Topics:
1. Using Formulas
2. Mathematical Properties
a. Commutative
b. Associative
c. Identity
3. Distributive Property
Using Formulas
Formulas are equations that show the way quantities relate to one another. We most commonly see these when shopping. Fruits and vegetables are going to be so much per pound, so you would use the formula cost = unit price x weight. The hardest part of this is making sure you put the numbers into the correct spot of the formula.
Sample Problem:
Orange juice costs $0.13 per ounce. What would it cost to purchase 8 oz. of orange juice?
To solve this, you would put the numbers from the problem into the sample formula cost = $0.13 x 8 making the cost $1.04.
That is the basics of the assignment. We are also going to do a task that combines the idea of putting numbers in for variables along with following the order of operations, allowing us to solve something like this:
If x=3 and y=4, then evaluate 4x²+ 3y.
If you put the numbers in for the variables, the expression becomes
4(3)²+3(4)=
4(9)+12=
36+12=
48
The one we will be doing in class, will have more steps, but will follow the same basic idea.
(Remember that a number right next to a parenthesis is just another way of saying multiply. Especially when we start working with variables, you will see the ‘x’ used less and less to represent multiplication. You may however see either of these two symbols as well * or •. Division will also have some new symbols. You may see the standard ÷, but you may also see a /. Division can also be represented as a fraction. ¼ = 1 ÷ 4 = 0.25.)
Mathematical Properties
The mathematical properties tell us things that are true under any situation. There are four of them we will be dealing with this year, but we will certainly be putting the most focus on the distributive property, which I will get to later in this blog.
Commutative Property -
The base word commute means to travel. The commutative property means the numbers can switch places and will still wield the same results. The commutative property works only with addition and multiplication.
Examples:
- 4 + 3 = 3 + 4 (They both equal 7)
- 7 x 5 = 5 x 7 (They both equal 35)
Non-examples:
- 4 – 3≠ 3 – 4 (1 ≠ -1)
- 8 ÷ 2 ≠ 2 ÷ 8 (4 ≠ 0.25)
Associative Property -
When we associate things, we connect them together. We may think of the people we associate with as our friends. This is the same with numbers. The associate property allows us to connect certain numbers together, without there being any change to the value of the expression. Again, this only works with addition and multiplication. Notice in the examples, we represent these “associations” by separating them with parenthesis.
Examples:
- 3 + (4 + 5) = (3 + 4) + 5 (They both equal 12)
- 2 x (5 x 6) = (2 x 5) x 6 (They both equal 60)
Identity Property -
The identity property takes a number and has it act as a mirror. There are two numbers. With addition and subtraction, the identity number is 0 because any number add or subtract 0 is still the same number.
3 + 0 = 3
7 – 0 = 7
The other identity number is for multiplication and division. It is 1 because any number multiplied or divided by 1 is still the same.
5 x 1 = 5
9 ÷ 1 = 9
Distributive Property
The distributive property get it’s own lesson because it is our main focus in 6th grade with the mathematical properties. The basic idea behind it is a number sharing itself through multiplication. Example:
3(7 + 4) =
3(7) + 3(4) =
21 + 12 =
33
where 3(7+4) = 3(11) = 33.
This property really comes of value when trying to multiply larger numbers, let’s say 3 x 112. The larger math is tougher to do, especially in your head, but if you break it apart, 112 = 100 + 12, these two numbers are a lot easier to multiply by 3.
3 x 112 =
3(100 + 12) =
3(100) + 3(12) =
300 + 36 =
336
You could also split apart other numbers. Maybe you’re not so good with your 8 times tables and there is a problem that requires you to do 6 x 8. You could split that apart just as easily:
6 x 8 =
6(5 + 3) =
6(5) + 6(3) =
30 + 18 =
48
These concepts can take some harder problems and really simplify the math you might have to do.