BOA Basic Operations & Applications
ACTCollege Readiness Standards Review
This document is a companion to the Comparison Table of Michigan Grade Level Content Expectations and High School Content Expectations that correlate to the ACT College Readiness Standards.
Because the ACT is taken during the junior year of high school, many of the Michigan Grade Level Content Expectations have not been reviewed or applied for three years (or more), based on a student’s individual curriculum and content covered within those coarse offerings. An ability to recall these skills may be problematic to many high school juniors. Failing to perform basic number operations, computation of probabilities, use of basic properties and concepts of numbers, and applications of geometric concepts may result in poor performance on the mathematics portion of the ACT and WorkKeys sections of the Michigan Merit Exam. Many of the concepts and skills required for students to score in the 13 to 27 range on the ACT are standards that correlate to Michigan Grade Level Content Expectations.
To help classroom educators better prepare their students for the mathematics portion of the ACT and WorkKeys sections of the MME, a review of these skills and concepts would be appropriate and beneficial for their students.
The following table is a list of Grade Level Content Expectations that should be reviewed, and a(n) sample assessment(s) that will help clarify whether students understand that content expectation. Teachers are also encouraged to develop or use their own assessments to evaluate each content expectation.
Many high school students could benefit from a continuous review of these skills and concepts.If a classroom teacher were to use these sample assessmentsas warm-up activities to begin there daily instruction, or as end-of-class activities, the refresher would most certainly be beneficial. Further, if a teacher were to find that many students demonstrated difficulty in a particular content expectation, then they could use this information to remediate and re-assess.
Other assessment items for individual GLCEs can be found at either;
(membership in MCTM required), or
Remember, research supports that a student’s ability to learn and retain knowledge is greater when they engage with;
- content at multiple levels (Wahlstrom and Webb)
- content in multiple exposures (Rogers)
- content through multiple styles (Gardner)
Bill Aten, Mathematics Consultant
Charlevoix-EmmetIntermediateSchool District
Charlevoix, MI
BOA – Basic Operations & Applications / Assessment SampleM.UN.06.01 Convert between basic units of measurement within a single measurementsystem, e.g., square inches to square feet. / Becky's backyard is 60 feet wide. How many yards wide is her backyard? Show how you found your answer. You may use pictures, words, and/or number sentences to explain your thinking.
N.MR.06.13 Solve contextual problems involving percentages such as sales taxes and tips. / Raymond's bill at the restaurant for his Grandmother and himself was $22.00. He wants to leave a 15% tip. How much should he leave?
N.FL.06.14 For applied situations, estimate the answers to calculations involving operations
with rational numbers. / Stamps cost 37 cents each. Alexia has $2.00 in cash. What is the greatest number of stamps she can buy?
N.FL.06.15 Solve applied problems that use the four operations with appropriate
decimal numbers. / Stan ran 3 7/8 miles in 3/4 hour. Julie ran 5.1 miles in 1 hour. Who ran the fastest average speed?
N.MR.07.02 Solve problems involving derived quantities such as density, velocity, and weightedaverages. / An object will float on water if its density is less than water's (1g/mL). Which of the following objects will float in water?
A)A rock with a mass of 50 grams and a volume of 54 mL.
B)A rock with a mass of 65 grams and a volume of 65 mL.
C)A piece of plastic with a mass of 30 grams and a volume of 25 mL.
D)A piece of plastic with a mass of 50 grams and a volume of 45 mL.
N.FL.07.03 Calculate rates of change including speed. / A swimming pool is being filled with water at a rate of 1 inch/minute. The owners started filling the poll at 6:00 a.m. What time was it when the water was 6 feet deep?
A)6:06 a.m.
B) 7:00 a.m.
C)7:06 a.m.
D)7:12 a.m.
N.FL.07.05 Solve proportion problems using such methods as unit rate, scaling, finding equivalent fractions, and solving the proportion equation a/b = c/d; know how to see patterns about proportionalsituations in tables. / Pose problems such as the following to your students:
A snail travels 3 cm in 40 seconds. You wonder how far the snail could travel in 3 minutes.
Part A: Write a proportion that could be used to model this situation.
Part B: How far could the snail travel in 3 minutes? Use your proportion to solve this problem.
N.MR.07.04 Convert ratio quantities between different systems of units, such as feet per
second to miles per hour. / In Germany, gas costs an average of $1.50 per liter. How many dollars would a gallon of gas cost in Germany?
A) $0.39
B) $3.00
C)$3.90
D)$5.70
N.FL.08.11 Solve problems involving ratio units, such as miles per hour, dollars per pound, or persons
per square mile. / If an individual can run the 100 meter dash in 6.5 seconds, how many miles per hour can she run? (1 mile is about 1609 meters)
A) 6.9
B) 290.5
C) 15.4
D) 34.4
N.MR.08.07 Understand percent increase and percent decrease in both sum and product
form, e.g., 3% increase of a quantity x is x + .03x = 1.03x. / The population of city is about 350,000. Over the next year, the population increased 13%. What is the new population of the city?
A) 455,000
B) 395,500
C) 350,013
D) 45,500
N.MR.08.08 Solve problems involving percent increases and decreases. / A store has a sale going on. A pair of jeans have been marked down from $35 to $28. Determine the percent that the jeans were marked down.
A) 25%
B) 20%
C) 11%
D) 7%
or,
In 1938 the minimum wage was $0.25 and by 1997 it had increased to $5.15 per hour. Find the percent of increase for the minimum wage from 1938 to 1997.
A) 4.9%
B) 19.6%
C) 95%
D) 1960%
N.FL.08.09 Solve problems involving compounded interest or multiple discounts. / A sweater had an original price of $59. It was marked 30% off. There was a special sale on Sunday that gave customers an additional 20% off the sale price. What was the final price of the sweater before sales tax?
N.MR.08.10 Calculate weighted averages such as course grades, consumer price indices,
and sports ratings. / A teacher counts tests as 60% of a student's final grade and homework as 40%. If a student has an average of 90% of tests and 70% on homework, what percent will a student receive for their final grade?
A) 82%
B) 80%
C) 78%
D) 80.6%
or,
In order for a student to earn an A in math class, they must have a 93% or better. The student has a 95% on tests and a 89% on homework. If the teacher counts tests as 70% of the grade and homework 30%, will the student receive an A in the class?
PSD – Probability, Statistics
& Data Analysis
D.PR.06.01 Express probabilities as fractions, decimals, or percentages between 0 and 1; know that 0 probability means an event will not occur and that probability 1 means an event will occur. / A bag holds the 26 letters tiles. Each tile is a different letter of the alphabet. If you choose 1 letter from the bag, what is the probability that you will choose a vowel?
D.RE.07.01 Represent and interpret data using circle graphs, stem and leaf plots, histograms,and box-and-whisker plots, and select appropriate representation to address specific questions. /
D.AN.08.01 Determine which measure of central tendency (mean, median, mode) best
represents a data set, e.g., salaries, home prices, for answering certain questions; justify the choicemade. / Here are the annual salaries of all the employees of a small company: President: $110,000; Vice President: $60,000; Senior Professionals: $50,000; $48,000; $48,000; $44,000; Junior Professionals: $36,000; $36,000; $36,000; $32,000; Clerical Staff: $22,000; $18,000; $14,000. (1.) What are the mean, median, and the mode of the salaries of the employees of this company?
or,
Bill is applying for a job at the video store and wants to know about how much money he can expect to make per hour. If the hourly wages of the current employees are $6.50, $7.20, $6.75, $7.40, $27.00, $27.00, $6.15, and $7.15, what measure of central tendency should he use to estimate typical hourly pay?
A) Mean because it is not affected by extreme values.
B) Median because it is not affected by extreme values.
C) Mode because it is not affected by extreme values.
D) There is not sufficient data to determine the mean, median or mode.
D.PR.06.02 Compute probabilities of events from simple experiments with equally likelyoutcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions. / You have a spinner that has 6 equal parts that are numbered 1, 2, 3, 4, 5, and 6. You spin that spinner once. What is the probability that you will spin an odd number?
A) 1/6
B) 1/2
C) 5/6
D) 1/3
D.AN.07.03 Calculate and interpret relative frequencies and cumulative frequencies for given data sets. / 50 students took an algebra test.
10 students got A's
22 students got B's
14 students got C's
4 students got D's
What is the relative frequency of B's
A) 0.08
B) 0.2
C) 0.28
D) 0.44
D.PR.08.03 Compute relative frequencies from a table of experimental results for a
repeated event. Interpret the results using relationship of probability to relative frequency. / When two regular octahedrons (8-faced polyhedra) with faces numbered 1 - 8 are rolled, which sum is most likely?
A) 8
B) 9
C) 12
D) 16
D.PR.08.06 Understand the difference between independent and dependent events, and
recognize common misconceptions involving probability, e.g., Alice rolls a 6 on a die three times in a row; she is just as likely to roll a 6 on the fourth roll as she was on any previous roll. / Luke's math teacher writes a number from 1 to 10, inclusive, in a notebook. She then asks students to guess the number. If Luke has guessed the number correctly three times in a row, what is the probability he will guess the correct number the next time?
A) 1 out of 4
B) 1 out of 6
C) 1 out of 8
D) 1 out of 10
NCP – Number, Concepts
and Properties
N.FL.06.10 Add, subtract, multiply and divide positive rational numbers fluently. / Kevin has 2/3 cups of flour. This is enough for 1/2 of a recipe of brownies. How much flour will she need for the whole recipe?
N.ME.06.16 Understand and use integer exponents, excluding powers of negative bases; express numbers in scientific notation. / What number is represented by 3.4 x 104?
A) 340,000
B) 3,400
C) 34,000
D) 0.00034
N.MR.07.06 Understand the concept of square root and cube root, and estimate using calculators.
N.FL.08.06 Find square roots of perfect squares and approximate the square roots of non-perfect squares by locating between consecutive integers, e.g., √130 is between 11 and 12. / The square root of 63 is between which two integers?
A) 31 and 32
B) 6 and 7
C) 7 and 8
D) 8 and 9
N.ME.08.01 Understand the meaning of a square root of a number and its connection
to the square whose area is the number; understand the meaning of a cube root and its
connection to the volume of a cube. / If the volume of a cube is 343 cubic cm, what is the length of a side of the cube?
- 7 cm
- 10 cm
- 101 cm
- 114.3 cm
XEI – Expressions, Equations & Inequalities
N.MR.06.03 Solve for the unknown in equations such as 1/4÷ x= 1, 3/4÷ x■= 1/4, and 1/2= 1x / 1/4 ÷ x = 1 Solve for x.
A) ¼
B) ½
C) 1
D) 4
N.ME.06.07 Understand that a fraction or a negative fraction is a quotient of two integers,
e.g., - 8/3is -8 divided by 3. / Write an expression for -5 divided by 25.
N.MR.06.13 Solve contextual problems involving percentages such as sales taxes and tips. / Alexis went shopping for clothes to wear on a job interview. She bought a skirt for $23.00, a blouse for $17.00, and shoes for $35.00. Alexis lives in Michigan where the sales tax is 6%. How much will her cost be for all items including tax?
A.PA.07.04 For directly proportional or linear situations, solve applied problems using graphs
and equations, e.g., the heights and volume of a container with uniform cross-section; height of water in a tank being filled at a constant rate; degrees Celsius and degrees Fahrenheit; distance and time under constant speed. / Mrs. Racine needed a loan of $10,000 to pay start-up costs in opening a Roller Rink. She uses $2.50 from each starting ticket sold to reduce the balance owed on the loan.
Part A: Create a table showing how much Mrs. Racine will still owe after selling 500, 1000, 1500, and 2000 tickets.
Part B: Write an algebraic expression the shows how Mrs. Racine's debit changes as she sells tickets.
Part C: Mrs. Racine sold 6000 tickets in her first month. Use your equation from Part B to find out what show owes on her loan after the first month. What does your answer mean?
A.FO.07.12 Add, subtract, and multiply simple algebraic expressions of the first degree,
e.g., (92x + 8y) – 5x + y, or x(x+2) and justify using properties of real numbers. / Which of the following is another way to represent this expression?
3x + (2x - 1) + 5(x +2)
A) 10x + 4
B) 10x + 6
C) 10x + 9
D) 11x + 4
or,
Simplify the expression.
-2x -(4 + 5y) - 5x + 2y
A.FO.07.13 From applied situations, generate and solve linear equations of the form
ax + b = c and ax + b = cx + d, and interpret solutions. / Carnival Cars automobile dealership pays its salespeople $650 monthly, plus a commission of $425 for each automobile sold. Which equation could be used to compute a salesperson's monthly pay?
A) (650+425)a = c
B) (650)a + 425 = c
C) 650 + (425)a = c
D) (650)a + (425)a = c
A.FO.08.07 Recognize and apply the common formulas:
(a + b)2= a2+ 2ab + b2
(a – b)2= a2– 2ab + b2
(a + b) (a – b) = a2– b2 ; represent geometrically. / Which is the correct factorization of x2 + 14x + 49?
- (x + 7)(x -7)
- (x + 7)²
- (x - 7)(x - 7)
- none of the above
A.FO.08.08 Factor simple quadratic expressions with integer coefficients, e.g.,
x2+ 6x + 9, x2 + 2x – 3, and x2– 4; solve simple quadratic equations, e.g., x2= 16 or x2= 5
(by taking square roots); x2 – x – 6 = 0, x2 – 2x = 15 (by factoring); verify solutions by evaluation. / If x2 = 64, then x =
- 2 and 32
- 4, 4 and 4
- 8 or -8
- 16 or -16
Factor x2 – 10x + 16
- (x - 10)(x + 16)
- (x - 8)(x + 2)
- (x - 4)(x - 4)
- (x - 8)(x - 2)
A.FO.08.09 Solve applied problems involving simple quadratic equations. / A garden plot 4 meters by 12 meters has one side along a fence (at the top of the drawing). The area of the garden is to be doubled by digging a border of a uniform width of x meters on the other three sides. What should the width of the border be?
A.FO.08.11 Solve simultaneous linear equations in two variables by graphing, by substitution, and by linear combination; estimate solutions using graphs; include examples with no solutions and infinitely many solutions. / Solve the following linear equations.
y = 2x - 6
y = 4x -10
or,
Solve the following linear equations by using a linear combination.
2y = 3x + 1
y = x + 5
A.FO.08.12 Solve linear inequalities in one and two variables, and graph the solution sets. / Which of the following is the solution set of 20 > -4x + 8 ?
- -3 > x
- x > 7
- 7 > x
- x > -3
A.FO.08.13 Set up and solve applied problems involving simultaneous linear equations and
linear inequalities. / A construction contractor has at most $33 to spend on nails for a project. The contractor needs at least 9 pounds of finish nails costing 60 cents per pound and at least 12 pounds of common nails costing 55 cents per pound.
- Write a system of three inequalities that describes this situation.
- Graph the system to show all possible solutions.
- Name a point that is a solution of the system.
- Name a point that is not a solution of the system.
GRE – Graphical Representations
A.RP.06.02 Plot ordered pairs of integers and use ordered pairs of integers to identify points
in all four quadrants of the coordinate plane. / Plot the ordered pairs
a. (3,5) (0,4)
b. (-3, 7) (4, -2)
c. (-3, -6)
N.ME.06.17 Locate negative rational numbers (including integers) on the number line; know that numbers and their negatives add to 0, and are on opposite sides and at equal distance from 0 on a number line. / Write an integer that represents each situation.
A) 10 degrees below 0 (Answer: -10 degrees)
B) a loss of $7 (Answer: -$7)
C) 3 feet under water (Answer: -3 feet)
D) sale price of $6 off (Answer: -$6)
A.PA.07.03 Given a directly proportional or other linear situation, graph and interpret the slope and intercept(s) in terms of the original situation; evaluate y = mx + b for specific x values, e.g., weight vs. volume of water, base cost plus cost per unit. / Your great Aunt Bertha offers to pay you $5.00 an hour to baby sit your twin 8 year old cousins.
Part A: Create a graph showing how much you make every hour you baby sit for your aunt.
Part B: What is the slope of the graph in Part A? Explain what the slope means in this situation.
Part C: What is the y-intercept? Explain what the y-intercept means in this situation.
A.PA.07.05 Recognize and use directly proportional relationships of the form y = mx, and distinguish from linear relationships of the form y = mx + b, b non-zero; understand that in a directlyproportional relationship between two quantities one quantity is a constant multiple of the otherquantity. / Given y = 2x + 3 and y = 2x, what is the slope for each equation? What is the y-intercept for each equation? Which equation is considered a directly proportional relationship and why?
A.PA.07.06 Calculate the slope from the graph of a linear function as the ratio of “rise/run”
for a pair of points on the graph, and express the answer as a fraction and a decimal; understand that linear functions have slope that is a constant rate of change. / What is the slope of a line that goes through the points (3, 4) and (7, 12)?
A.PA.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and
graph, interpreting slope and y-intercept. / For each table of x and y values, write the equation and show how you can find the slope and y-intercept in the table. Graph each set of data points on separate graphs. Show how you can find determine the slope and y-intercept from the graph.
A)
x / y
1 / 4
2 / 5
3 / 6
4 / 7
5 / 8
B)
x / y
1 / 3
2 / 6
3 / 9
4 / 12
5 / 15
A.FO.07.08 Find and interpret the x and/or y intercepts of a linear equation or function. Know that the solution to a linear equation of the form ax+b=0 corresponds to the point at which the graph of y=ax+b crosses the x axis. / What is the x-intercept for the following equation: y = 4 x + 4
A) (4,0)
B) (1,0)
C) (-1,0)
D) (0,-1)
G.LO.08.02 Find the distance between two points on the coordinate plane using
the distance formula; recognize that the distance formula is an application of the Pythagorean Theorem. / Find the distance between the ordered pairs A (2, 3) and B (10, 9).
A) 10
B) √(14)
C) 100
D) √(288)
PPF – Properties of Plane Figures
G.GS.06.01 Understand and apply basic properties of lines, angles, and triangles, including:
• triangle inequality
• relationships of vertical angles, complementary angles, supplementary angles
• congruence of corresponding and alternate interior angles when parallel lines are cut by a transversal, and that such congruencies imply parallel lines
• locate interior and exterior angles of any triangle, and use the property that an exterior angle of a triangle is equal to the sum of the remote (opposite) interior angles
• know that the sum of the exterior angles of a convex polygon is 360º. / Joan was making a box as a gift. She wanted the box top to be a rectangle. To see if the opposite sides of the rectangle were parallel, she drew one of the diagonals on the top of the box. The diagonal made several angles, as shown.
Which angles have to be congruent in order for the opposite sides to be parallel?
A) a and b
B) a and c
C) a and d
D) a and e
or,
In this triangle, what is the measure of angle g equal to in terms of k, I and h?
G.GS.08.01 Understand at least one proof of the Pythagorean Theorem; use the Pythagorean Theorem and its converse to solve applied problems including perimeter, area, and volume problems. / A carpenter needs to get a piece of wood into a house to build a wall. The rectangular piece of wood is 8 feet tall and 5 feet wide. The doorway is 7 feet tall and 4 feet wide. Will he be able to take it though the doorway without cutting it into smaller pieces?
MEA - Measurement
G.SR.08.03 Understand the definition of a circle; know and use the formulas for
circumference and area of a circle to solve problems. / Find the length of the diameter of a circle with an area of 63.585 square centimeters. Use 3.14 for the value of п.
A) 2.25 centimeters
B) 4.5 centimeters
C) 9.0 centimeters
D) 20.25 centimeters
G.SR.08.04 Find area and perimeter of complex figures by sub-dividing them into basic
shapes (quadrilaterals, triangles, circles). / Find the area of this polygon.
A) 25 m2
B) 50 m2
C) 100 m2
D) 150 m2
G.SR.08.05 Solve applied problems involving areas of triangles, quadrilaterals, and circles. / A bike tire has a circumference of 1.5 meters and spokes extending from the center to the outside edge. What is the length of one of these spokes?
A) 0.24 m
B) 0.69 m
C) 0.48 m
D) 1.05 m
or,
A baker needs to buy boxes to ship his pies. The diameter of each pie is 8 inches. If 2 pies are to be placed side by side in each box, what is the area of the bottom of the box needed to hold both pies?
G.SR.08.06 Know the volume formulas for generalized cylinders ((area of base) x height),
generalized cones and pyramids ( 1/3(area of base) x height), and spheres ( 4/3(radius)3) and apply them to solve problems. / Start with two identical sheets of tagboard (8.5 inches by 11 inches). Tape the long side of one sheet together to form a cylinder. Form a cylinder from the second sheet by taping the short sides together. Imagine that each cylinderhas a top and a bottom.
Have the students predict if the cylinders have equal volumes or if one cylinder has a greater volume than the other.
Use volume formulas to determine which has a greater volume.
G.SR.08.07 Understand the concept of surface area, and find the surface area of prisms,
cones, spheres, pyramids, and cylinders. / A packaging engineer is designing a box to hold 100 paper clips. The box has a length of 3 1/2 inches, a width of 2 3/4 inches and a height of 3/4 inches. What is the surface area of the box?
A) 7 in.
B) 7 7/32 in.
C) 14 5/16 in.
D) 28 5/8 in.
FUN - Functions
A.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inversely proportional relationships (y = k/x); cubics (y = ax3); roots (y = √x ); and exponentials (y = ax , a > 0); using tables, graphs, and equations. / What type of graph/function does the picture below represent?
- linear
- quadratic
- cubic
- exponential
Identify which graph represents the inverse function
- a
- b
- c
- d
A.PA.08.02 For basic functions, e.g., simple quadratics, direct and indirect variation,
and population growth, describe how changes in one variable affect the others. / Given y = 4x2 describe what happens to y if x is tripled?
- 3 times larger
- 4 times smaller
- 9 times smaller
- 9 times larger
Which of the following describes the relationship between x and y in the function:
- as x increases, y decreases
- as x increases, y increases
- as x increases, y is constant
- when x is a constant, y decreases
A.RP.08.05 Relate quadratic functions in factored form and vertex form to their graphs, and vice versa; in particular, note that solutions of a quadratic equation are the x-intercepts of the corresponding quadratic function. / Choose the equation whose graph is shown here:
- y = (x + 2) (x + 3)
- y = (x – 2) (x + 3)
- y = (x – 3) (x + 2)
- y = (x – 3) (x – 2)
For the quadratic equation y + 2 = 3(x – 4)², which point would be the location of the vertex?
- (4, -2)
- (3, 4)
- (2, 4)
- (-4, 3)
Wm. Aten, CharEm ISD, 2007Page 1