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Math Unit 3 (notes) Fractions and Decimals
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Math Unit 3 (notes) Fractions and Decimals
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| Learning Outcomes:
The student will:
Cuisenaire rods and pattern blocks helps one visualize equivalent fractions. Check out the following links:
http://teachertech.rice.edu/Participants/silha/Lessons/equivalent.html
Integer Bars (Cusenaire Rods): Learning Fractions
Pattern Blocks: Exploring Fractions with Shapes
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Review
Fraction Man
FUNBRAIN - Fresh Baked Fractions
Learning Planet - Fraction Frenzy
Quia! Matching Fractions
Quia! Equivalent Fractions
| Concept/Activity |
Example(s) |
Explanation/General Case |
| fraction ("part") |
1/2, 1/3, 2/3, 7/9,
5/4, 1 1/2, 3 1/4 |
means "part" of a whole
What is Fraction - Practice Game |
| fraction diagram |
blue portion = 3/5 |
3 parts out of 5 total pieces are blue. |
| numerator |
- 3
- 5
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The top number in a fracton is the numerator and represents the number of pieces being considered. |
| denominator |
The bottom number in a fraction is the denominator and represents the total pieces in one whole. |
| proper fraction |
1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 6/9 |
the numerator (top) is smaller than the denominator (bottom). |
| improper fraction |
3/3, 4/3, 5/4, 12/7 |
the numerator (top) is greater than or equal to the denominator (bottom). |
| mixed number |
1 1/2, 3 1/4, 5 3/7 |
natural number (1, 2, 3, ...) and proper fraction combined |
| basic fraction |
1/2, 1/3, 2/3, 7/9 |
a fraction reduced to lowest terms. The greatest common factor (GCF) of the numerator and denominator is one. |
| reducing to lowest terms |
12/18 = 6/9 = 2/3
15/20 = 3/4
1 5/10 = 1 1/2 |
divide numerator and denominator by the same number until the GCF is one. |
| equivalent fractions |
2/3 = 4/6 = 6/9 = 8/12 |
two or more fractions with the same basic fraction |
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2/3 = 4/6
2 x 6 = 3 x 4 |
two fractions with equal cross products
Equivalent Fractions Game |
| converting a mixed number to an improper fraction |
1 2/3 = (1 x 3 + 2)/3 = 5/3 |
- (natural number x denominator + numerator)
- denominator
or
- (side x bottom + top)
- bottom
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| converting an improper fractions to a mixed number |
7/4 = 1 3/4 |
Divide numerator by the denominator to get the natural number. The remainder of the quotient is placed above the denominator. |
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Learning Outcomes:
The student will:
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| Directions |
Example(s) |
| Decimal numbers:
3458.32 and 3458.318
Line up the ones digit for the numbers. Add zeros if necessary to give both numbers the same number of decimal places.
Compare the place value for each digit starting from the left. When you find the first difference, the number with the largest digit is the largest number.
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- 3458.320
- 3458.318
The ones digits are lined up and there are the same number of decimal places. Starting from the left, the first difference is in the tenths column.
Since 2 > 1, then 3458.320 > 3458.318 |
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| Directions |
Example(s) |
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When the denominators are the same, then the proper/improper fraction with the largest numerator is the largest fraction |
- 6/5 > 5/5
- 5/5 > 4/5
- 4/5 > 3/5
- 3/5 > 2/5
- 2/5 > 1/5
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When the numerators are the same, then the proper/improper fraction with the smallest denominator is the largest fraction. |
- 9/7 > 9/8
- 9/8 > 9/9
- 9/9 > 9/10
- 9/10 > 9/11
- 9/11> 9/12
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When proper/improper fractions have different numerators and denominators, they can be converted to equivalent fractions with the same denominators. |
1/2 = 3/6 1/3 = 2/6
Since 3/6 > 2/6, 1/2 > 1/3 |
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When proper/improper fractions have different numerators and denominators, the decimal equivalent of each can be calculated by dividing the numerator by the denominator. |
1/2 = 0.5 3/4 = 0.75
Since 0.75 > 0.5, then 3/4 > 1/2 |
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Proper/improper fractions can be compared using cross products. Start by review equal fractions.
a/b = c/d (equal fractions)
ad = bc (cross products are equal)
If the fractions are not equal then there are two possiblities.
If ad > bc, then a/b > c/d
If ad < bc, then a/b < c/d
If you are going to use this rule make sure the first fraction is a/b. |
Compare 4/5 and 2/3
Since 4(3) > 5(2), therefore 4/5 > 2/3
Algebra.help -- Fraction Inequality Calculator |
Compare Fractions
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| Directions |
Example(s) |
| When the whole numbers are different, you will not need to compare the fractions.
13 5/6 and 14 4/5
Line up the ones digit for the whole numbers.
Compare the place value for each digit starting from the left. When you find the first difference, the number with the largest digit is the largest number. |
- 13 5/6
- 14 4/5
The ones digits are lined up. Starting from the left, the first difference is in the tens column.
Since 4 > 3, then 14 4/5 > 13 5/6 |
| When the whole numbers are the same, then the mixed number with the largest fraction has the greatest value. Use the same rules as you did in:
Rules for Ordering Proper/Improper Fractions |
- 9/7 > 9/8
- 9/8 > 9/9
- 9/9 > 9/10
- 9/10 > 9/11
- 9/11 > 9/12
| Learning Outcomes:
The student will:
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Area of a rectangle can be calculated by multiplying the length and width of the rectangle. Any product can be shown as the area of a rectangle. The length and width are the factors and the product is the area. Let's look at an interactive example where we can change the length and width of the rectangle and the area created. In this model the dimensions are natural numbers. A similar model could be created that included decimal numbers for the dimensions of the rectangle.
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We will need some blocks to illustrate decimal products. You will need to understand the value of each type of block:
To illustrate the product:
3.2 x 2.4
we will need to combine blocks together
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length = 3.2 units |
width = 2.4 units |
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becomes
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3.2 units |
- 2.4
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Product = Area = 6x1 + 16 x 0.1 + 8 x 0.01 = 7.68
The following internet site(s) will give you the opportunity to build more rectangles to demonstrate decimal multiplication.
Base 10 Blocks: Exploring Whole Decimal Numbers with Blocks
or
Base 10 Blocks: Exploring Whole Decimal Numbers with Blocks
You have had an opportunity to visualize multiplication of decimals with rectangles. Now it is time to learn how to multiply decimal numbers without a visual aid.
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| Example |
Explanation |
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4.36 x 6.2
- 4.36
- 6.2
- 872
- 26160
- 27.032
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There will be 3 digits after the decimal in the answer, because there are 3 numbers after the decimal in the factors. | |
Line up the numbers as if there were no decimals.
Multiply as usual.
Count the total number of digits after the decimals in the factors.
The number of digits counted equals the number of digits after the decimal in the quotient.
Place decimal in answer. | Dividing decimals
Using Base-ten Blocks - HINT - (first divide long hand to get the second factor!!)
Divide 5.39 by 1.1
You are not dividing by a whole number, so you need to move the decimal point so that you are dividing by a whole number:
| move 1 |
| 5.39 |
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53.9 |
| 1.1 |
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11 |
| move 1 |
You are now dividing by a whole number, so you can proceed:
Ignore the decimal point and use Long Division:
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049
11 )539
5
0
53
44
99
99
0 |
Put the decimal point in the answer directly above the decimal point in the dividend:
The answer is 4.9
NOW you would draw your base ten blocks - 4.9 and 1.1 (see above drawing it just like multiplication)!!!
Percentages (%)
Percentage means parts per 100
When you say "Percent" you are really saying "per 100"
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So 50% means 50 per 100
(50% of this box is green) |
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A Percentage can also be expressed as a Decimal or a Fraction
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| A Half can be written... |
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| As a percentage: |
50% |
| As a decimal: |
0.5 |
| As a fraction: |
1/2 | |
Some Worked Examples
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Calculate 25% of 80 |
| 25% = 25/100 |
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(25/100) × 80 = 20 |
So 25% of 80 is 20
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A Skateboard is reduced 25% in price in a sale. The old price was $120. Find the new price |
Find 25% of $120 |
| 25% = 25/100 |
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(25/100) × $120 = $30 |
25% of $120 is $30 |
So the reduction is $30 |
| Take the reduction from the original price |
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$120 - $30 = $90 |
The Price of the Skateboard in the sale is $90 |
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The Word
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"Percent" comes from the latin Per Centum. The latin word Centum means 100, for example a Century is 100 years. |
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