New Jersey Center for Teaching and Learning Progressive

New Jersey Center for Teaching and Learning Progressive

New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org

6th Grade Number System 2012-08-08 www.njctl.org Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left.

In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 6 for an example.) Number System Unit Topics Opposites

Click on the topic to go to that section Absolute Value Comparing Integers Comparing and Ordering Rational Numbers Cartesian Plane Graphing Ordered Pairs Cartesian Plane Applications

Common Core Standards: 6.NS.5, 6.NS.6, 6.NS.7, 6.NS.8 Opposites Return to Table of Contents 1 Do you know what an integer is? Yes No

Define Integer Definition of Integer: The set of natural numbers, their opposites, and zero. Examples of Integers: {...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...} Integers on the number line Negative Integers -5

-4 -3 Positive Integers Zero -2 -1 0

1 2 Numbers to the left of zero are less than Zero is neither zero positive or negative 3 4

5 Numbers to the right of zero are greater than zero ` Classify each number as an integer, or not. 5 -6

-21 0 3 9 integer -65 -6.32

2.34437 x 10 not an integer 3.2 1 3 5

2 Which of the following are examples of integers? A 0 B -8 C -4.5

D 7 1 3 E 3 Which of the following are examples of integers?

B 1 4 6 C -4 D 0.75

E 25% A Integers In Our World Integers can represent everyday situations You might hear "And the quarterback is sacked for a loss of 7 yards." This can be represented as an integer: -7 Or, "The total snow fall this year has been 9

inches more than normal." This can be represented as in integer: +9 or 9 Write an integer to represent each situation: 1. Spending $6 Answer -$6 2. Gain of 11 pounds 3. Depositing $700

Answer 11 lbs. Answer $700 4. 10 degrees below zero -10 degrees Answer 5. 8 strokes under par (par = 0) 6. 350 feet above sea level

-8 Answer 350 ft Answer 4 Which of the following integers best represents the following scenario: The effect on your wallet when you spend 10 dollars. A -10

B 1 0 0 C D +/10 5

Which of the following integers best represents the following scenario: Earning $40 shoveling snow. A -40 B 40 C 0

D +/- 40 6 Which of the following integers best represents the following scenario: You dive 35 feet to explore a sunken ship. A -35

B 3 5 0 C D +/35 The numbers -4 and 4 are shown on the number line. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Both numbers are 4 units from 0, but 4 is to the right of 0 and -4 is to the left of zero. The numbers -4 and 4 are opposites. Opposites are the same distance from zero. There are 2 ways to read: -9 "negative nine" "the opposite of nine" So, saying "negative" and "the opposite of" are interchangeable. Remember, opposites are the same distance from zero, just

on different sides of the number line. 7 What is the opposite of -5? 8 What is the opposite of 25? 9 What is the opposite of 0?

10 What is the opposite of 18? 11 What is the opposite of -18? 12 What is the opposite of the opposite -18? What conclusions can you draw about the opposite of the opposite of a number?

It's the same as the original number! Let's look at the last three problems. 18 Opposite of 18 = -18 Opposite of the opposite of 18 = -(-18) = 18 This will be helpful to understand when we work with integer operations! 13 Simplify: - (- 9)

14 Simplify: - (- 12) 15 Simplify: - [- (-15)] Integers are used in game shows. In the game of Jeopardy you:

earn points for a correct response lose points for an incorrect response can have a positive or negative score When a contestant gets a $200 question correct: Score = $200 Then a $100 question incorrect: Score = $100 Then a $300 question incorrect: Score = - $200 How did the score become negative? 1. $0 + $200 = $200 2. $200 -Click

$100 = $100 for Answer 3. $100 - $300 = -$200 16 After the following 3 responses what would the contestants score be? $100 incorrect $200 correct $50 incorrect 17

After the following 3 responses what would the contestants score be? $200 correct $50 correct $300 incorrect 18 After the following 3 responses what would the contestants score be? $150 incorrect $50 correct $100 correct

To Review An integer is zero, any natural number, or its opposite. Number lines have negative numbers to the left of zero and positive numbers to the right. Zero is neither positive nor negative. Integers can be used to represent real life situations. Absolute Value Return to Table Of

Contents Absolute Value of Integers The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non-negative (positive or zero). -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the distance from 0 to 5? Absolute Value of Integers The absolute value is the distance a number is from zero

on the number line, regardless of direction. Distance and absolute value are always non-negative. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the distance from 0 to -5? Click5to Reveal We can use absolute value to describe the relative size of numbers. For example, if you dive 35 feet underwater, your depth is -35 feet. However, we often say 35 to describe the number of feet (|-35|).

If you owe someone $45, your debt is described as $45 rather than -$45. We use |-45| to describe the amount. Absolute value is symbolized by two vertical bars |4| This is read, "the absolute value of 4" -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the |4| ? 4

Click to Reveal Use the number line to find absolute value. |9| |-9| |-4| =

= = Move to 9 check Move 9to check Move to

4 check -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 19 Find |-7| 20 Find

|-28| 21 What is |56| ? 22 Find |-8| 23 Find |3|

24 What is the absolute value of the number shown in the generator? (Click for web site and set limits to -500 and 500) 25 Which numbers have 12 as their absolute value? A

-24 B -12 C 0 D 12

E 24 26 Which numbers have 50 as their absolute value? A -50 B -25

C 0 D 25 E 50 Comparing Integers

Return to Table of Contents Comparing Positive Integers An integer can be equal to; less than ; or greater than another integer. The symbols that we use are: Equals "=" Less than "<" Greater than ">"

For example: 4=4 4<6 4>2 When using < or >, remember that the smaller side points at the smaller number. 27 The integer 8 is ______ 9.

A = B < C > 28

The integer 7 is ______ 7. A = B < C > 29

The integer 3 is ______ 5. A = B < C >

Use the Number Line To compare integers, plot points on the number line. The numbers farther to the right are greater. The numbers farther to the left are smaller. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Place the number tiles in the correct places on the number line. Comparing Negative Integers The greater the absolute value of a negative integer, the smaller the integer. That's because it is farther

from zero, but in the negative direction. For example: -4 = -4 -4 > -6 -4 < -2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Comparing Negative Integers One way to think of this is in terms of money. You'd rather have $20 than $10. But you'd rather owe someone $10 than $20.

Owing money can be thought of as having a negative amount of money, since you need to get that much money back just to get to zero. Drag the appropriate inequality symbol between the following pairs of integers: < > 2) -237 -259 36

4) -10 -15 5) -6 -3 6) 127 172 7) -24

-17 8) -2 -8 9) 8 -8 10) -10 -7

1) -3 5 3) 63 30 The integer -4 is ______ -3. A =

B < C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 31 The integer -4 is ______ -5. A

= B < C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 32

The integer -20 is ______ -14. A = B < C >

33 The integer -14 is ______ -6 . A = B < C >

Comparing All Integers Any positive number is greater than zero and any negative number. Any negative number is less than zero and any positive number. 34 The integer -4 is ______ 6. A =

B < C > 35 The integer -3 is ______ 0. A =

B < C > 36 The integer 5 is ______ 0. A

= B < C > 37 The integer -4 is ______ -9. A

= B < C > 38 The integer 1 is ______ -54.

A = B < C > 39

The integer -480 is ______ 0. A = B < C > A thermometer can be thought

of as a vertical number line. Positive numbers are above zero and negative numbers are below zero. 40 If the temperature reading on a thermometer is 10, what will the new reading be if the , what will the new reading be if the temperature: falls 3 degrees? 41

If the temperature reading on a thermometer is 10, what will the new reading be if the , what will the new reading be if the temperature: rises 5 degrees? 42 If the temperature reading on a thermometer is 10, what will the new reading be if the , what will the new reading be if the temperature: falls 12 degrees? 43

If the temperature reading on a thermometer is -3, what will the new reading be if the , what will the new reading be if the temperature: falls 3 degrees? 44 If the temperature reading on a thermometer is 3, what will the new reading be if the , what will the new reading be if the temperature: rises 5 degrees? 45 If the temperature reading on a thermometer is

-3, what will the new reading be if the , what will the new reading be if the temperature: falls 12 degrees? Comparing and Ordering Rational Numbers Return to Table of Contents Use the Number Line To compare rational numbers, plot points on the number line. The numbers farther to the right are larger.

The numbers farther to the left are smaller. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 46 What is the position of the dot on the number line below? A B C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

47 What is the position of the dot on the number line below? A -5.5 B -6.5 C

-5.2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 48 What is the position of the dot on the number line below? A B C -1

0 1 2 3 49 What is the position of the dot on the number line below? -1

A -0.8 B -0.5 C -0.6 0

1 2 3 Where do rational numbers go on the number line? Go to the board and write in the following numbers: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Put these numbers on the number line.

-2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Which number is the largest? The smallest? Comparing Rational Numbers Sometimes you will be given fractions and decimals that you need to compare. It is usually easier to convert all fractions to decimals in order to compare them on a number line. To convert a fraction to a decimal, divide the numerator by the denominator. 0.75

4 3.00 -28 020 -20 0 Drag the appropriate inequality symbol between the following pairs of numbers: 1) < >

2) 3) 4) 5) 6) 7) 8)

9) 10) 50 A = B <

C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 51 A = B

< C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 52 A = B

< C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 53 A =

B < C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 54 A =

B < C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 55 A

= B < C > -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Cartesian Plane

Return to Table of Contents The development of the Coordinate or Cartesian plane is often credited to the French philosopher and mathematician, Rene Descartes. It is said that Descartes first came up with the idea for the plane as he lay in bed watching several flies crawl across his tiled ceiling; as he observed their movement he "realized that he could use the intersecting lines formed by the tiles to describe a flys location. Although historical evidence suggests that a contemporary of Descartes, Pierre de Fermat, did more to develop the coordinate system, Rene Descartes work certainly revolutionized mathematics by

describing the properties of the plane and using it as the first systematic link between Euclidean geometry and algebra. Cogito,ergo sum " (1,1) Pierre, bring me my fly swatter! x (-1,-1)

(2,-2) Rene Descartes 1596 - 1650 y The well known quote; "Cogito,ergo Fun Factsum" (I think,therefore I am) is attributed to Rene Descartes. 0

The coordinate plane is divided into four sections called quadrants. Each quadrant is numbered using the Roman numerals I through IV, in a counter-clock wise direction. Slide the "C" onto the coordinate plane C 0 The Coordinate plane is also called the Cartesian plane.

One way to remember how the quadrants are numbered is to write a big "C" on top of the plane. The "C" will begin in quadrant I and end in quadrant IV. y - axis 0 x - axis The quadrants are formed by two intersecting number lines called axes. The horizontal line is the x-axis. The vertical line is the y-axis.

0 Origin (0, 0) The point at which the x and y axes intersect is called the origin. The coordinates of the origin are (0, 0). 0 Points can be plotted on the plane using one coordinate from each of the axes. These sets are called ordered pairs. The x coordinate always appears first in these pairs. The y coordinate appears second.

(x, y) Each of the quadrants can be identified by the properties of the numbers that fall within their plane. Remember the ordered pairs are always of the form (x, y) (-,+) ( +,+) 0 (-,-)

(+,-) 56 What points are in quadrant II ? A B C D E F 57

What points are in quadrant I ? A B C D E F 58 What points are in quadrant IV ? A B

C D E F 59 What points are in quadrant III ? A B C D E F

60 What point is closest to the origin? A B C D E F Graphing Ordered Pairs Return to

Table of Contents To graph an ordered pair, such as (3,2): start at the origin (0,0) move left or right on the x-axis depending on the first number then move up or down from there depending on the second number plot the point To graph (-3, 4): Start at the origin and then move 3 left, up 4

To graph (-3, -2): Start at the origin and then move 3 left, down 2 To graph (5, -3): Start at the origin and then move 5 right, down 3 Place the star on (2,8) in quadrant I Place the triangle on (-4, 4) in quadrant II Place the square on (-7, -3)

in quadrant III Place the circle on (1, -4) in quadrant IV Place the circle on (-7,-5) Place the star on (4,9) Place the triangle on (-6,2) Place the square on (3,-9) In which quadrant is the circle?

Place the circle on (-4, -3) Place the star on (4,3) Place the triangle on (-4, 3) Place the square on (4, -3) What do you notice about the location of the points in relation to each other? What do you notice about the coordinates of the points? Click for answer

(-4, 3) (4,3) (-4, -3) (4, -3) When two ordered pairs differ only by their signs, the locations of the points are reflections of each other across one or both axes.

Move the letter to match it to the correct coordinate point. Then move the circle to check your answer. A F C B D E (-9,-4)

D (2,-2) E (9,0) B (0,6) F

(5,7) A (-3,2) C 61 The point (-5, 4) is located in quadrant_____. A

I B II C III D IV

62 The point (7, -2) is located in quadrant _____. A I B II C III

D IV 63 The point (4, 5) is located in quadrant ____. A I B

II C III D IV 64 The quadrant where the x & y coordinates are both negative is quadrant ___.

A I B II C III D

IV 65 When plotting points in the Cartesian Plane, you always start at ____. A the x - axis B the origin

C the y-axis D the Coordinate Plane E (0,0) List the coordinates of

each point A A (-2,2) B B (-2,-4) Click C here for

C (0,-3) answers D D (2,-1) E E (3,4) F F (5,0)

List the coordinates of each point A A (-4,3) B B (0,3) C C

(-5,-3) answers D D (0,-1) E E (3,2) F F (4,-4)

Click here for List the coordinates of each point A A (3,-2) B B (0,-4)

C here for C answers (0,5) D D (-4,0) E E (4,-1)

F F (1,3) Click 66 If the x-coordinate is positive, the point to be plotted will be in quadrant _____. A I

B I & II C I& IV II D 67

If the y-coordinate is positive, the point to be plotted will be in quadrant _____. A I B I & II C I& IV

II D 68 If the x - coordinate is negative and the ycoordinate is positive, the point to be plotted will be in quadrant _____. A I B

I & II C I& IV II D 69 If the x - coordinate is positive and the ycoordinate is negative, the point to be plotted will be in quadrant _____.

A I B II C III D

IV 70 Point A is located at (-3, 2) True False 71 Point A is located at (-5, 1) True False

72 Point A is located at (-2, 3) True False 73 Point A is located at (-2, 0) True False Cartesian Plane Applications

Return to Table of Contents Vocabulary Review Coordinate Plane: the two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. Also known as a coordinate graph and the Cartesian plane. II I III

IV Quadrant: any of the four regions created when the xaxis intersects the y-axis. They are usually numbered with Roman numerals. Vocabulary Review II

III I IV x-axis: horizontal number line that extends indefinitely in both directions from zero. (Right- positive Leftnegative) y-axis: vertical number line

that extends indefinitely in both directions from zero. (Up- positive Downnegative) Origin: the point where zero on the x-axis intersects zero on the y-axis. The coordinates of the origin are (0,0). 74 The points (-6, 2) and (3, 2) are plotted below. What is the distance between these two points? 75

The points (-5, 4) and (1, 4) are plotted below. What is the distance between these two points? 76 Plot and connect the following points: M(1,2), A(1,2), T (1,4), H (-1,4). What is the distance of MA? 77 Plot and connect the following points: A(-2,2), B(6,2), C (0,4) What is the distance of AC? Study the table below. What pattern do you see between the

set of points and the distance between them? Is there a way to find the distance between the two points without graphing them first on a coordinate plane? Points Distance (-6, 2) (3, 2) 9

(-5, 4) (1, 4) 6 (-2, 6) (-2, -4) 10 (-5, 7)

(-5, 3) 4 (3, -3) (8, -3) 5 If two points have either the same x- or y-coordinate, the distance between them can be as follows: If the different coordinates are either both positive or both negative, subtract their absolute values.

If the different coordinates are opposite signs, add their absolute values. Let's look at the table again to see how this works: Points Distance (-6, 2) (3, 2) |-6| + |3| = 6 + 3 = 9

(-5, 4) (1, 4) |-5| + |1| = 5 + 1 = 6 (-2, 6) (-2, -4) |6| + |-4| = 6 + 4 = 10 (-5, 7)

(-5, 3) |7 - 3| = |4| = 4 (3, -3) (8, -3) |3 - 8| = |-5| = 5 78 Find the distance between (-8, 4) and (-8, 9).

79 Find the distance between (6, 9) and (-2, 9). 80 Find the distance between (5, -7) and (5, -2). 81 Given the points A(-3,-3), B(2,-3), C (-3,0), D (2,0), what is the distance of CD? 82

Given the points X (-3, -2), Y (0,2), Z (3, -2), what is the distance of XZ?

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