Like Terms

We are now looking at like terms in algebra.

Firstly, we have to remember what a term is. A term is a number, a pronumeral, or a combination of both of these. For example, 4 is a term, x is a term, and 3abxyz is also a term. When we write an algebraic expression, terms are easy to spot because they are always seperated by a "+" or a "-". For example, in the expression;

2ab + 4c - 6 - x

our terms are 2ab, 4c, -6, and -x.

Now like terms are special terms. These are terms that have EXACTLY the same combination of pronumerals (letters). For example, 2a and 4a are like terms, because they both have just the pronumeral "a". However, 4b and 5bc are NOT like terms. They both do have a "b" in them, however the 5bc also contains a "c" so the combination of pronumerals is different.  

If we have a look at an expression now; 

4a + 2ab + 5 - a + 3ab - 1

We can go and collect like terms in this, so the like terms would be:

• 4a and -a
• 2ab and 3ab
• 5 and -1 

Once we have collected the like terms in an expression, we can do a process called simplification (or simplifying) where we combine these like terms and write the expression in a simpler, shorter way. So again looking at the expression above, if we were to combine the like terms we would get:

• 4a and -a becoming 3a, because we start with 4 lots of a and then take 1 lot of a away, leaving us with 3 lots of a, or 3a. 
• 2ab and 3ab becoming 5ab, because we start with 2 lots of ab and add 3 more lots of ab, leaving us with 5 lots of ab, or 5ab. 
• 5 and -1 becoming 4, because we start with 5 and take 1 away, leaving us with 4. 

We can then write our expression with the combined like terms instead of all of the original terms. The new expression would look like; 

3a + 5ab + 4

Which as you can see is much simpler and much shorter than the previous expression. We have now simplified the expression. 

Now a few points to remember when doing these questions is that when we are looking at these expressions and like terms; 

1. Don't worry about the coefficient (the number out the front of the term, so the 4 in 4ax). When we look at identifying like terms, we are just looking at the pronumerals. Once you have identified the like terms, you can then look at the coefficients and use these to combine your like terms. 
2. The order of the pronumerals does not matter. xy is the same as yx, so 4xy and 3yx would be like terms. If you really think about it, the combination of pronumerals in these terms is still the same. They both have an "x" and they both have a "y", so they must be like terms. 

I have uploaded a worksheet for this into the "Worksheets" tab. Work through this now for class work and homework. The first page is the most important work. The second page contains extra questions to work through if you have time. Remember to check your answers with the answers provided. 

If you have any questions feel free to comment on this post, or email or chat to me in person. 

Intro to Algebra - Continued

We are continuing to introduce Algebra and the processes we follow working with this. Be sure to familiarise yourself with the table of common algebra words and their definitions in the previous post. These are really important to understand, as they are used a lot in Algebra now and in the future.

One common misconception that I have be noticing come in when we are substituting. Substitution is the process where we replace the pronumerals (letters) with number values and actually solve algebraic expressions or equations. So for example, solve;

a + 3, when a = 6

We would simply replace the "a" with the value of 6 and solve, getting;

6 + 3 = 9

The misconception occurs when we have a problem like this. Solve; 

4b, when b = 7

We need to remember when we write 4b, what we are really saying is "4 lots of b" or "4 multiplied by b". This can be confusing, because if we replace the "b" with a 7 in this example, we could think the solution is actually 47, which is incorrect. This is why it is really important to remember that 4b really means "4 lots of b". So the solution for this example would be; 

4 x 7 = 28

You should be able to finish off the first worksheet for this algebra unit, which can be found in the "Worksheets" tab. We do not have maths again till Thursday the 29th of October, so this should give you plenty of time to finish this off. Remember to check your answers as you go. Some of the answers given by this worksheet are incorrect, however these are quite obviously wrong. 

If you have any questions on anything to do with this post or the worksheet, feel free to comment on this post, or email or chat to me in person. 

Introduction to Algebra

We have started a new unit this term. This unit will be on Algebra. When people here the word Algebra they often get quite nervous and think it changes everything in maths, making it heaps harder. This is not the case. Algebra still obeys all of the same laws that standard maths follows. The only difference is we are using symbols to represent information, instead of number like we have previously. The reason we use symbols is because sometimes values or information can change. If we look at the number 4 for example, this can only ever represent one value (4 units). However, if we use symbols, they can be used to represent any value we want. An example could be;

If we want to find the total number of people at the school, we can set up an equation like so;

Total number of people = a + b + c + d

where a = the number of students at school today, b = the number of teachers at school today, c = the number of support staff at school today (i.e. cleaners, office staff, etc), and d = the number of visitors at school today (i.e. parents, builders, etc). 

The values for a, b, c, and d can all change every day, depending on how many people are at school at any particular time.  This means that the equation we have above will work every single day, no matter what. 

Now when we are dealing with algebra, it is important to understand some key words. These are in the table below. Make sure you familiarise yourself with these words. 

Now, when we work with Algebra there are a few little tricks you need to remember. The first is that when we write say "4x" this really means "4 lots of whatever x is representing" or "4 multiplied by x". We don't put in the multiplication symbol. 

Also, we rarely will write "1w" or "1g" or "1z" because what we are really saying is "1 lot of whatever symbol we choose", which in the end is just that symbol. So instead of writing "1w" we would just write "w", and the same with "1g" being just "g", or "1z" being just "z". 

Now for some examples.

If we have a certain number, a, write an expression for the following; 

• This certain number (a) increased by 7:

To do this, think of what we would do with any normal number if we wanted to increase it by 7. We would simply take this number and add 7. We use the same idea with this question here. So to increase this certain unknown number (a) by 7 we would simply add 7 to a, so an expression for this would simply be:

a + 7

• This certain number (a) decreased by 21:

Again, think of what we would do to a normal number if we were decreasing it by 21. We would simply take 21 away from the number. We use the same idea with this question here. So to decrease this certain unknown number (a) by 21 we would simply take 21 from a, so an expression for this would simply be:

a - 21

• 4 lots of this certain number (a):

Again, we think of what we would do with a normal number, which would be to multiply 4 by this number. We do this with our certain unknown number (a). We would get the expression:

4 x a

Which we know we write as:

4a

• Increase a certain number (a) by 5 then decrease the result by 7:

This is a two stage problem. First we would increase a by 5 and get the expression:

a + 5

Next, we need to decrease this by 7, or subtract 7 from this. This gives us:

a + 5 - 7

Which we can simplify (i.e. solve 5 - 7) and write as:

a - 2

Hopefully this gives you an introduction into what algebra is and the basics of how we use it. I have uploaded a worksheet on this into the "Worksheets" tab. Work through this sheet, checking your answers as you go. 

If you do have any questions on this, feel free to comment on this post, or email or chat to me in person.