Understand Positive and Negative Numbers

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1 Lesson. Understand Positive and Negative Numbers Positive integers are to the right of on the number line. Negative integers are to the left of on the number line. Opposites are the same distance from, on opposite sides. What is the opposite of? Step Graph the integer is a negative integer. Graph it to the left of. Step Graph the integer and its opposite on a number line The opposite of is places to the right of. So, the opposite of is. Graph the integer and its opposite on the number line.. opposite:. opposite: opposite:. 7 opposite: Write the opposite of the opposite of the integer R Grade 6

2 Lesson. Compare and Order Integers Use a number line to compare and. Step Graph and. Both numbers are negative integers. Graph them to the left of Step Decide which number is greater. Numbers become greater as ou move to the right on a number line. is to the right of. So, is greater than. Write:.. Order these integers from least to greatest:, 7,. Step Graph the integers on a number line Step Write the numbers in order from left (least) to right (greatest). 7,, Compare the numbers. Write < or > is to the of 6 on the number line, so is than Order the numbers from least to greatest..,, , 7, Order the numbers from greatest to least. 8.,, R Grade 6

3 Lesson. Rational Numbers and the Number Line Graph.8 and. on the number line. Step Use positive and negative integers to help ou locate the decimals..8 is between and, so.8 is between and.. is between and. -.8 Step The number line is marked in tenths. There is a tick mark ever.. Count 8 tick marks to the left of for.8. Count tickmarks to the right of for Graph _ and _ on the number line. Step Use positive and negative integers to help ou locate the fractions. _ is between and. _ is between and, so _ is between and Step The number line is marked in tenths. There is a tick mark ever. Use equivalent fractions to help ou graph the points. Count tick marks _ 6 to the left of. Count 6 tick marks to the right of. Graph the number on the horizontal number line Grade 6 R

4 Lesson. Compare and Order Rational Numbers Compare. and using the number line. Step Graph the numbers. Use positive and negative integers to help ou locate the decimals.. is between and. is negative, so it is to the left of. Step As ou move right on the number line, numbers become greater So,... Compare _ and _ using the number line. Step Graph the numbers. Use positive and negative integers to help ou locate the fractions. _ is between and. _ is between and. Step As ou move left on the number line, numbers become less So, _, _. Compare the numbers. Write, or Order the numbers from least to greatest ,, 6..,.9, 7.,,. R Grade 6

5 Lesson. Absolute Value Absolute value is a number s distance from on a number line. Numbers and their opposites have the same absolute value. Find the absolute value of and. Step Graph the numbers Step Find each number s distance from. units units Step Write the absolute value. Find the absolute value of.7 and... units Step Graph the numbers..7 units Step Find each number s distance from Step Write the absolute value Find the absolute value.. is units from Grade 6 R

6 Lesson.6 Compare Absolute Values Use absolute value to epress an elevation less than meters as a depth. Step Elevation indicates distance from sea level. A negative elevation means a distance below sea level. is units below on the vertical number line. This shows that the absolute value of is. Step Depth indicates distance below sea level. It is alwas epressed as a positive number. Use the absolute value of to find the depth: Step List three elevations that are less than meters. Write the corresponding depths Elevation (m) Depth (m) So, an elevation less than meters is a depth greater than meters. Complete the table.. Elevations Greater than Depth feet feet 8 feet feet. Jordin s savings account balance is greater than $7. Use absolute value to describe the balance as a debt. Jordin s balance is a debt of than $7.. The table shows the changes in the weights of dogs. Which dog had the greatest decrease in weight? How much weight did the dog lose? Dog Weight Change (lb) Duff. Budd. Dinah. Grade 6 R

7 Rational Numbers and the Coordinate Plane A coordinate plane is formed b two intersecting lines on a grid. The horizontal line is the -ais. The vertical line is the -ais. The intersect at the origin. An ordered pair shows the horizontal and vertical distances a point is from the origin. Positive numbers in an ordered pair mean right for the first number and up for the second number. Negative numbers mean left for the first number and down for the second number. Lesson.7 -ais Origin -ais Write the ordered pair for point K. Step Place our finger at point K. Place our pencil tip at the origin. Step With our pencil tip, count how man units to the right or left of the origin point K is. Record that number. Point K is. units right of the origin, so the first number in the ordered pair is., or.. Step With our pencil tip, count how man units down from the origin point K is. Record that number. Point K is. units down from the origin, so the second number in the ordered pair is.. So, the ordered pair for point K is (.,.). K Write the ordered pair for each point.. point P R U. point Q. point R. point S P S. point T 6. point U T Q R6 Grade 6

8 Lesson.8 Ordered Pair Relationships You can tell which quadrant to graph a point in b looking at whether the coordinates are positive or negative. Find the quadrant for the point (, ). Step The -coordinate is, a positive number. So, the point must be in Quadrant I or IV. Quadrant II (-, +) Quadrant I (+, +) Step The -coordinate is, a negative number. So, the point must be in Quadrant III or IV. Step The onl quadrant that the - and -coordinates have in common is Quadrant IV. Quadrant III (-, -) Quadrant IV (+, -) So, the point (, ) is in Quadrant IV. Two points are reflections of each other if the -ais or -ais forms a line of smmetr for the two points. This means that if ou folded the graph along that ais, the two points would line up. (, ) and (, ) are reflected across the -ais. The -coordinates are the same. The -coordinates are opposites. (, ) and (, ) are reflected across the -ais. The -coordinates are the same. The -coordinates are opposites. - (, ) - (, ) (-, - - ) (, ) Identif the quadrant where the point is located.. (, ). (, ) -coordinate: Quadrant: or -coordinate: Quadrant: or -coordinate: Quadrant: or -coordinate: Quadrant: or The point is in Quadrant. The point is in Quadrant.. (, ). ( 6, 7). (8, ) 6. ( 7, ) Quadrant: Quadrant: Quadrant: Quadrant: The two points are reflections of each other across the - or -ais. Identif the ais. 7. (, 7) and (, 7) 8. (, ) and (, ) 9. (, 6) and (, 6). (8, ) and ( 8, ) ais: ais: ais: ais: R7 Grade 6

9 Lesson.9 Distance on the Coordinate Plane Find the distance between (, ) and (, ). Step Graph the points. Points with the same -coordinate are on the same vertical line. Think of the vertical line as a number line that shows the -coordinates. Step Use absolute value to find the distances between the -coordinates and. shows the distance from to. units shows the distance from to. units Step Since the points are in different quadrants, add to find the total distance between the -coordinates (, ) - (, ) So, the distance between (, ) and (, ) is units. Use the same steps when two points have the same -coordinates. Find the distance between the -coordinates to find the distance between the points. Graph the pair of points. Then find the distance between them.. (, ) and (, ) 6 The points are on the same horizontal line. Distance from to : Distance from to : Subtract to find distance from (, ) to (, ): - units. (, ) and (, ) units. (, ) and (, ) units. ( 6, ) and ( 6, ) units R8 Grade 6

10 Lesson. Problem Solving The Coordinate Plane Zachar is drawing a coordinate map of his town. He has graphed the police station at the point (, ). He is going to place the librar units up from the police station. What ordered pair shows where he will graph the librar? Read the Problem What do I need to find? I need to find the for the librar. What information do I need to use? The ordered pair for the is. The librar is units from the police station. How will I use the information? I can draw a diagram to the information on a coordinate plane. Solve the Problem Graph the point. 6 Label it. From this point, count units Graph the new point, and label it. So, the ordered pair for the librar will be. - Solve. Graph the pairs of points on the coordinate plane.. Zachar has graphed the middle school at ( 6, ). He has graphed the high school units to the right of the middle school. What is the high school s ordered pair? 8 6. Zachar will graph the apartment building units to the left and units down from the grocer store. He has graphed the grocer store at (7, 8). Give the ordered pair for the apartment building R9 Grade 6

The slope, m, compares the change in y-values to the change in x-values. Use the points (2, 4) and (6, 6) to determine the slope.

The slope, m, compares the change in y-values to the change in x-values. Use the points (2, 4) and (6, 6) to determine the slope. LESSON Relating Slope and -intercept to Linear Equations UNDERSTAND The slope of a line is the ratio of the line s vertical change, called the rise, to its horizontal change, called the run. You can find