# CAHSEE on Target UC Davis, School and University Partnerships

Save this PDF as:

Size: px
Start display at page:

## Transcription

1

2

4

5 Introduction to the CAHSEE The CAHSEE stands for the California High School Exit Exam. The mathematics section of the CAHSEE consists of 80 multiple-choice questions that cover 53 standards across 6 strands. These strands include the following: Number Sense (14 Questions) Statistics, Data Analysis & Probability (12 Questions) Algebra & Functions (17 Questions) Measurement & Geometry (17 Questions) Mathematical Reasoning (8 Questions) Algebra 1 (12 Questions) What is CAHSEE on Target? CAHSEE on Target is a tutoring course specifically designed for the California High School Exit Exam (CAHSEE). The goal of the program is to pinpoint each student s areas of weakness and to then address those weaknesses through classroom and small group instruction, concentrated review, computer tutorials and challenging games. Each student will receive a separate workbook for each strand and will use these workbooks during their tutoring sessions. These workbooks will present and explain each concept covered on the CAHSEE, and introduce new or alternative approaches to solving math problems. What is Number Sense? Number Sense is the understanding of numbers and their relationships. The Number Sense Strand concepts that are tested on the CAHSEE can be divided into five major topics: Integers & Fractions; Exponents; Word Problems; Percents; and Interest. These topics are presented as separate units in this workbook. 1

6 Unit 1: Integers & Fractions On the CAHSEE, you will be given several problems involving rational numbers (integers, fractions and decimals). Integers are whole numbers; they include... positive whole numbers {1, 2, 3,... } negative whole numbers { 1, 2, 3,... } and zero {0}. Positive and negative integers can be thought of as opposites of one another. A. Signs of Integers All numbers are signed (except zero). They are either positive or negative. When adding, subtracting, multiplying and dividing integers, we need to pay attention to the sign (+ or -) of each integer. Example: 5-3 = Example: = Example: = Whether it s written or not, every number has a sign: Example: 5 means +5 2

7 Signed Numbers in Everyday Life Signed numbers are used in everyday life to describe various situations. Often, they are used to indicate opposites: Altitude: The elevator went up 3 floors (+3) and then went down 5 floors (-5). Weight: I lost 20 pounds (-20) but gained 10 back (+10). Money: I earned \$60 (+60) and spent \$25 (-25). Temperature: The temperature rose 5 degrees (+5) and then fell 2 degrees (-2). Sea Level: Jericho, the oldest inhabited town in the world, lies 853 feet below sea level (-853), making it the lowest town on earth. Mount Everest is the highest mountain in the world, standing at 8850 meters (+8850), nearly 5.5 miles above sea level. Can you think of any other examples of how signed numbers are used in life? 3

8 i. Adding Integers When adding two or more integers, it is very important to pay attention to the sign of each integer. Are we adding a positive or negative integer? We can demonstrate this concept with a number line. Look at the two examples below. In the first example, we add a positive 3 (+3) to 2. Example: = In this second example, we add a negative 3 (-3) to 2. Example: 2 + ( 3) = As you can see, we get a very different answer in this second problem To add integers using a number line, begin with the first number in the equation. Place your finger on that number on the number line. Look at the value and sign of the second number: if positive, move to the right; if negative, move to the left. (If a number does not have a sign, this means it is positive.) With your finger, move the number of spaces indicated by the second number. Example: 1 + (-2) = 4

9 On Your Own -2 + (-3) = -6 + (3) = 3 + (-6) = = 5

10 Rules for Adding Signed Numbers (without a Number Line) A. Same Signs Find the sum Keep the sign B. Different Signs Find the difference Keep the sign of the larger number (# with larger absolute value) On Your Own -8 + (-7) = = (-13) + (-9) = (+13) + (+9) = 21 + (-21) = (-21) + 21 = = = Add -10 and -5: Add (-10), (+4), and (-16): 6

11 ii. Subtracting Integers We can turn any subtraction problem into an addition problem. Just change the subtraction sign (-) to an addition sign (+) and change the sign of the second number. Then solve as you would an addition problem. Example: 2 - (+ 3) = Turn it from a subtraction problem to an addition problem; then change the sign of the second number: 2 - (+ 3) = -2 + (-3) Now solve as you would an addition problem. We can show this on a number line. Place your finger on that number on the number line. Look at the value and sign of the second number: if positive, move to the right; if negative, move to the left. With your finger, move the number of spaces indicated by the second number: Let's look at another problem: Answer: Example: -2 - (-3) = -2 + ( ) Answer: 7

12 On Your Own CAHSEE on Target 6 (+3) = 6 + ( ) = 3 (-3) = 3 + ( ) = -5 (+1) = -5 + ( ) = 1 (+1) = 1 + ( ) = 8

13 Rules for Subtracting Signed Numbers (without a Number Line) Add its opposite! Draw the line and change the sign (of the second number), and follow the rules for addition. Example: 6 (-4) Steps: Draw the line (to turn the minus sign into a plus sign): 6 + Change the sign of the second number: 6 + (+ 4) Now you have an addition problem. Follow the rules of adding numbers: =10 On Your Own: Draw the line and change the sign. Then solve the addition problem. 19 (- 13) = -17 (-15) = 34 (-9) = = (-35) = 13 - (+15) = = (+35) = Subtract (-15) from (20): Subtract 4 from (-14): 9

14 Signed Numbers Continued Look at the following problem: Example: = We can represent this problem on a number line: We begin at 1, move 3 spaces backwards (to the left) and then 5 spaces forwards (to the right). We arrive at + 3. When we are given a problem with three or more signed integers, we must work out, separately, the addition and subtraction for each integer pair: = = -2 Work out the addition or subtraction for the 1st 2 integers = Take the answer from above & add it to the last integer. On Your Own = = = 10

15 iii. Multiplying and Dividing with Signed Numbers Multiplying The product of two numbers with the same sign is positive. Example: -5-3 = 15 The product of two numbers with different signs is negative. Example: -5 3 = -15 Dividing The quotient of two numbers with the same sign is positive. Example: = 5 The quotient of two numbers with different signs is negative. Example: = -5 On Your Own (+8) ( 4) = ( 7) (7) = ( 8) (-8) = (+7)(+8) = 36 (-3) = 36 3 = 11

16 B. Absolute Value The absolute value of a number is its distance from 0. This distance is always expressed as a positive number, regardless if the number is positive or negative. It is easier to understand this by examining a number line: The absolute value of 5, expressed as 5, is 5 because it is 5 units from 0. We can see this on the number line above. The absolute value of -5, expressed as -5, is also 5 because it is 5 units from 0. Again, look at the number line and count the number of units from 0. On Your Own: Complete the chart. How far from zero is the number? Number Absolute Value x -x

17 Finding the Absolute Value of an Expression On the CAHSEE, you may need to find the absolute value of an expression. To do this,... Evaluate the expression within the absolute value bars. Take the absolute value of that result. Perform any additional operations outside the absolute value bars. Example: = = = 10 On Your Own: Complete the chart = 5-5 = 5 5 = = = = = = = = = = = = = = = = = 13

18 Absolute Value Continued 2.5 While the absolute value of a number or expression will always be positive, the number between the absolute value bars can be positive or negative. Notice that in each case, the expression is equal to +8. You may be asked to identify these two possible values on the CAHSEE. Example: If x = 8, what is the value of x? For these types of problems, the answer consists of two values: the positive and negative value of the number. In the example above, the two values for x are 8 or -8. On Your Own 1. If y = 225, what is the value of y? or 2. If x = 1,233, what is the value of x? or 3. If m = 18, what is the value of m? or 4. If x = 12, what is the value of x? or 5. If y = 17, what is the value of y? or 14

19 C. Fractions A fraction means a part of a whole. Example: In the picture below, one of four equal parts is shaded: We can represent this as a fraction: 4 1 Fractions are expressed as one number over another number: 4 1 Every fraction consists of a numerator (the top number) and a denominator (the bottom number): A B Numerator Denominator Fractions mean division: B A = A B 1 = 1 4 = 4 1 = = 4 5 = 5 4 = = 1 2 = 2 1 =

20 i. Adding & Subtracting Fractions Same Denominator: Keep the denominator; add the numerators: Example: = We can represent this problem with a picture: Begin with the first fraction, 4 1, and add two more fourths ( 4 2 ): We now have three-fourths of the whole shaded: 4 3 On Your Own: Add the following fractions = = = = Rule: When adding and subtracting fractions that have common denominators, we just add or subtract the numerators and keep the denominator. It gets trickier when the denominators are not the same. 16

21 Different Denominator Example: Let's represent this with a picture: The first picture shows one whole divided into four parts. One of these parts is shaded. We represent this as a fraction: 4 1 The second picture shows one whole divided into eight parts. Three of these parts are shaded. We represent this as a fraction: 8 3 In order to add these two fractions, we need to first divide them up into equal parts. The first picture is divided into fourths but the second is divided into eighths. We can easily convert the first picture into eighths by drawing two more lines (i.e. divide each fourth by half): 17

22 Now let's see how the first fraction would appear once it is divided into eighths: 1 2 We can see, from the above picture, that is equal to. 4 8 Now that we have a common denominator (8), we can add the 2 3 fractions: +. Just keep the denominator and add the 8 8 numerators: = Let's look at another example: Can we add these two fractions in their current form? Explain. To add two fractions, we need a common denominator. We must therefore convert the fractions to ones whose denominator is the same. We can use any common denominator, but it is much easier to use the lowest common denominator, or LCD. One way to find the LCD is to make a table and list, in order, the multiples of each denominator. (Multiple means Multiply!) 18

23 Finding the Lowest Common Denominator (LCD) Look at the last problem again: Now list the multiples of each denominator until you reach a common number. Multiples of 3 Multiples of The lowest common denominator (LCD) is the first common number in both columns: 15. This will be the new denominator for both fractions. Since we changed the denominators, we must also change the numerators so that each new fraction is equivalent (or equal) to the original fraction. Let s start with the first original fraction: 2/3. Go back to the table. How many times did we multiply the denominator, 3, by itself? (Hint: How many rows did we go down in the first column?) Since we multiplied the denominator (3) by to get 15, we must also multiply the numerator (2) by. Our new fraction is 15 Now let s look at the second fraction: 4/*5. Since we multiplied the denominator (5) by, we do the same to the numerator: 4 =. Our new fraction is 15 Now add the new fractions. + = We have an improper fraction because the numerator > the denominator. We must change it to a mixed number: 22 = 15 19

24 Let's look at another example: Example: Add the following fractions: In order to add these fractions we must first find a common denominator. Make a table and list all of the multiples for each denominator until we reach a common multiple: Multiples of 4 Multiples of We have a common denominator for both fractions: 20. Since we changed the denominators for both fractions, we must also change the numerators so that each new fraction is equivalent to the original fraction. Let s begin with the first fraction: 4 3 = 20 Now let s proceed to the second fraction: 5 4 = 20 Now both fractions have common denominators; add them: = 20 If the sum is an improper fraction (i.e. numerator > denominator), we generally change it to a proper fraction: 20

25 On Your Own Example: Step 1: Make a table and list the multiples of each denominator until you reach a common denominator: Step 2: Convert each fraction to an equivalent fraction: Step 3: Add the fractions: Note: If you end up with an improper fraction, be sure to convert it to a mixed number. 21

26 Practice = ' 7 3 = = = = =

27 Prime Factorization Another way to find the lowest common denominator of two fractions is through prime factorization. First, let s learn more about prime numbers: Prime Numbers: A prime number has two distinct whole number factors: 1 and itself. Note: 1 is not prime because it does not have two distinct factors. Example: 6 is not prime because it can be expressed as 2 3. Example: 7 is prime because it can be expressed only as the product of two distinct factors: 1 7. Write the first 10 prime numbers: Composite Numbers A non-prime number is called a composite number. Composite numbers can be broken down into products of prime numbers: Example: 4 = 2 X 2 Example: 12 = 2 X 6 = 2 X 2 X 3 Example: 66 = 6 X 11 = 2 X 3 X 11 Example: 24 = 2 X 12 = 2 X 2 X 2 X 3 Example: 33 = 3 X 11 Example: 125 = 5 X 5 X 5 23

28 Practice: Circle all of the prime numbers in the chart below:

29 Prime Factor Trees CAHSEE on Target We can find the prime factors of a number by making a factor tree: Example: Find the prime factors of 18. Write your number: 18 Begin with the smallest prime number factor of 18 (i.e. the smallest prime number that divides evenly 18. This number is 2. Draw two branches: 2 and the second factor: \ 2 9 Continue this process for each branch until you have no remaining composite numbers. The prime factors of 18 are the prime numbers at the ends of all the branches: 18 \ 2 9 \ 3 3 The prime factored form of 18 is. 25

30 Example: Find the prime factors of 60 using the factor tree: 60 \ 2 30 \ 2 15 \ 3 5 The prime factors of 60 are the factors at the end of each branch:,, and. Helpful Guidelines: Start with the smallest numbers: first 2 s, then 3 s, and so on. If a number is even, it is divisible by 2. Note: An even number ends in 0, 2, 4, 6, and 8. Examples: If the digits of a number add up to a number divisible by 3, the number is divisible by 3. Example: 123 can be divided evenly by 3 because if we add all of its digits, we get 6: = 6 Since the sum of the digits of 123 is divisible by 3, so too is 123. If a number ends in 0 or 5, it is divisible by 5. Examples:

31 On Your Own CAHSEE on Target Find the prime factors of each number, using a factor tree:

32 Prime Factorization and the Lowest Common Denominator On the CAHSEE, you will be asked to find the prime factored form of the lowest common denominator (LCD) of two fractions: Example: Find the prime factored form for the lowest common denominator There are two methods we can use to solve this problem: Method I: Factor Tree and Pairing Steps: Make a factor tree for both denominators: 6 9 \ \ Pair up common prime factors: Multiply the common factor (counted once) by all leftover (unpaired) factors: LCD = 3 = 28

33 Let's look at another example: Example: Find the least common multiple of 72 and 24. Write the LCM in prime-factored form. Steps: Make a factor tree for each number: \ \ \ \ \ \ \ 3 3 Pair off common factors: 72 = = Count any common factor once! Multiply all common factors by all leftover (unpaired) factors: LCM = = 29

34 On Your Own: Solve the following problems, using the factor tree/pairing method. 1. What is the prime factored form of the lowest common denominator of ? 2. Find the least common multiple, in prime-factorization form, of 12 and 15. We will now look at the second method to find the prime factored from of the lowest common denominator (LCD) of two fractions. 30

35 Method II: Factor Tree and Venn Diagram To illustrate this second method, let's return to the original problem: Example: Find the prime factored form for the lowest common denominator of Use the factor tree method to find the prime factored form of 6: 6 \ 2 3 Use the factor tree method to find the prime factored from of 9: 9 \ 3 3 Use a Venn diagram to find the prime-factored form of the lowest common denominator: On the next page, we will learn how to fill out this diagram. 31

36 Venn Diagrams CAHSEE on Target Venn diagrams are overlapping circles that help us compare and contrast the characteristics of different things. We can use them to find what is common to two items (where the circles overlap in the middle) and what is different between them (what is outside the overlap on either or both sides). Here, we want to find out which prime factors are the same for two numbers and which factors are distinct, or different. 6 9 \ \ Steps: Since only one 3 is common to both numbers, we need to put it in the middle, where the two circles overlap: 6 Both 9 Continued on next page 32

37 Now find the prime factors that are left for 6 and place them in the part of the circle for 6 that does not overlap with the circle for 9. 6 Both 9 Next, find the prime factors that are left for 9 and place them in the part of the circle that does not overlap with the circle for 6. 6 Both 9 The lowest common denominator for 6 and 9 is the product of all of the numbers in the circles:, which is equal to Note: To write the LCD in prime-factored form, we do not carry out the multiplication; we just write the prime numbers: LDC of 6 and 9 = 33

38 On Your Own 1. What is the prime factored form of the lowest common denominator of 6 1 and10 3? Create separate prime factor trees for both denominators: 6 10 \ \ Organize the prime factors of both denominators, using a Venn diagram: 6 Both 10 What is the LCD? Write the LCD in prime factored form: 34

39 2. Find the prime factored form of the lowest common denominator for the following: Factor Trees: 8 Both 12 LCD: LCD in prime factored form: 35

40 ii. Multiplying Fractions Whenever you are asked to find a fraction of a number, you need to multiply. In math, the word of means multiply. Example: Find 2 1 of 2 1. This is a multiplication problem. It means, What is ? We can represent the problem visually. Here is the first part of the problem: 2 1 of the circle has been shaded. Taking 2 1 of a number means dividing it by 2. Now, if we take one-half of this again (divide it by 2 again), we get the following: of is equal to We end up with one-fourth of the circle. Note: We also could have solved the above problem by multiplying the numerator by the numerator and the denominator by the denominator: Numerator Numerator_ = 1 1 = 1 Denominator Denominator

41 When working these problems out during the CAHSEE, you will need to apply this rule: Numerator Numerator Denominator Denominator Look at the next problem: Find 2 1 of 24. In math, we can write this as follows: The first factor is a fraction and the second factor is a whole number. We can easily change the second factor to a fraction because any whole number can be expressed as a fraction by placing it over a 1: = because 24 means 24 ones We can rewrite the problem as follows: 2 1 Now, just follow the rule for multiplying two fractions: Numerator Numerator Denominator Denominator = = Note: Taking 2 1 of 24 means dividing 24 by 2. 37

42 Now look at the next example: 24 5 Example: = 1 6 There are two ways to solve this problem: 1. The hard way: Perform all operations Multiply numerators: 24 5 Multiply denominators: 1 6 Divide new numerator by denominator: = 120 = = The easy way: Simplify first, and then multiply: _ = Simplify by dividing out common factors! Look at the following problems: = 3,435 = 79 = Do you need to work out these problems, or do you already know the answers? Remember: If you divide both a numerator and denominator by a common factor, you can make the problem much simpler to solve. So save yourself the time and work, and recognize these types of problems right away. 38

43 Look at the next set of problems: What do you notice about the above problems? There is a lot of heavy multiplication involved in these problems. Is there a way to make your work easier? Explain: We can fractions by before solving. We can simplify these problems quite a bit before solving. This makes our job easier. Let s look at the first problem: We can divide out common factors in each fraction. These common factors become clear if we write each fraction as a product of prime factors. Let's begin with the first fraction: 4 = = Now do the second fraction on your own: 6 = 8 Now let's multiply the two reduced fractions; but first, can we simplify anymore? If so, simplify first, and then multiply: 39

44 On Your Own: Simplify and solve: = = = = = = = =

45 iii. Dividing Fractions When you divide something by a fraction, think, How many times does the fraction go into the dividend? Example: dividend This means, How many times does go into? We can represent this visually: Answer: Example: We can represent this visually: Answer: 41

46 On Your Own: Solve the next few problems, asking each time, How many times does the fraction go into the whole number? = = = Do you see a pattern? Explain. 42

47 Reciprocals As we saw in the previous exercise, each time we divide a whole number by a fraction, we get as our answer the product of the whole number and the reciprocal of the fraction. Reciprocal means the flip-side, or inverse. Example: The reciprocal of 5 4 is 4 5. On Your Own: Find the reciprocal of each fraction: Now let's find the reciprocal of a whole number. We know that any whole number (or integer) can be expressed as a fraction by placing it over 1: Example: 35 = 1 35 The reciprocal is the fraction turned upside down, or inverted: 1 Example: The reciprocal of 35 is 35 On Your Own: Find the reciprocal of each integer

48 Now we are ready to divide a whole number by a fraction Example: 2 = = = We can represent the above problem visually: means... If we count the number of little rectangles in the two big rectangles, we get. On Your Own = = = = = 44

49 Simplifying Division Problems Example: Remember the rule for dividing fractions: Rule: When dividing fractions, multiply the first fraction by the reciprocal of the second fraction! Steps: Multiplying the first fraction by the reciprocal of the second fraction, we get We can simplify this problem by dividing out common factors: Now, apply the rule for multiplication: Numerator Numerator = 1 2 = Denominator Denominator

50 On Your Own: Simplify and solve. 1 3 = = = = =

51 Unit Quiz: The following problems appeared on the CAHSEE ( ) = A. 3 1 B. 4 3 C. 6 5 D Which fraction is equivalent to ? 35 A. 48 B C D What is the prime factored form for the lowest common 2 7 denominator of the following: +12? 9 A. 3 X 2 X 2 B. 3 X 3 X 2 X 2 C. 3 X 3 X 3 X 2 X 2 D. 9 X 12 47

52 4. Which of the following is the prime factored form of the lowest 7 8 common denominator of +15? 10 A. 5 X 1 B. 2 X 3 X 5 C. 2 X 5 X 3 X 5 D. 10 X Which of the following numerical expressions results in a negative number? A. (-7) + (-3) B. (-3) + (7) C. (3) + (7) D. (3) + (-7) + (11) 6. One hundred is multiplied by a number between 0 and 1. The answer has to be. A. less than 0. B. between 0 and 50 but not 25. C. between 0 and 100 but not 50. D. between 0 and If x = 3, what is the value of x? A. -3 or 0 B. -3 or 3 C. 0 or 3 D. -9 or 9 48

53 8. What is the absolute value of -4? A. -4 B. C. 4 1 D The winning number in a contest was less than 50. It was a multiple of 3, 5, and 6. What was the number? A. 14 B. 15 C. 30 D. It cannot be determined 10. If n is any odd number, which of the following is true about n + 1? A. It is an odd number. B. It is an even number C. It is a prime number D. It is the same as n Which is the best estimate of 326 X 279? A. 900 B. 9,000 C. 90,000 D. 900,000 49

54 12. The table below shows the number of visitors to a natural history museum during a 4-day period. Day Number of Visitors Friday 597 Saturday 1115 Sunday 1346 Monday 365 Which expression would give the BEST estimate of the total number of visitors during this period? A B C D John uses 3 2 of a cup of oats per serving to make oatmeal. How many cups of oats does he need to make 6 servings? A B 4 C D If a is a positive number and b is a negative number, which expression is always positive? A. a - b B. a + b C. a X b D. a b 50

55 Unit 2: Exponents On the CAHSEE, you will be given several problems on exponents. Exponents are a shorthand way of representing how many times a number is multiplied by itself. Example: can be expressed as 9 4 since four 9's are multiplied together. Base 9 4 exponent The number being multiplied is called the base. The exponent tells how many times the base is multiplied by itself. 9 4 is read as 9 to the 4 th power, or 9 to the power of 4. Let's look at another example: 2 = = 32 On Your Own 2³ = 2 = 3² = 3³ = Power of 0 Any number raised to the 0 power (except 0) is always equal to 1. Example: = 1 On Your Own 7 0 = = (-131) 0 = 47 0 = 51

56 Power of 1 A number raised to the 1 st power (i.e., an exponent of 1) is always equal to that number. Example: = 100 On Your Own 7 1 = = (-131) 1 = 47 1 = Power of 2 (Squares) A number raised to the 2 nd power is referred to as the square of a number. When we square a whole number, we multiply it by itself. Example: 12² = = 144 The square of any whole number is called a perfect square. Here are the first 3 perfect squares: 1² = 1 1 = 1 2² = 2 2 = 4 3² = 3 3 = 9 On Your Own: Write the perfect squares for the following numbers: 4² = 5² = 6² = 7² = 8² = 9² = 10² = 11² = 20² = (2-8)² (3-7)² = 3² + 5² = 52

57 Square Roots CAHSEE on Target The square root ( ) of a number is one of its two equal factors. Example: 8² = Any number raised to the second power (the power of 2) can be represented as a square. That s why it s called squaring the number. The square above has 64 units. Each side (the length and width) is 8 units. The area of the square is determined by multiplying the length (8 units) by the width (8 units). The square root is the number of units in each of the two equal sides: 8 Note: 64 has a second square root: -8 (-8-8 = +64). However, when we are asked to evaluate an expression, we always take the positive root. Example: Find the square root of 36. Answer: 36 = 53

58 On Your Own = = = = = = = = = 10. Which is not a perfect square? A. 144 B. 100 C. 48 D

59 Power of 3 (Cubes) A number with an exponent of 3 (or a number raised to the 3 rd power) is the cube of a number. Example: 5³ = = 125 The cube of a whole number is called a perfect cube. Cubes of Positive Numbers The cube of a positive number will always be a positive number. 1³ = = 1 2³ = = 8 Cubes and Negative Numbers The cube of a negative number will always be a negative number. (-1)³ = (-1)(-1)(-1) = -1 (-2) 3 = (-2)(-2)(-2) = -8 On Your Own: Write the perfect cubes for the following numbers: 3³ = 4³ = 5³ = -3³ = -4³ = -5³ = 55

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Summer Math Packet. Number Sense & Math Skills For Students Entering Pre-Algebra. No Calculators!!

Summer Math Packet Number Sense & Math Skills For Students Entering Pre-Algebra No Calculators!! Within the first few days of your Pre-Algebra course you will be assessed on the prerequisite skills outlined

### Primes. Name Period Number Theory

Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### Word Problems. Simplifying Word Problems

Word Problems This sheet is designed as a review aid. If you have not previously studied this concept, or if after reviewing the contents you still don t pass, you should enroll in the appropriate math

### MATH Fundamental Mathematics II.

MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Florida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower

Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including

### Possible Stage Two Mathematics Test Topics

Possible Stage Two Mathematics Test Topics The Stage Two Mathematics Test questions are designed to be answerable by a good problem-solver with a strong mathematics background. It is based mainly on material

### Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers. with Integers Divide Integers

Page1 Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### RIT scores between 191 and 200

Measures of Academic Progress for Mathematics RIT scores between 191 and 200 Number Sense and Operations Whole Numbers Solve simple addition word problems Find and extend patterns Demonstrate the associative,

### Florida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper

Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,

### EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent?

EQUATIONS Introduction to Variables, Algebraic Expressions, and Equations (2 days) Overview of Objectives, students should be able to: Main Overarching Questions: 1. Evaluate algebraic expressions given

### Exponents, Factors, and Fractions. Chapter 3

Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 3-1 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Simply Math. Everyday Math Skills NWT Literacy Council

Simply Math Everyday Math Skills 2009 NWT Literacy Council Acknowledgement The NWT Literacy Council gratefully acknowledges the financial assistance for this project from the Department of Education, Culture

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### Fractions and Decimals (pages 62 66)

A Fractions and Decimals (pages 6 66) A decimal that ends, such as 0., is a terminating decimal. All terminating decimals are rational numbers. 0.,000 A decimal that repeats, such as 0. is a repeating

### 2013 Texas Education Agency. All Rights Reserved 2013 Introduction to the Revised Mathematics TEKS: Vertical Alignment Chart Kindergarten Algebra I 1

2013 Texas Education Agency. All Rights Reserved 2013 Introduction to the Revised Mathematics TEKS: Vertical Alignment Chart Kindergarten Algebra I 1 The materials are copyrighted (c) and trademarked (tm)

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9

HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

### Principles of Mathematics MPM1D

Principles of Mathematics MPM1D Grade 9 Academic Mathematics Version A MPM1D Principles of Mathematics Introduction Grade 9 Mathematics (Academic) Welcome to the Grade 9 Principals of Mathematics, MPM

### BASIC MATHEMATICS. WORKBOOK Volume 2

BASIC MATHEMATICS WORKBOOK Volume 2 2006 Veronique Lankar A r ef resher o n t he i mp o rt a nt s ki l l s y o u l l ne e d b efo r e y o u ca n s t a rt Alg e b ra. This can be use d a s a s elf-teaching

### OPERATIONS AND PROPERTIES

CHAPTER OPERATIONS AND PROPERTIES Jesse is fascinated by number relationships and often tries to find special mathematical properties of the five-digit number displayed on the odometer of his car. Today

### Clifton High School Mathematics Summer Workbook Algebra 1

1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

### Math Foundations IIB Grade Levels 9-12

Math Foundations IIB Grade Levels 9-12 Math Foundations IIB introduces students to the following concepts: integers coordinate graphing ratio and proportion multi-step equations and inequalities points,

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### SUMMER MATH PACKET 2016 FOR RISING EIGHTH GRADERS WHO WERE IN PRE-ALGEBRA

SUMMER MATH PACKET 2016 FOR RISING EIGHTH GRADERS WHO WERE IN PRE-ALGEBRA 1 DIRECTIONS FOR COMPLETING THE PACKET The following page contains the list of help topics available on the Glencoe website. Each

### SIXTH GRADE MATH. Quarter 1

Quarter 1 SIXTH GRADE MATH Numeration - Place value - Comparing and ordering whole numbers - Place value of exponents - place value of decimals - multiplication and division by 10, 100, 1,000 - comparing

### Whole Numbers and Integers (44 topics, no due date)

Course Name: PreAlgebra into Algebra Summer Hwk Course Code: GHMKU-KPMR9 ALEKS Course: Pre-Algebra Instructor: Ms. Rhame Course Dates: Begin: 05/30/2015 End: 12/31/2015 Course Content: 302 topics Whole

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### Supporting your child with maths

Granby Primary School Year 5 & 6 Supporting your child with maths A handbook for year 5 & 6 parents H M Hopps 2016 G r a n b y P r i m a r y S c h o o l 1 P a g e Many parents want to help their children

### Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

### Expressions and Equations

Expressions and Equations Standard: CC.6.EE.2 Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation Subtract y from 5 as 5 y.

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Math 016. Materials With Exercises

Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of

### Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975. Consumer Math

Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975 Consumer Math Consumer Math ENROLLED STUDENTS ONLY This course is designed for the student who is challenged by abstract forms of higher This math. course

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### MAT 0950 Course Objectives

MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### RESOURCE FOR THE COURSE 1 CARNEGIE TEXTBOOK. SECTION and PAGE NUMBER TOPIC. 1.1 Pages 5-14 Factors and Multiples

RESOURCE FOR THE COURSE 1 CARNEGIE TEXTBOOK SECTION and PAGE NUMBER TOPIC 1.1 Pages 5-14 Factors and Multiples Properties of Addition and Multiplication (Commutative, Associative and Distributive) Apply

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### ALGEBRA 1/ALGEBRA 1 HONORS

ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical

What five coins add up to a nickel? five pennies (1 + 1 + 1 + 1 + 1 = 5) Which is longest: a foot, a yard or an inch? a yard (3 feet = 1 yard; 12 inches = 1 foot) What do you call the answer to a multiplication

### Basic Pre Algebra Intervention Program

Basic Pre Algebra Intervention Program This 9 lesson Intervention Plan is designed to provide extra practice lessons and activities for students in Pre Algebra. The skills covered are basics that must

T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,

### Year 1 Maths Expectations

Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all

### Math 0306 Final Exam Review

Math 006 Final Exam Review Problem Section Answers Whole Numbers 1. According to the 1990 census, the population of Nebraska is 1,8,8, the population of Nevada is 1,01,8, the population of New Hampshire

### Pre-Algebra A: Full Course Summary

Pre-Algebra A: Full Course Summary Course Summary This is the first of two courses that comprise Pre-Algebra. In this course, the student will be introduced to basic algebraic principles. The student will

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to:

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 5-1 Lesson 5-2

### Maths for Nurses: Fractions and Decimals

Maths for Nurses: Fractions and Decimals This booklet will provide an overview of the basic numeracy skills for Nursing students. If you have any problems in answering the questions within the booklet

### Accuplacer Elementary Algebra Study Guide for Screen Readers

Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these

### Chapter 4 Fractions and Mixed Numbers

Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### Basic Laws. Associative Law for Addition and Multiplication. Commutative Law for Addition and Multiplication. Distributive Law : a*(b c) = a*b a*c

CHAPTER 1F Fraction 1 F1 The Basics Whole numbers 1,,,,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a

### MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

### eday Lessons Mathematics Grade 8 Student Name:

eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards- Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### ACCUPLACER Arithmetic Assessment Preparation Guide

ACCUPLACER Arithmetic Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

### Chapter 1. Foundations of Algebra

Chapter 1. Foundations of Algebra 1.1. Signed Numbers, Exponents and Order of Operations KYOTE Standards: CR 1, CR 2; CA 1 Exponents Multiplication can be viewed as repeated addition. For example, 3 4

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### Transition To College Mathematics

Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

### Numerator Denominator

Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### Standards-Based Progress Mathematics. Progress in Mathematics

SADLIER Standards-Based Progress Mathematics Aligned to SADLIER Progress in Mathematics Grade 5 Contents Chapter 1 Place Value, Addition, and Subtraction......... 2 Chapter 2 Multiplication....................................

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Introduction to Fractions

Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states

### Five 5. Rational Expressions and Equations C H A P T E R

Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

### The wavelength of infrared light is meters. The digits 3 and 7 are important but all the zeros are just place holders.

Section 6 2A: A common use of positive and negative exponents is writing numbers in scientific notation. In astronomy, the distance between 2 objects can be very large and the numbers often contain many

### 5 th Grade CCGPS Math Unit 2: Numbers and Operations in Base Ten Student Study Guide

5 th Grade CCGPS Math Unit 2: Numbers and Operations in Base Ten Student Study Guide Dear Student: Please review the Math standards below to prepare for the Georgia Milestones test. This will require that

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations

### Algebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills

McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.

### Pre-Algebra Curriculum Map 8 th Grade Unit 1 Integers, Equations, and Inequalities

Key Skills and Concepts Common Core Math Standards Unit 1 Integers, Equations, and Inequalities Chapter 1 Variables, Expressions, and Integers 12 days Add, subtract, multiply, and divide integers. Make

### SECTION P.5 Factoring Polynomials

BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

### Grade 6 Math Circles. Algebra

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Grade 6 Math Circles October 8/9, 2013 Algebra Note: Some material and examples from the Tuesday lesson were changed for the Wednesday lesson. These notes

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006

MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order

### Self-Directed Course: Transitional Math Module 2: Fractions

Lesson #1: Comparing Fractions Comparing fractions means finding out which fraction is larger or smaller than the other. To compare fractions, use the following inequality and equal signs: - greater than

### Unit 1, Review Transitioning from Previous Mathematics Instructional Resources: McDougal Littell: Course 1

Unit 1, Review Transitioning from Previous Mathematics Transitioning from previous mathematics to Sixth Grade Mathematics Understand the relationship between decimals, fractions and percents and demonstrate

### Section R.2. Fractions

Section R.2 Fractions Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions

### Category A: Grades 4 5 SOLUTIONS Round 6 Thursday, February 12, 2015

Category A: Grades 4 5 SOLUTIONS Round 6 Thursday, February 12, 2015 Question #1 A grocery store parking lot has space for 1000 cars. 2/5 of the spaces are for compact cars. On Monday, there were 200 compact

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Add Decimal Numbers LESSON 4

LESSON 4 Add Decimal Numbers In this lesson, we will begin using the algebra-decimal inserts to represent decimals. Turn a red hundred square upside down so the hollow side is showing, and snap the flat

### REVIEW SHEETS BASIC MATHEMATICS MATH 010

REVIEW SHEETS BASIC MATHEMATICS MATH 010 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts that are taught in the specified math course. The sheets

### Keystone National Middle School Math Level 8 Placement Exam

Keystone National Middle School Math Level 8 Placement Exam 1) A cookie recipe calls for the following ingredients: 2) In the quadrilateral below, find the measurement in degrees for x? 1 ¼ cups flour

### Course: Math 7. engage in problem solving, communicating, reasoning, connecting, and representing

Course: Math 7 Decimals and Integers 1-1 Estimation Strategies. Estimate by rounding, front-end estimation, and compatible numbers. Prentice Hall Textbook - Course 2 7.M.0 ~ Measurement Strand ~ Students

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots