Lesson 13 | Rational Numbers | 6th Grade Mathematics

Objective

Calculate vertical and horizontal distances on a coordinate plane using absolute value in real-world and mathematical problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

6.NS.C.7.C — Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value 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.

The Number System

6.NS.C.7.C — Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value 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.
6.NS.C.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.

The Number System

6.NS.C.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.

Foundational Standards

The foundational standards covered in this lesson

5.G.A.1

Geometry

5.G.A.1 — Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.G.A.2

Geometry

5.G.A.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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Find vertical and horizontal distances between two points on the same vertical or horizontal line by reasoning about absolute value.
Find the length of horizontal and vertical line segments in the coordinate plane when given two coordinate points.
Find possible endpoints of horizontal and vertical line segments when given one endpoint of the line segment and the length of the line segment.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Visual representations (maps or coordinate planes) can support student understanding of how distances are calculated and can enable students to verify by counting (MP.5). It is also important for students to understand what is happening beyond just counting the units. Ask students to represent their thinking using absolute value, and encourage them to try problems without using a visual once they’ve gotten the concept down.
The question of diagonal distance may come up. It isn’t until eighth grade that students learn how to use the Pythagorean Theorem to find distance between any two points in the coordinate plane; however, you could blow up a unit square and do a quick measurement activity to show students that the diagonal distance across a unit square is greater than 1 and thus can’t be counted in the same way as vertical or horizontal distances.

Lesson Materials

Graph Paper (2-3 sheets per student)

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Four friends are driving. They come to an intersection of two roads; the road they are on continues straight, and the other road is perpendicular to it. The sign at the intersection shows the distances to several towns.

a. Draw a map of the roads.

b. What is the distance between Albertsville and Dewey Falls? How can you represent this distance using absolute value?

c. What is the distance between Blossville and Cheyenne? How can you represent this using absolute value?

Guiding Questions

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References

EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 18 — Opening Exercise

Grade 6 Mathematics > Module 3 > Topic C > Lesson 18 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Consider the line segment with endpoints $${(-3, 3)}$$ and $${(-3, -5)}$$. Find the length of the line segment by finding the distance between the endpoints.

Guiding Questions

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References

EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 18 — Example 3

Modified by Fishtank Learning, Inc.

Problem 3

A horizontal line segment has one endpoint at the point $${(2, -3)}$$. If the line segment is $$7$$ units long, then what are possible coordinates for the other endpoint? Use the coordinate plane below if needed.

Guiding Questions

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References

EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 19 — Exercise 2

Grade 6 Mathematics > Module 3 > Topic C > Lesson 19 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

The coordinates of five points are shown below.

Point $$A$$: $$(-4,6)$$

Point $$B$$: $$(2,0)$$

Point $$C$$: $$(0,2)$$

Point $$D$$: $$(4,6)$$

Point $$E$$: $$(2,4)$$

a. Find two points that lie on the same horizontal line. Explain your reasoning.

b. Find two points that lie on the same vertical line. Explain your reasoning.

Student Response

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Problem 2

A line segment drawn in the coordinate plane has endpoints at $$(-3,4)$$ and $$(-3,-6)$$. What is the length of the line segment?

Student Response

An example response to the Target Task at the level of detail expected of the students.

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Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Illustrative Mathematics Nome, Alaska
Illustrative Mathematics Distances between Points
EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 18 — Exercises a-e and Problem Set
EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 14 — Exit Ticket (Add in question: If the grocery store is located 3 units from the Fire Dept., then name four ordered pairs that could represent the location of the grocery store.)
EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 14 — Exit Ticket (Add in questions to find the distance between the Fire Dept. and Hospital, the distance between the Fire Dept. and Bank, and the distance between the Bank and Police Station)
EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic C > Lesson 19 — Problem Set #1 only

Lesson 12

Lesson Map

Topic A: Understanding Positive and Negative Rational Numbers

Extend the number line to include negative numbers. Define integers.

6.NS.C.6 6.NS.C.6.C

Use positive and negative numbers to represent real-world contexts, including money and temperature.

6.NS.C.5

Use positive and negative numbers to represent real-world contexts, including elevation.

6.NS.C.5

Define opposites and label opposites on a number line. Recognize that zero is its own opposite.

6.NS.C.6.A 6.NS.C.6.B

Find and position integers and rational numbers on the number line.

6.NS.C.6.C

Topic B: Order and Absolute Value

Order integers and rational numbers. Explain reasoning behind order using a number line.

6.NS.C.6.C 6.NS.C.7.A

Compare and interpret the order of rational numbers for real-word contexts.

6.NS.C.6.C 6.NS.C.7.A

Write and interpret inequalities to compare rational numbers in real-world and mathematical problems.

6.NS.C.7.A 6.NS.C.7.B

Define absolute value as the distance from zero on a number line.

6.NS.C.7.C

Model magnitude and distance in real-life situations using order and absolute value.

6.NS.C.7.C 6.NS.C.7.D

Topic C: Rational Numbers in the Coordinate Plane

Use ordered pairs to name locations on a coordinate plane. Understand the structure of the coordinate plane.

6.NS.C.6.B 6.NS.C.6.C

Reflect points across axes and determine the impact of reflections on the signs of ordered pairs.

6.NS.C.6.B

Calculate vertical and horizontal distances on a coordinate plane using absolute value in real-world and mathematical problems.

6.NS.C.7.C 6.NS.C.8

Rational Numbers

Objective

Common Core Standards

Core Standards

The Number System

The Number System

Foundational Standards

Geometry

Geometry

Criteria for Success

Tips for Teachers

Lesson Materials

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Problem Set

Target Task

Problem 1

Student Response

Problem 2

Student Response

Additional Practice

Lesson Map

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