AStudent!andParent!Guide! ToHelpPrepare!Students!forthe...

A Student and Parent Guide To Help Prepare Students for the

Algebra 1 End-‐of-‐Course (EOC) Exam

12/17/10

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Table of Contents

GENERAL NOTES 3

VOCABULARY FIRST USED IN THE ALGEBRA 1 END-‐OF-‐COURSE EXAM 4

VOCABULARY FIRST INTRODUCED IN THE GRADES 3-‐8 MSP 6

MATHEMATICS SYMBOLS FIRST USED IN ASSESSMENT ITEMS 10

MEASUREMENT VOCABULARY 11

HIGH SCHOOL EOC EXAM FORMULA SHEETS 12

CORE CONTENT: SOLVING PROBLEMS 14

CORE CONTENT: NUMBERS, EXPRESSIONS, AND OPERATIONS 19

CORE CONTENT: CHARACTERISTICS AND BEHAVIORS OF FUNCTIONS 25

CORE CONTENT: LINEAR FUNCTIONS, EQUATIONS, AND INEQUALITIES 28

CORE CONTENT: QUADRATIC FUNCTIONS AND EQUATIONS 33

CORE CONTENT: DATA AND DISTRIBUTIONS 37

ADDITIONAL KEY CONTENT 42

CORE PROCESSES: REASONING, PROBLEM SOLVING, AND COMMUNICATION 46

ANSWERS TO EXAMPLE PROBLEMS 47

STRATEGIES FOR EFFECTIVE STUDENT LEARNING 49

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General Notes This document is not state (OSPI)-‐created or approved. This is not an exhaustive sampling of EOC Exam test items. This is intended to be a guide to give students an idea of what their level of understanding is for each of the Algebra 1 performance expectations (PEs) at the beginning of the school year and a month before the EOC Exam. The PEs will be assessed in multiple-‐choice (MC), completion (CP), and short answer (SA) formats. MC and CP items are worth one point each. SA items are worth two points. Calculators will be allowed on the Algebra 1 End-‐of-‐Course (EOC) exam. Knowing the following PEs and being able to answer problems in a contextual (story problem) situation should lead to success on the EOC exam. Students are expected to know all content, vocabulary, and processes in previous grade levels. These items may be assessed on the Algebra 1 EOC Exam. Students should still study the items not assessed on the EOC Exam since those items will be incorporated into future math courses. The EOC Exams will take place during the last three weeks of the school year. For more information about the EOC Exams, go to https://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf For more information about Washington’s math learning standards, go to https://www.k12.wa.us/Mathematics/Standards/K-‐12MathematicsStandards-‐July2008.pdf For questions about this document, contact Katelyn Hubert at [email protected].

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Vocabulary First Used in the Algebra 1 End-‐of-‐Course Exam Suggestion: Fill out this table at the beginning of the year so you know what to expect and can track your progress throughout the year. Term Where is this in my textbook? Check the box

once you know it. Approximate (as a verb) Arithmetic sequence Binomial Calculate Completing the square Consecutive Constant Correlation, negative Correlation, positive Correlation, strong Correlation, weak Cube (exponent) Cube root Direct proportion/Directly proportional

Direct variation Domain (function) Explicit (sequence, series) Exponential equation Exponential function Frequency Geometric sequence Initial Intersection Inverse proportion/Inversely proportional

Inverse variation Line that fits the data Line, point-‐slope form Line, slope-‐intercept form Line, standard form Model Monomial Nonnegative Nonzero Precise Precision Polynomial Quadrant Quadratic equation

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Quadratic formula Quadratic function Radical Range (function) Rate of change Real number Recursive (sequence, series)

Region (coordinate plane) Root (function) Satisfies (equation) Sequence Solution set Square (exponent) System (equations, inequalities)

Term (sequence, series) Trinomial Valid Variable, dependent Variable, independent You can access this information at http://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf on page 25.

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Vocabulary First Introduced in the Grades 3-‐8 MSP Suggestion: Fill out this table at the beginning of the year. You should already be familiar with these terms. If you are not, listen for them throughout the year. They will probably show up at some point. If not, be sure to review them before the EOC Exam. In parentheses is when the term is first assessed on the Measurement of Student Progress (MSP). Term Check the

box once you know it.

Term Check the box once you know it.

Absolute value (7) Add (3) Addition (3) Angle (3) Angle, acute (5) Angle, right (3) Angles, adjacent (8) Angles, complementary (8) Angles, corresponding (7) Angles, interior (8) Angles, obtuse (5) Angles, supplementary (8) Angles, vertical (8) Approximate (6) Area (4) Attribute/property (3) Axis, horizontal (5) Axis, vertical (5) Axis/axes (6) Base (geometry) (5) Bias (7) Box-‐and-‐whisker plot (8) Certain (4) Chart (3) Circle (3) Circle graph (7) Circumference (6) Clockwise (8) Closed figure (3) Cluster (8) Coin (3) Common denominator (4) Compare (3) Complement (probability) (6) Complete (3) Conclude (3) Conclusion (3) Cone (7) Congruent (4) Construct (6) Convert (4) Coordinate (4) Coordinate plane (8) Correct (3) Corresponding sides (7) Counterclockwise (8) Cube (6) Cylinder (7) Data (3) Decimal (4) Decompose (5) Denominator (3) Determine (5) Diagonal (3) Diagram (5) Diameter (6) Difference (3) Digit (3) Dilation (8) Dimensions (5) Disagree (3) Divide (3) Divisible (5) Division (3) Edge (6) Equal (3) Equally unlikely (4) Equation (3) Equivalent (4) Estimate (3) Evaluate (5) Event (6) Event, dependent (8) Event, independent (8) Events, mutually exclusive (8) Exponent (7)

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Expression (3) Face (6) Factor (4) Fewer than (3) Fewest (3) Figure (3) Formula (4) Fraction (3) Function (8) Function machine (5) Graph (3) Greater than (3) Greatest (3) Greatest common factor (5) Grid (4) Height (5) Height, slant (6) Hexagon (4) Hundreds (3) Hundredths (4) Hypotenuse (8) Identify (6) Image (8) Impossible (4) Improper fraction (4) Include (3) Inequality (3) Information (3) Integer (6) Intercept (8) Interquartile range (8) Intersecting lines (3) Interval (6) Justify (6) Key (graph) (3) Kite (3) Label (3) Law of exponents (8) Least (3) Least common multiple (5) Less likely (4) Less than (3) Likely (4) Line (3) Line graph (5) Line plot (3) Line segment (3) Linear (5) Linear equation (7) Linear function (8) Linear inequality (8) Linearly related (5) Location (3) Lowest terms (4) Maximum (7) Mean (5) Measure (3) Measure of center (7) Median (4) Metric system (7) Million (4) Minimum (7) Mode (4) Model (3) More likely (4) More than (3) Most (3) Most likely (4) Multiple (4) Multiplication (3) Multiply (3) Net (geometry) (6) Number (3) Number, composite (5) Number, even (3) Number, irrational (8) Number, mixed (4) Number, odd (3) Number, prime (5) Number, rational (7) Number, whole (3) Number line (3) Number pattern (3) Numerator (3) Octagon (4) Ones (3) Operation (3) Order (3) Order of operations (6) Ordered pair (4) Origin (5) Outcome (7) Outlier (7) Parallel (3) Parallelogram (3) Pattern (3)

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Pentagon (4) Per (6) Percent (6) Perfect square (of an integer) (8) Perimeter (3) Perpendicular (3) Pi (!) (6) Pictograph (3) Place value (3) Plot (4) Point (3) Polygon (4) Polygon, regular (4) Polyhedron (6) Polyhedron, regular (6) Population (3) Power (exponent) (8) Predict (7) Prime factorization (7) Prism (6) Probability (4) Probability, experimental (6) Probability, theoretical (6) Problem (3) Product (3) Property, distributive (6) Property, identity (6) Property/attribute (3) Proportion (6) Proportional (6) Pyramid (6) Pythagorean theorem (8) Quadrilateral (3) Quartile (8) Quartile, lower (8) Quartile, upper (8) Quotient (4) Radical (8) Radius/radii (6) Random sample (8) Range (4) Rate (6) Rate, unit (6) Ratio (6) Rectangle (3) Reflection (8) Relation (3) Relationship (6) Remainder (4) Represent (3) Rhombus/rhombi (3) Rotation (8) Round to the nearest (3) Ruler (3) Sample space (7) Scale (3) Scale (axis) (5) Scale (proportion) (7) Scale drawing (7) Scatter plot (8) Scientific notation (8) Semicircle (6) Set (3) Side (3) Similar figures (7) Simplify (with directions) (4) Slope (7) Solution (6) Solve (4) Square (3) Square root (8) Statement (3) Standard form (8) Straightedge (3) Student (3) Stem-‐and-‐leaf plot (7) Substitute (5) Subtract (3) Subtraction (3) Sum (3) Support (3) Surface area (6) Survey (3) Symbol (3) Symmetry (5) Table (3) Tally/tallies (3) Tens (3) Tenths (4) Tetrahedron (6) Thermometer (3) Thousands (3) Thousandths (5) Three-‐dimensional (6) Title (3) Total (3)

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Transformation (8) Translation (8) Transversal (8) Trapezoid (3) Tree diagram (7) Trend line (8) Triangle (3) Triangle, acute (5) Triangle, equilateral (5) Triangle, isosceles (5) Triangle, obtuse (5) Triangle, right (5) Triangle, scalene (5) Two-‐dimensional (6) Unlikely (4) U.S. Customary system (7) Value (3) Variability (7) Variable (5) Venn diagram (8) Verify (6) Vertex/vertices (3) Volume (6) Width (3) You can access this information at http://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf on pages 26-‐28. You can get more information about the grades 3-‐5 MSP at http://www.k12.wa.us/Mathematics/pubdocs/TestItemSpecGrade3-‐5.pdf. You can get more information about the grades 6-‐8 MSP at http://www.k12.wa.us/Mathematics/pubdocs/TestItemSpecGrade6-‐8.pdf.

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Mathematics Symbols First Used in Assessment Items

You can access this information at http://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf on page 21.

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Measurement Vocabulary

You can access this information at http://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf on page 24.

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High School EOC Exam Formula Sheets

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You can access this information at http://www.k12.wa.us/Mathematics/pubdocs/ItemSpec_A1.pdf on pages 22-‐23.

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Core Content: Solving Problems Students learn to solve many new types of problems in Algebra 1, and this first core content area highlights the types of problems students will be able to solve after they master the concepts and skills in this course. Students are introduced to several types of functions, including exponential and functions defined piecewise, and they spend considerable time with linear and quadratic functions. Each type of function included in Algebra 1 provides students a tool to solve yet another class of problems. They learn that specific functions model situations described in word problems, and so functions are used to solve various types of problems. The ability to determine functions and write equations that represent problems is an important mathematical skill in itself. Many problems that initially appear to be very different from each other can actually be represented by identical equations. Students encounter this important and unifying principle of algebra – that the same algebraic techniques can be applied to a wide variety of different situations. Performance expectation A1.1.A

I can select and justify functions and equations to model and solve problems.

Where is this in my textbook?

Note: This PE most likely shows up in several chapters/units. It may not be possible to cite one specific lesson where you will find it.

Example problems

• Joe gave away one dollar on day 1. Each day after that, he gave away twice as many dollars as he had given away on the previous day. Let f(n) represent the number of dollars given away on day n. Which function models this situation? A. f(n) = 2n B. f(n) = 2n-‐1 C. f(n) = 2n – 1 D. f(n) = (n – 1)2

• Keisha is planning to rent a van for her trip to Mt. Rainier. Two of her friends each rented the same type of van from the same car rental company last week. This is what they told her.

John: “The cost of my rental was $240. The company charged me a certain amount per day and a certain amount per mile. I had the rental for five days and I drove it 200 miles.” Katie: “The cost of my rental was only $100. I drove it for 100 miles and had it for two days.”

Keisha plans to get the same type of van that John and Katie had, from the same car rental company. Keisha estimated her trip would be 250 miles, and she would have the vehicle for four days. Let C = cost, M = miles, and D = days. Which of the following equations could Keisha use to figure out how much her rental would cost? A. C = 40.00M + 0.20D B. C = 40.00D + 0.20M C. C = 20.00M + 0.40D D. C = 20.00D + 0.40M

In what form(s) will the test questions be like on the EOC Exam?

Multiple-‐choice Short answer

INITIAL SELF-‐ASSESSMENT: Draw yourself on the line of understanding.

___________________________________________________________________________________________________________ No clue Totally get it

APRIL SELF-‐ASSESSMENT: Draw yourself on the line of understanding.

____________________________________________________________________________________________________________ No clue Totally get it

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Performance expectation A1.1.B

I can solve problems that can be represented by linear functions, equations, and inequalities.


Note: This PE most likely shows up in several chapters/units. It may not be possible to cite one specific lesson where you will find it.

Example problems

• The Acme Recycling Company has three salary options for its part-‐time summer employees. The total money earned is related to the amount of cans recycled and an optional hourly wage. Option 1: $0.25 a can plus $1.00 an hour Option 2: $0.05 a can plus $5.00 an hour Option 3: $0.40 a can and no hourly wage Jamal wrote an equation for each salary option to see what he could make per hour. Option 1: y = 0.25x + 1.00 Option 2: y = 0.05x + 5.00 Option 3: y = 0.40x Jamal estimates that he can recycle a minimum of 20 cans per hour. Based on these equations and Jamal’s estimate, which option will allow Jamal to make the most money? Show your work using words, numbers, and/or diagrams.

• Jay earns $16.42 per hour. He earns 1.5 times his hourly wage for every hour he works over 40 hours each week. He earns 2 times his hourly wage on Sunday. Jay worked 3 hours on Sunday and earned a total of $903.10 for the week. How many total hours did Jay work last week? Show your work using words, numbers, and/or diagrams.

• Dorian is saving money to buy a bicycle. Currently he has saved !! of the money he needs to buy

the bicycle. He earns $14.50 more mowing lawns and now has !! of the money he needs to buy

the bicycle. Determine the cost of the bicycle. • The assistant pizza maker makes 6 pizzas an hour. The master pizza maker makes 10 pizzas an

hour but starts baking two hours later than his assistant. Together, they must make 92 pizzas. How many hours from when the assistant starts baking will it take?

• A swimming pool holds 375,000 liters of water. Two large hoses are used to fill the pool. The first hose fills at the rate of 1,500 liters per hour and the second hose fills at the rate of 2,000 liters per hour. How many hours does it take to fill the pool completely?


Completion Short answer





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Performance expectation A1.1.C

I can solve problems that can be represented by a system of two linear equations or inequalities.


Example problems

• Two hundred items were sold at a snack stand for a total of $130.00. The only items sold were cans of pop for $0.50 and bags of popcorn for $0.75. How many of each item were sold?

• Only chocolate and vanilla ice cream cones are sold at an ice cream store. In one day, the number of chocolate cones sold was 1 more than 4 times the number of vanilla cones sold. A total of 121 cones were sold that day. Let c = the number of chocolate cones sold Let v = the number of vanilla cones sold Write two equations to determine the number of chocolate cones sold that day. Then use the equations to determine the number of chocolate cones sold that day. Show your work using words, numbers, and/or diagrams.

• A 40% dye solution is to be mixed with a 70% dye solution to get 210 L of a 50% solution. How many liters of the 40% and 70% solutions will be needed?

• If a plane can travel 440 mph against the wind and 500 mph with the wind, find the speed of the wind and the speed of the plane in still air.


Completion Short answer





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Performance expectation A1.1.D

I can solve problems that can be represented by quadratic functions and equations.


Example problems

• Find the solutions to the simultaneous solutions y = x + 2 and y = x2. • If you throw a ball straight up (with an initial height of 4 feet) at 10 feet per second, how long

will it take to fall back to the starting point? The function h(t) = -‐16t2 + v0t + ho describes the height, h, in feet, of an object after t seconds, with initial velocity v0 and initial height h0.

• What two consecutive negative numbers, when multiplied together, give the first number plus 16?


Completion





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Performance expectation A1.1.E

I can solve problems that can be represented by exponential functions and equations.


Example problems

• E. coli bacteria reproduce by a simple process called binary fission – each cell increases in size and divides into two cells. In the laboratory, E. coli bacteria divide approximately every 15 minutes. A new E. coli culture is started with 1 cell. Write a function that models the E. coli population size at the end of each 15-‐minute interval. After what 15-‐minute interval will you have at least 500 bacteria?

• Estimate the solution to 2x = 16,000. In what form(s) will the test questions be like on the EOC Exam?

Multiple-‐choice Completion





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Core Content: Numbers, expressions, and operations Students see the number system extended to the real numbers represented by the number line. They work with integer exponents, scientific notation, and radicals, and use variables and expressions to solve problems from purely mathematical as well as applied contexts. They build on their understanding of computation using arithmetic operations and properties and expand this understanding to include the symbolic language of algebra. Students demonstrate this ability to write and manipulate a wide variety of algebraic expressions throughout high school mathematics as they apply algebraic procedures to solve problems. Performance expectation A1.2.A

I know the relationship between real numbers and the number line, and compare and order real numbers with and without the number line.


Example problem

Order the following numbers from least to greatest. 3!, 62, 8.7 x 10!, !"!

A. !"!, 3!, 8.7 x 10!, 62

B. 62, 8.7 x 10!, 3!, !"!

C. 8.7 x 10!, 3!, !"!, 62

D. 3!, 62, !"!, 8.7 x 10!


Multiple-‐choice





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I recognize the multiple uses of variables, can determine all possible values of variables that satisfy prescribed conditions, and can evaluate algebraic expressions that involve variables.


Example problems

• For what values of a and n, where n is an integer greater than 0, is an always negative? • For what values of a is !

! an integer?

• For what values of a is 5 − ! defined? • For what values of a is –a always positive?







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I can interpret and use integer exponents and square and cube roots, and apply the laws and properties of exponents to simplify and evaluate exponential expressions.


Example problems • Write the expression !"

!" in simplest radical form.

• The expression !!!!

!!∙!!!

!

simplifies to the form !!, for all nonzero values of x. Determine the value of n.



INITAL SELF-‐ASSESSMENT: Draw yourself on the line of understanding.




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I can determine whether approximations or exact values of real numbers are appropriate, depending on the context, and justify the selection.


Example problems

• Your math teacher gives you the equation !! − 12 = 0 and asks you to solve for x. Is the answer 2 3 or 3.46 the most appropriate answer in this situation? Explain your answer.

• Your math teacher asks you to solve the following problem: If you throw a ball straight up (with initial height of 4 feet) at 50 feet per second, how long will it take to fall back to the starting point? The function ℎ ! = −16!! + !!! + ℎ! describes the height, h, in feet, of an object after t seconds, with initial velocity v0 and initial height h0. Is the answer !!

! seconds or about 3 seconds the most appropriate answer in this situation?

Explain your answer. In what form(s) will the test questions be like on the EOC Exam?

Not assessed





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I can use algebraic properties to factor and combine like terms in polynomials.


Example problems

• Which term is a factor of 3!! + 12!? A. 3a B. 4a C. 3a2 D. 4a2

• Factor the polynomial 36!! − 25!!. A. 6! − 5! ! B. 6! + 5! !! C. (6! − 5!)(6! − 5!) D. (6! − 5!)(6! + 5!)


Multiple-‐choice





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Performance expectation A1.2.F

I can add, subtract, multiply, and divide polynomials.


Example problems

• Add the polynomials. 3!! − 4! + 5 + −!! + ! − 4 + (2!! + 2! + 1) • Subtract the polynomials. 2!! − 4 − (!! + 3! − 3) • Multiply the polynomials. (5! − 1)(!! + 9! − 2) • Divide the polynomials. !

!!!!!!!!!

, when ! ≠ −1


Not assessed





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Core Content: Characteristics and behaviors of functions Students formalize and deepen their understanding of functions, the defining characteristics and uses of functions, and the mathematical language used to describe functions. They learn that functions are often specified by an equation of the form y=f(x), where any allowable x-‐value yields a unique y-‐value. While Algebra 1 has a particular focus on linear and quadratic equations and systems of equations, students also learn about exponential functions and those that can be defined piecewise, particularly step functions and functions that contain the absolute value of an expression. Students learn about the representations and basic transformations of these functions and the practical and mathematical limitations that must be considered when working with functions and when using functions to model situations. Performance expectation A1.3.A

I can determine whether a relationship is a function and identify the domain, range, roots, and independent and dependent variables.


Example problems

• A function ! ! = 60! is used to model the distance in miles traveled by a car traveling 60 miles per hour in n hours. Identify the domain and range of this function.

• What is the domain of following function? ! ! = 5 − ! • Which of the following equations determines y as a function of x?

A. ! = 5 B. ! = 5 C. !! + !! = 1 D. !! = !!







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I can represent a function with a symbolic expression, as a graph, in a table, and using words, and make connections among these representations.


Example problems • Which function best represents the graph below?

A. ! ! = ! B. ! ! = ! C. ! ! = !! D. ! ! = !

• Which function best represents the values in the table below? x f(x) -‐3 -‐27 -‐1 -‐1 0 0 2 8 5 125 A. ! ! = !! B. ! ! = ! C. ! ! = !

!

D. ! ! = ! In what form(s) will the test questions be like on the EOC Exam?






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I can evaluate !(!) at ! (i.e., !(!)) and solve for x in the equation ! ! = !.


Example problems

• Roses-‐R-‐Red sells its roses for $0.75 per stem and charges a $20 delivery fee per order. This can be represented as the function ! ! = 0.75! + 20, where x represents the number of roses sold and f represents the cost of the delivery. How many roses can you have delivered for $65?

• Roses-‐R-‐Red sells its roses for $0.75 per stem and charges a $20 delivery fee per order. This can be represented as the function ! ! = 0.75! + 20, where x represents the number of roses sold and f represents the cost of the delivery. How much does it cost to have 24 roses delivered?







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Core Content: Linear functions, equations, and inequalities Students understand that linear functions can be used to model situations involving a constant rate of change. They build on the work done in middle school to solve sets of linear equations and inequalities in two variables, learning to interpret the intersection of the lines as the solution. While the focus is on solving equations, students also learn graphical and numerical methods for approximating solutions to equations. They use linear functions to analyze relationships, represent and model problems, and answer questions. These algebraic skills are applied in other Core Content areas across high school courses. Performance expectation A1.4.A

I can write and solve linear equations and inequalities in one variable.


Example problems

• The equation 13 − 2 ! + 3 = 5 has two real solutions. Determine the negative solution of the equation.

• Write an absolute value equation for all the numbers 2 units from 7. • Solve for x in 2 ! − 3 + 4! = 15 + 6!. • Solve for x in 8 < 3! + 2 ≤ 9.







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I can write and graph an equation for a line given the slope and the y-‐intercept, the slope and a point on the line, or two points on the line, and translate between forms of linear equations.


Example problems

• Which equation below describes a line that has a slope of 5 and a y-‐intercept of -‐9? A. ! = −9! + 5 B. ! = 5! − 9 C. ! = 9! − 5 D. ! = −5! + 9

• Which equation below describes a line that has a slope of 2 and goes through the point (1, 5)? A. ! = 2! + 5 B. ! = 2! − 9 C. ! = 2! + 1 D. ! = 2! + 3

• Which equation below describes a line that goes through the points (-‐3, 5) and (6, -‐2)? A. ! − 2 = − !

!(! + 6)

B. ! + 2 = − !!(! − 6)

C. ! + 6 = − !!(! − 2)

D. ! − 6 = − !!(! + 2)

• Write the equation 3! − 2! = 12 in slope-‐intercept form. A. ! = !

!! − 6

B. ! = − !!! − 6

C. ! = !!! + 6

D. ! = − !!+ 6

• Describe what the graph of the equation ! − 6 = 3(! + 1) looks like. A. The line has a slope of 3 and goes through the point (1, -‐6). B. The line has a slope of 3 and goes through the point (-‐1, 6). C. The line has a slope of 3 and goes through the point (-‐6, 1). D. The line has a slope of 3 and goes through the point (6, -‐1).


Multiple-‐choice





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I can identify and interpret the slope and intercepts of a linear function, including equations for parallel and perpendicular lines.


Example problems

• The graph below shows the relationship between time and distance from a gas station for a motorcycle and a scooter. What can be said about the relative speed of the motorcycle and scooter that matches the information in the graph?

A. The scooter’s speed was always greater than the motorcycle’s speed. B. The scooter started out slower than the motorcycle and then went faster than the

motorcycle. C. The motorcycle traveled farther than the scooter. D. The motorcycle’s speed increased and then the motorcycle traveled at a steady rate.

• A 1,500-‐gallon tank contains 200 gallons of water. Water begins to run into the tank at the rate of 75 gallons per hour. When will the tank be full but not overflowing? A. 7 hours, 8 minutes B. 17 hours, 20 minutes C. 20 hours D. 22 hours, 40 minutes

• Write an equation of the line that is perpendicular to ! = !!! + 8 and goes through (-‐4, 5).

A. ! = − !!! + 3

B. ! = !!! + 7

C. ! = −2! + 8 D. ! = −2! − 3


Multiple-‐choice





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I can write and solve systems of two linear equations and inequalities in two variables.


Example problems

• Solve the following system of linear equations. −2! + ! = 2 ! + ! = −1

• Graph the solution of the following system of linear inequalities. ! < 2! + 1 ! < −3! + 3







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I can describe how changes in the parameters of linear functions and functions containing an absolute value of a linear expression affect their graphs and the relationships they represent.


Example problems

• Describe how the graph of ! ! = 3! will change if the constant of variation changes to 4. • Compare the functions ! = 3 ! and ! = − !

!! . Make at least two comparison statements.







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Core Content: Quadratic functions and equations Students study quadratic functions and their graphs, and solve quadratic equations with real roots in Algebra 1. They use quadratic functions to represent and model problems and answer questions in situations that are modeled by these functions. Students solve quadratic equations by factoring and computing with polynomials. The important mathematical technique of completing the square is developed enough so that the quadratic formula can be derived. Performance expectation A1.5.A

I can represent a quadratic function with a symbolic expression, as a graph, in a table, and with a description, and make connections among the representations.


Example problem

Look at the table of values below. Describe the type of function that the values represent.

x -‐2 -‐1 0 1 2 3 y -‐5 1 3 1 -‐5 -‐15

A. Quadratic function that opens up B. Quadratic function that opens down C. Linear function that rises D. Linear function that falls


Multiple-‐choice





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I can sketch the graph of a quadratic function, describe the effects that changes in the parameters have on the graph, and interpret the x-‐intercepts as solutions to a quadratic equation.


Example problems

• Determine the roots of the quadratic function ! = !! + 5! + 6. A. (2, 0) and (3, 0) B. (-‐2, 0) and (-‐3, 0) C. (1, 0) and (5, 0) D. (-‐1, 0) and (-‐5, 0)

• Which of the following observations are true about ! ! = !! + 1 and ! ! = !! − 1. A. f and g each have 2 x-‐intercepts B. f and g each have 1 x-‐intercept C. f has 0 x-‐intercepts, g has 2 x-‐intercepts D. f has 2 x-‐intercepts, g has 0 x-‐intercepts


Multiple-‐choice





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I can solve quadratic equations that can be factored as (!" + !)(!" + !) where a, b, c, and d are integers.


Example problems

• Determine the solution(s) of 2!! + ! − 3 = 0. • Determine the solution(s) of 4!! + 6! = 0. • Determine the solution(s) of 36!! − 25 = 0. • Determine the solution(s) of !! + 6! + 9 = 0.







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I can solve quadratic equations that have real roots by completing the square and by using the quadratic formula.


Example problems

• Determine the exact solution(s), if any, of !! + 4! − 13 = 0. • Determine the exact solution(s), if any, of 4!! − 2! − 5 = 0.







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Core Content: Data and distributions Students select mathematical models for data sets and use those models to represent, describe, and compare data sets. They analyze data to determine the relationship between two variables and make and defend appropriate predictions, conjectures, and generalizations. Students understand limitations of conclusions based on results of a study or experiment and recognize common misconceptions and misrepresentations in interpreting conclusions. Performance expectation A1.6.A

I can use and evaluate the accuracy of summary statistics to describe and compare data sets.


Example problems

• The local minor league baseball team has a salary dispute. Players claim they are being underpaid, but managers disagree. Bearing in mind that a few top players earn salaries that are quite high, would it be in the players’ best interest to cite their median or mean team salary as proof that they are underpaid? Explain your answer using words, numbers, and/or diagrams.

• Each box-‐and-‐whisker plot below shows the prices of used cars (in thousands of dollars) advertised for sale at three different car dealers. If you want to go to the dealer whose prices seem least expensive, which dealer would you go to? Use statistics from the displays to justify your answer.







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I can make valid inferences and draw conclusions based on data.


Example problem

Mr. Shapiro found that the amount of time his students spent doing mathematics homework is positively correlated with test grades in his class. He concluded that doing homework makes students’ test scores higher. Is this conclusion justified? Explain any flaws in Mr. Shapiro’s reasoning.







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I can describe how linear transformations affect the center and spread of univariate data.


Example problem

Due to budget constraints at a particular company, every employee receives a 2% decrease to their salary. What impact does this salary decrease have on the mean and on the range of employee salaries at the company? A. The mean and range both decrease. B. The mean and range do not change. C. The mean does not change but the range decreases. D. The mean decreases but the range does not change.


Multiple-‐choice





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I can find the equation of a linear function that best fits bivariate data that are linearly related, interpret the slope and y-‐intercept of the line, and use the equation to make predictions.


Example problem







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I can describe the correlation of data in scatterplots in terms of strong or weak and positive or negative.


Example problem

Which term best describes the scatterplot below?

A. Positive correlation B. Negative correlation C. Zero correlation D. Perfect correlation


Multiple-‐choice





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Additional Key Content Students develop a basic understanding of arithmetic and geometric sequences and of exponential functions, including their graphs and other representations. They use exponential functions to analyze relationships, represent and model problems, and answer questions in situations that are modeled by these nonlinear functions. Students learn graphical and numerical methods for approximating solutions to exponential equations. Students interpret the meaning of problem solutions and explain limitations related to solutions. Performance expectation A1.7.A

I can sketch the graph for an exponential function of the form ! = !!! where n is an integer, describe the effects that changes in the parameters ! and b have on the graph, and answer questions that arise in situations modeled by exponential functions.


Example problems

• Graph A is the graph of ! = 4(5)! and graph B is the graph of ! = 5(4)! . Which statement about the two graphs is true? A. Both graphs A and B rise at the same rate. B. Graph A rises at a faster rate than graph B. C. Graph B rises at a faster rate than graph A. D. The y-‐intercept of graph A is above the y-‐intercept of graph B.

• You have won a door prize and are given a choice between two options: $150 invested for 10 years at 4% compounded annually. $200 invested for 10 years at 3% compounded annually. If you want to receive the most money, which prize would you choose? Support your answer using words, numbers, and/or diagrams.







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I can find and approximate solutions to exponential equations.


Example problem

Solve the equation 3! = 729. A. ! = 5 B. ! = 6 C. ! = 243 D. ! = 726


Multiple-‐choice





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I can express arithmetic and geometric sequences in both explicit and recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.


Example problems

• Write a recursive formula for the arithmetic sequence 5, 9, 13, 17, … • Given that ! 0 = 3 and ! ! + 1 = ! ! + 7 when n is a positive integer, determine !(5). • Determine the 25th term for the geometric sequence 5, 10, 20, 40,…







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I can solve an equation involving several variables by expressing one variable in terms of the others.


Example problems

• Kent is using the scale to compare the weight of various solids.

How many spheres will balance one cube? A. 2 spheres B. 3 spheres C. 4 spheres D. 5 spheres

• Solve ! = ! + !"# for p. A. ! = ! − !"# B. ! = !

!"

C. ! = !!!!"

D. ! = !!!!"


Multiple-‐choice





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Core Processes: Reasoning, problem solving, and communication Students formalize the development of reasoning in Algebra 1 as they use algebra and the properties of number systems to develop valid mathematical arguments, make and prove conjectures, and find counterexamples to refute false statements, using correct mathematical language, terms, and symbols in all situations. They extend the problem-‐solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students formalize a coherent problem-‐solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. The mathematical thinking, reasoning, and problem-‐solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways and more and more occupations and fields of study rely on mathematics. The following core processes performance expectations will manifest themselves throughout the core content performance expectations. A1.8.A – I can analyze a problem situation and represent it mathematically. A1.8.B – I can select and apply strategies to solve problems. A1.8.C – I can evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problems. A1.8.D – I can generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems. A1.8.E – I can read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics. A1.8.F – I can summarize mathematical ideas with precision and efficiency for a given audience and purpose. A1.8.G – I can synthesize information to draw conclusions, and evaluate the arguments and conclusions of others. A1.8.H – I can use inductive reasoning about algebra and the properties of numbers to make conjectures, and use deductive reasoning to prove or disprove conjectures.

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Answers to Example Problems A1.1.A • B • B

A1.1.B • Option 3 • 48 hours • $108.75 • 7 hours • A little over 107 hours

A1.1.C • 80 cans of pop, 120 bags of popcorn • ! = 4! + 1, ! + ! = 121, 97

chocolate cones sold • 80 L of 40% solution, 40 L of 70%

solution • Wind speed = 30 mph, speed of plane

in still air = 470 mph A1.1.D • ! = 2 !" ! = −1 • 5

8 of a second • −4 and −3

A1.1.E • ! ! = 2! !", where f is the number

of bacteria and x is the number of minutes; ! 135 = 512, when x is 135 minutes

• ! ≈ 14

A1.2.A B

A1.2.B • When ! < 0 and n is odd • −1 ≤ ! ≤ 1 • ! ≤ 5 • ! < 0

A1.2.C • ! !

!

• −30

A1.2.D • 2 3 because 3.46 is an

approximation and approximations are usually not appropriate when solving equations without a context.

• About 3 seconds because !"! of a

second is not an answer you would give in a real-‐world context.

A1.2.E • A • D

A1.2.F • 4!! − ! + 2 • !! − 3! − 1 • 5!! + 44!! − 19! + 2 • ! − 3

A1.3.A • Domain: ! ≥ 0, range: !(!) ≥ 0 • Domain: ! ≤ 5 • A

A1.3.B • D • A

A1.3.C • 60 roses • $38

A1.4.A • ! = −7 • ! − 7 = 2 • No solution • 2 < ! ≤ !

!

A1.4.B • B • D • B • A • B

A1.4.C • A • B • D

A1.4.D • −1, 0 • See graph below

A1.4.E • ! ! = 4! will be a steeper line

than ! ! = 3! because the constant of variation (slope) is greater.

• ! = 3 ! opens up while ! = − !

!! opens down. ! = 3 !

is narrower than ! = − !!! .

A1.5.A B

A1.5.B • B • C

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A1.5.C • ! = − !

! !" ! = 1

• ! = − !! !" ! = 0

• ! = − !! !" ! = !

!

• ! = −3

A1.5.D • −2 ± 17 • !± !"

!

A1.6.A • The median. Example 1: For the

values 1, 2, 3, 4, 5, the median is 3 and the mean is 3. Example 2: For the values 1, 2, 3, 4, 20, the median is 3 and the mean is 6.

• Example: I would go to Yours Now because the lowest 50% of its prices are less than the lowest 50% of the other two dealers.

A1.6.B Correlation is not the same as causation. One example is that students could spend more time doing math homework and have lower test scores because they do the homework incorrectly.

A1.6.C A

A1.6.D A

A1.6.E B

A1.7.A • B • Prize 2 would result in the most

money. Prize 1: ! = 150 1.04 !" =$222.04. Prize 2: ! = 200(1.03)!" =$268.78.

A1.7.B B

A1.7.C • !! = 4! + 1 • ! 5 = 38 • 83,886,080

A1.7.D • B • C

Answer to question 2 for A1.4.D

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Strategies For Effective Student Learning Ten Strategies for Students to Learn Math In the Classroom

1. Be to class on time!

2. Have a positive attitude. You can do math!

3. Take notes.

4. Complete all of your classwork and homework assignments.

5. Show your work on assignments and assessments. That will help you and your teacher identify any

mistakes that you make.

6. Participate in classroom activities.

7. Take all assessments seriously. Do not leave anything blank.

8. Let your teacher know (when s/he is not talking or helping someone else) when there is something

you do not understand. Be as specific as you can about what part of the problem you do not

understand.

9. Ask a classmate for help (when it is okay for you to talk in class).

10. Do not give up! Take a deep breath and then try again. Perseverance is an essential skill in all areas of

life.

Ten Strategies for Students to Learn Math Outside of the Classroom

1. Read aloud your class notes and the section in the book you are working on.

2. Redo the examples from your class notes.

3. Do an internet search to see if there are descriptions and example problems about the topic you are

working on.

4. Do a youtube search to see if there is someone who has posted a lesson about the topic you are

working on.

5. Ask your teacher questions before or after school.

6. Ask a friend for help.

7. Ask another teacher for help.

8. Be sure to find out, right away, about any assignments or activities you missed when absent.

9. If you get stuck on a problem in your homework assignment, take a short break, and then return to

the assignment later.

10. Never, ever, ever say, “I hate math. I’m not good at math.” Having a negative attitude will make it

harder to be successful in the class and on the EOC Exam.

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Ten Strategies for Parents to Support Their Child’s Learning of Mathematics

1. Never, ever, ever say, “I was never any good at math in school.”

2. Never, ever, ever say, “I didn’t like math when I was in school.”

3. Never, ever, ever say, “I know math is boring and difficult, but you need to pass the class so do your

homework!”

4. Ask your child’s math teacher if your child is ever not paying attention in class, not participating in

classroom activities, not turning in assignments, or failing assessments.

5. Ask your child each day what s/he worked on in math class that day and if s/he has any homework. If

you notice that s/he frequently tells you s/he does not have homework or finished it in class, check

with his/her math teacher to verify that s/he is indeed completing and turning in all assignments.

6. Encourage your child to refer to the previous Ten Strategies sections if s/he is having difficulties with

math, inside or outside of the classroom.

7. Have your child write down on your home calendar, or post somewhere in the house, when

assessments are.

8. Make sure your child finds out about any missing assignments and activities when absent and

completes them right away.

9. Check with your child’s math teacher to find out if the school has a before-‐school or after-‐school study

hall where students have access to math teachers and/or students in advanced math classes. Make

sure that there are enough teachers/tutors to meet the needs of your child. Also make sure the study

hall is not just a place for students to socialize; the expectation of the students in the study hall should

be that they are doing schoolwork.

10. If you use a tutor, silently observe a tutoring session to be sure the tutor is not just doing the work for

your child. Good tutors ask guiding questions and do not write down the work for the students. Good

tutors do not focus just on that day’s assignment, but review lessons previously covered and preview

upcoming lessons. Make spending up to $50/hour for a math tutor a worthwhile investment.

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