Math With Sheppard



Unit #1: Numeracy

Name: ____________________________________________

Unit Outline:

|Date |Topic |Assignment Completed |

| |1.0 Introduction and Grade 9 Diagnostic | |

| |1.1 Integers Operations | |

| |1.2 Powers and Exponents | |

| |1.3 Order of Operations and Substitution | |

| |Mid Unit Test #1 – Lessons 1.1 to 1.3 | |

| |1.4 Introduction to Rational Numbers | |

| |1.5 Multiplication and Division of Rational Numbers | |

| |1.6 Addition and Subtraction of Rational Numbers | |

| |Mid Unit Test #2 – Lessons 1.4 to 1.6 | |

| |1.7 Order of Operations with Rational Numbers | |

| |1.8 Numeracy Review | |

| |Final Numeracy Unit Test on 1.1 to 1.8 | |

Please note: Use of Calculators in this Unit

Although calculators are able to quickly add, divide, multiply and subtract, calculators should not be used as complete fallback plan – you still need to know how to do math long-hand. Understanding the math behind the calculators will be very beneficial for future success in algebra and advanced mathematics. I will encourage you to not use a calculator in this unit, but you will be allowed to use a calculator if you wish.

1.1 Integer Operations

Warm-up:

|a) Represent +5 using a diagram |c) Give 5 situations where we use integers in our daily life |

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|b) Represent -4 using a number line | |

|d) Another day, the temperature started at 3 C, increased 5 C, and dropped 15 C by|e) What is an integer? Explain using words. |

|midnight. Use a number line to represent this situation. What was the temperature | |

|by midnight? | |

|f) Every integer has an opposite integer. Use a number line to describe and define|e) List any rules that you can remember about integers from grade 8. |

|opposite integers | |

Learning Goals: By the end of this lesson, you will be able to:

Adding Integers: To add integers using a number line, start at the location of the first integer and move the number of spaces (positive to the right, negative to the left) indicated by the next integer in the expression.

Practice Problems:

a) 10 + (-3)

b) -4 + (-8)

c) -7 + 5

Reflect: When an integer is written without a sign in front, how is this integer marked on the number line?

Reflect: Why do some integers need brackets around them, while others do not?

Subtracting Integers:

Subtracting Integers Adding the Opposite

2 – 2 = 2 + (-2) =

2 – 1 = 2 + (-1) =

2 – 0 = 2 + (0) =

2 – (-1) = 2 + (+1) =

To subtract an integer, _______________________________________________________________.

Practice Problems: Rewrite each subtraction sentence as an addition, then evaluate.

a) – 5 – 3 b) 4 – (-2) c) -6 – (-3)

Reflect: When you are subtracting two negative integers, is the result going to be positive, negative, or could it be both? Explain your answer with an example.

Multiplying Integers: Let’s look at some patterns in multiplication

|4 x 3 = |(-5) x 3 = |

|4 x 2 = |(-5) x 2 = |

|4 x 1 = |(-5) x 1 = |

|4 x 0 = |(-5) x 0 = |

|4 x (-1) = |(-5) x (-1) = |

|4 x (-2) = |(-5) x (-2) = |

|4 x (-3) = |(-5) x (-3) = |

a) From the patterns what is the sign of a product of a positive integer and a positive integer?

b) From the patterns what is the sign of a product of a positive integer and a negative integer?

c) From the patterns, what is the sign of the product of a negative integer and a positive integer?

d) From the patterns, what is the sign of the product of a negative integer and a negative integer?

Reflect: How can we remember the rules for multiplying integers?

Dividing Integers: Since multiplication and division are inverse operations, the rules for division of integers are the same as the rules for multiplication of integers.

Practice Problems:

a) 21 ÷ (-7) b) (-12) ÷ (-4) c) (-9) ÷ (-1)

Reflect: Explain what a negative integer divided by a positive integer gives a negative answer. Use an example to support your explanation.

Assignment 1.1 Integer Operations

*Please review the homework assignment criteria page before completing this assignment

1. Put in order from least to greatest.

a) -2, 0, 1, -3, -1, 3, 2 b) -7, 0, -3, -13, 5, -10, 7

2. Add

a) 4+(-2) b) 3+(-5) c) 2+(-6) d) -3+(+7)

e) -3+(-2) f) -4+(+4) g) -7+(-8) h) 5+(-7)

i) 0+(-9) j) 10+(-1) k) 7+(+8) l) -10+(-1)

m) -2+(-5)+(+6) n) 9+(-3)+(-7) o) -1+(+5)+(-8)

3. Subtract

a) 6-(+12) b) 2-(+6) c) 5-(-2) d) 3-(+7)

e) -5-(+3) f) -2-(+5) g) 4-(-8) h) -3-(-7)

i) 0-(-6) j) -8-(-2) k) -7-0 l) 0-(+4)

m) 6-(-1) n) -3-(-2) o) 0-(+3) p) 3-(-9)

4. Multiply

a) 3(+4) b) 2(-3) c) 4(-5) d) -3(-1)

e) -2(4) f) -5(+2) g) (+7)(+2) h) (0)(-7)

i) 6(0) j) -6(5) k) -5(+6) l) (+7)(-3)

m) (-5)(-4) n) -2(+3)(-4) o) (+3)(+4)(-5) p) -4(-2)(-3)

5. Divide

a) [pic] b) [pic] c)[pic] d) [pic]

e) [pic] f) [pic] g) [pic] h) [pic]

i) [pic] j) [pic] k) [pic] l) [pic]

6. Evaluate

a) 5 + 9 – 7 b) -3 + 8 – 1 c) 2 – 6 – 3 + 1

d) – 1 – 2 + 9 e) 5 – 3 – 7 +12 f) – 8 + 4 – 10 – 2

7. Carla’s credit card statement showed that she owed $350. She made a payment of $200, then she charged $23 for gasoline. She returned a sweater she had charged earlier for a refund of $35. Then she charged $15 at the bookstore. What is her new balance? 9)

8. In Detroit the high temperatures in degrees Celsius for five days in January were -12°, -8°, -3°, 6°, -15°. What was the average temperature for these five days?

9. A football team gained 6 yards on a first down, lost 15 yards on the second down, and gained 12 yards on the third down. How many yards do they need to gain on the fourth down to have a 10 yard gain from their starting position?

1.2 Powers and Exponents

Warm-up:

|Evaluate: |Write in expanded from, and then evaluate |

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|(-3)(-3)(-3)(-3)(-3) |(-5)4 |

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|(-2)(-2)(-2)(-2)(-2)(-2)(-2) |36 |

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|One day, a student started a rumour. The very next day, the student told two |Suppose the trend continues, how many new people will know the rumour on the 5th day? |

|of her friends. The day after that, those two friends each told two more | |

|friends. Draw a diagram to represent the situation. | |

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| |Suppose the trend continues, how many new people will know the rumour on the 14th day?|

|Exponents are often used in measurement on units. Explain the difference |List 3 situations where you use exponents in real life. |

|between… | |

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|cm cm2 cm3 | |

Learning Goals: By the end of this lesson, you will be able to:

Review: Powers

A power consist of a __________________________ and an _________________________________

BaseExponent The exponent tells us how many times the base is used as a factor.

The exponent is also called the power of the base.

Example:

Exponential Form:

| Factored Form: | | |

| Standard Form: | | | |

Powers of 2: Complete the chart below

|Exponential |Factored |Standard |

|Form |Form |Form |

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|22 = | | |

|23 = | | |

|24 = | | |

|25 = | | |

|26 = | | |

|27 = | | |

Reflect:

Add 21 to you chart, what generalization can we make about any base raised to the exponent of 1?

your generalization with your calculator.

Reflect:

Add 20 to you chart, what generalization can we make about any base raised to the exponent of 0?

Check your generalization with your calculator.

Powers with Integer Bases

Practice Problems: Write the expanded form of each of the following, then evaluate.

a) (-5)2 b) (-3)5 c) (-1)7

Reflect: If m is 1, 2, 3, 4 and so on, how can you predict the sign of (-2)m?

Identifying the base

The base of any power is defined as the number that is directly to the left of the exponent, unless there are brackets, then the brackets enclose the base.

Practice Problems: Identify the base in each of the following, then write the expanded form of each of the following, and then evaluate

a) [pic] b) [pic] c) [pic] d) [pic]

e)[pic] f) [pic] g)[pic] h) [pic]

i) [pic] j) [pic] k) [pic] m) [pic]

Reflect: Why is the value of -2m never positive for any value of m?

Assignment 1.2 Powers and Exponents

*Please review the homework assignment criteria page before completing this assignment

1. Use repeated multiplication to show that 23 and 32 are not equivalent.

2. Use repeated multiplication to show that 23 and 2(3) are not equivalent.

3. State the base.

a) 26 b) (- 5)2 c) - 14 d) -93

4. State the exponent.

a) - 25 b) 42 c) (- 4)0 d) -5

5. Complete the following table.

|Power |Base |Exponent |Repeated Multiplication |Standard Form |

|25 | | | | |

|(-3)3 | | | | |

| |10 |4 | | |

| | | |-(2 x 2 x 2 x 2) | |

| |(-1) |5 | | |

6. Write in exponential form.

a) (- 4)(- 4)(- 4) b) (- 3)(- 3)(- 3)(- 3)(- 3) c) -(3)(3)(3)(3)(3)(3)

d) –(-5)(-5)(-5)(-5) e) [pic][pic][pic] f) [pic][pic][pic][pic][pic][pic]

7. Write each as repeated multiplication and then in standard form.

a) 32 b) (- 3)2 c) (- 1)4 d) [pic]

e) - 26 f) (- 2)6 g) [pic] h) - 32

8. A student was told that -23 and (-2)3 were the same. Is this correct? Use repeated multiplication and standard form to support your answer above.

9. Evaluate the following powers.

a) 63 b) 27 c) –42 d) (–2)6 e) 112 f) [pic]

g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic]

m) [pic] n) [pic] [pic] o) [pic] p) [pic] q) [pic] r) [pic]

s) [pic] t) [pic] r) [pic] t) [pic] u) [pic] v) -(-3)3

w) -(-2)4 x) 40 y) -100 z) (-5)0

1.3 Order of Operations and Substitution

Warm Up:

|a) Evaluate each power. |Explain what the B in BEDMAS represents. Use examples in your calculations. |

|(-2)2 (-2)3 (-2)4 (-2)5 | |

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| |Which operation would you perform first in each of the following? |

|b) Examine the signs of your answers. What pattern do you notice? | |

| |a) [pic] |

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|c) Explain how you can tell the sign of the answer when a power has a negative |b) [pic] |

|base. | |

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|Evaluate from left to right: |Determine the value of each of the following if x = -3 |

|[pic] | |

| |2x |

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| |x2 |

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| |x – 5 |

|Evaluate using the correct order of operations: | |

|[pic] | |

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| |5 – x |

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| |4x + 2 |

Learning Goals: By the end of this lesson, you will be able to:

Order of Operations

When you have more than one operation, the order of operations tells you which operation to use first.

Order of Operations

1. Simplify the expressions inside grouping symbols, like brackets.

2. Find the value of all powers.

3. Multiply and divide in order from left to right.

4. Add and subtract in order from left to right.

Practice Problems: Underline the operation you are going to do before you write the next line.

a) 18 - (-12 - 3) b) -92 + (7 + 4)3

c) 20 + -4 (32 - 6) d) -3 + 2(-6 ÷ 3)2

e) (-2)3 + (-16) ÷ 42 • 5 - (-3) f) 4(-6) + 8 - (-2)

15 – 7 + 2

Substitution with Integers

*Whenever we substitute an integer, we always replace the variable with brackets, then use the correct order of operations to evaluate.

Practice Problems:

Evaluate the following for a= -1, b = 3

i) a + 3b ii) (b – a)2

iii) ab – b iv) a2 + b2

Evaluate the following for x=5, y=-2, z=-1

i) x – y(3 – z) ii) xyz-y2

iii) xy – (x2 + y2) iv) z – y – x

Assignment 1.3 Order of Operations and Substitution

1. Evaluate

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

f) [pic]

g) [pic]

h) [pic]

i) [pic]

j) [pic]

k) [pic]

l) [pic]

m) [pic]

n) [pic]

o) [pic]

p) [pic]

q) [pic]

r) [pic]

s) [pic]

t) [pic]

u) [pic]

v) [pic]

w) [pic]

x) [pic]

y) [pic]

z) [pic]

aa)

2. Find the error in each solution. Explain what was done incorrectly. Redo the solution, making the necessary corrections.

a) [pic] b) [pic]

3. Evaluate each expression for the given values of the variables.

a) 3x4 x = 2 b) 2x2 + 5 x = 3

c) 4r2 – r r = -1 d) t2 – 2t t = -4

e) m2 + m – 4 m = -3 f) x2 – y2 x = -7, y = -5

4. Evaluate the expression [pic] when a = 5 and b = 3.

5. Evaluate the expression [pic] when x = 4 and y = 9.

6. Evaluate for x = -2 and y = 3.

a) x3 b) 5y4 c) x2 + y2 d) (x – y)3 e) (y – x)2 f) - 6xy g) 4x3 – 5y

1.4 Introduction to Rational Numbers

Warm-up:

|Explain the difference between a mixed number and an improper fraction. |List the first 10 factors of the following numbers |

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| |4: |

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|Convert the following mixed number to an improper fraction: |5: |

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| |6: |

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| |7: |

|Determine the unknown value for the equivalent fraction. Explain your reasoning|Compare [pic] and [pic]. Which fraction is larger? Explain your reasoning. |

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|a) [pic] | |

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|b) [pic] | |

|For every fraction, there are an infinite number of |Explain what it means to “reduce a fraction to lowest term”. Use an example in |

|equivalent fractions. Give 5 equivalent fractions to… |your explanation. |

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|[pic] | |

Learning Goals: By the end of this lesson, you will be able to:

What is a Rational Number?

Rational numbers: numbers that can be written in the form [pic], where a and b are _______________ with b ≠ ___________. We call a the ______________________ and b the __________________________.

Examples of Rational Numbers: [pic]

Non-repeating, non-terminating decimals like [pic]cannot be written as fractions and are examples of irrational numbers.

Negative Rational Numbers

Practice Problems: Convert each of the following rational numbers into decimals

a) [pic] b) [pic] c) [pic] d)[pic] e) [pic]

Reflect: What conclusions can we make about negative rational numbers?

|Practice Problems: Simplify the signs |

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|a) |[pic] |b) |[pic] |c) |[pic] |d) |[pic] |

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Review of Rational Numbers (Fractions):

|Converting Back-and-Forth Between Mixed Numbers and Improper Fractions |

|Sometimes it may be more appropriate to have a mixed number as an improper fraction, or vice versa. So we will have to be able to convert back and |

|forth between the two forms. |

|Mixed Numbers |Number composed of a ___________________ and a _____________________. |

| |If we have 4 whole parts and [pic] of another whole then the mixed number that represents this is |

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| |[pic] | |Numerator |

| | | |Denominator |

| | |Whole Number |

|Improper Fraction |Sometimes it is not helpful to have two different parts of a number and it is better to represent it as just a fraction. |

| |In these cases the numerator will be ______________________than the denominator because we will have more than one whole |

| |part, these are called __________________________________. |

| |[pic] | |Numerator |

| | | |Denominator |

Converting a Mixed Number to an Improper Fraction

Practice Problems: Write each of the following as an improper fraction

a) [pic] * b) [pic]

Converting an Improper Fraction to a Mixed Number

Practice Problems: Write each of the following as a mixed number.

a) [pic] b) [pic]

Equivalent Fractions

For every fraction, there are an infinite number of equivalent fractions.

Ex. [pic]

Multiplying or dividing both the numerator and the denominator by the same value will always yield an equivalent fraction.

Ex. [pic]

Practice Problems: Determine an equivalent fraction

|a) |

|e) |

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|1. |[pic] |2. |[pic] |3. |[pic] |4. |[pic] |

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|5. |[pic] |6. |[pic] |7. |[pic] |8. |[pic] |

Converting decimals to fractions:

Convert 0.625 into a fraction, and then reduce to lowest terms

(Hint: Your denominator will have the same number of zeros as there are decimal digits in the decimal number you started with ( 0.625 has three decimal digits so the denominator will have three zeros before reductions.)

|Practice Problems: Convert each of the following decimals into fractions |

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|a) |0.45 |b) |-0.8163 |c) |4.6 |d) |12.12 |

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Assignment 1.4 Introduction to Rational Numbers

*Please review the homework assignment criteria page before completing this assignment

A. Write as an improper fraction.

1. 1[pic] 2. 4[pic] 3. -1[pic] 4. -2[pic]

5. -6[pic] 6. -2[pic] 7. 1[pic] 8. 3[pic]

B. Write as a mixed number.

1. -[pic] 2. [pic] 3. [pic] 4. [pic]

5. -[pic] 6. [pic] 7. [pic] 8. [pic]

C. Write in lowest terms.

1. -[pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. -1[pic] 10. 2[pic] 11. 5[pic] 12. -3[pic]

D. Find the missing numerator

1. [pic] = [pic] 2. [pic] = [pic] 3. [pic] = [pic] 4. [pic] = [pic]

E. Convert the following decimals to fractions.

1. 0.225 2. 0.375 3. 0.0175 4. -0.95 5. 0.256 6. -0.45

1.5 Multiplying and Dividing Rational Numbers

Warmup:

|Evaluate: |Reduce the following rational numbers to lowest terms: |

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|a) -3(5) |-[pic] |

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|b) 4(-3) | |

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|c) (-10)(-5) | |

| |[pic] |

|d) 21÷ 7 | |

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|e) -12 ÷ 6 | |

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|f) (-36) ÷ (-6) | |

| |[pic] |

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|In a few sentences, jot down your strategies for multiplying and dividing | |

|integers: | |

| |Explain how to convert a mixed number into an improper fraction. Use the |

|What sign do you expect for the answer of [pic]? |following mixed number in your explanation. |

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| |[pic] |

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|What sign do you expect for the answer of [pic]? | |

|If it takes [pic] cups of flour to make a cake, then how much flour will you |Is dividing a number by 2 equivalent to multiplying a number by ½? Explain. |

|need to make [pic] of the recipe? | |

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Learning Goals: By the end of this lesson, you will be able to:

Multiplying Fractions

There are 2 ways to multiply fractions:

1) Multiply the numerators and multiply the denominators, and THEN reduce the final answer to lowest form (HARDER)

2) Reduce first, and THEN multiply the numerators and multiply the denominators (EASIER)

Practice Problem:

[pic] Method #1: Method #2:

[pic] Method #1: Method #2:

[pic] Method #1: Method #2:

Reciprocals of a Fraction

The reciprocal of a fraction is when the numerator becomes the denominator and the denominator becomes the numerator. This process is also called reciprocating.

Note that the product of a fraction and its reciprocal is always 1.

Practice Problems: Reciprocate the following fractions

a. [pic] b. [pic] c. [pic] d. [pic] f. [pic]

Dividing Fractions

DIVIDING by a fraction is the same as MULTIPLYING by the ________________________

o For example, [pic] is exactly the same as [pic]

o So to divide a fraction, we multiply by the reciprocal of the second fraction.

Practice Problems: Divide

a) [pic] ÷ [pic] = b) [pic] ÷ [pic] = c) [pic] ÷ [pic] =

Think about it: - Subtracting is equivalent to adding the opposite

- Dividing is equivalent to multiplying by the reciprocal

Multiplying and Dividing Mixed Numbers:

In most cases, it is easiest to convert all mixed numbers into improper numbers before multiplying.

Answers can be left in improper form.

Practice Problems:

a) 6[pic] × 1[pic] b) 2[pic] ÷ - 4[pic]

Assignment 1.5 Multiplying and Dividing Rational Numbers

*Please review the homework assignment criteria page before completing this assignment

A. Multiply.

1. [pic] × [pic] = 2. [pic] × [pic] = 3. [pic] × (-4)

4. [pic][pic] = 5. [pic] × [pic] = 6. [pic] × [pic] =

7. [pic] × 10 = 8. -1[pic] × [pic] = 9. [pic][pic] =

10. [pic] × 1 [pic] = 11. -18 × 1[pic] = 12. 16 × - 2[pic] =

13. 6[pic] × 1[pic] = 14. 2[pic] × 4[pic] × 3[pic] = 15. 4[pic] × 4[pic] × 2[pic]=

16. [pic] ÷ [pic] = 17. [pic] ÷ [pic] = 18. [pic] ÷ [pic] =

19. [pic] ÷ [pic] = 20. 4 ÷ [pic] = 21. [pic] ÷ [pic] =

22. [pic] ÷ [pic] = 23. [pic] ÷ [pic] = 24. [pic] ÷ 4 =

25. -15 ÷ [pic] = 26. 1[pic] ÷ 1[pic] = 27. 3[pic] ÷ 5 =

28. 6[pic] ÷ 2[pic] 29. 5[pic] ÷ [pic] = 30. [pic] ÷ 1[pic] ÷ [pic]=

31. A farmer made a square chicken coop with a length of [pic]m. Determine the area of the chicken coop

32. Gave made a patio area out of square patio stones. The area of his patio is [pic]sq ft and the length is [pic] ft. Determine the width of his patio.

1.6 Adding and Subtracting Rational Numbers

Warm Up:

|Evaluate: |Determine the Lowest Common Multiple (LCM) for the following numbers. |

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|a) -3 + 5 |a) 2,3 |

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|b) 4 + (-3) |b) 12,4 |

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|c) (-1) + (-5) |c) 5,8, 3 |

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|d) 3 – 7 |d) 24,16 |

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|e) (-2) – 6 |e) 15, 20 |

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|f) 3 – (-6) |f) 11,13 |

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| |Reflect: |

|Reflect: |In a few sentences, jot down your strategies for finding a lowest common |

|In a few sentences, jot down your strategies for adding and subtracting |multiple. |

|integers. | |

|Draw a diagram to represent the sum below |Draw a diagram to represent the sum below |

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|[pic] + [pic] |[pic] + [pic] |

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Learning Goals: By the end of this lesson, you will be able to:

Adding and Subtracting Fractions

Step 1 – Find a common denominator (a number that both denominators will go into)

Step 2 – Raise each fraction to higher terms as needed

Step 3 – Add or subtract the numerators only as shown

Step 4 – Carry denominator over

Step 5 – Reduce answer to lowest terms (leave all answers in improper form)

Practice Problems: Add or subtract.

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic] + 6

g) [pic] h) 4[pic] – 1[pic]= i) [pic]

Assignment 1.6 Adding and Subtracting Rational Numbers

*Please review the homework assignment criteria page before completing this assignment

1. [pic] + [pic] = 2. [pic] + [pic] = 3. [pic] + [pic] = 4.[pic] + [pic] =

5. [pic] - [pic] = 6. [pic] + 1[pic] = 7. [pic] - [pic] = 8. -2[pic] + 1[pic]=

9. 1[pic] + [pic] = 10. 2[pic] + [pic] = 11. [pic] – [pic] = 12. [pic] – [pic] =

13. [pic] – [pic] = 14. [pic] – [pic] = 15. [pic] – [pic] = 16. 1[pic] – [pic] =

17. 5[pic] – [pic] = 18. 3[pic] – [pic] = 19. 2[pic] – [pic] = 20. 4[pic] – 1[pic]=

21. If an [pic] inch nail is hammered through a board [pic] inches thick and into a support beam, how far into the support beam does the nail extend?

22. Sofia spent [pic] weeding her garden on Monday and[pic]on Tuesday.

a) How many hours did she spend weeding her garden altogether?

b) How many more hours did she spend weeding her on Monday than on Tuesday?

23. Alexis left her house at 7:45 pm to go shopping for clothes. She returned at 10:30 pm.

a) Express the time in hours (as a fraction) that Alexis spent away from home.

b) If Alexis spent [pic]hours shopping for clothes, then how much time did she spend doing other things?

1.7 Order of Operations with Rational Numbers

Warm Up:

|Evaluate: |

|a) [pic] b) [pic] c) [pic] |

|Write out the steps to follow when you are… |

|Multiplying rational numbers |

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|Dividing rational numbers |

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|Adding or subtracting rational number |

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|Evaluate : |

|a) [pic] b) [pic] c) [pic] |

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Learning Goals: By the end of this lesson, you will be able to:

Fractions raised to an Exponent

Practice Problems: Write each power in standard form then evaluate

a) [pic] b) [pic] c) [pic]

Order of Operations with Fractions

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

Assignment 1.7 Order of Operations with Rational Numbers

*Please review the homework assignment criteria page before completing this assignment

1. Evaluate

|[pic] |[pic] |[pic] |[pic] |

2. Evaluate if x = [pic] and y = [pic]

a) x2 b) xy c) 4y3 d) 3x + 2y

e) 6x2 f) (4x)(3y) g) y – xy h) -2y + 9x2

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Step 1 - Determine the place value of the last number in the decimal; this becomes the denominator.

Step 2hã=èhptB*[pic]OJQJmH phsH

h“ vhr&®%hã=èhR7B*[pic]OJQJmH phsH %hã=èhr&®B*[pic]OJQJmH phsH

h“ vhpt(hã=èh'^?5?B*[pic]OJQJmH phsH (hã=èhpt5?B*[pic]OJQJmH phsH hã=èh:ilOJQJhã=èhdirOJQJhã=èhÇ – Make the decimal number your numerator.

Step 3 - Reduce your answer.

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