Sixth  Grade  Mathematics                                                    

Unit  #3:  Extending   the  Number  System:       Positive     and     Negative  Numbers   Pacing:    25  Days   Unit Overview

This unit builds on students’ earlier study of systems of numbers (natural numbers, whole numbers, and positive fractions and decimals) as students investigate values less than zero. They formalize an understanding of integers and their relationship to the set of rational numbers. Students develop an understanding of how integers are used in real-world contexts, including the meaning of absolute value. In Grade 7, students learn to operate with positive and negative rational numbers. By the end of this unit, students expand their understanding of rational numbers by representing them in all four quadrants of the coordinate plane. Students will apply their understanding of absolute value to determine distances in the coordinate plane.

Prerequisite Skills 1) Plot and identify points in the first quadrant of the coordinate plane 2) Distinguish between the x and y axis 3) Plot values on a number line with precision 4) Compare and order positive whole numbers, fractions and decimals

Vocabulary absolute value, axis (plural - axes) coordinate pair coordinate plane coordinate system coordinates greater than (how to read >) inequality integers less than (how to read , −5 (c) −5 > −3 (d) −5 < −3 (e) −3 < −5 Find and label the numbers 4/3, 5/4, -2/3 and -3/4 on the number line below.

Engage NY Module 3 Lesson 9 (Appendix C) “Representing Rational Numbers on the Number Line” (Appendix C)     http://www.ixl.com/m ath/grade-­‐6/compare-­‐ rational-­‐numbers   Engage NY Module 3 Lesson 10 (Appendix C)

Is 4/3 > 5/4 , or is 4/3 < 5/4 ? Is -2/3 > -3/4, or is -2/3 < -3/4?

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Express the absolute value of a positive or negative quantity in a real world context

• Students understand that the order of positive/negative numbers is the same as the order of their absolute values. • Interpret absolute value as the magnitude of a positive or negative quantity in a real-world situation.

Is -3/4 closer to 0 or is 5/4? Explain how you know. In your own words, explain why the absolute value of two opposites are the same. (i.e. The absolute value of -6 and 6 are 6)

Engage NY Module 3 Lesson 11 (Appendix C) “What’s Your Sign? Part II” (Appendix C) My Math

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Chapter 5 Lesson 2 13

Compare and order positive and negative numbers by their actual and absolute values

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Students understand that negative numbers are always less than positive numbers.

Write 6 of the numbers from the box into the blanks to create three true mathematical statements. You may not use any number more than once.

Engage NY Module 3 Lesson 12 (Appendix C) My Math Chapter 5 Lesson 3

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Make sense of and persevere in solving real world scenarios involving absolute value

Identify the phrase below that cannot be described by the same absolute value as the other three. Explain your answers using words or pictures. 1. A debit of $15 2. A credit of $15 3. 15 feet above sea levels 4. $15 less than $25 5. 15 degree drop in temperature

Engage NY Module 3 Lesson 13 (Appendix C) “Illustrating Values” (Appendix C)

Flex Day (Instruction Based on Data) Recommended Resources: “Learning Task: Absolute Value and Ordering” (Appendix C) “Comparing Temperatures” (Appendix C) “What’s Your Sign? Part III” (Appendix C) http://www.mathgoodies.com/standards/alignments/grade6.html https://grade6commoncoremath.wikispaces.hcpss.org/file/detail/6.NS.7d%20Lesson%20Illustrating%20Values.doc

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Identify and name points on a coordinate grid using coordinate pairs

Use the graph below to answer the following questions:

Engage NY Module 3 Lesson 14 (Appendix C) My Math Chapter 5 Lesson 6

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Graph and identify points in all four quadrants of the coordinate plane

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•

Students must understand that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Students understand that changing the sign of one or both numbers in the ordered pair will create a reflection of the point. Students label the components of the coordinate plane (Quadrant (+, +), Quadrant (-,+) , Quadrant (-,-) Quadrant (+, -), x and y axes, origin)

1. Identify the coordinate pairs: a. K: _____ b. F: _____ c. E: _____ 2. Identify each letter: a. (4, 8): _____ b. (6, 4): _____ c. (10, 9): _____ 3. Jackie said that the letter C is represented by the coordinate pair (7, 6). Is she correct? Explain Label the coordinate pair for each point:

http://www.ixl.com/math /grade-6/coordinategraphs-review

Engage NY Module 3 Lesson 15 (Appendix C) My Math Chapter 5 Lesson 7

Marysol is graphing points on the grid above. He graphs his first point, D at (3,3). She then graphs the reflection across the y-axis to the coordinate pair (3, -3). What mistake did she make? Explain your answer using words then identify the correct coordinate pair.

http://www.ixl.com/math /grade-6/graph-points-ona-coordinate-plane

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Determine when a set of ordered pairs are a reflection of one another. Given a coordinate pair, identify another pair that reflects this point

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Pacing: up to 2 days Students must understand that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Students understand that changing the sign of one or both numbers in the ordered pair will create a reflection of the point. Attend to precision when determining the appropriate scale for axes Sample problem for practice:

The coordinates of point P are (−6, 5). Point R is a reflection of point P across the x-axis. The  coordinates  of  point  Q  are  (−1,  0).   Point  T  is  a  reflection  of  point  Q  across   the  y-­‐axis.     Plot  and  label  points  P,  Q,  R,  and  T  on   the  coordinate  plane  below.  

Engage NY Module 3 Lesson 16 (Appendix C) http://www.ixl.com/math /grade-6/reflectionsgraph-the-image

Part A: Plot point Q so that it is a reflection of point P across the y-axis. What are the coordinates of point Q? Explain your thinking. Part B: The coordinates of point S are Explain how point S is related to point P. Part C: Point R is plotted on the coordinate graph so that line segments SR and PQ are a reflection of each other. What would have to be the coordinates of point R? Explain your thinking.     11  |  P a g e    

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Construct and label a coordinate plane. Explain the relationship between number lines and coordinate axes.

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Determine the distance between points on a coordinate plane

Students understand that by plotting ordered pairs on a coordinate plane it represents the relationship between quantities that can be analyzed using the absolute value between the ordered pairs.

The map of a town is placed on a coordinate grid with each whole number distance north (N), south (S), east (E), or west (W) representing 1 block. A grocery store has the coordinates (−2, −4). The owners of the grocery store plan to build an additional grocery store at a location that is 5 blocks to the east and 3 blocks to the north of the original store. Plot the location of the additional grocery store on the coordinate grid.

Engage NY Module 3 Lesson 17 (Appendix C)

Plot four unique points on the coordinate grid that are each 5 units from the point (1, 2). Each point must contain coordinates with integer values and there must be a point in all four quadrants.

My Math Chapter 5 Inquiry Lab Pages 411 – 414 Engage NY Module 3 Lesson 18 (Appendix C) http://www.ixl.com/math /grade-6/distancebetween-two-points

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Use the coordinate plane to graph points, line segments and geometric shapes in the various quadrants and then use the absolute value to find the related distances.

Students understand that by plotting ordered pairs on a coordinate plane it represents the relationship between quantities that can be analyzed using the absolute value between the ordered pairs. • Suggested application problem: The map of a town will be placed on a coordinate plane. City Hall will be located at the origin of the map. The locations of five other buildings that will be added to the coordinate plane are: • Bank: (−8, 5) • School: (−8, −6) • Park: (4, 5) • Post Office: (−9, 5) • Store: (−9, −6) •

The coordinates of point A are (−6, 4). The coordinates of point B are (3, 4). Which expression represents the distance, in units, between points A and B? A |−6| + |3| B |3| − |−6| C |−6| + |−4| D |4| − |−6|

Engage NY Module 3 Lesson 19 (Appendix C) https://learnzillion.com /lessons/1295-­‐graph-­‐ and-­‐mathematical-­‐ problems-­‐using-­‐a-­‐ coordinate-­‐plane    

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Graph and solve real world problems using a coordinate plane

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Work with your students to solve realworld problems involving points on a coordinate grid both mathematically and by plotting the points on a grid. Students should practice solving word problems by drawing points on a coordinate grid and then checking mathematically to ensure their new points are correct and vice versa by first solving mathematically and then plotting points on a grid to check their work. Students understand, when solving problems about distance on a coordinate plane, why absolute value is used to record the answer.

City planners are creating a neighborhood map on a coordinate grid. The table shows the locations of the neighborhood library and school on a coordinate grid. Library (-4.-6) School (5,-6) Create a coordinate grid to represent and solve this problem. If the distance between each gridline represents 1 mile, what is the distance, in miles, between the library and the school on the grid?

“Graphing on the Coordinate Plane” (Appendix C) https://learnzillion.com/le ssons/1149-graph-andsolve-real-worldproblems-using-acoordinate-plane

Flex Day (Instruction Based on Data) Recommended Resources: “Planning a Field Trip” (Appendix C) “Sounds of the Band” (Appendix C) My Math Chapter 5 Review and Reflect (Pages 417 – 420) http://www.opusmath.com/common-core-standards/6.ns.8-solve-real-world-and-mathematical-problems-by-graphing-points-in-all-four http://www.mathgoodies.com/standards/alignments/grade6.html

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MCLASS Beacon Unit 6.3 Assessment (Appendix B) *Note: This assessment will be administered online

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Appendix A: Unpacked Standards Guide Source: Public Schools of North Carolina NCDPI Collaborative Workspace  

Common Core Cluster Apply and extend previous understandings of numbers to the system of rational numbers. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: rational numbers, opposites, absolute value, greater than, >, less than, –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

6.NS.7 Students use inequalities to express the relationship between two rational numbers, understanding that the value of numbers is smaller moving to the left on a number line. Common models to represent and compare integers include number line models, temperature models and the profitloss model. On a number line model, the number is represented by an arrow drawn from zero to the location of the number on the number line; the absolute value is the length of this arrow. The number line can also be viewed as a thermometer where each point of on the number line is a specific temperature. In the profit-loss model, a positive number corresponds to profit and the negative number corresponds to a loss. Each of these models is useful for examining values but can also be used in later grades when students begin to perform operations on integers. Operations with integers are not the expectation at this level. In working with number line models, students internalize the order of the numbers; larger numbers on the right (horizontal) or top (vertical) of the number line and smaller numbers to the left (horizontal) or bottom (vertical) of the number line. They use the order to correctly locate integers and other rational numbers on the number line. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between two numbers. Case 1: Two positive numbers

5>3 5 is greater than 3 3 is less than 5 Case 2: One positive and one negative number

3 > -3 positive 3 is greater than negative 3 negative 3 is less than positive 3 Case 3: Two negative numbers

-3 > -5 negative 3 is greater than negative 5 negative 5 is less than negative 3 18  |  P a g e    

Example 1: Write a statement to compare – 4 ½ and –2. Explain your answer. Solution: – 4 ½ < –2 because – 4 ½ is located to the left of –2 on the number line

b. Write, interpret, and

explain statements of order for rational numbers in real-world contexts. For example, write –3oC > –7oC to express the fact that –3oC is warmer than –7oC.

Students recognize the distance from zero as the absolute value or magnitude of a rational number. Students need multiple experiences to understand the relationships between numbers, absolute value, and statements about order. Students write statements using < or > to compare rational number in context. However, explanations should reference the context rather than “less than” or “greater than”. Example 1: The balance in Sue’s checkbook was –$12.55. The balance in John’s checkbook was –$10.45. Write an inequality to show the relationship between these amounts. Who owes more? Solution: –12.55 < –10.45, Sue owes more than John. The interpretation could also be “John owes less than Sue”. Example 2: One of the thermometers shows -3°C and the other shows -7°C. Which thermometer shows which temperature? Which is the colder temperature? How much colder? Write an inequality to show the relationship between the temperatures and explain how the model shows this relationship. Solution: • The thermometer on the left is -7; right is -3 • The left thermometer is colder by 4 degrees • Either -7 < -3 or -3 > -7 Although 6.NS.7a is limited to two numbers, this part of the standard expands the ordering of rational numbers to more than two numbers in context. Example 3: A meteorologist recorded temperatures in four cities around the world. List these cities in order from coldest temperature to warmest temperature: Albany 5° Anchorage -6° Buffalo -7°

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c. Understand the absolute

value of a rational number as its distance from 0 on the number line; interpret absolute as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. d. Distinguish comparisons

of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

Juneau Reno

-9° 12°

Solution: Juneau Buffalo Anchorage Albany Reno

-9° -7° -6° 5° 12°

Students understand absolute value as the distance from zero and recognize the symbols | | as representing absolute value. Example 1: Which numbers have an absolute value of 7 Solution: 7 and –7 since both numbers have a distance of 7 units from 0 on the number line. Example 2: 1 2

What is the | –3 |? Solution: 3

1 2

€

In real-world contexts, the absolute value can be used to describe size or magnitude. For example, for an ocean depth of 900 feet, write | –900| = 900 to describe the distance below sea level. € When working with positive numbers, the absolute value (distance from zero) of the number and the value of the number is the same; therefore, ordering is not problematic. However, negative numbers have a distinction that students need to understand. As the negative number increases (moves to the left on a number line), the value of the number decreases. For example, –24 is less than –14 because –24 is located to the left of –14 on the number line. However, absolute value is the distance from zero. In terms of absolute value (or distance) the absolute value of –24 is greater than the absolute value of –14. For negative numbers, as the absolute value increases, the value of the negative number decreases.

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6.NS.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.

6.NS.8 Students find the distance between points when ordered pairs have the same x-coordinate (vertical) or same y-coordinate (horizontal). Example 1: What is the distance between (–5, 2) and (–9, 2)? Solution: The distance would be 4 units. This would be a horizontal line since the y-coordinates are the same. In this scenario, both coordinates are in the same quadrant. The distance can be found by using a number line to find the distance between –5 and –9. Students could also recognize that –5 is 5 units from 0 (absolute value) and that –9 is 9 units from 0 (absolute value). Since both of these are in the same quadrant, the distance can be found by finding the difference between the distances 9 and 5. (| 9 | - | 5 |). Coordinates could also be in two quadrants and include rational numbers. Example 2: 1 2

What is the distance between (3, –5 ) and (3, 2

1 )? 4

1 2 €

1 3 ) would be 7 units. This would be a vertical line since the x4 4 € 1 1 coordinates are the same. The distance can be found by using a number line to count from –5 to 2 or by 2 4 1 1 recognizing that the distance€(absolute value) € from –5 €to 0 is 5 units and the distance (absolute value) from 0 to 2 2 1 1 1 1 3 2 is 2 units so the total distance would be 5 + 2 or 7 units. € € 4 4 2 4 4

Solution: The distance between (3, –5 ) and (3, 2

€

 

€

€

Students graph coordinates for polygons and find missing vertices based on properties of triangles and quadrilaterals.

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