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Understanding Integers N.FL.07.07 Solve problems involving operations with integers. N.FL.07.08 Add, subtract, multiply, and divide positive and negative rational numbers fluently.* N.FL.07.09 Estimate results of computations with rational numbers. Unwrapping the GLCE’s: Concepts (what students need to know): Absolute value Rules for adding, subtracting, multiplying and dividing integers Multiplicative inverse/reciprocal Skills (what students need to be able to do): Fluently add, subtract, multiply and divide positive and negative rational numbers Solve contextual problems with integers Prior Knowledge: Students should have mastered basic facts in all operations. Students should understand order of operations. Big Ideas: 1. Absolute value is the distance a number is from zero on number line. 2. The sign of an integer determines how rules of operations are applied to a given situation. 3. Addition & subtraction, and multiplication & division are opposites of each other. Essential Questions: 1. What is absolute value? 2. What is an integer? 3. How does the sign of a number impact the four operations? 4. How can integers be used to solve real world problems? Pre-Assessment Questions: PreAssessment Answers: Answer: B Answer: A Answer: A Answer: B Answer: A Answer: C Answer: C INSTRUCTIONAL ACTIVITIES & LESSON PLANS: Introductory Activity: Human Number Line: Give each student an index card with an integer on it. These should range from -100 to 100, randomly. Students should arrange themselves in order from least to greatest. This activity should be done without talking. Integer/Absolute Value PowerPoint: See attached slides for example or use one at: http://tahquamenon.eup.k12.mi.us//66947015173235/lib/66947015173235/integers_in_the_real_world.doc Students can take notes or just view as teacher explains. INTEGERS: Positively OUR Friends OR…..The Negative Influences of Absolute Value Absolute Value: The distance a number is away from zero on the number line without regard for its sign 0 -7 7 The absolute value of +7 and -7 are both SEVEN! +7 is 7 spaces to the right of zero. -7 is 7 spaces to the left of zero. Both have an absolute value of 7. | | is the symbol used for absolute value |1|=1 | -1 | = 1 What is the | 8 |? What is the | -8 | ? INTEGERS Positive Integers Negative Integers Whole numbers greater than zero – often written without +sign Whole numbers less than zero – ALWAYS has a - sign ZERO IS NEUTRAL – IT HAS NO SIGN! Opposites: Give the opposites of the following: Top Forward Increase Positive Add Multiply OPPOSITES in INTEGERS: Two integers are opposites if they are the same distance away from zero on a number line. EXAMPLES: +1, -1 +5, -5 +76, -76 ELEVATION: Distance a place is above or below sea level Mt. McKinley – highest elevation in United States – 20,320 feet above sea level Death Valley – lowest elevation in United States – 282 feet below sea level Sea Level – 0 feet elevation What is the difference in elevation of Mt. McKinley and Death Valley? Mt. McKinley is 20,320 feet above sea level. Death Valley is 282 feet below sea level. 20320 + 282 = 20602 feet is the difference in elevation Mt. McKinley is 20,320 feet above sea level. Represent as the integer +20,320 Death Valley is 282 feet below sea level. Represent as the integer -282 To find the difference, 20,320 - -282= How do you subtract a negative? REAL LIFE & NEGATIVE NUMBERS Spending & earning $$$$ Rising & falling temps Stock market gains and losses Gaining and losing yards in football Intro examples: Example 1 Write an integer for the situation. a 10-yard loss The integer is –10. Example 2 Write an integer for the situation. 5 above normal The integer is +5. Examples Compare Two Integers Replace each with <, >, or = to make a true sentence. Use the integers graphed on the number line below. Example 3 5 –3 5 is greater than –3, since it lies to the right of –3. Write 5 > –3. Example 4 –6 –2 –6 is less than –2, since it lies to the left of –2. Write –6 < –2. Example 5 Order Integers WEATHER The table below shows the record low temperatures for selected states. Order these temperatures from least to greatest. State AZ IL NJ NY OH PA RI SC WA Temperature (°F) −40 −36 −34 −52 −39 −42 −25 −19 −48 Source: The World Almanac Graph each integer on a number line. Write the numbers as they appear from left to right. The temperatures –52, –48, –42, –40, –39, –36, –34, –25, and –19 are in order from least to greatest. Example 6 Expressions with Absolute Value Evaluate 6 . 6 = 6. The graph of –6 is 6 units from 0. Example 7 Expressions with Absolute Value Evaluate 3 5 . 3 5 = 3 + 5 The absolute value of 3 is 3. =3+5 The absolute value of –5 is 5. =8 Simplify. Example 8 Expressions with Absolute Value Evaluate 3 + x if x = −7. 3 + x = 3 + 7 =3+7 = 10 Replace x with –7. 7 = 7. Simplify. Once teacher has gone through the powerpoint and examples with students, they can complete online quiz at: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-8296358&chapter=1&lesson=3&headerFile=4&state=mi Activity #2: Adding Integers: Use counters to explore adding positive and negative numbers. Use number lines to explore adding positive and negative numbers. Examples: Example 1 Add Integers with the Same Sign Find –6 + (–3). Method 1 Use a number line. Start at zero. Move 6 units left. From there, move 3 units left. So, –6 + (–3) = –9. Method 2 Use counters. Example 2 Add Integers with the Same Sign Find –10 + (–16). –10 + (–16) = –16 Add 10 and 16 . Both numbers are negative, so the sum is negative. Example 3 Add Integers with Different Signs Find 7 + (–5). Method 1 Use a number line. Start at zero. Move 7 units right. From there, move 5 units left. So, 7 + (–5) = 2. Method 2 Use counters. Example 4 Add Integers with Different Signs Find –3 + 2. Method 1 Use a number line. Start at zero. Move 3 units left. From there, move 2 units right. So, –3 + 2 = –1. Method 2 Use counters. Example 5 Add Integers with Different Signs Find –16 + 7. –16 + 7 = –9 To find –16 + 7, subtract 7 from 16 . The sum is negative because 16 > 7 . Example 6 Add Three or More Integers Find –3 + (–10) + 5. –3 + (–10) + 5 = [–3 + (–10)] + 5 = –13 + 5 or –8 Associative Property Simplify. Example 7 Add Three or More Integers Find –7 + 6 + (–3) + 10. –7 + 6 + (–3) + 10 = –7 + (–3) + 6 + 10 Commutative Property = [–7 + (–3)] + (6 + 10) Associative Property = –10 + 16 or 6 Simplify. Example 8 Use Integers to Solve a Problem MONEY The starting balance in a checking account is $100. What is the balance after checks for $15 and $65 are written? Writing a check decreases the account balance, so the integers are –15 and –65. Add these integers to the starting balance to find the new balance. 100 + (–15) + (–65) = 100 + [(–15) + (–65)] = 100 + (–80) = 20 Associative Property –15 + (–65) = –80 Simplify. The balance is now $20. Students can now use the online quiz at: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-829635-8&chapter=1&lesson=4&headerFile=4&state=mi Independent Practice: Activity #3: Subtracting integers is typically one of the most difficult concepts for students to grasp. By using counters, number lines, and a variety of real life examples, helping them understand these concepts can be easier. Use the following to show students how to solve subtraction problems using counters and on number lines. Example 1 Subtract a Positive Integer Find 8 – 10. 8 – 10 = 8 + (–10) = –2 To subtract 10, add –10. Add. Example 2 Subtract a Positive Integer Find –3 – 7. –3 – 7 = –3 + (–7) = –10 To subtract 7, add –7. Add. Example 3 Subtract a Negative Integer Find 6 – (–12). 6 – (–12) = 6 + 12 = 18 To subtract –12, add 12. Add. Example 4 Subtract a Negative Integer Find –12 – (–22). –12 – (–22) = –12 + 22 = 10 To subtract –22, add 22. Add. Examples : Evaluate Algebraic Expressions Evaluate each expression if x = 8, y = –7, and z = –3. Example 5 15 – y 15 – y = 15 – (–7) = 15 + 7 = 22 Replace y with –7. To subtract –7, add 7. Add. Example 6 z–x z – x = –3 – 8 = –3 + (–8) = –11 Replace z with –3 and x with 8. To subtract 8, add –8. Add. Independent Practice: For a review of adding and subtracting integers, students can go through the steps online at: http://www.math.com/school/subject1/lessons/S1U1L11GL.html Game to play: Arrange students into groups of two or more. Have students deal out as many cards as possible from a deck of cards, so that each student has an equal number of cards. Put aside any extra cards. Explain to students that every black card in their pile represents a positive number. Every red cards represents a negative number. In other words a black seven is worth +7 (seven), a red three is worth –3 (negative 3). Note: If this game is new to students, you might want to discard the face cards prior to dealing. If students are familiar with the game, or if you want to provide an extra challenge, leave the aces and face cards in the deck. In that case, explain to students that aces have a value of 1, jacks have a value of 11, queens have a value of 12, and kings have a value of 13. At the start of the game, have each player place his or her cards in a stack, face down. Then ask the player to the right of the dealer to turn up one card and say the number on the card. For example, if the player turns up a black eight, he or she says “8”. Continue from one player to the next in a clockwise direction. The second player turns up a card, adds it to the first card, and says the sum of the two cards aloud. For example, if the card is a red 9, which has a value of -9, the player says “8 + (-9) = (-1)” The next player takes the top card from his or her pile, adds it to the first two cards, and says the sum. For example, if the card is a black 2, which has a value of +2, the player says “(-1) + 2 = 1.” The game continues until someone shows a card that, when added to the stack, results in a sum of exactly 25. Extra Challenging Version To add another dimension to the game, you might have students always use subtraction. Doing that will reinforce the skill of subtracting negative integers. For example, if player #1 plays a red 5 (-5) and player #2 plays a black 8 (+8), the difference is -13: (-5) (+8) = -13 If the next player plays a red 4, the difference is -9: (-13) - (-4) = -9. [Recall: Minus a minus number is equivalent to adding that number.] Adapting for Special Students For students who find the game too challenging, you might change the sum you're aiming for to a number less than 25. The game will end more quickly. As students become more comfortable with the game, you can gradually increase the numeric goal. For more practice: Circle 21 http://nlvm.usu.edu/en/NAV/frames_asid_188_g_3_t_1.html Circle 99 http://nlvm.usu.edu/en/NAV/frames_asid_269_g_3_t_1.html Activity #4: A little humor: Q: What does the zero say to the the eight? A: Nice belt! Here are ‘The Rules’ and ‘The Reasons’, ‘The How’ and ‘The Why’ for Multiplying Integers. The Rules for Multiplying and Dividing Integers. Next, using counters, look at Why the Rules for Multiplying Integers Work*.This lesson is called “Why is a negative times a negative a positive?” and slowly builds up to this at the end of the lesson. *It is very important to have pre-taught the concept of zero before this lesson, (the same negative and positive number together cancel each other out: together -4 and +4 = 0). But what about division you might ask? I find this harder to show with counters so I usually explain that every multiplication question has two equivalent, related division questions: If So if, 3 x 4 = 12 -3 x -4 = +12 Then Then 12 ÷ 4 = 3 +12 ÷ - 4 = -3 and and 12 ÷ 3 = 4 +12 ÷ -3 = -4 This makes further sense to students when they realize that multiplying two integers with opposite signs = negative, and they can see that the same rings true for division as well. Example 1 Multiply Integers with Different Signs Find 5(–6). 5(–6) = –30 The factors have different signs. The product is negative. Example 2 Multiply Integers with Different Signs Find –7(3). –7(3) = –21 The factors have different signs. The product is negative. Example 3 Multiply Integers with the Same Sign Find –3(–5). –3(–5) = 15 The factors have the same sign. The product is positive. Example 4 Multiply More than Two Integers Find –4(3)(–2). –4(3)(–2) = [–4(3)](–2) = –12(–2) = 24 Associative Property –4(3) = –12 –12(–2) = 24 Example 5 Divide Integers Find –22 2. –22 2 = –11 The dividend and the divisor have different signs. The quotient is negative. Example 6 Divide Integers Find –56 (–8). –56 (–8) = 7 The dividend and the divisor have the same sign. The quotient is positive. Example 7 Evaluate Algebraic Expressions Evaluate –3x – y if x = –2 and y = –4. –3x – y = –3(–2) – (–4) = 6 – (–4) =6+4 = 10 Replace x with –2 and y with –4. The product of –3 and –2 is positive. To subtract –4, add 4. Add. Example 8 Find the Mean of a Set of Integers WEATHER In Smalltown, the low temperatures for one week in January were 10F, 15F, – 5F, 25F, –15F, –10F, and 29F. Find the mean low temperature for the week. To find the mean of a set of numbers, find the sum of the numbers. The divide the result by how many numbers there are in the set. Find the sum of the numbers. 10 15 (5) 25 (15) (10) 29 49 7 7 =7 Divide by the number in the set. Simplify. The average low temperature for the week was 7F. Independent practice: For a review of multiplying and dividing integers, students can follow the steps online at: http://www.math.com/school/subject1/lessons/S1U1L12GL.html Post Assessment Items: For each of the following, write an expression to describe the situation, then solve: 1.A team gains 8 yards on one play, then loses 5 yards on the next. 2. A scuba diver dives 125 feet. Later, she rises 46 feet. 3.You get on an elevator in the basement of a building, which is one floor below ground level. The elevator goes up 7 floors. 4.The temperature outside is -2° F. The temperature drops by 9°. 5.On Mercury, the temperatures range from 805° F during the day to -275° at night. Find the drop in temperature from day to night. 6.For every 1 kilometer increase in altitude, the temperature drops 7°C. Find the temperature change for a 5 kilometer altitude increase. 7.Draw a number line. Graph 2 numbers on your number line which have the same absolute value. 8. Give an example of a number that is not an integer. Post assessment key: 1.A team gains 8 yards on one play, then loses 5 yards on the next. 8 + -5 = 3 The team has a net gain of 3 yards. 2.A scuba diver dives 125 feet. Later, she rises 46 feet. -125 + 46 = -79 The diver is now 79 feet below the surface. 3.You get on an elevator in the basement of a building, which is one floor below ground level. The elevator goes up 7 floors. -1 + 7 = 6 You are now on the 6th floor. 4.The temperature outside is -2° F. The temperature drops by 9°. -2 + – 9 = -11 The temperature drops 11 degrees. 5.On Mercury, the temperatures range from 805° F during the day to -275° at night. Find the drop in temperature from day to night. 805 - -275 = 1080 The temperature dropped 1080°. 6.For every 1 kilometer increase in altitude, the temperature drops 7°C. Find the temperature change for a 5 kilometer altitude increase. -7 * 5 = -35° or a drop of 35° 7.Most people lose 100-200 hairs per day. If you were to lose 150 hairs per day for 10 days, what would be the change in the numbers of hairs you have? -150 * 10 = -1500 hairs or 1500 hairs lost 8.Draw a number line. Graph 2 numbers on your number line which have the same absolute value. -2 0 2 9. Give an example of a number that is not an integer. ½, .75 RESOURCES: http://glencoe.com/sec/math/msmath/mac04/course3/index.php/mi/2004 http://datruss.wordpress.com/2007/03/24/multiplying-integers-why-is-3-x-4-12/ http://www.education-world.com/a_tsl/archives/03-1/lesson001.shtml http://nlvm.usu.edu/en/NAV/category_g_3_t_1.html United Streaming videos: The Zany World of Basic Math, Module 1: Integers and Addition The Zany World of Basic Math, Module 2: Subtracting Integers The Zany World of Basic Math, Module 3: Multiplying Integers The Zany World of Basic Math, Module 4: Dividing Integers