INTEGERS Integers are whole numbers. They can be positive, negative or zero. They cannot be decimals or most fractions. Let us look at some examples: Examples of integers: +4 0 9-302 Careful! This is a square root, but it can be simplified into a whole number (3), so it is in fact an integer! 7 1 This fraction is tricky! This is a fraction, but it can be simplified into a whole number (+7), so it is in fact an integer! Example of NON-integers: +4.1-3.5 10 100,000.5 7 5 Now you try to complete the table below! Which of the following examples are integers? Number Integer? (Yes or No) -5 12 6 12 5 601 5.32 4 5 Page 1
FACTORS Integers can be multiplied by each other. For example, 6 7 = 42 In this example, 6 & 7 are multiplied together to give us a product of 42. Numbers 6 & 7 are called FACTORS of 42. FACTOR: A number that divides EXACTLY into another number. Before we start investigating factors, let us complete a multiplication table of integers 1-12. This will help us find factors of numbers later. Multiplication Table of Integers 1-12: Complete the multiplication table below: 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Check your work with a calculator. Page 2
Factors Factors of integers 1-25 Examine the chart below. It shows the factors of each whole number from 1 to 9. Complete the chart for numbers 10-25. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 2 5 2 7 2 3 4 3 4 9 6 8 Answer the following questions: Which numbers have EXACTLY 2 factors? What do you notice about these numbers? What are these numbers called? Which numbers have an odd number of factors? Which numbers are square numbers? What seems to be true about the factors of square numbers? Here are some ways to tell if a number is a square number: A number times itself gives us a square product. For example, 3 3 = 9. Therefore, 9 is a square number. Square numbers have factors that occur in pairs, like 3 and 3 in the example above. Nine has 3 factors: 1, 3 and 9. Since there are odd number of factors, one of the factors will occur in pairs (3 and 3) When a number has an odd number of factors, it is a square number. When we multiply a number by itself, we are squaring the number. Can you complete the questions below? 1 1 = 1 2 = 1 7 7 = = 2 2 = 2 2 = 4 8 8 = = 3 3 = 3 2 = 9 9 9 = = 4 4 = = 10 10 = = 5 5 = = 11 11 = = 6 6 = = 12 12 = = A factor that occurs twice is only written once in the list of factors. Page 3
The opposite of squaring a number is taking a square root. That is, square roots undo squaring. Can you find square roots of the following integers? 1 = 1 49 = 4 = 2 64 = 9 = 3 81 = 16 = 100 = 25 = 121 = 36 = 144 = Extra practice: 1. List the factors of number 48. 2. The factors of each number are listed below. Number Factors 225 1 3 5 9 15 25 45 75 225 500 1 2 4 5 10 20 25 50 100 125 250 500 324 1 2 3 4 6 9 12 18 27 36 54 81 108 162 324 a) Which numbers are square numbers? b) How do you know? 3. Use a calculator to find the square root of each palindromic number: a) 121 = b) 12 321 = c) 1234 321 = d) 123 454 321 = A palindromic number is a number that reads the same forward and backward. e) Continue the pattern. Write the next 3 palindromic numbers & their square roots. Page 4
Numbers can be positive or negative: ADDING & SUBTRACTING POSITIVE & NEGATIVE INTEGERS Shooting bricks In basketball, the term shooting a brick means a missed basket which bounces off a rim or backboard. Negative numbers (-) Bricks are negative Positive numbers (+) 0 Basketballs are positive If a number has no sign it means it is positive. Example: 7 is really +7. There are 4 rules when adding & subtracting positive & negative integers: RULE 1. Adding positive numbers: This is simple addition: 7 + 4 = 11 What we are saying is: positive 7 plus positive 4 equals positive 11. Rule 1. Adding basketballs If you add basketballs, you are adding positive values, so your score goes upwards (positive) We could write is as: +7 + (+4) = +11 RULE 2. Subtracting positive numbers: This is simple subtraction: 7 4 = 3 What we are saying is: positive 7 minus positive 4 equals positive 3. We could write is as: +7 (+4) = +3 Rule 2. Removing basketballs If you remove basketballs, you are removing positive values, so your score goes down (negative) RULE 3. Subtracting negative numbers: This is double subtraction: 7 ( 4) = 11 What we are saying is: positive 7 minus negative 4 equals positive 11. We could write is as: +7 ( 4) = +11 NOTICE: Subtracting a negative is the same thing as adding! Rule 3. Removing bricks If you remove bricks, you are removing negative values, so your score goes up (positive) RULE 4. Adding negative numbers: This is also subtraction: 7 + ( 4) = 3 What we are saying is: positive 7 plus negative 4 equals positive 3. We could write is as: +7 + ( 4) = +3 Rule 4. Adding bricks If you add bricks, you are adding negative values, so your score goes down (negative) Page 5
Let s say you are practicing right and left-handed layups. You are working with a partner and you record the following: Your LEFT Opponent s LEFT Your RIGHT missed Opponent s RIGHT successful layups: successful layups: layups: missed layups 13 9 5 7 We can use these numbers to figure out your total score based on the equation below: + 8 + (+ ) (+ ) ( ) + ( ) = Your LEFT successful layups Opponent s LEFT successful layups Your RIGHT missed layups Opponent s RIGHT missed layups Rule 1. Adding basketballs Rule 2. Removing basketballs Rule 3. Removing bricks Rule 4. Adding bricks If you add basketballs, you are adding positive values, so your score goes If you remove basketballs, you are removing positive values, so your score goes If you remove bricks, you are removing negative values, so your score goes If you add bricks, you are adding negative values, so your score goes What s the final answer? You can use a number line to help you out: Negative numbers (-) Positive numbers (+) Extra Practice 1. Devon and Marques are practicing their shooting skills. Every time they make a basket, they get one point. Every time they miss (a brick), they add a negative point. Coach Victor is recording their scores. a) If Devon shot 21 baskets & got 8 bricks, what is his total score? COMPLETE ANSWER: What we are saying is: We start at 21 points. 8 bricks means we are adding 8 negative points. We can write: +21 + ( 8) = +13 Page 6
b) Marques looks at his score sheet. It says he made 13 baskets & got 18 bricks! What is his total score? PARTIAL ANSWER (you have to fill in the box): What we are saying is: Start at 14 points. 18 bricks means we are adding 18 negative points. We can write: Fill this box with the correct math expression. c) Marques doesn t think this score is correct. He shows it to coach Victor. Coach Victor tells him that 8 of the bricks were actually shot by Devon. How would you calculate Marques corrected score? What we are saying is: We can write: 2. Akua is playing a basketball game and gets fouled as she attempts a 3 point basket. She scores all 3 free throws, which brings her team to 31 points. However, the referee says there was a shooting violation and takes away all 3 free throws! a) State a mathematical expression to show the removal of 3 points Akua scored. b) What is the score after the removal of the 3 points? 3. Solve each equation: A. ( 19) + ( 7) = F. ( 14) (+5) = B. ( 19) (+7) = G. (+4) + (+13) = C. ( 18) + ( 39) = H. ( 4) + ( 9) = D. (+24) ( 43) = I. ( 5) ( 11) = E. ( 10) + (+44) = J. ( 1) (+10) = Page 7
First Number Multiplying & Dividing Positive & Negative Integers There are 3 rules to remember (NOTE: the rules are same for multiplication & division!): When we multiply or divide 2 positive numbers, the answer is positive. Ex. (+6) (+3) = +18 When we multiply or divide a positive and a negative number, the answer is always negative. Ex. (+6) ( 3) = 18 When we multiply or divide 2 negative numbers, the answer is always positive! Ex. ( 6) ( 3) = +18 Complete the multiplication table below using the rules for multiplication: Second Number -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10-10 -9 45-8 -7-6 -5-25 -4-3 -2-1 0 1 0 2-16 3 4 5 35 6 7 8 9 10 Page 8
Extra practice: 1. Will each product be positive or negative? How do you know? a) ( 3) (+7) c) (+6) (+8) b) ( 4) ( 11) d) (+5) ( 6) 2. Replace with an integer to make the equation true. a) (+5) = +30 b) ( 9) = +45 c) (+3) = 21 d) ( 7) = 42 e) ( 8) = 63 3. Few of your friends are playing basketball at MLSE LaunchPad. Each basket is worth +3 points. Each missed basket is worth -2 points. Here is a table of results: Player Name Made baskets (+3 points each) Missed baskets (-2 points each) Total score Nidani 7 3 Deon 4 5 Jorge 6 1 Keon 9 6 Christian 5 2 Pablo 3 0 Bemnte 11 7 Amarion 8 4 Calculate each player s total score & record in the last column. Page 9
Division of Integers When we divide integers, we use the fact that division is opposite of multiplication. We know that (+6) (+3) = +18 So, (+18) (+6) = +3 and (+18) (+3) = +6 dividend divisor quotient We know that ( 6) (+3) = 18 So,, ( 18) ( 6) = +3 and, ( 18) (+3) = 6 Extra Practice A division can be written with a division sign: ( 18) (+3) or as a fraction: 18 +3 1. Here are some divisibility rules (there are few others) 2 The last digit is even (0, 2, 4, 6, 8) 3 The sum of the digits is divisible by 3 5 The last digit is 0 or 5 Based on the divisibility rules, decide what number (2, 3 or 5) each of the following numbers are divisible by: Number Divisible by: How do you know? 4156 3147 41240 5553 2. Ismail scored 8 points every game. He now has a total of 56 points. How many games did he play? 3. Match each word with the correct letter. Dividend Quotient Divisor A. 56 7 = 8 B. 56 7 = 8 C. 56 7 = 8 Page 10
Order of Operations with Integers Order of operations B E D M A S Brackets first How many different ways are there to solve: 7 5 + 36 9 3? There is only ONE correct way to solve this problem, using BEDMAS! Since there are no brackets or exponents, we can do division: 7 5 + 4 3 Next, we can do the multiplication: 35 + 4 3 To add & subtract, we go from left to right: 39 3 Finally we get: 36 This expression can also be written as: 7(5) + 36 9 3 You are given the following integers: 8, +4, 2, 1, 5, 24, 1 Replace each with an integer to get the greatest possible value. Each integer can be used only once! Try at least 4 possible ways to place the numbers in the boxes. ( )( ) + ( ) ( ) ( ) Equation 1: Equation 2: Equation 3: Equation 4: Page 11
Extra Practice: 1. A few QSLA players are practising their shots at the MLSE LaunchPad. They shoot from a FG line (2 points for each basket) and from the 3-point line (3 points for each basket). Every time they miss a shot, they get -3 points! Player Name FG baskets (+2 points each) 3-point baskets (+3 points each) Missed baskets ( 3 points each) Total score (use this space to calculate each person s total score) Tenisha 9 4 6 Nate 8 5 3 Emmanuel 7 3 5 a) Calculate each person s total score. b) Who won the game? 2. Solve: a) 4 + 9 3 25 5 b) 3 + 7 4 48 8 c) 3 + 7(8 + 1) 6 d) 6(5 7) 4 e) 6 2(2 + 3) 2 + 5 Page 12
Pythagorean Theorem Pythagorean theorem is a formula that allows us to calculate the measure of an unknown side in a right angle triangle if we are given the measures of 2 other sides. Be careful! Pythagorean theorem ONLY applies to right angled triangles. hypotenuse The Pythagorean theorem formula is: a 2 + b 2 = c 2 where a and b are the lengths of 2 shorter sides and c is the longest side, across from the right angle. This side is always the longest side and it has a special name, hypotenuse. Example 1. You dribbled along 2 sides of a right triangle & recorded the time along each side. The shorter side took you 4 seconds and the longest side took you 7 seconds to dribble (see diagram on the right). You need to use the Pythagorean theorem to determine the 3 rd side length. Answer: We start with writing the Pythagorean theorem formula & then we can replace 2 side lengths with given values (4 & 7 seconds). 4 sec a 2 + b 2 = c 2 a 2 + 4 2 = 7 2 a 2 = 7 2 4 2 a 2 = 49 16 a 2 = 33 a = 33 Extra practice Use the Pythagorean theorem to find the unknown side lengths for the 3 triangles shown below. 7 5 6 13 Page 13