Indicators1. Use models and visual representation to develop the concept of ratio as part-to-part and part-to-whole, and the concept of percent as part-to-whole..
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ExamplesRatios show the relationship of one group to another.
Part to part: Of the shapes above, you could say there are 3 red squares to every 7 blue circles. Part to whole: Of the shapes above, you could also say that 3 out of 10 shapes are red squares. Example: 3/10 Part to whole as percent: Since 3 out of 10 shapes are red squares, you could state that 30% of the shapes are red squares. Example: 3/10= 30/100= .30= 30 percent |
2. Use various forms of “one” to demonstrate the equivalence of fractions.
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One can be shown as fractions:
Example: 7/7 =1 2/2 = 1 14/14 = 1 Remember, any number multiplied by 1 is still that number. Example: 5x1= 5 The same is true for numbers that are fractions. Example: 3/5 x 2/2 = 6/10 Remember, any number divided by 1 is still that number. Example: 5 divided by 1 = 5 The same is true for numbers that are fractions. Example: 6/10 divided by 2/2 = 3/5 Why is it important? Understanding this helps us find common denominators when we add or subtract fractions. Understanding this also helps us reduce fractions to lowest terms. |
3. Identify and generate equivalent forms of fractions, decimals and percents.
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Percent means "out of 100." We use the percent symbol (%) to express percent. Percents are used everywhere in real life, so you'll need to understand them well. Here are some examples of ways to write the same thing:
1/4 = 25/100 .25 25% 3/5= 60/100 .60 60% 4/10= 40/100 .40 40% 1/3 (1 divided by 3) .33.. 33.3..% 2/3 (2 divided by 3) .66.. 66.6..% 3/8 (3 divided by 8) .375 37.5% |
4. Round decimals to a given place value and round fractions (including mixed numbers) to the nearest half.
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To round decimals
41.267 (7 is more than 5) 41.270 To round fractions
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5. Recognize and identify perfect squares and their roots.
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Squaring a number means taking a number times itself, such as 5 x 5:
52 = 5 x 5 = 25 Finding the square root of a number is the opposite (inverse operation) of squaring a number. 25 = 5 The array representing a number squared would be square, for example: 5 x 5 |
6. Represent and compare numbers less than 0 by extending the number line and using familiar applications; e.g., temperature, owing money.
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Negative numbers are numbers that are less than 0.
We use negative numbers to show when the temperature goes below 0 on a thermometer. We use negative numbers to show that we owe money, like when you don't have any money in your wallet and you have to borrow lunch money for $2.00, you might say you have -$2.00 (you owe more than you have!). We even use negative numbers on game shows like Jeopardy, such as when a contestant has 0 points and misses a question, they then have -200 points. |
7. Use commutative, associative, distributive, identity and inverse properties to simplify and perform computations.
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There is an inverse (opposite) relationship between addition and subtraction.
Example: 3 + 7 = 10.
Example: 10 - 3 = 7
It's important because we can use this to solve problems. Example: ? + 7 = 15 (Work backwards and use the inverse operation!) 15 - 7 = 8 ? = 8 The same thing is true for multiplication and division. |
8. Identify and use relationships between operations to solve problems.
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Commutative Property - An operation is commutative if you can change the order of the numbers involved without changing the result. Addition and multiplication are both commutative. Examples:
Addition: 2 + 1 = 1 + 2 Multiplication: 5 × 9 = 9 × 5 NOTE: Subtraction and division are not commutative. Examples: 4 - 3 is not equal to 3 - 4 6 ÷ 2 is not equal to 2 ÷ 6 Associative property - An operation is associative if you can group numbers in any way without changing the answer. It doesn't matter how you combine them, the answer will always be the same. Addition and multiplication are both associative. Addition: (3 + 2) + 1 = 3 + (2 + 1) Multiplication: (4 × 5) × 9 = 4 × (5 × 9) NOTE: Subtraction and division are not associative: (4 - 3) - 2 is not equal to 4 - (3 - 2) (12 ÷ 2) ÷ 3 is not equal to 12 ÷ (2 ÷ 3) Distributive Property - When you distribute something, you give pieces of it to many different people. The most common distributive property is the distribution of multiplication over addition. It says that when a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to the other two numbers and multiplied by each of them separately. 3 × (2 + 1) = (3 × 2) + (3 × 1) |
9. Use order of operations, including use of parentheses, to simplify numerical expressions.
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Using Order of Operations
Parentheses Exponents Multiplication and Division (left to right) Addition and Subtraction (left to right) A popular method for remembering Order of Operations is by thinking about this mnemonic: Please Excuse My Dear Aunt Sally |