6.4 SUPPORTING PATH calculations with letters

Theory

Like terms

apple, apples, like terms
apple, apples, like terms

Apples and apples are the same --> They are like terms

dog, dogs, like terms
dog, dogs, like terms

Dogs and dogs are the same --> They are like terms

tomato, tomatoes, like terms
tomato, tomatoes, like terms

Tomatoes and tomatoes are the same --> They are like terms

letter A, letters A, like terms
letter A, letters A, like terms

Letter A and letter A are the same --> They are like terms

Letter C, letters c, like terms
Letter C, letters c, like terms

Letter c and letter c are the same --> They are like terms

number 1, numbers, like terms
number 850, numbers like terms

numbers and numbers are the same --> They are like terms 

!Important: numbers do not have to be the same to be like terms!

You may only add and subtract like terms! 3a + 2a = 5a and 6b+b = 7b. 

It is not possible to add or subtract non like terms. 3a + 2b = 3a + 2b you can not simplify this any further. 

The sum of like terms 

You may only add or subtract like terms. 

Imagine Sophie buying 3 apples

apples, like terms, product, sum

She could say that Sophie bought an  apple + apple + apple (sum) or with her abbreviations a + a + a.

She can write this as a product so, 3 apples = 3 . 1a = 3a 

Simplifying products 

Unfortunately, using pictures or making stories with simplifying products makes no sense. Therefore, you are going to learn how to simplify products in a more abstract way. 

7a .3z = 21az 

There is an invisible multiplication sign between the numbers and the letters. 

Therefore, you can also write the calculation as 7 . a . 3 . z 

You are allowed to swap the numbers and the letters because it is multiplication: 2 . 3 gives the same answer as 3 . 2. 

You will get 7 . 3 . a . z  or 3 . 7 . a . z or a different order.  Now, it is easier to calculate because the numbers and the letters are next to each other. 

a . a = a 2 ,  x . x = x 2 ,  l . l = l 2 ,  f . f = f 2  ...

If you are going to  multiply a letter by itself, the answer is going to be squared. 

This can be explained by looking at the area of a square.

square 4 by 4
x times x = x squared

Area square = length . width

Area square = 4 . 4 = 4 2  

The same is with letters. 

square h by h

Area square = length . width

Area square = h . h= h2 

You can also give the edges of the square a different letter and you will see that your answer is going to be squared. 

Examples

  •    a + a = 2a 

a - a = 0 

a . a = a2

  • 7ab + ab = 8ab 

(you may only add or subtract like terms)

  • 5t + 8q - t + 2q = 4t + 10q 

(you may only add or subtract like terms)

  • 4 . p . k . 5 = 20kp 

(multiply the numbers + write the numbers in alphabetical order) 

  • 7be . 2b = 14b2e 

(a number times itself is the number squared) 

Exercises 

Calculate 

a) 5a . 2b 

b) 7b . 9b 

c) 18y . 0 . 3b 

d) -s . u 

e) 3a + 7a 

f) 10pq + pq 

g) 5ab + 6a + 7b + 3ab + 2a 

h) 5gh . g . 4 . a 

i) 6a + 7 . 2a - 5 

 

order of operations, order of operations pyramid