MATH GRADE 6 UNIT 1 RATIONAL NUMBERS ANSWERS FOR EXERCISES

Transcription

1 MATH GRADE UNIT RATIONAL NUMBERS FOR EXERCISES

2 LESSON : THE OPPOSITE OF A NUMBER.NS..a. a. b. 7.5.NS..a. 0.NS..a 3. B.NS..a. C 5 5.NS..a 5. A 33.NS..a NS..a NS..a NS..a Copright 05 Pearson Education, Inc.

3 LESSON : THE OPPOSITE OF A NUMBER.NS.. Negative Number Integer Rational Number None of These a. 3 b. 0 c.. d. 9.5 e. 9 a.ns.. The number 0 is rational because, where a = 0 and b 0. b = 0.NS..a NS..a NS..a Challenge Problem.NS..a.NS.7.c 5. The distance between an number and its opposite is. Copright 05 Pearson Education, Inc. 3

4 LESSON 3: ABSOLUTE VALUE.NS.7.c. B 3.NS.7.c. 0.NS.7.c 3. or 5 3.NS.7.c..3 3.NS.5 5. C Januar.NS.7.c. C 0.09.NS.7.c NS.7.c. or 3 7.NS.5.NS.7.b.NS.7.d.NS.5.NS.7.b.NS.7.d 9. a. The highest boiling point was C. b. The lowest boiling point was 5 C. c. The difference between the highest and lowest boiling points is.. The distance between the highest and lowest freezing points is. First, I figured out the highest and lowest freezing points recorded b the class. The highest freezing point is higher than 0 C, or C. The lowest freezing point is lower then 0 C, or C. Net, I created a number line to show where the two numbers are located. Then I counted the number of jumps from C to C. (continues) Copright 05 Pearson Education, Inc.

5 LESSON 3: ABSOLUTE VALUE.NS.5.NS.7.b.NS.7.d. (continued) Challenge Problem.NS.7.c. In general, there are two values of a that have the same absolute value: a and its opposite. The onl eception is a = 0, since 0 is its own opposite. Copright 05 Pearson Education, Inc. 5

6 LESSON : OPPOSITE AND ABSOLUTE VALUE.NS.7.c. B.NS..a.NS.7.c. Number Opposite Absolute Value NS.7.d 3. B.NS.7.d. D 0.NS..a 5. A 55.NS.7.c. and.ns.7.c 7. 5.NS.7.c. The answer is.ns..a.ns.7.c 9. A number and its opposite are both positive. Never true The absolute value of a number is never negative. Alwas true The absolute value of a number is 0. Sometimes true The opposite of a number s absolute value is greater than 0. Never true Copright 05 Pearson Education, Inc.

7 LESSON : OPPOSITE AND ABSOLUTE VALUE Challenge Problem.NS..a.NS.7.c. Emma is incorrect, because the opposite of an negative number is equal to its absolute value. Zero is, therefore, the greatest number with an opposite that is equal to its absolute value, since zero is its own opposite. Copright 05 Pearson Education, Inc. 7

8 LESSON 5: ORDERING AND COMPARING.NS.7.a. a. > 3 b. < 5 c. > d. 7 < e. < 0.NS.7.a. C,.0,.0,,.NS.7.a 3. The order of the dives from deepest to shallowest is third dive, second dive, fourth dive, and first dive. 90, 0.5, 5, 30.NS.7.a > 50.7.NS.7.a.NS.7.c 5. B Mia.NS.7.a. B <.NS.7.a A.REI. 7. D 0. >.NS.7.b. Carlos won the tournament: Carlos ( 5), Jan (), Emma ( 3), Jason ( ), Denzel (), Mia ()..NS.7.b 9. The score with the greatest opposite wins. The absolute value of Mia s score is greater than the absolute value of Carlos s score. Eplanations will var. Here is one possible answer. Golfer Score Opposite Absolute Value Jason Emma Denzel Carlos Mia Jan Copright 05 Pearson Education, Inc.

9 LESSON 5: ORDERING AND COMPARING.NS.7.b. D 007: 75 south latitude Challenge Problem.NS.7.a.EE.. a. Inequalities will var. However, the resulting inequalit will be true, whether a is negative or positive. b. The resulting inequalit is true if a is positive, but false if a is negative. Copright 05 Pearson Education, Inc. 9

10 LESSON : PUTTING IT TOGETHER.NS.5. Negative Numbers in Everda Life Elevations of locations below sea level Temperatures below zero degrees Diving under water Depths of roots below the soil Golf scores Latitudes south of the equator, longitudes west of the prime meridian Mone that ou owe (debt) An overdrawn bank account balance Discounts on items for sale In video games: loss of life, damage, penalt, or using up a resource Negative statistics in sports (e.g., errors in baseball, technical fouls in basketball, face-offs lost in hocke) Race times in sports competitions (e.g., downhill skiing, swimming, and running) showing the current racer compared to the leader (negative if the current racer completes the race faster than the leader) Lap times in Formula racing the difference between the previous lap and the lap just completed (negative if the lap just completed was faster than the previous lap) Penalt minutes in hocke Differences in time of da between time zones Copright 05 Pearson Education, Inc. 50

11 LESSON : PUTTING IT TOGETHER.NS.5.NS.7 3. Definitions and eamples will var. Here are some eamples. Word or Phrase Definition Eamples integer A whole number Can be positive, negative, or zero 5, 0 5 positive number A number that is greater than zero.5 Numbers to the right of zero on the number line negative number A number that is less than zero Written with a minus sign in front of the number 3 0. Numbers to the left of zero on the number line opposite of a number absolute value The opposite of a number is the same distance awa from zero but on the opposite side of the number line. The opposite of the opposite of a number is the number itself. The distance a number is from zero on a number line Number: Opposite: opposite 5 0 distance to zero = 5 distance to zero = number Absolute value is alwas either 0 or positive. Number: Absolute Value: Written as a number inside two lines: 3 Copright 05 Pearson Education, Inc. 5

12 LESSON 9: THE COORDINATE PLANE.NS..b..NS..b. B A E C D 3. D ( 3, ). C (, ) 5. B (, ). NORTH WEST EAST SOUTH Copright 05 Pearson Education, Inc. 5

13 LESSON 9: THE COORDINATE PLANE 7. C Quadrant III. D C Challenge Problem.NS..b. Sign (-coordinate) Sign (-coordinate) Quadrant + + I IV + II III Copright 05 Pearson Education, Inc. 53

14 LESSON : DRAWING FIGURES.NS.. B Point B.NS..b. A Point A and C.NS..b 3. A Point A.NS.. The distance between point A and point D is 7 units. Strategies will var. Here are two possible answers. Eample : Points A and D share the same -value and are on opposite sides of the -ais. It s as if the are on a horizontal number line on opposite sides of zero. The distance between the two points is the distance between their -coordinates, 7 units. Eample : absolute value = 3 A absolute value = D I plotted points A and D in the coordinate plane and then found the distance to zero, which is the absolute value. For point A, the distance is 3 units. For point D, the distance is units. The -coordinate of point A is 3. 3 = 3 units The -coordinate of point B is. = units Then, I added the two absolute values together to find the total distance of 7 units. Copright 05 Pearson Education, Inc. 5

15 LESSON : DRAWING FIGURES.NS. 5..NS..G.3..NS. 7. A units.ns.. Copright 05 Pearson Education, Inc. 55

16 LESSON : DRAWING FIGURES.NS..G.3 9. Strategies will var. Here is one possible answer The two given vertices form the end points of the diagonal of the square: (3, ) and (3, ). Because the have the same first coordinate, I can picture them as being on a vertical number line with one point at and the other point at. The distance between them is the length of the diagonal. 3 The distance from to 0 is its absolute value:. The distance from to 0 is its absolute value:. Adding the two absolute values together, I find the length of the diagonal. + = 5 The diagonal is 5 units long Copright 05 Pearson Education, Inc. 5

17 LESSON : DRAWING FIGURES.NS.. The hospital is 9 blocks east of police station and 9 blocks west of police station. Strategies will var. Here is one possible answer. WEST Police Station The hospital is at (, ). NORTH Hospital 9 9 SOUTH Police Station EAST The two police stations share the same -value as the hospital. The distance between them is the distance along the -ais. From police station (, ) to police station (, ), I counted units. Half of is 9. So, the hospital is 9 units from each of the police stations. From police station (, ) to the hospital, I counted 9 units to the left (west) on the -ais and arrived at (, ). From police station (, ) to the hospital, I counted 9 units to the right (east) on the -ais and arrived at (, ). Challenge Problem.NS..G.3. a. The -coordinate must be 0, and the -coordinate can be an number ecept 0. (continues) Copright 05 Pearson Education, Inc. 57

18 LESSON : DRAWING FIGURES.NS..G.3. (continued) b. The -coordinate must be either or, and the -coordinate can be an number ecept 0. c. The -coordinate is either less than or greater than, and the -coordinate can be an number ecept 0. Copright 05 Pearson Education, Inc. 5

19 LESSON : CREATING MIRROR IMAGES.NS..b. A.NS..b. B.NS..b.NS. 3. a. ( 5, ), ( 9, ), ( 7, ) b. Reflecting a point across the -ais results in the opposite of the -coordinate. The -coordinate remains the same..ns..b.ns.. Copright 05 Pearson Education, Inc. 59

20 LESSON : CREATING MIRROR IMAGES.NS..b.NS. 5..NS..b.NS..G.3. Ask a classmate to check our work. Here is one possible solution. a. units units units I made a quick sketch. I counted and found that the given points, (, 5) and (, ), are units apart. The other sides of the square need to be the same length. From (, 5), I counted units to the left in the coordinate plane to find point ( 5, 5). That means the other point is at ( 5, ). Points: ( 5, 5) and ( 5, ) b. The line of reflection is =. All four points are the same distance (3 units) from the line of reflection. Copright 05 Pearson Education, Inc. 0

21 LESSON : CREATING MIRROR IMAGES.NS..G.3 7. Challenge Problem.NS..b.NS..G.3. Ask a classmate to check our work. Copright 05 Pearson Education, Inc.

22 LESSON : PUTTING IT TOGETHER.NS..b.NS. 3. Definitions and eamples will var. Here are some eamples. Word or Phrase non-integer Definition A number that is not a whole number Eamples Can be positive or negative coordinate plane A plane made of a horizontal number line (-ais) and a vertical number line (-ais) that cross at the coordinates (0, 0) also called the Cartesian plane II quadrant 5 3 III quadrant vertical ais origin (0, 0) I quadrant horizontal ais 3 IV quadrant 5 5 coordinates A set of two numbers that show where a point is located in the coordinate plane: (, ) To plot coordinates in the coordinate plane:. Use the first number (-coordinate) to move right or left along the horizontal number line (-ais).. Use the second number (-coordinate) to move up or down along the vertical number line. Eamples: (3, 5) is located at = 3 and = 5. (, ) is located at = and =. 5 (3, 5) (, ) 3 5 Copright 05 Pearson Education, Inc.