positives cancel with two negatives, leaving three negatives. Therefore, the answer is      -3.

+ +

- - - - -

= -3

2. Treat it as an inverted difference

Think of it as the larger number minus the smaller number.

5 – 2 = 3

BUT, though you subtract 2 from 5 to get 3, you must remember that in the original problem, it was 2 – 5, where the second number was larger than the first and the answer is negative (-3).

Either method is fine, as long as it is easy for you to remember and use.

When the second number is a negative number, as in 1 - (-2) = ?

The minus/negative signs cancel each other out. You can then think of it as adding a positive of that number.

1 – ( - 2) can be written as 1 + (+2) and it becomes a simple addition problem: +1 + (+2) = +3

MULTIPLICATION/DIVISION

Here is a chart that makes it simple:

Multiplication is basically repeated addition performed more quickly. For example, 4 + 4 + 4 = 12 can also be written as 4 x 3 = 12. The larger the numbers that you are dealing with, the more useful multiplication becomes. The symbol x is called the times sign, and in the same way that the plus sign indicates addition and the minus sign indicates subtraction, the times sign indicates that multiplication is the required function.

Division is actually the inverse of multiplication. Instead of adding multiple numbers together, in division we are subtracting multiple numbers. The language of division also has a few basic terms, the first one being the word divide, which means to separate into groups. The function divide is represented by the division symbols    /    or   ÷

• The dividend is the number to be divided by the divisor.
• The quotient is the number that results from the division of one number by another.

Division can be compared to exhaustive subtraction. Let’s say you want to divide 15 by 3. 15 is the dividend and 3 is the divisor. You would subtract 3 from 15 repeatedly until you cannot subtract anymore. The number of times you are able to subtract 3 gives you the quotient, or the answer. Since 3 can be subtracted 5 times, you say that 15÷3 = 5

Remember that division can be seen either as inverted multiplication or repeated subtraction.

Let’s do a word problem together:

You’re throwing a party for 33 children, and you have 3 cakes. If you want to seat an equal number of children around each cake, how many children are there per cake?

In order to figure this out, you need to divide 33 children by 3 cakes. Then, rewrite the dividend as 3 tens and 3 ones. Divide each part of the dividend by 3 ones, which gives you a quotient of 1 ten and 1 one, or 11.

Therefore, you would have 11 children per cake.

You can use this same method regardless of the size of the dividend. Always begin by dividing the number furthest to the left, then continue dividing as you move towards the right until there are no numbers left to be divided.

Sometimes your division problem will not be as simple, numerically, as the preceding one.

If the divisor does not divide evenly into the first number on the left, you must divide the divisor into the first TWO numbers on the left. Let’s do an example of this right now:

Given 255 ÷ 5, we see that 5 does not divide evenly into 2 hundreds, so we use the first two numbers 25 tens, and divide it by 5.

This equals 5 tens.

Place the number 5 in the quotient over the tens number. 

Then divide the 5 into 5 ones left in the dividend. 5 ÷ 5 = 1, so place 1 in the quotient over the ones number.

Your quotient is 51.

HELPFUL TIP

On standard grid problems, the answer is never negative.

a. If you get two results, one negative and one positive, use the positive

b. If you only get a negative result, go back and check your work

DECIMALS, FRACTIONS, PROPORTIONS, PERCENT

1. Converting

A fraction is one number on top of another: The top number is called the numerator, and the bottom number is called the denominator.

Fractions are like compact division problems. The line that divides the two numbers of a fraction signifies division. For example, 1/4 is the same as 1 ÷ 4.

Fractions are also comparisons between two numbers. This comparison is called a ratio.

Comparing 1 to 4 is written as 1/4 or as 1 : 4.

Ratios come in handy all the time. For instance, when you are cooking, a recipe may call for 2 parts of butter to 4 parts of milk. This can be written as a ratio of 2 : 4, or 2/4.

The following example is a visual example of the ration 1 : 4, “one part of four” or “a fourth of one”:

A decimal number like 0.2 is called “two tenths.”

It represents the fraction 2/10, which can also be thought of as a quotient or division problem

2/10 = two divided by 10 = 0.2

The decimal number 3.25 is the same as 3 and 25/100.

The fractional part, 25/100, can be reduced to its “lowest terms.” The lowest terms for 25/100 is ¼.

Therefore, 3.25 is another way of writing 3 ¼.

MIXED NUMBERS AND IMPROPER FRACTIONS

3 ¼ is called a mixed number because it contains both a whole number and a fraction.

It can be changed into a fraction without a whole number.

Multiply the whole number, 3, by the denominator of the fraction, 4 .

3 x 4 =12. Then, add your result, 12, to the numerator of the original fraction:

1 + 12 = 13 This is your new numerator. Place your new numerator over the original denominator of the fraction. Your new improper fraction is 13/4.

A fraction like this, where the numerator is larger than the denominator, is known as an “improper fraction.”

There are two kinds of fractions, the proper fraction and the improper fraction. Simply, proper fractions have a numerator that is less than the denominator. Therefore, the value of any proper fraction will be less than 1. An improper fraction has a numerator equal to or greater than the denominator, giving the fraction a value equal to or greater than one.

REPEATING DECIMALS

The fraction 1/3 happens to equal 0.333333333… and on and on forever.

That is called a repeating decimal and is expressed by placing a line over the repeating parts.

The fraction 2/3 = 0.6666666666666…, which can also be written as 0.667.

This process is called “rounding up.” If the number that continues is 5 or greater, you round up to the next number. If the number is smaller than 5, you do not round up.

HELPFUL TIP

On standard grid questions:

a. If you get a mixed number, convert it to a decimal or improper fraction.

b. If you get a repeating decimal, round it off or convert it into a fraction.

Otherwise your answer will not be accepted as correct.

Back: Math Approach | Next: Math Lesson 1B

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