Download Unit 1 – The Number System Class Notes Date Greatest Common

Unit 1 – The Number System
Class Notes
Date
Greatest Common Factor (GCF)
Learning Target: I can find the greatest common factor of two numbers.
Key Terms


Factor: A number that is multiplied by a number to get a product. For example, the factors of 6 are 1, 2, 3 and
6.
Greatest Common Factor: The largest common factor of two or more numbers. This is sometimes referred to
as the GCF.
Methods
1. List the factors.
2. Use a Ladder Diagram
Examples
1) Find the GFC of 16 and 24.
Method 1: List the factors
2) Find the GFC of 12 and 18.
Method 2: Use a Ladder Diagram
3)
There are 12 boys and 18 girls in Mr. Smith’s science
class. The students must form lab groups. Each
group must have the same number of boys and the same number of girls. What is the greatest number of groups Mr.
Smith can make it every student must be in a group?
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The GCF of 12 and 18 is 6, so Mr. Smith students can create 6 lab groups.
Try This
Write the GCF of each set of numbers using Method 1.
1) 10 and 35
2) 28 and 70
3) 36 and 72
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Page 1
Write the GCF of each set of numbers using Method 2.
4) 60 and 84
5) 45 and 75
6) 42 and 56
Write the GCF of each set of numbers using either method.
7) 66 and 88
8) 14 and 17
9) 28 and 42
Solve the following word problems.
10) The local recreation center held a scavenger hunt. There were 15 boys and 9 girls at the event. The group was
divided into the greatest number of teams possible with the same number of boys and each team and the same number
of girls on each team. How many teams were made if each person was on a team?
11) Ms. Kline makes balloon arrangements. She has 32 blue balloons and 24 yellow balloons. Each arrangement must
have the same number of each color. What is the greatest number of arrangements that Ms. Kline can make if every
balloon is used?
12) Peter has 18 oranges and 27 pears. He wants to make fruit baskets with the same number of each fruit in each
basket. What is the greatest number of fruit baskets he can make?
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Page 2
Date
Least Common Multiple (LCM)
Learning Target: I can find the least common multiple of two numbers.
Key Terms


Least Common Multiple: The smallest number, other than zero, that is a multiple of two or more numbers.
Also referred to as the LCM.
Multiple: The product of any number and a whole number is a multiple of that number. For example, multiples
of 2 include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20….
Methods
1. List the Multiples.
2. Use a Ladder Diagram
Examples
Find the LCM of 5 and 6.
Method 1: List the multiples
Find the LCM of 8 and 12
Method 2: Use a ladder diagram.
5: 5, 10, 15, 20, 25, 30, 35, 40, 45…
6: 6, 12, 18, 24, 30, 36, 42, 48, 54…
LCM: 30
Try This
Write the LCM of each set of numbers using Method 1.
1) 6 and 20
2) 5 and 8
3) 10 and 15
Write the LCM of each set of numbers using Method 2.
4) 6 and 7
5) 22 and 55
6) 14 and 6
Write the LCM of each set of numbers using either method.
7) 21 and 63
8) 6 and 16
9) 28 and 10
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Page 3
Solve the following word problems.
10) Two students are stacking blocks, one on top of the other. Reece’s blocks are 5 cm high and Maddy’s blocks are 8
cm high. How tall will their stacks be when they are the same height for the first time?
11) During a promotion, a music store gives a free CD to every fifteenth customer and a free DVD to every fortieth
customer. Which customer will be the first to get both gifts?
12) Pencils are sold in packs of 12 and erasers are sold in packs of 9. Mr. Jones wants to give each of 36 students a
pencil and an eraser. What is the least number of packs he should buy so that there are no left overs?
13) Tony wants to make 36 party bags. Glitter pens come in packs of 6. Stickers come in sheets of 4, and balls come in
packs of 3. What is the least number of each package he should buy to have 1 of each item in every party bag, and no
supplies left over?
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Page 4
Date
Dividing Fractions by Fractions
Learning Target: I can divide fractions.
Important Term:
 Reciprocal: One of two numbers whose product is one. For example, 2/3 and 3/2 are reciprocals.
Important Information
 You DO NOT need common denominators.
 To find the reciprocal of a fraction, flip it.
Ex. ¾ is 4/3
2/9 = 9/2
1/10 = 10/1
Steps
1. Multiply by the reciprocal.
a. Change the division sign to a multiplication sign.
b. “Flip” the fraction that comes after it.
2. Look to see if you can simplify before multiplying. Remember you can simplify any numerator with any
denominator as long as they have a common factor.
3. Multiply numerators.
4. Multiply denominators.
5. Make sure your answer is in simplest form.
 Don’t forget to ask yourself the following questions before moving on to the next problem. If you answer YES to
either of these questions you MUST fix that before moving on!!!
o Is this an improper fraction?
o Can it be reduced?
Example:
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Try This:
1) 2/3 ÷ 1/6
2) 5/8 ÷ 3/4
3) 9/10 ÷ 2/5
4) 9/16 ÷ 7/12
5) 11/12 ÷ 5/6
6) 11/15 ÷ 3/5
7) 7/8 ÷ 3/4
8) 5/6 ÷ 2/9
9) 7/8 ÷ 1/6
10) Julia has ¾ of a yard of felt. How many puppets can she make if she uses ¼ of a yard of felt for each puppet?
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Page 6
Date
Dividing Fractions by Whole Numbers & Mixed Numbers
Learning Targets:
 I can divide fractions by whole numbers.
 I can divide fractions by mixed numbers.
Important Information
 You DO NOT need common denominators.
 To find the reciprocal of a fraction, flip it. The reciprocal of…
Ex. ¾ is 4/3
2/9 is 9/2
1/10 is 10/1
 To find the reciprocal of a whole number or mixed number, make it a fraction first then flip it.
Ex. 8 = 8/1 is 1/8
2 ½ = 5/2 is 2/5
Steps
1. Look for any whole number of mixed numbers and make them fractions.
2. Multiply by the reciprocal.
a. Change the division sign to a multiplication sign.
b. “Flip” the fraction that comes after it.
3. Look to see if you can simplify before multiplying. Remember you can simplify any numerator with any
denominator as long as they have a common factor.
4. Multiply numerators.
5. Multiply denominators.
6. Make sure your answer is in simplest form.
 Don’t forget to ask yourself the following questions before moving on to the next problem. If you answer YES to
either of these questions you MUST fix that before moving on!!!
o Is this an improper fraction?
o Can it be reduced?
Examples:
1)
2)
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Page 7
Try This:
1) 9/16 ÷ 6
2) 12 ÷ 2 ¼
3) 7 ÷ 1 ¾
4) 1 3/7 ÷ 7/12
5) 15/16 ÷ 1 5/6
6) 3 3/11 ÷ 1 4/5
7) 1 ½ ÷ ¾ ∙ ⅖
8) 2 ¾ ÷ 1⅔ ÷ 5
9) 11/12 ∙ 9/10 ÷ 1 ¼
10) A washing machine holds 36 gallons of water. It fills at a rate of 8 ⅖ gallons a minute. How long does the machine
take to fill with water?
11) A roll contains 80 ½ feet of ribbon. It takes 3 ⅚ feet of ribbon to make a bow. How many bows can be made?
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Page 8
Date
Locating a Point and Its Opposite on a Number Line
Learning Target: I can locate a point and its opposite on a number line.
Important Terms:
 Positive Number: Any number greater than 0. They may be written with a positive sign (+) but they are usually
without it. Ex 8 or +8
 Negative Number: Any number less than 0. They are always written with a negative sign (-). Ex. -8
 Opposite: Numbers that are the same distance from 0 on a number line but on different sides of 0. Ex. 3 and -3
 Integer: A set of all whole numbers and their opposites.
o Whole Numbers: Zero and the counting numbers. Ex. 0, 1, 2, 3, 4, …
Examples:
Try This:
Graph each integer and its opposite on the number line.
1)
7
2)
-6
3) 0
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Page 9
Date
Describe Quantities Using Positive and Negative Numbers
Learning Target: I can describe quantities using positive and negative numbers.
Important Information:


Positive Number: Any number greater than 0. They may be written with a positive sign (+) but they are usually
without it. Ex 8 or +8
Negative Number: Any number less than 0. They are always written with a negative sign (-). Ex. -8
Examples:
Name a positive or negative number to represent each situation.
Write a situation that each integer could represent.
A. 15
saving $15.00 dollars
B. -9
a loss of 9 yards in football
C. 18
an increase of 18 points
D. 0
zero degrees outside
Try This:
Name a positive or negative number for each situation.
1) Earning $50
3) 7 feet above sea level
2) 20° below zero
4) a decrease of 39 points
5) a house sinking 5 inches
6) elevator going up 5 floors
Write a situation that each integer could represent.
7) +49
9) -83
8) -7
10) +15
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Page 10
Date
Absolute Value
Learning Target: I can identify and explain the absolute value of a number.
Important Terms:
 Absolute Value: The distance a number is from 0 on a number line. The symbol for absolute value is ││.
Ex. │-5│ = 5 and │3│ = 3
Important Information:
 Absolute values are never negative.
 Opposite integers have the same absolute value. Ex. │-5│ = 5 and │5│ = 5
 The absolute value of zero is zero. Ex. │0│ = 0
Examples:
Try This:
Use the number line to find the absolute value of each integer.
1) │-9│
3) │6│
2) │3│
4) │-2│
Find each absolute value.
5) │419│
8) │96│
6) │-295│
9) │723│
7) │-806│
10) │-150│
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Page 11
Date
Converting Between Decimals and Fractions
Learning Target: I can convert between decimals and fractions.
Important Terms:
 Mixed Number: A number that contains both a whole number greater than zero and a fractions, such as 1 ½.
 Terminating Decimal: A decimal number that has a finite number of decimal places, a decimal that ends or
terminates, such as 0.75.
 Repeating Decimal: A decimal number that has a block of one or more digits that repeats continuously, such
as 0.6666…
Important Information:
Examples:
Part A: Writing Fractions as Decimals
Part B: Writing Decimals as Fractions
Try This:
Write each fraction or mixed number as a decimal.
1) 2/5
2) 2 7/8
3) 2 5/6
4) 17/20
Write each decimal as a fraction or mixed number.
5) 0.15
6) 2.6
7) 0.625
8) 3.375
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Page 12
Date
Compare Rational Numbers
Learning Target: I can compare rational numbers, including integers, fractions and decimals.
Important Information:
 >, is the symbol for greater than.
 <, is the symbol for less than.
 =, is the symbol for equal to.
Examples:
Part A:
Comparing Decimals
Part B:
Ordering Decimals
A. 8.04
8.403
<
When comparing the tenths place, the 0 in 8.04
is smaller than the 4 in 8.403. As a result, 8.04 is
less than 8.403.
B. 0.0906
-0.6801
>
When comparing a positive and a negative number
the positive number is always the greater one,
regardless of the numbers. As a result, 0.0906 is
greater than -0.6801.
Comparing Fractions
Ordering Fractions
(you can also change all fractions to decimals and then follow the steps above)
Comparing Integers
Ordering Integers
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Page 13
Try This:
Compare. Write <, > or =.
1) 3.578
3.758
2) 3/5
2/21
3) -8
-9
4) 0.78
7/9
5) 0.65
- 2/3
6) -21
-1
Order the fractions, decimals and integers from least to greatest.
7) 32.525, 32.52524. 31.6257
8) 2/5, 4/9, 11/15
9) -18, -9, -31, 15, 0
Order the fractions, decimals and integers from greatest to least.
10) -0.38, ¾,- 1/5
11) 2/5, 0.7, -2/3
12) -67, 82, -73, -10, 20
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Page 14
Date
Graph/Identify Points
Learning Target: I can graph and identify point on the coordinate plane in all four quadrants.
Important Terms:
 Coordinate grid (plane): A plane formed by the intersection of a horizontal line called the x-axis and a vertical
line called the y-axis.
 Ordered pair: A pair of numbers that can be used to locate a point on a coordinate plane.
 Origin: The point where the x-axis and y-axis intersect on the coordinate plane. (0,0)
 Quadrant: The x-axis and y-axis divide the coordinate plane into four regions. Each region is called a quadrant.
 x-coordinate: The first number in an ordered pair. It tells the distance to move right or left from the origin.
 y-coordinate: The second number in an ordered pair. It tells the distance to move up or down from the origin.
Important Information:
Examples:
Part A: Identifying Quadrants
Part B: Locating Points on a Coordinate Plane
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Page 15
Part C: Graphing Points on a Coordinate Plane
Try This:
Name the quadrant where each point is located.
1. Q
2. X
3. Y
4. Z
Give the coordinates of each point.
5. P
6. R
7. T
8. H
Graph each point on a coordinate plane.
9. L (0, 3)
10. M (3, -3)
11. V (-4, 3)
12. N (-2, -1)
13. T (-5, 0)
14. S (1, -4)
Name the quadrant or axis each ordered pair is located.
15. (3, -1)
16. (0, 5)
17. (-4, -4)
17. (-2, 3)
18. (-12, 0)
19. (21, 13)
20. (0, 0)
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