Topic 12: Directed Numbers

12.1 What is an Integer? An integer is any positive or negative whole number and 0 such as 1, -2, 10, 7, 216, -155. Integers have a direction, positive or negative, therefore they are also known as directed numbers.Positive (+) Integers are all around us, we use them every day and all the time.

She is 40 years old; This bottle contains 500ml of water; He is 156cm tall; There are 365 days in a year, unless it is a leap year in which case there will be 366 days and February will have 29 days instead of 28.A positive number is a number above 0. We may denote a positive number by a plus sign in front of the number, however it is common to put no sign at all: 5, 12, 254Negative (−) Integers ?? ……..we need to think about this one!

o Bank Accounts – Below €0o Temperature – Below 0˚C o Elevators – Below ground level o Depth – Below sea level o Golf – Under par

A negative number is a number that goes below 0. We denote a negative number by a minus sign: -2, -5, -19

12.1.1 The Number Line We can illustrate integers on a number line. It helps us understand positive and negative numbers better!

12.2 Adding and Subtracting Integers

SMALLER

BIGGER

Ms Greta Notes Senior 1

The number line helps us to add and subtract any two integers. The 1st number is your starting point The sign of the 2nd number tells you the direction (positive or negative) The 2nd number tells you how many steps in that direction you must takeLet’s say you have 3+4, now we know that the answer to this sum is 7 because we are used to adding positive integers. Put your finger on the first number ‘3 ’ and move ‘ 4 ’steps in the positive direction. Where do you land? 7!

+3+4=7

What about −6+3 ? Put your finger on −6 and move 3 steps in the positive direction. Where do you land? -3! −6+3=−3

What about 3−7? Put your finger on 3 and move 7 steps in the negative direction. Where do you land? -4! 3−7=−4

What about −7−3? Put your finger on −7 and move 3 steps in the negative direction.Where do you land? -10! −7−3=−10

In SummaryBoth negative? Answer will be negative −3−4=−7 (keep walking in the negative direction)

Both positive? Answer will be positive +3+4=+7 (keep walking in the positive direction)

Different signs? If the bigger number has a negative sign the answer will be negative, if the bigger number has a positive sign the answer will be positive, always find the difference between the numbers and put the sign:

−3+4=1−4+3=−1

Such questions may also be reasoned out by considering debt which is money that you owe. Consider the negative numbers as money that you owe and the positive numbers as money that you have in your pocket:

−6+3=−3 I am 6 euros in debt (-6) and I have 3 euro in my pocket (+3), I can pay 3 euro but I will still be 3 euro in debt (-3)

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Ms Greta Notes Senior 1

3−7=−4 I have 3 euros in my pocket (+3) and I am 7 euro (-7) in debt, I can pay 3 euro but I will still be 4 euro in debt (-4)

−7−3=−10 I am 7 euro in debt (-7) and another 3 euro in debt (-3), so in all I am 10 euro in debt (-10)

12.2.1 When the Signs are Touching

Sometimes, we might have two signs next to each other. In such a case these two signs must become one sign and then we work out the sum as in section 12.2 above.Same signs become a positive: 4−−4=4+4=8

4++4=4+4=8

Different signs become a negative: 4+−4=4−4=0

4∓4=4−4=0

Examples:a) 4−−3=4+3=7

b) −6∓5=−6−5=−11

c) 4±3=4−3=1d) 3+−9=3−9=−6

e) 2+−9−1=2−9−1=−7−1=−8

12.3 Multiplying and Dividing Integers

What does 2 x 4 mean? (we, know that the answer is 8)

2 for 4 times 2 + 2 + 2 + 2 = 8or

4 for 2 times 4 + 4 = 8 2

Don’t not go (−−¿)

means GO (+¿)

Don’t go (∓¿)

means don’t (−¿)

Ms Greta Notes Senior 1

Here we are remembering that multiplication is repeated addition

So what does -2 x 4 mean?

-2 for 4 times - 2 - 2 - 2 - 2 = -8−2×4=−8

Therefore, we can say that multiplying a negative number by a positive number and vice versa will give a negative answer

So, using this new information we are now able to multiply:Positive x Positive = PositivePositive x Negative = NegativeNegative x Positive = NegativeSo, we just need to figure out what answer a Negative x Negative will give. Let us consider the ‘negative’ 3 times table:

3×−3=−9

2×−3=−63

1×−3=−3

0×−3=0

−1×−3=3

−2×−3=6

−3×−3=9

−4×−3=12

In this pattern we are adding 3 each time, so we can see that multiplying two negative numbers will give us a positive answer. There is a more complicated explanation for why this is so, but for the moment we shall allow this pattern to suffice.

In summary therefore we can say: Multiplying numbers having SAME SIGNS gives a POSITIVE

ANSWER

3

−×+¿−¿

+×−¿−¿

+3

+×+¿+¿

−×−¿+¿

+3

+3

+3

+3

+3

+3

Ms Greta Notes Senior 1

Multiplying numbers having DIFFERENT SIGNS gives a NEGATIVE ANSWER

These same rules apply to dividing directed numbers as division is repeated subtraction:−4÷2 means divide −4 between 2 people then each person will get −2. If two people are 4 euro in debt (−4) and they share the debt equally they each need to give 2 euro (−2).In conclusion, to multiply or divide directed numbers, you must multiply or divide the two numbers normally to get the answer and then you must decide what sign to put in front of the answer - remembering that same signs give a positive answer whilst different signs give a negative answer.Examples:a¿−8×−2=16¿

b¿−8×2=−16¿

c ¿−24 ÷−6=4 ¿

d ¿24÷−6=−4¿

To conclude, here is a summary on operations with negative numbers:

Topic 13: Introducing Algebra

Remember when in junior school you used to fill in the empty boxes?

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Ms Greta Notes Senior 1

In algebra, the same thing happens but we use letters instead of boxes to hold a place for numbers that we do not know yet.

Algebra is when we use a letter to represent (stand for) a number

But why use a letter when we can use a number?

Unfortunately, we can’t always put a number or value to an amount, so that’s when we use letters.

There are 12 eggs in this box! There are x smarties

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Ms Greta Notes Senior 1

If I take a smartie, there will be x−1 smartiesConsider this container of paperclips, we can’t tell how many paper clips there are in all, so we use a letter to represent this number. We can however, use this letter to say things about the paper clips:

Let’s say there are n paper clips

If I take 2 paper clips out, there will now be: n−2 paper clips

If I add 10 paper clips to the n we started with there will now be: n+10 paper clips

If I add another container with the same amount of paper clips, there will be: n+n=2×n=2n paper clips

n−2, n+10and 2n are called expressions.

Expressions are a combination of numbers (1,2,3. .), letters (a ,b , c …) and operators (+,−,× . .) WITHOUT an EQUALS sign. Expressions represent a number.

Important Points on Expressions• When a letter and a number are next to each other, there is a

‘hidden’ × (times) n+n+n+n=4×n which we write as 4 n

• In expressions it is common practice to write the number in front of the letter not the other way round: 4 n NOT n4

13.1 Collecting like terms

• A term is part of an expression

• Like terms contain the same letter

• You can simplify an expression by collecting like terms:y+ y+ y=3 y

b+b+b+b+b=5b

a+a+a+b+b=3a+2b

g+g−g−5 r=g−5 r

Note that, when there is no number in front of a letter as in the last example, it means that there is a hidden ‘1’ g=1g

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Ms Greta Notes Senior 1

When the expressions get longer, simplifying by collecting like terms gets trickier. I would suggest to circle, underline or mark the like terms making sure to include the signs. Collect each set separately, then put the answers to each set together to formulate the simplified expression.Example 1: Simplify the expression 5a+3b+3– 2a+b−5

5a+3b+3– 2a+b−5

a terms 5a –2a=3a

b terms 3b+b=4b

number terms 3−5=−2

Ans :3a+4 b –2

Example 2: Simplify the expression 4 h+3−h+3 f −9h−5+h−1 4 h+3−h+3 f −9h−5+h−1

h terms 4 h−h−9h+h=−5h

f terms just 3 f →nothing ¿collect !

number terms 3−5−1=−3

Ans :−5h+3 f – 3

13.2 Substitution

Picture a football match. We have players on the pitch and players on the bench. At any time during the game the coach may choose to substitute a player. What does this mean? It means that a player on the bench will take the place of a player on the pitch.

Substitution in Maths is exactly the same, only instead of players on the pitch we have letters and instead of players on the bench we have numbers. In substitution we replace letters by given numbers to find the value of an expression.

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Ms Greta Notes Senior 1

Example 1: Evaluate the expression 2 y+7 when y=4

The letter y goes out and is replaced by the number 4. Now, remember that when a number and a letter are next to each other there is a hidden times.

2 y+7

2(4)+7

2×4+7

8+7

Ans :15

Example 2: Evaluate the expression 3a−2b when a=10∧b=4

The letter a goes out and is replaced by the number 10 and the letter b goes out and is replaced by the number 4. Now, remember that when a number and a letter are next to each other there is a hidden times.

3a−2b

3 (10 )−2 (4)

3×10−2×4

30−8

Ans :22

Example 3: Evaluate the expression pq−4 when p=7∧q=2

When a letter and a letter are next to each other there is a hidden times too.

pq−4

(7)(2)−4

7×2−4

14−4

Ans :10

13.3 EquationsAn equation is like a balance

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Ms Greta Notes Senior 1

An equals sign tells us that one side balances the other side exactly

To find the value of x, we must get it on its own on one side of the balance (equals)

To get x on its own we must add or remove the same amount from both sides to keep the scale in balance

In the example above, to get x on its own we must remove 3 from the left hand side (LHS). To keep the scale in balance, we must also remove 3 from the right hand side (RHS). We are then left with x on the LHS and 2 on the RHS, which implies that x=2. We have solved an equation!This year we will be solving simple equations which we can categorize into three types. Let us look at how to solve each type in turn using the balancing method.

Equation Type 1

A+2=7

A+2−2=7 – 2

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Ms Greta Notes Senior 1

A=5

Equation Type 23 x=12

3 x÷3=12÷3

x=4

Equation Type 3 = Type 1 + Type 2

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Ms Greta Notes Senior 1

2B+1=5

(Type 1) 2B+1−1=5−1

2B=4

(Type 2) 2B÷2=4÷2

B=2

More examples

Type 1 Type 2 Type 3

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Ms Greta Notes Senior 1

m + 4 = 11m + 4 – 4 = 11 – 4

m = 7

2b = 62b ÷ 2 = 6 ÷ 2

b = 3

2e + 3 = 72e + 3 – 3 = 7 – 3

2e = 4

2e ÷ 2 = 4 ÷ 2

e = 2

p – 6 = 10p – 6 + 6 = 10 + 6

p = 16

3h = 93h ÷ 3 = 9 ÷ 3

h = 3

4c – 5 = 114c – 5 + 5 = 11 +

5 4c = 16

4c ÷ 4 = 16 ÷ 4

c = 4

To solve an equation, you always want to get the letter(unknown value) on its own. To do this you must get rid of the numbers attached to the letter by doing the reverse operation (- if +, + if -, ÷ if x). This reverse operation must be done to both sides, to keep the equation in balance.The best thing about solving equations is that you can always check whether your answer is correct.

How?

By using substitution

When we get our answer we can substitute it into our starting equation and check whether the equation balances.

p – 6 = 10 p = 16

16 – 6 = 10

3h = 9 h = 3

3 x 3 = 9

4c – 5 = 11 c = 4

4 x 4 – 5 = 1116 – 5 = 11

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