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Unformatted text preview: Number Sense Number Sense Statistics, Data Analysis, Probability Algebra Measurement Math and and Reasoning Functions Geometry Algebra I Fourteen of the 80 CAHSEE math questions are based on 10 selected standards from the grade 7 Number Sense strand. WHAT DO THE NUMBER SENSE STANDARDS ASK ME TO DO? The CAHSEE Number Sense strand includes basic arithmetic calculations with whole numbers, fractions, and decimals, all done without using a calculator. The CAHSEE Number Sense questions will ask you to: solve problems with fractions, decimals, percentages, and integers compare and order numbers understand percents, including percents less than 1% and greater than 100% use ratios and proportions to solve problems understand the meaning of numbers written in scientific notation find and use multiples, factors, and primes add, subtract, multiply, and divide numbers, and use the relationships among these operations including inverse, commutative, associative, and distributive properties estimate square roots of whole numbers to the nearest whole number Vocabulary The following words have appeared on the CAHSEE previously. If any of these words are unfamiliar to you, look them up in the CAHSEE Math Vocabulary list in the appendix at the back of this Study Guide, or check with your math teacher. absolute value integer simple interest compound interest prime square decreased by scientific notation square root equivalent expression 33 NUMBER SENSE WHY IS NUMBER SENSE IMPORTANT? As an adult, you will use all the basic skills of arithmetic included in the Number Sense strand. You'll use your number sense skills as a consumer (nutrition choices, shopping for credit, choosing the best buy), as a U.S. citizen (taxes, voting issues), and as an employee (handling money, cost estimates, profit and loss, quality control). Many large companies give screening tests that include number sense knowledge for entry-level jobs. The anchor problem for this strand, Emergency Refrigerator!, presents a situation that you may encounter when you rent your first apartment. But before we try Emergency Refrigerator!, let's first look at some sample CAHSEE questions, with solutions, that apply to this strand. Number Sense HOW WILL THE CAHSEE TEST MY KNOWLEDGE OF NUMBER SENSE? The CAHSEE tests 10 of the 12 standards from the Number Sense strand for grade 7. Let's start by looking at 8 of these standards and some actual CAHSEE questions based on them. Each box that follows contains one of the standards, a sample question based on that standard, and a solution with explanation. 34 How Will the CAHSEE Test My Knowledge of Number Sense? 35 7NS1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation. [1 question] Sample CAHSEE Question The radius of the earth's orbit is 150,000,000,000 meters. What is this number in scientific notation? A B C 1.510- 11 1.51011 151010 M00213 Number Sense D 150109 Mathematical Solution The correct answer is B. Please refer to the next column for a description of the solution. Descriptive Solution Scientific notation is a short way to write very large or very small numbers using powers of 10. Here are some examples of the same numbers written first in the usual way and then in scientific notation: 2000 = 2 103 143, 000 = 1.43105 0.0000234 = 2.34 10- 5 A number in scientific notation is always written as a number greater than or equal to 1 but less than 10, times a power of 10. Looking at the possible answers for this CAHSEE question, you can see that options C and D are both incorrect because 15 and 150 are larger than 10. Also, in option A, the 10 has a negative exponent, so it would be a very small number. The correct answer must be B. But why? You can rewrite 150,000,000,000 as 1510, 000, 000, 000, which is 151010. But 151010 is not yet in scientific notation because 15 is not a number between 1 and 10. So think of 15 as 1.5101. Then 151010 = (1.5101 )1010 = 1.51011 , choice B. 36 NUMBER SENSE 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. [3 questions] Sample CAHSEE Question Number Sense 11 1 1 - + = 12 3 4 A B C D 1 3 3 4 5 6 9 5 M02048 Mathematical Solution 11 1 4 1 3 - i + i = 12 3 4 4 3 11 4 3 11 4 + 3 = - + = - 12 12 12 12 12 11 7 11 - 7 4 - = = 12 12 12 12 Descriptive Solution One way to do this problem is to first find the least common denominator for the three fractions. Notice that the numbers 3, 4, and 12 all divide evenly into 12, so 12 is the least common denominator. Next, find equivalent 1 1 fractions for and for , each with 12 as the 3 4 denominator: 1 1 4 4 1 1 3 3 = i = and = i = 4 4 3 12 3 3 4 12 Finally, the numerators of these fractions can be combined to get the solution. You could write your work out like this: 11 1 1 11 4 3 11 7 4 - + = - + = - = 3 4 12 12 12 12 12 12 12 But 4 isn't an answer choice! You need to reduce 12 4 1 to get , so the correct answer is A. 12 3 Reduce the fraction. 44 1 = 12 4 3 Therefore, the correct answer is A. How Will the CAHSEE Test My Knowledge of Number Sense? 37 7NS1.6 Calculate the percentage of increases and decreases of a quantity. [1 question] Sample CAHSEE Question The price of a calculator has decreased from $12.00 to $9.00. What is the percentage of decrease? Number Sense A 3% B 25% C 33% D 75% Mathematical Solution 12 - 9 i 100 = 12 3 i100 = 12 1 i100 = 4 M02868 25 Therefore, the correct answer is B. Descriptive Solution A price change from $12 down to $9 is a net decrease of $3. To find the percentage of decrease (or percentage of increase), the base is always the original or starting number, in this case $12. So, the correct percentage of decrease is 3 12 = 25%, choice B. Notice that 3 9 = 33%, option C, is not correct because $9 is the ending price, not the starting price. 38 NUMBER SENSE 7NS2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. [1 question] Sample CAHSEE Question Number Sense 10-2 = 10-4 A 10-6 B 10-2 C 102 D 108 Mathematical Solution 10- 2 10 4 = 10- 4 102 10 4 = 10(4-2) = 102 102 Or 10- 2 = 10- 2-(- 4) 10- 4 10- 2-(- 4) = 102 Therefore, the correct answer is C. M02832 Descriptive Solution When calculating with numbers written in scientific notation, it's important to know how to multiply and divide powers of ten. Here are a few powers of 10 and their equivalents written in the usual way: 10 4 = 10, 000 103 = 1000 102 = 100 101 = 10 100 = 1 10-1 = 0.1 = 1 10 1 100 1 10- 3 = 0.001 = 1000 10- 2 To simplify - 4 , one way is to rewrite 10 1 -2 1 1 1 10, 000 10 = 100 = = = 100 = 102 -4 1 100 10, 000 100 1 10 10, 000 10- 2 = 0.01 = Therefore, the correct answer is 102 , choice C. How Will the CAHSEE Test My Knowledge of Number Sense? 39 7NS2.1 Sample CAHSEE Question cont'd Another way is to remember that a negative power of ten is just the reciprocal of the positive power of ten. Using this 1 1 idea, 10- 2 = 2 and - 4 = 10 4 , therefore: 10 10 10- 2 10 4 = = 102. 10- 4 102 A third way is to remember the "law of exponents" for dividing powers of the same base: am = a(m-n). So for this problem, n a 10- 2 = 10(- 2-(- 4)) = 10(- 2+4) = 102. -4 10 Number Sense 40 NUMBER SENSE 7NS2.2 Add and subtract fractions by using factoring to find common denominators. [1 question] Sample CAHSEE Question Number Sense Which of the following is the prime factored form of the lowest common denominator 7 8 of + ? 10 15 A 5 i1 B 2 i3i 5 C 2 i 5i3i 5 D 10 i 15 Mathematical Solution Looking at the prime factors of the denominators, 10 = 2 i 5 and 15 = 3 i 5. Combining the smallest set of prime factors to both, you get 2 i 3 i 5. Therefore, the correct answer is B. M02826 Descriptive Solution The denominators of these two fractions are 10 and 15. In order to add or subtract these fractions you would need to find a common denominator (a number that both 10 and 15 divide into evenly). One way to do this is to list the multiples of the larger number, 15, until you get a multiple that the smaller number also divides into evenly. Multiples of 15 are 15, 30, 45, 60, and so on. The lowest number in this list that 10 also divides into evenly is 30. Therefore, 30 is the least common denominator. So, which of the multiple choice answers multiplies out to 30? Only choice B, which is the correct answer. But notice that standard 7NS2.2 says that you are to "use factoring to find the common denominator." According to the California standards, the proper way to do this problem is to first find the prime factorization of each denominator: 10 = 2 i 5 and 15 = 3 i 5. Then the common denominator is the product of the smallest set of prime factors that are common to both prime factorings, in this case 2 i 3 i 5. Notice that option C is incorrect. Even though 2 i 5 i 3 i 5 = 150, which is a common denominator, it is not the least common denominator; the factor 5 doesn't need to be included twice. How Will the CAHSEE Test My Knowledge of Number Sense? 41 7NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules. [1 question] Sample CAHSEE Question ( 38 ) A B C D 2 = Number Sense 34 36 310 316 M02406 Mathematical Solution (38 ) 2 = 38 i 2 = 316 Therefore, the correct answer is D. Descriptive Solution Sometimes it's difficult to remember how to use exponents. But you can answer these questions correctly if you understand how to use them. The exponent tells you how many times the base number is multiplied by itself. So, (38 ) 2 = (38 )(38 ) = (3 i 3 i 3 i 3 i 3 i 3 i 3 i 3) i (3 i 3 i 3 i 3 i 3 i 3 i 3 i 3) = 316. Choice D is correct. 7NS2.4 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. [1 question] Sample CAHSEE Question The square root of 150 is between-- A 10 and 11. B 11 and 12. C 12 and 13. D 13 and 14. M02666 Mathematical Descriptive Solution Solution The square root of a number is a number that can be multiplied by itself to get the original number. Some numbers have square roots The correct that are integers, for example, 49. The square root of 49 is 7, because answer is C. 7 i 7 = 49. But most numbers, like 150, do not have square roots that Please refer to the next column are integers. So, instead of figuring out the square root of 150, let's for a description look at the squares of the answer choices for this question: of the solution. 10 i 10 = 100; 11 i 11 = 121; 12 i 12 = 144; 13 i 13 = 169; 14 i 14 = 196. Because 150 is between 144 and 169, the square root of 150 must be between 12 and 13. The correct answer is C. 42 NUMBER SENSE 7NS2.5 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. [1 question] Sample CAHSEE Question Number Sense If x = 3, what is the value of x? A - 3 or 0 B - 3 or 3 C 0 or 3 M02122 D - 9 or 9 Mathematical Solution The correct answer is B. Please refer to the next column for a description of the solution. Descriptive Solution The absolute value of a number is its distance away from 0 on the number line. So if x = 3, then x must be 3 units away from 0. 3 units 3 units -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 The number line above shows there are two such numbers, 3 and - 3, so the correct answer is B. Using Number Sense Standards in a Real-Life Situation USING NUMBER SENSE STANDARDS IN A REAL-LIFE SITUATION You might be asking yourself, "When will I ever need to use this stuff ?" To help you get the "big picture," the remaining two Number Sense standards are illustrated by a real-life problem Emergency Refrigerator! that a person might encounter after high school. Even though the CAHSEE doesn't include problems like this one, you might find it easier to remember one large problem--an "anchor problem"--in which many of the skills are combined, rather than trying to recall each of the standards individually. Try to do this problem before you look at its solution on the following pages. Number Sense Emergency Refrigerator! You're just getting comfortable living in your first apartment. Then-- emergency! The old refrigerator left to you by the previous tenants stops working. You need to get a new refrigerator immediately. You decide on the brand and model of refrigerator you want and you have the following three options for purchasing it: Option #1 Wagmen's Department Store has the refrigerator for $240, but this week only it is marked as part of the "red sticker sale." The red sticker means it will be sold at "14 off " the list price. Delivery is $20. Option #2 The same refrigerator is advertised at Big Box Discount Appliances for $210. But you are lucky; this weekend only they have a coupon in the paper for 15% off the total cost of any item with a regular price over $100. Delivery costs $30. Option #3 A friend works for Mike's Furniture and can get the same refrigerator for you for 20% over the wholesale price of $180. Your friend can use the company truck to deliver it for free. You'll need to figure out the cost of each option before deciding what to do. Try to work out the cost of each option before going on to the next page. Remember, no calculators are allowed! 43 NUMBER SENSE Emergency Refrigerator! Solution and Standards Are you ready to check your answers to the Emergency Refrigerator! anchor problem? In order to figure the cost of the options, you need to use two of the Number Sense standards that are tested on the CAHSEE. The standards that apply in this situation, along with the number of questions on the CAHSEE that are based on that standard, appear to the left in the small print. To decide which option has the lowest price, you'll need to calculate the cost of each option. Option #1 One-fourth of $240 is $60, the amount to be taken off. The sale price would be $240 $60 $180. But to get the refrigerator, you'll have to get it delivered for $20. So the total cost for Option #1 would be $200. Option #2 Because the refrigerator costs more than $100, you can use the 15% off coupon to discount the selling price by 15%. One method is to first find 15% of $210. One way to think of 15% is "15 cents per dollar." So, 15% of $210 is 0.15 $210, which is $31.50. Then subtract $31.50 from the advertised price of $210, giving a sale price of $178.50. Another way to figure the sale price is to realize that if you get 15% off, that means you need to pay only 85% of the advertised price. 85% of $210 is $178.50. Finally, you need to add the $30 delivery charge, giving a total cost of $208.50. Option #3 In order to figure the cost of this option, you need to find your price by adding on a 20% markup to the wholesale price of $180. Markups can be done in two ways. One way is to find 20% of $180, which is $36, and then add the $36 to the $180 to get the selling price of $216. Another way is to realize that if 20% is to be added on, then the selling price is 120% of the wholesale price. Then 120% of $180 is $216. Either way you do it, that's the total cost for Option #3 because there is no delivery charge. You know that Option #1 costs $200, Option #2 costs $208.50, and Option #3 costs $216. Now that you've done the math, which option would you choose? You've seen the 10 Number Sense standards; now you are ready for some additional practice. Answer the sample questions in the next section and then check your answers using the answer key provided in the appendix at the back of this Study Guide. (Note: The CAHSEE questions used as examples throughout this Study Guide are questions that were used on prior CAHSEEs. These items will not be used in future CAHSEEs.) Number Sense 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. [2 questions] 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. [2 questions] 44 Additional Number Sense Sample Questions 45 ADDITIONAL NUMBER SENSE SAMPLE QUESTIONS 1. 3.6 X 10 2 = A B C 3.600 36 360 M00036 4. D 3,600 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? A 10% B 20% C 25% D 40% M02158 Number Sense 2. 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 B $90 $95 5. C $450 D $500 M00331 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? A B C $16.00 $24.00 $48.00 M02119 3. If Freya makes 4 of her 5 free throws in a basketball game, what is her free throw shooting percentage? A 20% B 40% C 80% D 90% M00223 D $160.00 6. 43 i 42 = A B 45 46 C 165 D 166 M02661 46 NUMBER SENSE 7. The square of a whole number is between 1500 and 1600. The number must be between-- A 30 and 35. Number Sense 9. What is the absolute value of 4? A 4 1 B 4 C M00313 B 35 and 40. C 40 and 45. D 45 and 50. 1 4 4 M02667 D 8. 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 ...
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