Embed Size (px)
Citation preview
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 Vocabulary Mathematical Practices
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
absolute value, axis (plural - axes) coordinate pair coordinate plane coordinate system coordinates greater than (how to read >) inequality integers less than (how to read <) line of symmetry line symmetry magnitude size
negative numbers number line opposites ordered pair origin positive numbers quadrants rational number signed number value x-axis x-coordinate y-axis y-coordinate
MP.1: Make sense of problems and persevere in solving them
MP.2: Reason abstractly and quantitatively
MP.3: Construct viable arguments and critique the reasoning of others
MP.4: Model with mathematics
MP.5: Use appropriate tools strategically
MP.6: Attend to precision
MP.7: Look for and make use of structure
MP.8: Look for and express regularity in repeated reasoning
2 | P a g e
Common Core State Standards Progression of Skills According to the PARCC Model Content Framework, Standard 3.NF.2 should serve as an opportunity for in- depth focus:
According to the PARCC Model Content Framework,
standard 6.NS.8 represents an opportunity for in-depth focus:
“When students work with rational numbers in the coordinate plane to solve problems, they combine and consolidate elements from the other standards in
this cluster.” The work of this unit is deeply connected to their future work in mathematics: “Plotting rational numbers in the coordinate plane (6.NS.8) is part of analyzing
proportional relationships (6.RP.3a, 7.RP.2) and will become important for studying linear equations (8.EE.8) and graphs of functions (8.F).”
5th Grade 6th Grade 7th Grade
N/A
6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
7.EE.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.
5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
7.NS.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
N/A
6.NS.7: Understand ordering and absolute value of rational numbers.
7.NS.1D: Apply properties of operations as strategies to add and subtract rational numbers.
5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
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.
7.NS.3: Solve real-world and mathematical problems involving the four operations with rational numbers
6.NS.5: Contextual Positive and Negative Rational Numbers
6.NS.6: Rational Numbers on the Number Line 6.NS.7: Ordering and Absolute Value of Rational
Numbers 6.NS.8: Graph and Find distance Between Coordinate
Points
Major Standards (70%)
Additional Standards (10%)
Supporting Standards (20%)
3 | P a g e
Big Ideas Students Will… • What does the position of a
number on a number line have to do with its value? How do you determine the relationship of two numbers by comparing them on a number line?
• How does the opposite of a number differ from its absolute value? How does the distance of a number from zero differ from the distance of the opposite of that same number? Why is it important to relate absolute value to real world situations?
• Why is the absolute value of a
number never negative? How does this impact ordering numbers written in absolute value?
• How do you plot points in all 4 quadrants? How can plotting points on a coordinate plane be used to determine specific locations in real life?
• How would you find the distance
between two vertical or horizontal points on a coordinate plane?
Know/Understand Be Able To… • that positive and negative numbers are used together to
describe quantities having opposite directions or values. • the concept of a negative number and how it can be used
in real-life contexts • how the meaning of 0 can change in different situations. • the relationship between number line diagrams and
coordinate axes. • a rational number as a point on the number line. • opposite signs of numbers indicate locations on opposite
sides of 0 on the number line. • that when two ordered pairs differ only by signs, the
locations of the points are related by reflections across one or both axes.
• that rational numbers are any numbers that are the quotient or fractions of two integers when the denominator is not 0.
• the absolute value of a rational number is the non-negative value of the number.
• the absolute value of a rational number as its distance from 0 on the number line.
• that rational numbers may be ordered by their value or by their absolute value.
• that rational numbers found in real-world contexts can be ordered and interpreted.
• 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.
• how points on a coordinate plane can represent different things, such as distance.
• when solving problems about distance on a coordinate plane, why absolute value is used to record the answer.
• that a coordinate plane can be used to solve mathematical problems.
• use negative and positive numbers to describe quantities with opposite directions or values.
• use positive and negative numbers together to describe real-life situations and explain the value of 0.
• plot negative numbers on number line diagrams and negative coordinates on coordinate axes.
• identify non-linear and linear functions and can explain the difference.
• find and position integers and other rational numbers on a horizontal or vertical number line diagram.
• extend number line diagrams to include negative numbers.
• label the components of the coordinate plane (Quadrant (+,+), Quadrant (-,+) , Quadrant (-,-) Quadrant (+, -), x and y axes, origin)
• interpret absolute value as the magnitude of a positive or negative quantity in a real-world situation.
• explain the difference between comparisons of absolute value of rational numbers from comparisons based on order on a number line.
• place negative and positive numbers on a number line. • compare rational numbers using >,<, and = symbols. • express rational numbers in terms of their overall value
and their absolute value. • order rational numbers. • graph ordered pairs in all four quadrants of the
coordinate plane. • plot ordered pairs to solve mathematical problems
involving distance, within one quadrant or two different quadrants.
• solve mathematical problems involving distance on a coordinate plane by finding the difference between ordered pairs using their absolute value.
4 | P a g e
Unit Sequence Student Friendly Objective
SWBAT… Key Points/
Teaching Tips Sample Assessment Item from
Exit Ticket Instructional
Resources 1 Use positive and negative
numbers together to describe real-life situations and explain the value of 0.
• Work with your students to place negative numbers on a number line based on a problem. Then support students in using the number line to solve addition and subtraction problems with positive and negative numbers. Check out this LearnZillion video for additional ideas.
• Students must understand the concept of a negative number and how they can be used in real-life contexts. Students must also understand how the meaning of 0 is dependent on the situation it is describing.
1) Write an integer for each situation. a. Loss of $20.
__________________ b. 10-‐yard gain.
_________________ 2) Denver, Colorado is called "The Mile High City" because its elevation is 5280 feet above sea level. Someone tells you that the elevation of Death Valley, California is −282 feet. (a) Is Death Valley located above or below sea level? Explain. (b) Plot the elevation of Death Valley on a number line
“Contest Winner” (Appendix C) My Math Chapter 5, Lesson 1
2 Represent situations where positive and negative numbers are used together using real-world tools such as a thermometer, balance sheet, etc. Compute changes (gains or losses).
• Pacing: 2 days • Work with your students to identify
situations in which you have negative amounts of money. Ask them to construct a series of situations in which this occurs and then present scenarios where they will need to add or subtract from their original amount. Check out this LearnZillion video for additional ideas.
1) The temperature rose 3°F in the afternoon and fell 3°F in the evening. What was the total change in temperature at the end of the day? 2) Leroy was evaluating the total change between -5 and 3. He said the total change was 2. Was Leroy correct? If he was explain the steps he took to find the answer. If he was incorrect explain his mistake and how you would find the correct answer.
Engage NY Module 3 Lessons 2 & 3 (Appendix C) 3
5 | P a g e
4 Use negative and positive numbers to describe quantities with opposite directions or values. Use a number line to compute sums and differences of positive and negative numbers.
1) Germain owed his brother Todd $15. Germain then earned $15 delivering newspapers. After Germain pays Todd what he owes him, how much money will Germain have? 2) The bottom of a swimming pool is 10 feet below the surface of the water in the pool. The surface of the water is represented by the number 0, and the bottom of the pool is represented by the number –10. The pool’s diving board is the same distance above the surface of the water as the bottom of the pool is below the surface of the water. What number represents the location of the diving board?
Engage NY Module 3 Lesson 4 (Appendix C)
5 Precisely locate and position opposite numbers on a number line
Jose graphed the opposite of the opposite of 3 on the number line. First, he graphed point 𝑃 on the number line 3 units to the right of zero. Next, he graphed the opposite of 𝑃 on the number line 3 units to the left of zero and labeled it 𝐾. Finally, he graphed the opposite of 𝐾 and labeled it 𝑄.
Is his diagram correct? Explain. If the diagram is not correct, explain his error and correctly locate and label
Engage NY Module 3 Lesson 5 (Appendix C) http://www.mathgoodies.com/standards/alignments/grade6.html
𝟎
K
Q
P
6 | P a g e
point 𝑄
6 Find and position integers and other rational numbers on a horizontal or vertical number line diagram. Use an appropriate scale when graphing rational numbers on the number line
• Students understand a rational number as a point on the number line.
1) Points Q and R on the number line below each represent a real number.
Which of the following numbers is located between points Q and R on the number line? A 3.84 B 3.88 C 3.94 D 3.98 2) Which number best represents the location of point E on the number line below?
A −1.8 B −1.6 C −1.5 D −1.3
Engage NY Module 3 Lesson 6 (Appendix C) http://www.mathgoodies.com/standards/alignments/grade6.html
7 Flex day (Instruction Based on Data) Recommended Resources:
“What’s Your Sign? Part I” (Appendix C) http://www.mathgoodies.com/standards/alignments/grade6.html
7 | P a g e
8 Write, interpret, and explain statements of order for rational numbers in real-world contexts.
• Suggested problem to use for hook, heart of the lesson or closing application practice:
The level of the top of the water in the ocean is considered to be at an altitude of zero (0) feet. • The ocean floor at a particular dive site
is −20 feet. • A diver is located at −5 feet at that same
site. • The captain of a boat is located at an
altitude of 15 feet, directly above the diver.
Which of the following statements are true? Justify and explain each choice:
a. The distance from the captain to the diver is greater than the distance from the top of the water to the ocean floor
b. The distance from the captain to the top of the water is the same as the distance from the diver to the ocean floor.
c. When the diver swims to -10 feet, the diver will be the same distance below the top of the water as the captain is above the top of the water
d. When the diver swims to -10 feet, the diver’s distance to the ocean floor will be equal to the diver’s distance to the top of the water
Here are the low temperatures (in Celsius) for one week in Juneau, Alaska:
(a) Arrange these temperatures in order from coldest to warmest temperature. (b) On a winter day, the low temperature in Anchorage was 23 degrees below zero (in °C) and the low temperature in Minneapolis was 14 degrees below zero (in °C). Sophia wrote, "Minneapolis was colder because −14 < −23." Is Sophia correct? Explain your answer.
Engage NY Module 3 Lessons 7 - 8 (Appendix C) My Math Chapter 5 Lesson 5
9
8 | P a g e
10 Compare and order rational numbers and integers using a number line
Find and label the numbers −3 and −5 on the number line below.
For each of the following, state whether the inequality is true or false. Use the number line diagram to help explain your answers. (b) −3 > −5 (c) −5 > −3 (d) −5 < −3 (e) −3 < −5
Engage NY Module 3 Lesson 9 (Appendix C) “Representing Rational Numbers on the Number Line” (Appendix C)
http://www.ixl.com/math/grade-‐6/compare-‐rational-‐numbers
11 Write, explain and justify inequality statements involving rational numbers
• Students will define absolute value as a number’s distance from 0 on a number line
• Students must be able to explain why the absolute value of two opposites (i.e. -5 and 5 are the same, even though their actual values are different)
Find and label the numbers 4/3, 5/4, -2/3 and -3/4 on the number line below.
Is 4/3 > 5/4 , or is 4/3 < 5/4 ? Is -2/3 > -3/4, or is -2/3 < -3/4? Is -3/4 closer to 0 or is 5/4? Explain how you know.
Engage NY Module 3 Lesson 10 (Appendix C)
12 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.
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
9 | P a g e
Chapter 5 Lesson 2
13 Compare and order positive and negative numbers by their actual and absolute values
• 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
14 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)
15 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
10 | P a g e
16 Identify and name points on a coordinate grid using coordinate pairs
Use the graph below to answer the following questions:
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
Engage NY Module 3 Lesson 14 (Appendix C) My Math Chapter 5 Lesson 6 http://www.ixl.com/math/grade-6/coordinate-graphs-review
17 Graph and identify points in all four quadrants of the coordinate plane
• 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)
Label the coordinate pair for each point:
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.
Engage NY Module 3 Lesson 15 (Appendix C) My Math Chapter 5 Lesson 7 http://www.ixl.com/math/grade-6/graph-points-on-a-coordinate-plane
11 | P a g e
18 Determine when a set of ordered pairs are a reflection of one another. Given a coordinate pair, identify another pair that reflects this point
• 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:
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.
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/reflections-graph-the-image
19
12 | P a g e
20 Construct and label a coordinate plane. Explain the relationship between number lines and coordinate axes.
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)
21 Determine the distance between points on a coordinate plane
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/distance-between-two-points
13 | P a g e
22 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
14 | P a g e
23 Graph and solve real world problems using a coordinate plane
• Work with your students to solve real-world 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/lessons/1149-graph-and-solve-real-world-problems-using-a-coordinate-plane
24 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
25 MCLASS Beacon Unit 6.3 Assessment (Appendix B)
*Note: This assessment will be administered online
15 | P a g e
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, <, greater than or equal to, ≥, less than or equal to, ≤, origin, quadrants, coordinate plane, ordered pairs, x-axis, y-axis, coordinates
Common Core Standard Unpacking What does this standard mean that a student will know and be able to do?
6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.5 Students use rational numbers (fractions, decimals, and integers) to represent real-world contexts and understand the meaning of 0 in each situation. Example 1:
a. Use an integer to represent 25 feet below sea level b. Use an integer to represent 25 feet above sea level. c. What would 0 (zero) represent in the scenario above?
Solution:
a. -25 b. +25 c. 0 would represent sea level
16 | P a g e
6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs
of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., – (–3) = 3, and that 0 is its own opposite
6.NS.6 In elementary school, students worked with positive fractions, decimals and whole numbers on the number line and in quadrant 1 of the coordinate plane. In 6th grade, students extend the number line to represent all rational numbers and recognize that number lines may be either horizontal or vertical (i.e. thermometer) which facilitates the movement from number lines to coordinate grids. Students recognize that a number and its opposite are equidistance from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. For example, – 4 could be read as “the opposite of 4” which would be negative 4. In the example, – (–6.4) would be read as “the opposite of the opposite of 6.4” which would be 6.4. Zero is its own opposite.
Example 1:
What is the opposite of 2 ? Explain your answer?
Solution:
- 2 because it is the same distance from 0 on the opposite side. €
12
€
12
17 | P a g e
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Students worked with Quadrant I in elementary school. As the x-axis and y-axis are extending to include negatives, students begin to with the Cartesian Coordinate system. Students recognize the point where the x-axis and y-axis intersect as the origin. Students identify the four quadrants and are able to identify the quadrant for an ordered pair based on the signs of the coordinates. For example, students recognize that in Quadrant II, the signs of all ordered pairs would be (–, +). Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the y-coordinates differ only by signs, which represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been reflected across both axes. Example1: Graph the following points in the correct quadrant of the coordinate plane. If the point is reflected across the x-axis, what are the coordinates of the reflected points? What similarities are between coordinates of the original point and the reflected point?
Solution:
The coordinates of the reflected points would be . Note that the
y-coordinates are opposites. Example 2:
Students place the following numbers would be on a number line: –4.5, 2, 3.2, –3 , 0.2, –2, . Based on number
line placement, numbers can be placed in order. Solution: The numbers in order from least to greatest are:
–4.5, – 3 , –2, 0.2, 2, 3.2,
Students place each of these numbers on a number line to justify this order.
⎟⎠
⎞⎜⎝
⎛ −213 ,
21
⎟⎠
⎞⎜⎝
⎛ −− 3 ,21
€
12
, 312
⎛
⎝ ⎜
⎞
⎠ ⎟
€
−12
, 3⎛
⎝ ⎜
⎞
⎠ ⎟
€
0.25, 0.75( )
€
35
€
112
€
35
€
112
18 | P a g e
6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of
inequality as statements about the relative position of two numbers on a number line. For example, interpret –3 > –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 profit-loss 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
19 | 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 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.
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 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°
20 | P a g e
Juneau -9° Reno 12° Solution: Juneau -9° Buffalo -7° Anchorage -6° Albany 5° Reno 12°
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.
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:
What is the | –3 |?
Solution: 3
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.
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.
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.
€
12
€
12
21 | P a g e
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:
What is the distance between (3, –5 ) and (3, 2 )?
Solution: The distance between (3, –5 ) and (3, 2 ) would be 7 units. This would be a vertical line since the x-
coordinates are the same. The distance can be found by using a number line to count from –5 to 2 or by
recognizing that the distance (absolute value) from –5 to 0 is 5 units and the distance (absolute value) from 0 to
2 is 2 units so the total distance would be 5 + 2 or 7 units.
Students graph coordinates for polygons and find missing vertices based on properties of triangles and quadrilaterals.
€
12
€
14
€
12
€
14
€
34
€
12
€
14
€
12
€
12
€
14
€
14
€
12
€
14
€
34
