math 8 catalina foothills school district overarching understandings...

MATH 8 CATALINA FOOTHILLS SCHOOL DISTRICT

Overarching Understandings for the Course: Students will understand Number Systems, Percents, Expressions, Equations, and Inequalities, Exponent Rules, Functions, Systems of Equations, and Statistics.

Number Systems: (No Calculators and Show All Work.) 1.) Complete the diagram by filling in the following words in the appropriate spot:

Natural Integer Real Irrational Rational Whole

2.) Complete the chart. Make a check in the box if the number is a member of the set described, leave it blank if it is not.

Natural Whole Integer Rational Irrational Real 1.3 – 5 8

€

5 12

€

13

€

− 25 3.) Explain why all integers are also rational numbers. Use a negative integer, zero, and a positive integer as examples to justify your response. 4.) Give the name of the set of real numbers that is not rational. Describe the two characteristics of such numbers. Then provide two numerical examples.

5.)

€

130 is between which two consecutive 6.) Put a point on the number line for an integers? approximate value of

€

15 . Simplify: 7.)

€

13− 9+16 8.)

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144 + 3 ⋅ 16 9.)

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−36+ 15(5)÷ 3 ________________ ________________ _________________ 10.) Find the side length of a square with an area of 169 square meters. Label your answer. Work: Answer: _________________ For Problems 11 – 14: Show your work to estimate (approximate) the square roots of the following numbers. Then plot the points on the given line to show their relative positions. 11.)

€

− 5 12.)

€

11 13.)

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8 14.)

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3 | | | | | | | –2 –1 0 1 2 3 4 For Problems 15 and 16: Calculate the approximate square root (showing all your work): 15.) Estimate

€

19 to the nearest hundredth. Show your iterative process. Work: Answer: _________________

16.) Estimate

€

50 to the nearest tenth. Show your iterative process. Work: Answer: _________________

For Problems 17 and 18: Solve the following equations: 17.)

€

y2 = 484 18.)

€

t 2 = 42 − 6 ⋅ 4 +18 ___________________ ___________________ 19.) The period of a pendulum is the time that it takes to swing from one side to the other and back. A pendulum’s period

€

P (in seconds) and its length

€

l (in feet) are related by the equation

€

P =1.1 l . If a pendulum has a length of 42 feet, what is the approximate time of its period? Answer: _________________ 20.) One attraction at an amusement park ride is a free-fall drop ride. The equation

€

d =16t 2 can be used to determine the time, t, (in seconds) that it takes a dropped object to fall a distance of

€

d feet. How far do you fall if the time of free-fall is

€

52 seconds?

Answer: _________________ 21.) Convert each of the following into their equivalent decimal form:

a.)

€

58

= ________ b.)

€

23

= ________

22.) Express each of the following decimal numbers as the ratio of two integers in lowest terms. a.)

€

−0.625 = ________ b.)

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0.4 = ________ c.)

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0.24 = ________



Percents: (No Calculators and Show All Work.) For Problems 1 – 4, write the equivalent fraction (simplified), decimal, and/or percent. Fraction Decimal Percent 1.) ______ ______ 0.68%

2.)

€

58

______ ______

3.) ______ 5.6 ______

4.)

€

16

______ ______

For Problems 5 – 7, use a proportion to solve the percent problem. 5.) What percent of 125 is 15? 6.) What is 50% of 78? 7.) 15 is 20% of what number? In problems 9 - 11, use an equation to solve the percent problem. 8.) What percent of 78 is 26? 9.) What is 7% of $56.00? 10.) 23 is 1/6 of what number?

Show all work to show how you solved the following problems: 11.) From 1980 to 1990, the population of Cook County, Illinois increased from 5,000,000 to 5,300,000. By

what percent did the population change from 1980 to 1990? Show all work! 12.) We are currently feeding our country on 920 million acres of farmland. This is about 40% of the

nation’s total land mass. What is the total number of acres of land in the United States?

13.) The average ant weighs 0.3 grams and can carry up to 15 grams. What percent of its own body weight

can the average ant carry? 14.) Explain (in words), why the percent change from 1 to 2 is not equal to the percent change from 2 to 1. ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________

15.) Use the given information to find the total cost. Round up to the nearest cent. Food bill: $65 Sales tax: 8% Tip: 20% (on food bill only) 16.) a.) A salesperson makes $11.00/hr. After a training period, the worker will receive a 5% increase on

his/her hourly wage. If the person then works 35 hrs per week, how much should the worker be paid per week?

b.) If the government withholds 20% of the paycheck for tax purposes, how much should the worker

actually receive in their paycheck? 17.) The iPhone6s Plus is on sale for 15% off the original price. The sale price is $901. What was the original price? (Hint: The sale price is always a part of the whole). 18.) Suppose you deposit $600 into an account that earns simple annual interest of 5%. After 2 years, what

is the account balance?



Expressions, Equations, and Inequalities: (No Calculators and Show All Work.) For Problems 1 – 9, simplify the following expressions:

1.)

€

84 ÷ 19 −16( ) ⋅2[ ] 2.)

€

4 +23−45⋅34

⎛

⎝ ⎜

⎞

⎠ ⎟ ÷ 0.7 3.)

€

8 − 4( )2 + 5 −12 ÷ 6( )

________________ ________________ _________________ 4.)

€

3 ⋅ 11⋅9 − 62( ) +17 5.)

€

22+14 − 4( ) ÷ 4 + 52 6.)

€

16 + 9 − 3 ________________ ________________ _________________ 7.)

€

33 + 4 − 2( )2 8.)

€

33 + (4 − 22 ) 9.)

€

7( )2 + 2 ⋅ 169 ________________ ________________ _________________ 10.) Find the length of each side of a square whose area is 81

€

cm2 . Answer: _________________ 11.) Find the perimeter of the square whose area is 144

€

in2 . Answer: _________________

For Problems 12 – 17, solve each equation for the variable x, showing all work: 12.)

€

3x − 9 = 6 13.)

€

5 − 6x = −61 14.)

€

19 = 7 − 3x ________________ ________________ _________________

15.)

€

1= 5+12x 16.)

€

−0.3+ 0.8x = −0.5 17.)

€

35x − 5 = 25

________________ ________________ _________________

For Problems 20 – 25, solve each inequality, then graph its solution set on the number line provided.

18.) Solve; then list the property you used. Then check your solution.

€

3x+15 = 48 Property Check:

19.) Solve; then list the property you used. Then check your solution.

€

16 =25x − 24 Property

Check:

20.)

€

−8 + x < −14 21.)

€

x5

< −2

Answer: ______________________ Answer: ______________________ 0 0

22.)

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x − 73

≥12 23.)

€

−2 > −8x − 4

Answer ______________________ Answer ______________________ 0 0

24.)

€

−20 − 5x >10 25.)

€

−4 + x5

< − 3

Answer ______________________ Answer ______________________ 0 0

MATH 8

CATALINA FOOTHILLS SCHOOL DISTRICT Overarching Understandings for the Course: Students will understand Number Systems, Percents, Expressions, Equations, and Inequalities, Exponent Rules, Functions, Systems of Equations, and Statistics.

Exponent Rules & Scientific Notation: (No Calculators and Show All Work.) For Problems 1 – 8, simplify. Write your answer using exponents.

1.)

€

23 ⋅2 ⋅25 2.)

€

26

22 3.)

€

−2w2 ⋅5w

4.)

€

15c 8

3c 5 5.)

€

2mn( ) −3m4n2( ) 6.)

€

30x11y13

12x 6y 9

7.)

€

3x 3 • 4x 4

36x 2 8.)

€

5d3 • 20d4

25d5

For Problems 9 and 10, find the area of each of the following geometric figures: 9.) A rectangle with sides of

€

x 2 inches and

€

x5 inches 10.) A right triangle with a base of

€

4x cm and height of

€

5x cm For Problems 11 – 20, simplify. Write your answer using positive exponents only. 11.)

€

130 12.)

€

x 2y −8 13.)

€

3−3 14.)

€

4y −3

15.)

€

250 p−3 16.)

€

−18x5y−3

⎛

⎝ ⎜

⎞

⎠ ⎟ −8x−4y5( ) 17.)

€

12r−4

4r6

18.)

€

9g3

36g−5 19.)

€

−4x4( ) 3x10( )4x8

20.)

€

3r−5( ) −5rp4( )4 p−4

21.) Explain why

€

20 =1.

__________________________________________________________________________________ __________________________________________________________________________________ For Problems 22 – 25, write each number in scientific notation. 22.) 95,645 23.) 0.00164 24.) 0.000003 25.) 9,850,000,000 ______________ ______________ ______________ ______________ For Problems 26 – 29, write each number in standard notation. 26.)

€

5×10−3 27.)

€

2.445×105 28.)

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4.31×104 29.)

€

1.25×10−3 ______________ ______________ ______________ ______________ 30.) Multiply

€

4.2×105( ) 2×10−3( ). Write your answer in scientific notation. Answer: __________________

31.) Divide

€

2.5×10−4( )2×10−3( )

. Write your answer in scientific notation.

Answer: __________________


Overarching Understandings for the Course: Students will understand Number Systems, Percents, Expressions, Equations, and Inequalities, Exponent Rules, Functions, Systems of Equations, and Statistics. Functions: (No Calculators and Show All Work.) 1.) What is the slope of the line graphed at the right?

a.) m = 2 b.) m =

€

12

c.) m =

€

24

d.) m =

€

−12

2.) What is the slope in the equation

€

y = −2x − 5? a.) m = –2 b.) m = 2 c.) m = 5 d.) m = –5 3.) What is the slope of the line passing through the points

€

3, 0( ) and

€

−2, −15( ) ? (Show all work!) a.) m = 1/3 b.) m = 3 c.) m = –5 d.) m = –15 4.) What is the slope a line passing through the points in the table at the right?

a.) m = –2 b.) m =

€

12

c.) m = 2 d.) m =

€

−12

5.) What is the slope of the line graphed at the right?

a.) m =

€

−32

b.) m = 0

c.) m =

€

−23

d.) m =

€

32

x y 1 –2 3 –3 5 –4

6.) What is the slope of the line passing through the points

€

4, 2( ) and

€

2, 7( )?

a.) m =

€

52

b.) m =

€

25

c.) m =

€

−52

d.) m =

€

−25

7.) In the situation described below, identify the slope: “At 2:00 pm., John stands at the top of a 200-foot building and drops a ball off the building. The ball is traveling at a rate of 20 feet per second. How far off the ground will the ball be after 8 seconds?” a.) m = 2 b.) m = 200 c.) m = –20 d.) m = 8 For Problems 8 – 10, graph the following: 8.)

€

y = −5 9.)

€

x = 3

10.)

€

y = −2x − 3 11.) Write the equation of the line graphed below:

12.) Sketch a graph of a line that has 0 slope: 13.) Sketch a graph of a line with undefined slope:

14.) Choose the equation that has the steepest slope. Circle your answer.

A.)

€

y = 2x −15 C.)

€

y =65x − 5

B.)

€

y =34x − 5 D.)

€

y = −3x − 5

15.) Given the graph at the right, identify which line has a steeper slope and why? ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ 16.) Find the slope of each line segment: C a.)

€

AB : ___________________________ b.)

€

BC : ___________________________ B c.) Use similar triangles to explain why the slope is the same from any two points on the line for #16. A

For Problems 17 and 18, identify the independent and dependent variables, write an equation in slope-intercept form to represent the situation, and answer the question about the problem, showing all work. 17.) Mack drove to the pumpkin patch to pick out a pumpkin for Halloween. He spent a total of $3.50 on gas and the pumpkins cost $5 each. Let p be the number of pumpkins Mack picked out and let m be the total amount of money (in dollars) he spent that day for the pumpkin trip. _______ = ____________________________________________ (independent variable) _______ = ____________________________________________ (dependent variable) Equation: _____________________________________________ Mack drove to the pumpkin patch and purchased four pumpkins. Use your equation to find out how much total money Mack spent that day for the pumpkin trip. (Show all work!) 18.) Jack bought a printer for $75. He noticed that ink cartridges cost $35 each for the printer he purchased. Let c be the purchase price of both printer and cartridges, and let i be the number of in cartridges he purchased. _______ = ____________________________________________ (independent variable) _______ = ____________________________________________ (dependent variable) Equation: _____________________________________________ Use your equation to find the total cost to purchase the printer and 12 ink cartridges.

Determine the domain & range of each relation and identify if it is a function. Circle either ‘Yes’ or ‘No’ 19.) 20.) { (–4, 3), (–3, –1), (0, –1), (2, –4) } Domain :_________________ Domain :_________________ Range:___________________ Range:___________________ Function: Yes or No Function: Yes or No 21.) 22.)

Domain :_________________ Domain :_________________ Range:___________________ Range:___________________ Function: Yes or No Function: Yes or No 23.) 24.)

Is the graph a function? Yes or No Is the graph a function? Yes or No

25.) Johnny says the graph at the right is not a function. Do you agree or disagree? Justify your response. ________________________________________________________ ________________________________________________________

x y 5 –10 0 –6 5 4

x y –2 4 0 0 2 4 4 20

Fill in the t-chart for each equation, graph the equation, and identify whether it is linear or non -linear function. 26.)

€

y = 2x − 3 27.)

€

y = −3x + 4 28.)

€

y = x 2 + 2

Circle One: Linear or Non-Linear Linear or Non-Linear Linear or Non-Linear Read the statement to determine if the dependent variable goes up or down. 29.) As the number of hours you use your air conditioner goes up, the amount of your air conditioning bill goes up or down. 30.) As the amount of yearly rainfall goes down, the fire danger in Arizona goes up or down. 31.) As the number of hours you study goes up, your grade point average goes up or down. For 32 and 33, determine if the equation is linear. 32.)

€

y = −4x 2 + 5 Circle one: This equation is / is not linear.

33.)

€

y =13x − 5

3 Circle one: This equation is / is not linear.

34.) Define a function: _______________________________________________________________ ________________________________________________________________________________

x

€

2x − 3 y

–2 2(–2) – 3

–1

0

1

2

x

€

−3x + 4 y –2

–1

0

1

2

x

€

x 2 + 2 y –2 (–2)2 + 2

–1

0

1

2

35.) Give an example of a graph that is not a function: Determine if the independent and dependent labels are placed in the appropriate column. Fill in the last column with YES or NO whether they are correctly placed. Independent Dependent Is the placement

correct? YES or NO

36.) X Y

37.) Output Input

38.) Domain Range

39.) Number of Square Feet Amount of Carpet Purchased

40.) Cost of Restaurant Bill Number of People

41.) Population growth Number of Years

42.) Length of class Number of Problems Practiced

For Exercises 1-5, consider the following situation.

The deli in the grocery store gives each customer a free cup of coffee worth $1.25.

1. Write a function to represent the situation. 1.

2. Make a function table to find the total cost of the coffee if 3, 6, 9, or 12 customers come in. 2.

3. Graph the function. 3.

4. State the domain and range of the function. 4.

5. Is the function continuous or discrete? Explain. 5.

For Exercises 6 and 7, find each function value.

6. f(−4) if f(x) = −2x + 7. 6.

7. f(5) if f(x) = −6x – 1. 7.

9. Compare the functions’ rates of change.

9.

12. Sketch a qualitative graph that represents a cup of hot coffee cooling down to room temperature quickly.

12.

13. Graph y = −x2 − 1. 13. y

O x

8. A circle has a radius of r inches. The area of a circle is represented by the expression 3.14r2. The area of a circle is a function of the radius. Does this situation represent a linear or nonlinear function? Explain. 8.

For Exercises 9-11, consider the following situation.

The total cost of renting a carpet cleaner from Carpets Inc. is represented by the function y = 10x + 80, where x represents the number of days and y represents the total cost. The cost of renting a carpet cleaner from Clark Cleaners is shown in the table.

11. Which company should you use if you rent the carpet cleaner for 6 days?

11.

10. Find and interpret the initial value of renting from Clark Cleaners.

10.

Number of Days

Cost ($)

2 60 3 85 4 110 5 135

MATH 8

CATALINA FOOTHILLS SCHOOL DISTRICT Overarching Understandings for the Course: Students will understand Number Systems, Percents, Expressions, Equations, and Inequalities, Exponent Rules, Functions, Systems of Equations, and Statistics.

Systems of Equations: (No Calculators and Show All Work.)

1.) Using the Number of Solutions below, graph an example of the three types of systems possible with two linear equations. Number of Solutions: One Solution No solution Infinite solutions For Problems 2 – 4, what is the solution to each system of graphed equations? 2.) 3.) 4.) Solution: ______________ _______________ _______________ For Problems 5 and 6, use the graphing method to solve the system of equations.

5.)

€

y = −2x+1y = 3x − 3⎧ ⎨ ⎩

6.)

€

y = −4x+ 3y = 2x − 3⎧ ⎨ ⎩

Solution: __________________ Solution: __________________

7.) Sedona says that the ordered pair

€

2, 1( ) is a solution to the system

€

y = 3x − 5y = −2x+ 5⎧ ⎨ ⎩

. Is she correct? Check

by showing her work.

Is Sedona correct? Yes or No For Problems 8 and 9, solve the system by elimination.

For Problems 10 and 11, solve the system by substitution.

€

y = 3x − 5

€

y = −2x + 5

8.)

€

5x+ 3y = 22x − 3y = −16⎧ ⎨ ⎩

Solution: _______________________

9.)

€

4x − 3y = 63x+ 2y = 30⎧ ⎨ ⎩

Solution: _______________________

10.)

€

x+ 3y = 92x − y = −10⎧ ⎨ ⎩

Solution: _______________________

11.)

€

y = 2x+ 4y = −4x − 2⎧ ⎨ ⎩

Solution: _______________________

For Problems 12 – 15: The graph shows the lengths, l, of two candles (in inches) after h hours. 12.) What was the height of each candle after an hour? Green Candle:_____________________ Blue Candle: ______________________ 13.) After how many hours are the candles the SAME length? What is that length? ___________________________________ ___________________________________ 14.) What is the meaning of the point

€

10, 0( ) in the context of the problem? ______________________________________ ______________________________________ _____________________________________ 15.) What is the meaning of the y-intercept of the green line in the context of the problem? ____________________________________________________________________________ ____________________________________________________________________________



Statistics: (No Calculators and Show All Work.) 1.) Create a scatter plot for the following data set and then draw in a regression line.

a.) Find the equation of your best-fit line.

b.) Find and interpret (in context) the slope of your line. c.) Does the x-intercept have any meaning in context? d.) Given that you are 65 inches tall, based on your linear model, how much would you weigh? Do you think this is realistic? Why or why not? e.) Based on the scatter-plot, a student concludes that a person’s height causes them to weigh more. What is wrong with this student’s thinking?

Height (inches) Weight 60 98 63 115 70 165 68 150 62 105 66 132 64 127 71 172 66 125 67 155

math 8 catalina foothills school district overarching understandings...

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