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Unit One
- Number Relationships: Represent, order and compare rational numbers; expanded form; factors, multiples, prime and composite numbers; squares; square roots; applying order of operations in expressions with brackets
GCF & LCM.
Multiple tells you that you have to multiply. Factor tells you that you have to divide.
The two basic functions that you will be performing will be called the GCF (greatest common factor) and the LCM (least common multiple).
Even though you are looking for the GREATEST common factor, your number will still be small since you are dividing.
Even though you are looking for the LEAST common multiple, your number will be large since you have to multiply.
To find the GCF of 15, 70, and 100: I have to list all of the factors (numbers that divide into) 15, 70, 100
15--- (1, 3, 5, 15)
70---(1, 2, 5, 7, 10, 14, 35, 70)
100--(1, 2, 4, 5, 10, 20, 25, 50, 100)
I can see the largest number that is common is 5.
To find the LCM we will do the same thing, we just list the multiples of a set of numbers and find the smallest number.
5---(5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60)
20--(20, 40, 60, 80)
30--(30, 60, 90)
I can see the smallest number between the three sets of numbers is 60. This is the LCM.
Exponents
Exponents are just a way of telling how many times a number will be multiplied by itself.
When you see a number and exponent, this is what it really means.
54 5x5x5x5 (You will be seeing the number 5 multiplied 4 times)
The sum of 54 is 625.
(5x5 = 25, 25x5 = 125, 125x5= 625)
Square Roots
What does this have to do with perfect squares?
Perfect squares are nothing but a number times itself.
Below are the following perfect squares: 22 32 42 52 62 72
These are perfect squares: 2x2=4 3x3=9 4x4=16 5x5=25 6x6=36 7x7=49
Prime Factorization
Prime factorization breaks down a number's factors into prime numbers. This can be done using factor trees.
Order of Operations
Remember BEMDAS. We work from left to right in the direction that we read.
Using order of operations
Each of the letters in BEMDAS stands for a math operation that must be performed.
B- brackets (work out anything inside of these first)
E- exponents
M- multiplication D- division (What ever comes first)
A- addition S- subtraction (What ever comes first)
All that we do is sweep across the problem solving one operation at a time. Look at the example below.
73 + (20-2) – 12 ÷ 3
1. Moving left to right, I solve the brackets first.
73 + 18 -12 ÷ 3
2. Next I solve the exponents
343 + 18 -12 ÷ 3
3. Next I divide 12 by 3.
343 + 18 -4
4. Next I add 343 and 18.
361 -4 = 357
5. I am left with my basic problem of 361 – 4
357 is the answer
Ratios Rates & Proportions
Ratio: A comparison between two different amounts.
There are 3 ways to write ratios
8 to 3
8:3
8/3
A ratio is usually a part-to-part comparison, but it can be a part to whole comparison.
Example: The score was 15 to 4.
There are two parts being compared - the score of one team being compared to the score of the other team.
Proportion: Two ratios that are equal to each other.
Example: 4 cats/3 dogs = 24 cats/8 dogs
Proportions are used when two things are being compared and one of the parts is missing.
Example: Margaret knows that she can serve 7 people with 2 cans of green beans. She will be feeling 84 people at the luncheon. How many cans of green beans will she need to buy? 2 cans/7 people = N cans/84 people
N = 24 cans
Rate: A ratio comparing 2 amounts measured in 2 different units.
Example: The ratio below is comparing minutes to kilometers. These are two different units of measurement so this ratio is a rate.
23 min/5 km
Unit Rate: A unit rate is the amount for 1 item
Example: The car gets 32 miles per gallon of gasoline. This is a unit rate because we are talking about 1 gallon of gasoline
32 miles/1 gallon
A proportion can be used to find a unit rate.
Example: A bottle of shampoo cost $3.99 for 13.5 ounces. Find the unit rate.
$3.99/13.5 oz = N dollars/1 oz
N = about $0.30 per ounce
To find the %:
Divide the numerator (top) by the denominator (bottom)
Sofia served 36 out of 40 volleyballs over the net, what percent of the serves were good?
36 = 0.9 0.9 x 100 =90%
40
To find the % of a number:
1. Change the % into a decimal
2. Multiply the decimal by the whole number
Example: Sorcha went out for dinner and her bill came to $80. It is customary to leave a 15% tip. How much should she leave for the server?
1. 15%/100 = 0.15
2. 0.15 x 80 = 12
3. She should leave a $12 tip.
To find the % of a number when you know the end result:
75% of ____ = 12
Step 1: change the percent into a decimal 75%/100 = 0.75
step 2: divide the end result into the decimal 12 / .75
(DISCOUNT)
1. Convert percent into a decimal.
2. Multiply decimal by the original price.
3. Subtract amount off the original price.
E.G. Trevon wants to buy a pair of Air Jordans.
They cost $134.00. They are on sale for 33% off. What is the total cost?
33% is .33 as a decimal.
.33 X $134 = $44.22 (discounted amount)
$134 - $44.22 = $89.78 (total amount)
Discount is an amount taken off the price of an object.
Therefore we multiply by a number less than 1.
Tax is an amount added to the price of an object.
Therefore we multiply by a number greater than 1.
To solve tax:
1. Change the percent into a decimal
2. add 1 to the decimal
3. Multiply the price by the decimal plus 1
eg. Joel buys a Smart Car for $17995.00.
This price does not include the 15% sales tax.
What is the toal price for his car?
17995 x 0.15 = 2699.25
17995 + 2699.25 = $20694.25
17995 x 1.15 = $20694.25
Interest Rate
Interest = Price x Rate x Time
Cheat Sheet - Integers.pdf
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