7th-Ch 2 Study Guide-Big Ideas.doc

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Ch 2 Big Ideas Study Guide

Rational Numbers

Vocabulary:

 Rational Number – a number that can be written as a ratio in the form of a/b (where a and b are integers and b ≠ 0)

 Terminating Decimal – a decimal that ends (has a final digit).

 Repeating Decimal – a decimal that has a pattern that repeats. Can be written with bar notation. 1.3333… = 1.3

Lesson 1-

:

Numbers are either rational or irrational.

Rational Numbers -

A rational number is a number that can be written as a ratio in the form of a/b. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are integers.

 The number 8 is a rational number because it can be written as the fraction 8/1.

 Likewise, 3/4 or 2-1/2 are a rational numbers because they can be written as a fraction (2 ½ = 5/2).

 Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

 Rational numbers also include all terminating decimals (decimals that have a final digit) or repeating decimals (decimals that use bar notation because the repeat the same digits over and over).

Every whole number or integer is a rational number, because any they can be written as a fraction. For example, 4 can be written as 4/1, -65 can be written as -65/1, and 3,867 can be written as 3,867/1.

Irrational Numbers -

All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592…

= 1.414213… (in fact, any number that is not a perfect square is irrational)

Although irrational numbers are not often used in daily life, they do exist on the number line.

Writing Rational Numbers as Decimals:

If the # is a fraction, divide d n ( ¾ = 3 4 = 0.75)

If the # is a mixed #, change to improper 1st_{, then divide d n (-2 ¼ = -9/4 = -9 4 = -2.25)}

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Writing a Decimal as a Fraction:

1. Write the decimal as it is read. Examples: 0.125 = 125 = 1

 The #s before the decimal will be whole #s 1000 8

 The #s after the decimal will be the numerator -2.60 =

-2

60/100 =

-2

3/5

 The place value will be the denominator 2. Simplify

Comparing/Ordering Rational #s:

1. Write all rational #s in fraction decimal form. If fractions or mixed #s, divide (d n) 2. Compare/order decimal #s using a number line or place value chart.

 If using a # line use tick marks to represent tenths/hundredths

 If using a place value chart, line up the decimals/place values and compare from left to right. Example: Order the following from lest to greatest

5 , -1 ¾ , 13 , and -0.6 (a) 5 = Compare from left to right

9 8 9 9 5 = 0.555… = 0.5 (b) -1 ¾ = -7 =

4 4 -7 = -1.75 (c) -13 =

8 8 -13 = -1.625 (d) -0.6

Answer: -1 ¾ , - 13 , -0.6, 5 8 9

Lesson 2-3:

When you add/subtract rational numbers, use the same rules you would use to +/- integers. If they are decimals, don’t forget to line up the decimal points/place values to +/-. If they are fractions or mixed #s, don’t forget to find common denominators 1st_.

+/- Decimals:

1) Write the #’s vertically (up & down)

2) Line up the decimal point (the ones with the ones, the tenths with the tenths, etc…)

3) Blank spots represent a zero. If you would like, fill in any missing digits with zeros (or x’s) 4) Add or subtract beginning at the right and regroup as necessary (carry/borrow)

5) Decimal point is carried straight down into the answer. Example: 0.159 + 2.3 0.159

+ 2.3xx (Line up the decimals and fill in the blank spots with zeros or x’s)

2.459 (add like normal and bring the decimal straight down)

+/- Fractions:

1. If denominators are different, find a common denominator (use x__ = ___)

 This is a multiple that both #s share. If you can’t find the least common denominator multiply both denominators together.

2. Once denominators are the same, +/- the numerator and keep the denominator. 3. Simplify

Examples: (a) 2 + 1 = 3 (b) LCD Example: 1 + 3 1 x ___ = 2 (multiply by 2) 5 5 5 6 4 6 12

LCD = 12 3 x ___ = 9 (multiply by 3) 4 + 12

11

ones

_.

10th 100th 1000th

0 . 5 5 5

-1 . 7 5

-1 . 6 2 5

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12 (c) CD Example: 1 + 3 1 x ___ = 4 (multiply by 4) 6 4 6 24

(6 x 4 = 24 so CD = 24) 3 x ___ = 18 (multiply by 6) 4 + 24

22 = 11 (simplify) 24 12

+/- Mixed #s

1. Add the whole #’s (or change the mixed #s to improper fractions 1st₎

2. Follow the directions above for adding the fractions

3. Simplify (any improper fractions can be simplified and added to the whole #)

Example: 1 ½ LCD = 6 1 ½ x -- =

1

3/6 (multiply by 3) or 9/6 +

2

2/6 +

2

2/6 =

2

2/6 + 14/6

3

5/6

23/6 =

3

5/6

Adding Integer Rules:

Same Sign

1. Keep the common sign 2. Add the absolute values

Examples: 2 + 4 = 6 (Keep the positive and add) -6 + (-2) = -8 (keep the negative and add) Different Sign

1. Keep the sign of the integer with the bigger absolute value 2. Subtract the lesser value from the greater value.

Examples: 3 + (-2) = 1 (sign of the bigger and subtract) 2 + (-5) = -3 (sign of the bigger and subtract)

Subtracting Integers: COPY, CHANGE, CHANGE

(or) add the opposite of the 2nd_#

1. Copy the 1st_number _{Example : -6 - 4}

2. Change subtract to add -6 + (-4) = -10

3. Change the 2nd_{number to it’s opposite}

4. Follow addition rules for same or different sign.

+/- Rational #s:

1. If variables are used, substitute the given values in for each variable. 2. Follow the steps for +/- the decimal or fraction

3. Follow the integer rules for

+/-4. If necessary, use order of operations (PEMDAS)

Examples: (a) 4 + -3 4 x ___ = 8 (b) -3.92

5 10 5 10 + (-6.89) (same sign-keep the sign and add)

-3 (different sign-sign -10.81 + 10 of the bigger and subtract)

5 = 1 10 2

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3 6 =

-14

5/6 (same sign-keep the sign and add)

6

1 - 6

Lesson 4:

When you multiply/divide rational numbers, use the same rules you would use to x/divide integers.

Multiplying Decimals:

1. Write the #’s vertically (top to bottom), placing the # with more digits on top. 2. Multiply as if the decimals were not there.

3. Count the # of decimals places in the factor s to place the decimal in the product. Note: You may need to add a zero to the front of the product if there are not enough places in the product.

4. Estimate to check.

Example: 2.53 (2 decimal places)

x .002 (3 decimal places)

.00506 (answer needs 5 decimal places)

Dividing Decimals:

1. If there is a decimal in the divisor, move it all the way to the right. Move the decimal the same # of places in the dividend.

2. Use long division as you would with whole #s 3. Carry the decimal straight up from the dividend Examples:

(a) 3.1 (b) 5.9 4 12.4 0.6 3.54 06 35.4 -12 -30 0 4 54

- 4 -54

Multiply Fractions/Mixed #s:

1. Change any mixed #’s to improper fractions 2. Multiply the numerator

3. Multiply the denominator

4. Simplify (change any improper fractions back to mixed #’s) Fractions Example: 4 x 2 = 8 or 1 (simplified)

6 4 24 3

Mixed # Example: 2 ½ x 1 ¾

5 x 7 = 35 or

4

3/8 (simplified) 2 4 8

Dividing Fractions / Mixed #s:

Copy, Change, Flip 1. Change any mixed #’s to improper fractions

2. Change the division sign to multiplication 3. Use the Reciprocal of the 2nd_{fraction (flip it)}

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1 6

15 x 6 = 90 = 90 (copy, change, flip) 1 1 1

Mixed # Example: 2 ½ 1 ¾

5 7 (change mixed #s to improper fractions 1st₎

4 4

5 x 4 (copy, change, flip) = 20 = 5 (simplified) 4 7 28 7

Multiply/Divide IntegerRules:

E

xamples: 8 x 2 = 16 (same signs = positive)  Same sign = positive (-8) (-2) = 16 (same signs = positive)  Different Sign = negative 8 x (-2) = -16 (different signs = negative)

(-8) 2 = -16 (different signs = negative)

x/Divide Rational #s:

1. If variables are used, substitute the given values in for each variable. 2. Follow the steps for x/dividing the decimal or fraction

3. Follow the integer rules for x/divide

4. If necessary, use order of operations (PEMDAS) Examples: (a) (-4.8) (b) 3 ¼ (-1 ⅛)

x 0.2 13 -9 (change mixed to improper) -0.96 4 8

13 x -8 (copy, change, flip) 4 9

= -104 =

-2

32/36 =

-2

8/9 (simplify)

36 (c) 10 x

(-3

3/5) x (-3)

3

10 x (-18) x (-3) 3 5

-180 x (-3) 15

(-12) x (-3) = 36

Finding the Mean:

1. Add all the #s in the set

2. Divide by how many #s in the set

Example: Find the mean of (-25.63), 37.15, 18.92, and (-44.25) (-25.63) + 37.15 + 18.92 + (-44.25) = -13.81 =