California High School Exit Examination Mathematics Released Test Questions California Department of Education October 2008 © 2008 California Department of Education (CDE) Permission is granted in advance for reproduction of these resources for educational purposes The content must remain unchanged and in its entirety as published by the California Department of Education To obtain permission to reproduce the information (text or graphics) contained in this document or any other CDE materials for any commercial purpose, submit the specifics of your request in writing to the Copyright Program Office, California Department of Education, CDE Press, 1430 N Street, Suite 3207, Sacramento, CA 95814 Fax: 916-324-9787 Mathematics Released Test Questions Introduction All California public school students must satisfy the California High School Exit Examination (CAHSEE) requirement, as well as all other state and local requirements, in order to receive a high school diploma The CAHSEE is divided into two parts: English-language arts and mathematics All questions on the CAHSEE are evaluated by committees of content experts, including California educators, teachers, and administrators, to ensure the questions’ appropriateness for measuring the designated California academic content standards in English-language arts and mathematics In addition to content, all items are reviewed and approved to ensure their adherence to the principles of fairness and to ensure no bias exists with respect to characteristics such as gender, ethnicity, and language This document combines released test questions that have appeared on the mathematics part of the CAHSEE since the 2000–2001 school year, and contains new test questions from the 2007–2008 school year A similar document for English-language arts is also available The questions are grouped by strand (e.g., Number Sense) At the beginning of each strand section is a list of the specific standards assessed on the CAHSEE Following a group of questions is a table that gives the correct answer for each question, the content standard each question is measuring, and the year each question originally appeared on the CAHSEE The following table lists each strand, the number of items that appear on the exam, and the number of released test questions that appear in this document STRAND • • • • • • Number Sense (NS) Statistics, Data Analysis, and Probability (PS) Algebra and Functions (AF) Measurement and Geometry (MG) Mathematical Reasoning (MR) Algebra I (1A) TOTAL NUMBER OF QUESTIONS ON EXAM NUMBER OF RELEASED TEST QUESTIONS 14 12 17 17 12 38 30 40 45 24 39 80 216 In selecting test questions for release, three criteria are used: (1) the questions adequately cover the content standards assessed on the CAHSEE; (2) the questions demonstrate a range of difficulty; and (3) the questions present a variety of ways each standard can be assessed These released test questions not reflect all of the ways the standards may be assessed Released test questions will not appear on future tests For more information about the CAHSEE, visit the CDE’s Web page at http://www.cde.ca.gov/ta/tg/hs/ —1— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education Mathematics Released Test Questions NUMBER SENSE The following ten California mathematics academic content standards from the Number Sense strand are assessed on the CAHSEE by 14 test questions and are represented in this booklet by 38 released test questions These questions represent only a few of the ways in which these standards may be assessed on the CAHSEE GRADE — NUMBER SENSE Standard Set 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms: 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers Convert fractions to decimals and percents and use these representations in estimations, computations, and applications Calculate the percentage of increases and decreases of a quantity Solve problems that involve discounts, markups, commissions, and profit, and compute simple and compound interest 1.2 1.3 1.6 1.7 Standard Set 2.0 Students use exponents, powers, and roots, and use exponents in working with fractions: 2.1 Understand negative whole-number exponents Multiply and divide expressions involving exponents with a common base Add and subtract fractions by using factoring to find common denominators Multiply, divide, and simplify rational numbers by using exponent rules 2.2 2.3 2.4 2.5 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers —2— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense The radius of the earth’s orbit is 150,000,000,000 meters What is this number in scientific notation? 3 B A 1.5 ×10−11 A B 1.5 ×1011 C 15 ×1010 D 150 ×109 M00213 3.6 ×10 = A 3.600 B 36 C 360 11 ⎜⎛ 1 ⎞⎟ −⎜ + ⎟ = 12 ⎜⎝ 4⎠⎟ C D M02048 D 3,600 M00036 Which of the following numerical expressions results in a negative number? A (–7) + (–3) Which expression represents 0.0000007 in scientific notation? A B (–3) + (7) C (3) + (7) 7×10−9 D (3) + (–7) + (11) B 7×10−7 C M00116 7×107 D 7×109 M20956 A less than The five members of a band are getting new outfits Shirts cost $12 each, pants cost $29 each, and boots cost $49 a pair What is the total cost of the new outfits for all of the members? A $90 B $95 One hundred is multiplied by a number between and The answer has to be— B between and 50 but not 25 C between and 100 but not 50 D between and 100 M00275 C $450 D $500 M00331 —3— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense of a cup of oats per serving to make oatmeal How many cups of John uses oats does he need to make servings? A 11 Some students attend school 180 of the 365 days in a year About what part of the year they attend school? A 18% B 50% C 75% D 180% M00047 B C 12 What number equals ? A 0.267 D B 0.375 M23015 C 2.67 D 3.75 ⎛ ⎞2 What is the value of ⎜⎜ ⎟⎟⎟ ? ⎜⎝ ⎠ A 64 B 32 C 16 D M13470 of all students at a school 16 participated in the science fair About 13 Last year what percentage of the students participated? A 18% B 23% M10014 C 44% D 56% 10 If Freya makes of her free throws in a basketball game, what is her free throw shooting percentage? M32378 A 20% B 40% C 80% D 90% M00223 —4— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense 14 The cost of an afternoon movie ticket last year was $4.00 This year an afternoon movie ticket costs $5.00 What is the percent increase of the ticket from last year to this year? 17 On Monday, Lisa’s fish bowl contained gallon of water On Friday, the fish bowl contained 0.75 gallon of water By what percentage did the amount of water in Lisa’s fish bowl decrease? A 10% A 0.25% B 20% B 0.75% C 25% C 25% D 40% D 75% M12141 M02158 18 Sally puts $200.00 in a bank account Each year the account earns 8% simple interest How much interest will be earned in three years? 15 The weekly sales of a magazine increased from 500,000 to 600,000 By what percentage did the magazine sales increase? A 17% A $16.00 B 20% B $24.00 C 83% C $48.00 D $160.00 D 120% M02119 M11242 19 A pair of jeans regularly sells for $24.00 They are on sale for 25% off What is the sale price of the jeans? 16 Traditions Clothing Store is having a sale Shirts that were regularly priced at $20 are on sale for $17 What is the percentage of decrease in the price of the shirts? A A $6.00 B $18.00 3% C $20.00 B 15% D $30.00 C 18% M02870 D 85% M30820 20 A CD player regularly sells for $80 It is on sale for 20% off What is the sale price of the CD player? A $16 B $60 C $64 D $96 M02425 —5— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense 24 21 Jana bought a car for $4200 and later sold it for a 30% profit How much did Jana sell the car for? 10−2 = 10−4 A 10−6 A $1260 B $2940 B 10−2 C $5460 C 102 D $7140 D 108 M02832 M10580 25 Which of the following is equivalent to 7−6 i ? 22 A salesperson at a clothing store earns a 2% commission on all sales How much commission does the salesperson earn on a $300 sale? A 7−24 A $6 B 7−10 B $15 C 7−2 C $60 D 72 D $150 M12679 M20470 26 Which fraction is equivalent to −4 23 Which number equals ( 2) A D A 35 48 B C 20 21 D 41 24 −8 B − C ? 16 16 + ? M12713 M10015 —6— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense 30 i = 27 Which of the following is the prime factored form of the lowest common A 45 + ? denominator of 10 15 B 46 A ×1 D 166 C 165 B 2×3×5 M02661 C 2×5×3×5 31 What is i 2 ? D 10 ×15 A 32 B 48 M02826 28 What is A B 3 − ? C 144 D 256 M22029 32 What is the value of A 12 B 10 11 D 12 D 32 C 26 i 24 ? 25 C 16 M25206 M13552 33 The square root of 150 is between— 29 ( 38 ) = A 10 and 11 B 11 and 12 A 34 C 12 and 13 B 36 D 13 and 14 C 310 M02666 D 316 M02406 —7— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Number Sense 37 What is the absolute value of − 4? 34 The square of a whole number is between 1500 and 1600 The number must be between— A −4 A 30 and 35 B 35 and 40 B − C D C 40 and 45 D 45 and 50 M00313 35 Between which two integers is the value of 61 ? M02667 A and 38 Which number has the greatest absolute value? B and C and A −17 D and 10 B −13 M22059 36 If x = 3, what is the value of x? C 15 D 19 M12795 A −3 or B −3 or C or D −9 or M02122 —8— This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Mathematical Reasoning Question Number Correct Answer Standard Standard School Year of Exam 154 155 156 157 158 159 160 161 162 163 164 A C C C B B A C C C B 7MR1.1 7MR1.1 7MR1.1 7MR1.1 7MR1.2 7MR1.2 7MR1.2 7MR1.2 7MR2.1 7MR2.1 7MR2.1 7MG1.3 7NS1.2 7MG1.3 7MG2.1 7AF1.1 7MG1.1 7AF1.1 7NS1.2 7NS1.2 7NS1.2 7NS1.7 2001–2002 2000–2001 2005–2006 2006–2007 2001–2002 2000–2001 2003–2004 2005–2006 2002–2003 2000–2001 2003–2004 165 166 167 168 169 170 171 172 173 174 175 176 177 C D C D A B A B C A C D D 7MR2.1 7MR2.1 7MR2.3 7MR2.3 7MR2.3 7MR2.3 7MR2.4 7MR2.4 7MR2.4 7MR3.3 7MR3.3 7MR3.3 7MR3.3 7AF4.2 7AF4.2 7AF1.5 7PS1.2 7AF3.3 7AF3.4 7AF1.1 6PS2.5 7NS1.2 7AF4.1 7NS1.2 7MG2.2 7NS2.5 2004–2005 2005–2006 2000–2001 2001–2002 2004–2005 2007–2008 2002–2003 2006–2007 2000–2001 2002–2003 2001–2002 2004–2005 2007–2008 — 72 — This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education Mathematics Released Test Questions ALGEBRA I The following ten California mathematics academic content standards from the Algebra I strand are assessed on the CAHSEE by 12 test questions and are represented in this booklet by 39 released test questions These questions represent only a few of the ways in which these standards may be assessed on the CAHSEE ALGEBRA I Standard Set 2.0 Standard Set 3.0 Standard Set 4.0 Standard Set 5.0 Standard Set 6.0 Standard Set 7.0 Standard Set 8.0 Standard Set 9.0 Standard Set 10.0 Standard Set 15.0 Students understand and use such operations as taking the opposite, finding the reciprocal, and taking a root, and raising to a fractional power They understand and use the rules of exponents.* Students solve equations and inequalities involving absolute values Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4) They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).* Students verify that a point lies on a line, given an equation of the line Students are able to derive linear equations by using the point-slope formula.* Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related Students are able to find the equation of a line perpendicular to a given line that passes through a given point.* Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets Students add, subtract, multiply, and divide monomials and polynomials Students solve multistep problems, including word problems, by using these techniques Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems * The crossed-out portion of this standard is not assessed on the CAHSEE, but is still included in grade-level standards — 73 — This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Algebra I 178 If x = −7, then −x = 181 If x is an integer, what is the solution to x − < 1? A −7 B − C D {−3} B {−3, −2, −1, 0, 1} C {3} D {−1, 0, 1, 2, 3} A M03035 182 If x is an integer, which of the following is the solution set for x = 15? M02863 {0, 5} B {−5, 5} C {−5, 0, 5} D {0, 45} A 179 The perimeter, P, of a square may be ⎛ 1⎞ found by using the formula ⎜⎜ ⎟⎟⎟ P = A , ⎜⎝ 4⎠ where A is the area of the square What M00059 is the perimeter of the square with an 183 What are all the possible values of x such that 10 x = 2.5 ? area of 36 square inches? A inches A 0.25 and − 0.25 B 12 inches B C 24 inches and − C 4.5 and − 4.5 D 72 inches M00057 D 25 and − 25 M12992 ax ? 180 What is the reciprocal of y A − ax y B − y ax C ax y D y ax M13174 — 74 — This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Algebra I 184 Which of the following is equivalent to ( x + 5) − ( x + 2) = 14? 187 Which of the following is equivalent to − x > ( x −2) ? A 1− x > 3x − A x + 20 − x − = 14 B B 1− x > 3x − x + − x + = 14 C 1− x > 3x − C x + − x + = 14 D 1− x > 3x − D x + 20 − x − = 14 M02231 M02936 185 Which of the following is equivalent to − x > ( x − 1) ? 188 Which equation is equivalent to x + x −1 = ? A 13 < 11x B 13 > 11x A 5x + = 16 x −1 C 10 > 11x B x + 15 = 16 x − D 6x > C 8x + = 10 x − M02531 D x + 24 = 10 x − M13117 20 = x x −5 189 Which equation is equivalent to x + −4 x = ( x −2) ? 186 Which of the following is equivalent to the equation shown above? A − x + = x −12 A x ( x − 5) = 80 B − x + = x −2 B 20 ( x − 5) = x C 20x = ( x − 5) D 24 = x + ( x − 5) C x + = x −12 D x + = x −2 M13109 M02403 — 75 — This is a sample of California High School Exit Examination questions This is NOT an operational test form Test scores cannot be projected based on performance on released test questions Copyright © 2008 by the California Department of Education California High School Exit Examination Algebra I 190 Colleen solved the equation 192 Which inequality represents the solution ( x + 5) = using the following of (11x + 2) + ( x + 4) + ( x + 5) >90? steps Given: ( x + 5) = Step 1: x + 10 = A x> 79 18 x> 79 17 B Step 2: x = −2 Step 3: x = − To get from Step to Step 3, Colleen— A divided both sides by C x> 101 18 D x> 101 17 M20669 B subtracted from both sides C added to both sides 193 What is the y-intercept of the line x − y = 12? D multiplied both sides by A (0, −4) M03139 B (0, −3) 191 Solve for x C (2, 0) D (6, 0) ( x − 3) − x < M02591 A x < −1.5 B x < 1.5 194 What are the coordinates of the x-intercept of the line x + y = 12? C x