By the end of this unit we should be able to work most of the following algebra below beginning with adding like terms.
ALGEBRA PART ONE
Fig.2 shows a list of algebra skills that are essential for achieving up to Excellence in YR11. We will focus on the Achievement section for YR9 and YR10.
BEDMAS - PRE ALGEBRA
Understanding BEDMAS is essential to the understanding of algebra, because BEDMAS teaches us about the structure and order of mathematics. It provides us with a set of rules. Without these rules our maths makes no sense. Take a look at this example, which do you think is the correct answer?
a) 7 + 3 x 5 b) 7 + 3 x 5
= 10 x 5 =7 + 15
= 50 = 22
Now let's list the order of operations as agreed by Mathematicians.
These are the order of operations
brackets (parentheses)
exponents
multiply/divide
add/subtract (left to right in the order they appear)
So if we look at our first example we would do the following:
7 + 3 x 5 (multiply first)
=7 + 15 (then add)
=22
so answer B was correct.
try these examples
1) 8 + (2 x 5)
2) 12 - 4 + 3
3) (8 + 4) x 2
4) 3 x 4 + 5
5) 4 x 3 + 1 x 2
6) 20 ÷ 5 x 9 ÷ 3
7) (7 + 3) x 4 ÷ 2 - 5 x 6
2) 12 - 4 + 3
3) (8 + 4) x 2
4) 3 x 4 + 5
5) 4 x 3 + 1 x 2
6) 20 ÷ 5 x 9 ÷ 3
7) (7 + 3) x 4 ÷ 2 - 5 x 6
Answers: 1)18 2)11 3)24 4)17 5)14 6)12 7)20-30= -10
Let's take a closer look at problem 2)
12 - 4 + 3
If we use the rules of BEDMAS, when it comes to addition and subtraction we work in the order they appear. In other words in this example we work from left to right.
12 - 4 + 3 a common mistake is 12 - 4 + 3
= 8 + 3 = 12 - 7
= 11 = 5
BEDMAS #1
Use the rules of BEDMAS to answer the following. It might be a good idea to do just one column if done in class to allow time to do BEDMAS 2. The answers are provided for guidance. If you are not sure about a question, check the answer and work backwards. For example,
1) 7 + 6 x 2 =
7 + 12 = 18 (multiply first)
As above there are a lot of examples to do here and you won't have time to do them all in class. Hopefully the answer guide will assist you with that.
1) 7 + 6 x 2 =
7 + 12 = 18 (multiply first)
As above there are a lot of examples to do here and you won't have time to do them all in class. Hopefully the answer guide will assist you with that.
BEDMAS #2
In BEDMAS 2, put brackets in the correct places so that the statements are true. For example,
(3 + 1) x 5 = 20
4 x 5 = 20 "It is known"
(3 + 1) x 5 = 20
4 x 5 = 20 "It is known"
answers
literacy activity
The aim here is to write out word statements that have been carefully shrouded in symbols. Some of them you may recognise because they have become a part of our literacy or everyday speech. This is a good literacy exercise with difficulty ranging from easy to "excuse me?"
WRITE OUT THE SENTENCES
1). 60 s in a M
2). The 10 C
3). A the W in 80 D
4). 12 I makes 1 F
5). 0 D C is 32 D F
6). 7 D in 1 W
7). S W and the 7 D
8). 1760 Y in a M
9). K H the 8th had 6 W
10). 24 H in a D
11). 26 L in the A
12). The A N of H is 1
13). Half a G is 4 P
14). 2Q=1C and 2C=1M
15). 7U
16). 2 of H and 1 of O is W
17). J had 12 D
18). 32 T in your M
19). 8 P is a G
20). A S in T S 9
21). 5 C of the W
22). A B and the 40 T
23). 12 M in a Y
24). The first P N is 2
25). 5 S is 25
26). 30 D in A
27). 1 out of 5 is 20 P
28). 100 Y in a C
29). The 4th of J is I D
30). 18 H on a G C
31). A S has 8 L
32). B is 7, P is 6, B is 5.......
33). 3 M in a B
34). The M is 26 m 385 y long
35). There are 11 P in a F T
36). 50 Y is a G W, 25 is a S W
37). 0-0,15-0,30-0,40-L is M P
38). 3 B M
39). 52 C in a P
40). 21 C in the A
41). 10 Y is a D
42). 4 T get to the S F of the F A C
43). The S R of 100 is 10
44). 10 T on your F
45). 101 D
46). 366 D in a L Y
47). 144 is a G
48). 6 B in an O
49). 5/8 of a M is 1 K
50). 50 Q in this Q
1). 60 s in a M
2). The 10 C
3). A the W in 80 D
4). 12 I makes 1 F
5). 0 D C is 32 D F
6). 7 D in 1 W
7). S W and the 7 D
8). 1760 Y in a M
9). K H the 8th had 6 W
10). 24 H in a D
11). 26 L in the A
12). The A N of H is 1
13). Half a G is 4 P
14). 2Q=1C and 2C=1M
15). 7U
16). 2 of H and 1 of O is W
17). J had 12 D
18). 32 T in your M
19). 8 P is a G
20). A S in T S 9
21). 5 C of the W
22). A B and the 40 T
23). 12 M in a Y
24). The first P N is 2
25). 5 S is 25
26). 30 D in A
27). 1 out of 5 is 20 P
28). 100 Y in a C
29). The 4th of J is I D
30). 18 H on a G C
31). A S has 8 L
32). B is 7, P is 6, B is 5.......
33). 3 M in a B
34). The M is 26 m 385 y long
35). There are 11 P in a F T
36). 50 Y is a G W, 25 is a S W
37). 0-0,15-0,30-0,40-L is M P
38). 3 B M
39). 52 C in a P
40). 21 C in the A
41). 10 Y is a D
42). 4 T get to the S F of the F A C
43). The S R of 100 is 10
44). 10 T on your F
45). 101 D
46). 366 D in a L Y
47). 144 is a G
48). 6 B in an O
49). 5/8 of a M is 1 K
50). 50 Q in this Q
ANSWERS LATER
Students often get confused between multiplying numbers and powers of numbers. A common mistake is multiplying a number and power together. They are completely separate operations.
The 3 properties of number
1) The commutative property
The word commutate means to change or transfer. Electric generators contain a commutator, which rotates on a shaft called the armature. The commutator turns alternating current into direct current or in other words, AC into DC. You may be more familiar with the context of commuting from one place to another either by bus or by train etc.
In the commutative property this means we can exchange or swap the numbers around and the answer will still be the same.
For example, in addition:
2 + 8 = 8 + 2 because 10 = 10
so in algebra, a + b = b + a
For example, in multiplication:
5 x 6 = 6 x 5 because 30 = 30
so in algebra, a x b = b x a
However, in subtraction:
2 - 8 does not equal 8 - 2 because -6 does not equal 6
Like wise in division:
5/6 does not equal 6/5 Check with your calculator.
For example, in addition:
2 + 8 = 8 + 2 because 10 = 10
so in algebra, a + b = b + a
For example, in multiplication:
5 x 6 = 6 x 5 because 30 = 30
so in algebra, a x b = b x a
However, in subtraction:
2 - 8 does not equal 8 - 2 because -6 does not equal 6
Like wise in division:
5/6 does not equal 6/5 Check with your calculator.
2) The associative property
The associative property makes use of the fact that it doesn't matter how we group or associate the numbers when we add and multiply. Let's look at an example in addition:
2 + (3 + 4) = (2 + 3) + 4, both of which gives 9
This is important because if you wanted to add the following numbers:
4 + 17 + 16 + 3 = you can rearrange the numbers (mentally or on paper) so they are easier to add.
4 + 16 + 17 + 3 = 20 + 20 = 40 but you probably knew that already.
And an example in multiplication:
2 x (3 x 4) = (2 x 3) x 4, both of which gives 24
So a + (b + c) = (a +b) + c and a x (b x c) = (a x b) x c
Once again subtraction and division don't work with these rules. An example of subtraction:
6 - (4 - 2) = 4 and (6 - 4) - 2 = 0 Certainly not the same.
an example with division:
(16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2) because (16 ÷ 4) ÷ 2 = 2 and 16 ÷ (4 ÷ 2) = 8 Certainly not the same.
Let's move on to the next property, its probably the most important in algebra. The distributive property.
2 + (3 + 4) = (2 + 3) + 4, both of which gives 9
This is important because if you wanted to add the following numbers:
4 + 17 + 16 + 3 = you can rearrange the numbers (mentally or on paper) so they are easier to add.
4 + 16 + 17 + 3 = 20 + 20 = 40 but you probably knew that already.
And an example in multiplication:
2 x (3 x 4) = (2 x 3) x 4, both of which gives 24
So a + (b + c) = (a +b) + c and a x (b x c) = (a x b) x c
Once again subtraction and division don't work with these rules. An example of subtraction:
6 - (4 - 2) = 4 and (6 - 4) - 2 = 0 Certainly not the same.
an example with division:
(16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2) because (16 ÷ 4) ÷ 2 = 2 and 16 ÷ (4 ÷ 2) = 8 Certainly not the same.
Let's move on to the next property, its probably the most important in algebra. The distributive property.
3) the distributive property
Let's say we want to multiply 3 lots of (2 + 4). Some of you might be thinking why don't you just add the 2 and 4 together in the bracket and times that by 3 to get 3 x 6 = 18. Well yes that makes sense to do that. However, algebra statements usually contain a letter or letters that make that a little more difficult. How can you multiply 3 lots of (2 + a) if you don't know what a is? It could be 4, but what if it wasn't? But there is a plan that we can use. First we need to know that 3 lots of (2 + 4) is the same as 3 lots of 2 PLUS 3 lots of 4. Mathematically:
3 x (2 + 4) = (3 x 2) + (3 x 4) Notice how the multiplier, the 3 has distributed itself across the (2 + 4)
= 6 + 12
= 18
Check out the next example:
6 x 204 = 6 x (200 + 4) = 6 x 200 + 6 x 4
= 1200 + 24
= 1224
and another:
6×2 + 2×2 + 3×2 + 5×2 + 4×2 Notice the 2 has distributed itself across the expression. We can write this as:
2 x (6 + 2 + 3 + 5 + 4) = 2 x 20
= 40 so we call this the distributive property.
3 x (2 + 4) = (3 x 2) + (3 x 4) Notice how the multiplier, the 3 has distributed itself across the (2 + 4)
= 6 + 12
= 18
Check out the next example:
6 x 204 = 6 x (200 + 4) = 6 x 200 + 6 x 4
= 1200 + 24
= 1224
and another:
6×2 + 2×2 + 3×2 + 5×2 + 4×2 Notice the 2 has distributed itself across the expression. We can write this as:
2 x (6 + 2 + 3 + 5 + 4) = 2 x 20
= 40 so we call this the distributive property.
replacing numbers with letters (variables)
From the distributive property we learned that:
4 x (5 + 2) = 4 x 5 + 4 x 2
= 20 + 8
= 28
Now use the distributive property to multiply and 'expand' 2 x (x + 4) usually written 2(x + 4) because you don't have to write the times symbol in front of the bracket.
4 x (5 + 2) = 4 x 5 + 4 x 2
= 20 + 8
= 28
Now use the distributive property to multiply and 'expand' 2 x (x + 4) usually written 2(x + 4) because you don't have to write the times symbol in front of the bracket.
Just to remind you, we don't know what 'x' is. It can stand for any number we choose. More about this property when we come to the 'expanding algebra' topic in Algebra 5 Expanding and simplifying.