ABSOLUTE VALUE 2013 Judo Math Inc.
6 th grade Number Sense Discipline: Orange Belt Training Order of Mastery: Absolute Value (6NS7-8) 1. Inequality to show position 2. Explain statements of order in the real world 3. Absolute value as the distance from zero 4. problems on the coordinate plane Welcome to the Orange Belt Absolute Value The absolute value of a number is that number s distance from zero. But here is a very important question: Can a distance be negative? Think about it. Can you drive -52 miles? Or can you walk -21 feet south? If you answered no to this question than you are exactly right. Distance can only be positive and as a result, the absolute value of a number is always positive as well. Whenever we talk about absolute value, it will be important to think of a number line. For now we will look at absolute value very simply, but eventually you will be able to graph absolute value functions and they look like the graph to the right. Crazy huh? I hope you ABSOLUTELY enjoy this belt on absolute value! Good Luck Grasshopper. 6.NS.C.7 Understand ordering and absolute value of rational numbers. 6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. 6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. 6.NS.C.7d Distinguish comparisons of absolute value from statements about order. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 2013 Judo Math Inc.
1. Statements of order Cold temps!: It is very chilly in Alaska! Here are the low temperatures (in Celsius) for one week in Juneau, Alaska: Monday Tuesday Wednesday Thursday Friday Saturday Sunday 5-1 -6-2 3 7 0 a. Arrange them in order from coldest to warmest temperature (use a number line if you need to!) b. On a winter day, the low temperature in Anchorage was 23 degrees below zero (in C ) and the low temperature in Minneapolis was 14degrees below zero (in C ). Sophia wrote, Minneapolis was colder because 14< 23. Is Sophia correct? Explain your answer. c. The lowest temperature ever recorded on earth was 89 C in Antarctica. The average temperature on Mars is about 55 C. Which is warmer, the coldest temperature on earth or the average temperature on Mars? Write an inequality to support your answer. 1
When working with positive and negative numbers, it is very important to think about the number line so that you can make correct statements about which number is larger and which number is smaller. Important notes about position on the number line! Any time a number is further to the RIGHT on Any time a number is further to the LEFT on the number line, it is larger. the number line, it is smaller We will be using < and > to show which number is larger remember, the alligator mouth always eats the bigger number! (6>4 AND -6<-4) 1. What does it mean to say: 3 > 7 Explain using words and a number line. 2. Is it true to say that -4<0? Explain using words and by drawing a number line. 3. Is it true that -22<-24? Explain why or why not using a number line. 4. Write at least 4 inequality statements where both of the numbers are negative. 2
5. Below is a number line with a few numbers filled in. a. Find and label the numbers -3 and -5 on the number line. b. For each of the following, state whether the inequality is true or false. Use the number line diagram to help explain your answers. a. 3> 5 b. 5> 3 c. 5< 3 d. 3< 5 6. On the number line below, label where negative 1 would be and then fill in at least 10 fractions or decimals between -1 and 1. 3
7. Describe in words this chart and give an example of what it would mean to have more or less debt 8. If you borrow $5.25, show on a number line how much you need to earn to get out of debt. 9. If New York is -15, and Chicago is -10, which city is colder? Justify your answer with a drawing. 10. If the temperature in Alaska was -12 F at 8:00am and it was 22 F at noon. What was the temperature difference? Draw a picture and solve. 4
11. Design a number line where you Find and label the numbers 43, 54, 23, and 34 on the number line. a. For each of the following, state which inequality is true. Use the number line diagram to help explain your answers. a. Is 43>54, or is 43<54? b. Is 23> 34, or is 23< 34? c. Is 34 closer to 0 or is 54? Explain how you know. d. Write a situation that could be modeled/solved using your number line and these given numbers. 5
2. Graphing on the number line Graphing an inequality on a number line: An inequality uses a greater than or less than symbol, and all that we have to do to graph an inequality is find the number, '3' in this case and color in everything above or below it. Just remember: if the symbol is ( or ) then you fill in the dot, like the top two examples in the graph below if the symbol is (> or <) then you do not fill in the dot like the bottom two examples in the graph below 6
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3. Absolute value as the distance from zero Absolute value indicates a number s distance from zero. As we discussed on the first page of this packet, distance can always be positive therefore the absolute value of a number is always positive. Example: -7 = 7, because it is 7 steps away from zero or 7 = 7, because it is 7 steps away from zero -7 = 7 Steps away from zero 7 = 7 steps away from zero -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 Fill in the following chart to practice the most basic absolute value problems: Symbolically In Words Solution 3 Distance that three is from zero 3-7 Distance that negative seven is from zero 7-12 47-23.1 1/7 20.4-5/7 -x -m Analyze this: 0 = 8
1. Fill in the blank and prove your solution using at least 3 examples on a number line: As the value of a negative number decreases, its absolute value. 2. Fill in this blank and prove your solution using at least 3 examples on a number line: As the value of a positive number decreases, its absolute value. 9
For the following problems, pair up with a classmate and discuss the problem, then record your conclusions below the problem. 3. When is -4.75 more than -2.25? 4. Jenny and Tracy are running a race. Jenny is 8.5 yards from the finish line and Tracy is 3.75 yards from the finish line. Show who is further from the finish line using absolute value and a number line. 5. You are $3.50 in debt. Write your debt using absolute value. How far from being out of debt are you? 10
6. Your friend Hunter says that he has an account balance of 30 dollars in his bank account because he accidentally bought a new iphone when he couldn t afford it. He tells you that he writes 30 = 30 to describe the size of the debt in dollars. Do you agree with his mathematical representation? Say yes or no and then explain. 7. Jimmy and Paul were placing numbers on a number line. Jimmy said My number is greater than yours. Paul agreed but added, My number has a greater absolute value than yours. What could Paul and Jimmy s numbers be? Explain your answers using a number line. 11
8. The table below shows the lowest elevation above sea level in three American cities. City State Elevation relative to sea level Distance from sea level Barkley Delaware 220 ft New Orleans Louisiana -8 Seattle Washington 0 Lakewood Florida 145 ft Death Valley California -282 ft Indio California -20 ft Baltimore Maryland 20 ft Finish filling in the table as you think about the following statements. Decide whether each of the following statements is true or false. Explain your answer for each one. a) Which city has the lowest elevation? b) Which city is farthest from sea level? c) What is Seattle s distance from sea level? d) Which city is farther from sea level, Indio or New Orleans? e) List the cities in order from lowest to highest elevation f) How does distance from sea level relate to Absolute Value? 12
9. (game) Absolute Value War a) Using small pieces of paper, write one rational number on each paper. Use both positive and negative numbers. Be sure to include fractions and decimals. b) Challenge a friend to a game of War. c) Shuffle the cards, and deal them out equally. d) Each person turns a card over. Say the absolute value of your number. e) Whoever has the greater absolute value keeps the cards. f) The player with the most cards wins. 10. (game 2) Absolute value ordering a) Create a set of integer flashcards. b) Draw five cards at random. c) Line the cards up from least to greatest as integers. Document your results here: d) Line the cards up from least to greatest as absolute values. Document your results here: 11. Ordering Oceans (mini project) Using the Internet, find the depth of several oceans. Create a picture that shows the depth of each ocean. Order the oceans from deepest to shallowest. Share your picture with your class 13
4. Problems on the coordinate plane 1. How far apart are these points? Explain using words and by drawing on this coordinate plane. 2. What is the distance between the two coordinate pairs A (2,-3) and B (0,-3)? Use absolute value and the coordinate plane above to solve the problem. 14
3. What is the distance between the two coordinate pairs C (-5,0) and D (-5,-9)? Use absolute value and the coordinate plane to solve the problem 4. Which of the following coordinate pairs has a distance that is 8 units long? 15
5. Draw a rectangle on the coordinate plane below where one corner is in each quadrant. The AREA of the rectangle should be 36. Label the four corners and identify each quadrant. Explain how absolute value could be used to help find the area here. 16
6. Sink their battleship game 17
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ANSWERS Section 1- Statements of order 1. a. -6, -2, -1, 0, 3, 5, 7 b. No, she is incorrect c. Mars is warmer. -55 > -89 P.2 Number Line 1. -3 is greater than -7 2. Yes 3. No 4. -2 > -6, -22 < -18, -3 > -5, -10 < -3 5. a. -6-5 -4-3 -2-1 0 1 2 3 4 5 6 b. a. True b. False c. True d. False 6. -1-7/8-6/8-5/8-4/8-3/8-2/8-1/8 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 1 7. More debt means you owe more money, therefore you have less money. For example, if you are $5 in debt, you have -$5.00. If you owe $2.00, then you have - $2.00, which is more than -$5.00 8. -6-5 -4-3 -2-1 0 1 2 3 4 5 6 You need to earn $5.50 to get out of debt 19
9. New York is colder 10. 34 degrees difference (see number line below) -14-13-12-11-10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11. -40-30 -20-10 0 10 20 30 40 50 60 Section 2. Graphing on a Number Line 1. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 2. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 3. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 20
4. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 5. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 6. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 7. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 9. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 21
10. > < -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 11. X -3 12. X 4 13. X < -3 14. X < -6 15. X 0 16. X 0 Section 3. Absolute Value as the Distance from Zero Table- Example- -12 - Distance that -12 is from zero, which is 12 12 Steps -12-11 -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 -Try the rest on your own! Analyze This- 0 - How many steps away from 0 is 0? 1. Increases- Example -3 = 3, -6 = 6, -12 = 12 2. Decreases- Example 10 = 10 7 = 7 2 = 2 3-7. Discuss with a partner! 8.a. Death Valley b. Death Valley c. 0 ft d. Indio e. Death Valley, Indio, New Orleans, Seattle, Baltimore, Lakewood, Barkley f. Think about it! Section 4. Problems on the Coordinate Plane 1. 6 steps away 2. 2 steps away 3. 9 steps away 4. C 5. Try it! 22