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Unit 4 Integers and the Coordinate Plane Math 6 ACC

LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 12/21/17 2017-2018

Unit Goals – Stage 1 Number of Days: 28 days 1/8/18 – 2/16/18 Unit Description: Students will extend the number line to include negative numbers using both horizontal and vertical number lines. Students will understand the absolute value of a number and will graph ordered pairs in all four quadrants as well as drawing polygons given coordinates for the vertices. In addition, students will also learn seventh grade standards of operations with rational numbers. They learn to add, subtract, multiply, and divide integers, while maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. Students will then apply their understanding to perform the four operations on positive and negative rational numbers in the context of real-world situations. Materials: coordinate grid cling, number line, graph paper, two-color counters, Desmos

Standards for Mathematical Practice SMP 1 Make sense of problems

and persevere in solving them.

SMP 2 Reason abstractly and quantitatively.

SMP 3 Construct viable arguments and critique the reasoning of others.

SMP 4 Model with mathematics. SMP 5 Use appropriate tools

strategically. SMP 6 Attend to precision. SMP 7 Look for and make use

of structure. SMP 8 Look for and express

regularity in repeated reasoning.

Standards for Mathematical Content Clusters Addressed [m] 6.NS.C Apply and extend

previous understandings of numbers to the system of rational numbers.

[s] 6.G.A Solve real-world and mathematical problems involving

Transfer Goals

Students will be able to independently use their learning to…

Make sense of never-before-seen problems and persevere in solving them.

Construct viable arguments and critique the reasoning of others.

Making Meaning

UNDERSTANDINGS Students will understand that…

The number system includes negative numbers. Positive and negative numbers are used together to describe quantities having opposite directions or values.

Each positive number has an opposite, a corresponding negative number that is the same distance away from zero on the left side of zero.

Understand the absolute value of a rational number as its distance from 0 on the number line.

Integer operations can be represented in multiple ways, such as using manipulatives, number lines, and real-world situations.

Ordered pairs can be used to identify points on a coordinate plane.

ESSENTIAL QUESTIONS Students will keep considering…

What real world situations can be represented with negative and positive numbers?

How are adding and subtracting integers related?

How are multiplying and dividing integers related?

Acquisition

KNOWLEDGE Students will know…

The definition of academic vocabulary words, such as absolute value, additive inverse, axis, coordinate plane, integers, opposites, origin, and quadrants.

Opposite signs of numbers indicate locations on opposite sides of 0 on the number line.

The sum of opposite numbers is zero.

SKILLS Students will be skilled at and/or be able to…

Locate and position integers and other rational numbers on a vertical and horizontal number line.

Use rational numbers to represent quantities in real-world contexts.

Write and interpret statements about the order of rational numbers in real-world contexts.

Find the absolute value of a rational number.



Unit Goals – Stage 1 area, surface area, and volume.

[m] 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Any subtraction problem can be written as an addition problem by adding the additive inverse, p – q = p + (–q).

A coordinate plane has an x-axis and a y-axis.

Add, subtract, multiply, and divide rational numbers to solve real-world and mathematical problems.

Apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers.

Plot ordered pairs of integers and other rational numbers on a coordinate plane.

Solve real-world and mathematical problems by using coordinates and absolute value to find the distance between points with the same first coordinate or same second coordinate.

Recognize that if the coordinates only differ by the signs, the points are reflections across one or both axes.

Draw polygons in the coordinate plane, given coordinates for the vertices in the context of solving real-world and mathematical problems.



Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.

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.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. 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. 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 diagram. 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.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3C > –7C to

express the fact that –3C is warmer than –7C. 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.

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.

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.

[s] 6.G.A Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with

the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.



Assessed Grade Level Standards [m] 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two

constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is

positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational

numbers by describing real‐world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two

rational numbers on the number line is the absolute value of their difference, and apply this principle in real‐world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the

properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or

eventually repeats. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters



Evidence of Learning – Stage 2 Assessment Evidence

Unit Assessment Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 6.NS.C

The student uses positive and negative numbers to represent quantities in real-world contexts.

The student can identify the location of ordered pairs on the coordinate plane based on the signs of the numbers in an ordered pair.

The student locates and positions integers and other rational numbers on a number line.

The student positions ordered pairs of integers and other rational numbers on a coordinate plane.

The student writes and interprets statements about the order of rational numbers in real-world contexts.

The student represents the absolute value of a rational number as the distance from zero on a number line.

The student can make comparisons of absolute value from statements about order.

The student solves real-world and mathematical problems by graphing ordered pairs on a coordinate plane and using coordinates and absolute value to find the distance between points with the same first coordinate or same second coordinate.

6.G.A

The student draws polygons in the coordinate plane, given coordinates for the vertices in the context of solving real-world and mathematical problems. 7.NS.A

The student interprets rational number values on a number line, including modeling addition and subtraction expressions.

The student applies properties of operations as strategies to add and subtract rational numbers.

The student applies properties of operations as strategies to multiply and divide rational numbers.

The student converts from a fractional form of rational numbers to a decimal form of rational numbers.

The student solves real-world and mathematical problems involving the four operations with rational numbers. Claim 2: Students can solve a range of well-posed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2:

6.NS.C

6.G.A

7.NS.A

Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3:

6.NS.C

7.NS.A

Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4:

6.NS.C

6.G.A

7.NS.A



Other Evidence

Formative Assessment Opportunities

Informal teacher observations

Checking for understanding using active participation strategies

Exit slips/Summaries

Tasks

Modeling Lessons (SMP 4)

Formative Assessment Lessons (FAL)

Quizzes/Chapter Tests

SBAC Interim Assessment Blocks

Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents

http://www.lbschools.net/Asset/Files/Curriculum/Math/Courses/HS/15-16/Using%20FA%20to%20Address%20Learning%20Needs%20of%20Stu%20in%20Subgroups%20MATH.pdf



Learning Plan – Stage 3 Suggested Sequence of Key Learning Events and Instruction

Days Learning Target Expectations Big Ideas Math Advanced 1

(Activities and Lessons) Supplemental Resources

1 day

I will explore integers and the coordinate plane by participating in the Opening Task.

OPENING TASK (choose one)

“Extending the Number Line” is an activity in which students explore negative numbers on a number line. Teachers: Prepare the transparencies and number lines. (SMP 7, 8)

“Using a Coordinate Grid” is an online video with a corresponding activity. Have students work in pairs on the discussion questions. Share as a whole group. (SMP 3)

“A Fly on the Ceiling” is a children’s book in which students explore the coordinate plane.

Conceptual Understanding:

Extending the Number Line Exploration

Horizontal Number Lines

Vertical Number Lines

“Using a Coordinate Grid” Video

Questions to “Using a Coordinate Grid” Video

“A Fly on the Ceiling”

2 days

I will explore numbers that are less than zero by… 6.NS.5, 6.NS.6,

6.NS.6a

Extending vertical and horizontal number lines to include negative numbers.

Explaining that a rational number is a point on a number line. (SMP 3)

Using positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of zero. (SMP 3)

Recognizing opposite signs of numbers as indicating locations on opposite sides of zero.

Recognizing that the opposite of the opposite of a number is the number itself.

Answering questions such as… o How can you represent numbers that are less

than zero? o Describe real-life examples that use numbers

that are less than zero. o Is zero considered positive or negative? Explain

your reasoning. o Synergy Item Bank: Item ID 53035, 53204,

55355, 56104, 51268, 56090, 59773

Section 6.1 (Activities 1, 2, and 3; Examples 1, 2, and 3)


Positive and Negative Numbers in Real Life

Integers on the Number Line Task

Procedural Skills and Fluency:

Opposite Directions

Khan Academy: Intro to negative numbers

Khan Academy: Number opposites

Application:

Mile High Task

Lakes of North America Task






1-2 days

I will compare and order integers by… 6.NS.5, 6.NS.6c, 6.NS.7a, 6.NS.7b

Finding and positioning integers and other rational numbers (fractions and decimals) on horizontal and vertical number lines.

Interpreting statements of inequality as statements about the relative position of two numbers on a number line.

Writing, interpreting, and explaining statements of order for rational numbers in real-world contexts. (SMP 2, 3)

Answering questions such as… o Explain how to use a number line to compare

two integers. (SMP 3, 5) o How can you use a number line to compare

positive and negative fractions and decimals? o Does every number have an opposite? Give

examples to justify your reasoning. o “The farther right a number is on the number

line, the greater it is.” Do you agree or disagree with this statement? Explain your reasoning.

o “The farther left a number is on the number line, the less it is.” Do you agree or disagree with this statement? Explain your reasoning.

o Synergy Item Bank: Item ID 53290, 55348, 55349, 55361, 55363, 55571, 56108

Section 6.2 (Activities 1 and 2; Examples 1, 2, 3, 4, and 5)

Section 6.3 (Activities 2 and 3; Examples 2 and 3)

STEM Video: Tuning a Guitar


Guess My Integer Procedural Skills and Fluency:

Plotting Rational Numbers

Khan Academy: Negative decimals & fractions on the number line

Khan Academy: Comparing negative numbers

Desmos: Order Rational Numbers

Desmos: Ordering Integers Application:

Fractions on the Number Line Task

STEM Video Performance Task: Tuning Guitars






2 days

I will explore absolute value by… 6.NS.7c, 6.NS.7d

Explaining that the absolute value of a rational number is its distance from 0 on the number line. (SMP 3)

Interpreting absolute value as magnitude for a positive or negative quantity in a real-world situation.

Comparing and ordering absolute value expressions.

Answering questions such as… o What does the absolute value of a number

represent? o “Absolute values are always positive.” Do you

agree or disagree with this statement? Explain your reasoning.

o Synergy Item Bank: Item ID 52921, 53042, 53222, 55351

Section 6.4 (Activities 1 and 2; Examples 1, 2, and 3)


Absolute Value Task


Khan Academy: Absolute value

Application:

Above and Below Sea Level Task

3 days

I will add integers by…

7.NS.1a-d

Using manipulatives to create conjectures about the rules for adding integers. (SMP 5, 7, 8)

Using a horizontal and vertical number line diagram.

Showing that a number and its opposite have a sum of zero (called additive inverses).

Explaining p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. (SMP 3)

Interpreting sums of integers by describing real-world contexts. (SMP 2)

Answering questions such as… o Why do additive inverses have a sum of zero? o How can you add integers using multiple

methods? o How is adding integers similar to adding whole

numbers? How is it different? o Synergy Item Bank: Item ID 53043, 53059,

53334, 53337

Section 11.2 (Activities 1, 2, 3, and 4; Examples 1, 2, and 3)

STEM Video: Freezing Solid


Adding Integers with Number Lines

Adding Two-Digit Numbers Task

Additive Inverse Video

myPD Course #2572: Adding Integers With Two Color Counters


Integer Addition – Beat the Computer Drill

Adding Integers Tic-Tac-Toe

Khan Academy: Intro to adding negative numbers

Application:

STEM Video Performance Task: Melting Matters






3 days

I will subtract integers by…

7.NS.1a-d

Using manipulatives to create conjectures about the rules for subtracting integers. (SMP 5, 7, 8)

Showing that the distance between two integers on a number line is the absolute value of their difference.

Explaining subtraction of integers as adding the

additive inverse, )( qpqp . (SMP 3)

Interpreting differences of integers by describing real-world contexts. (SMP 2)

Answering questions such as… o Why do you add the inverse when subtracting? o How can you subtract integers using multiple

methods? o How is subtracting integers similar to

subtracting whole numbers? How is it different? (SMP 7)


Section 11.3 (Activities 1, 2, 3, and 4; Examples 1, 2, and 3)


Subtracting Integers Using Two-Color Counters PowerPoint

Subtracting Two-Digit Numbers Task

Integer Operations on the Number Line – Which One Doesn’t Belong?


Adding and Subtracting Integers Comic Strip Activity

Top This Game

Adding and Subtracting Integers Math-O

Adding and Subtracting Integers Tic-Tac-Toe

Khan Academy: Intro to subtracting negative numbers

Khan Academy: Adding & subtracting with negatives on the number line

Application:

Cliff Diving Task

Differences of Integers Task

Modeling: How Much Did the Temperature Drop? (SMP 4)






2-3 days

I will check my understanding of adding and subtracting of integers by participating in the FAL.

OPTION 1: FORMATIVE ASSESSMENT LESSON

Adding and Subtracting Directed Numbers (SMP 1, 2, 3, 5, 6, 7, 8)

2 days

I will multiply integers by…

7.NS.2a-d

Using manipulatives to create conjectures about the rules for multiplying integers. (SMP 5, 7, 8)

Explaining why the product of two negative numbers must be positive using properties, i.e. (–1)(–1) = 1. (SMP 3)

Interpreting products of integers in real-world contexts. (SMP 2)

Answering questions such as… o How can you multiply integers using multiple

methods? o How is multiplying integers similar to multiplying

whole numbers? How is it different? o Synergy Item Bank: Item ID 53058, 53339,

53354, 53444, 53449, 57147

Section 11.4 (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4)


myPD Course #2574: Multiplying Integers with Two Color Counters

Multiplying Two-Digit Numbers Task

Why is a Negative Times a Negative Always Positive? Task


Integer Multiplication – Beat the Computer Drill

Right on Target Game

Add, Subtract, Multiply Integer Operations – How Do They Fit?

Integers Product Game

Khan Academy: Multiply & divide negative numbers

Khan Academy: Exponents with negative bases

Application:

Multiply to Make 64 Task

The Running Videos Task






2 days

I will divide integers by…

7.NS.2a-d

Using manipulatives to create conjectures about the rules for dividing integers. (SMP 5, 7, 8)

Explaining that integers can be divided as long as the divisor is not zero and know that every quotient of integers is a rational number. (SMP 3)

Interpreting quotients of integers in real-world contexts.

Answering questions such as… o Compare and contrast multiplying and dividing

integers. o How can you divide integers using multiple

methods? o How is dividing integers similar to dividing whole

numbers? How is it different? (SMP 7) o Synergy Item Bank: Item ID 53315, 53442,

53446, 53448, 64215

Section 11.5 (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4)


Dividing Positive and Negative Numbers Video

Dividing Two-Digit Numbers Task

Missing Factor Model for Dividing Integers

myPD Course #2467: Dividing Integers with Two Color Counters


Integer Division – Beat the Computer Drill

Multiplying and Dividing Integers Connect 4

Mixed Operations with Integers Math-O

Substituting Negative Values into Algebraic Expressions

Khan Academy: Multiply & divide negative numbers

Khan Academy: Negative signs in fractions






2 days

I will apply operations of rational numbers by…

7.NS.3

Converting a fraction into a decimal.

Adding, subtracting, multiplying, and dividing positive and negative decimals, fractions, and mixed numbers.

Solving real-world and mathematical problems involving the four operations with positive and negative rational numbers.

Applying properties and order of operations as strategies to add, subtract, multiply, and divide positive and negative rational numbers.

Answering questions such as… o When would you need to add, subtract, multiply,

or divide rational numbers in a real-world context?

o How is adding rational numbers similar to adding integers? How is it different?


Section 12.1 (Example 1)

Section 12.2 (Activities 1 and 2; Examples 1, 2, 3, and 4)

Section 12.3 (Activities 1 and 2; Examples 1, 2, 3, and 4)

Section 12.4 (Examples 1, 2, and 4)

STEM Video: Carpenter or Joiner


Converting Rational Numbers Video

Decimal Expansions of Fractions Task

Operations on the Number Line Task

Rational Numbers – Which One Doesn’t Belong?

Multiplying Fractions 3 Task Procedural Skills and Fluency:

Converting Between Fractions and Decimals Task

Decimal Maze

Multiplying Rational Numbers Product Game

Khan Academy: Adding & subtracting negative fractions

Khan Academy: Multiply & divide negative fractions

Application:

Differences and Distances Task

Who is the Better Batter? Task

STEM Video Performance Task: Precisely Perfect






3 days

I will explore the coordinate plane by… 6.NS.6b, 6.NS.6c,

6.NS.8

Identifying the components of the coordinate plane, such as the axes, origin, and quadrants.

Plotting ordered pairs in all quadrants, including ordered pairs with rational numbers.

Identifying the coordinates of an ordered pair in a graph.

Identifying the location of an ordered pair based on the signs of the numbers in the ordered pair.

Recognizing that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one of both axes.

Describing how the signs of ordered pairs are affected by reflections across the x-axis and/or y-axis.

Answering questions such as… o Explain how to locate points on the coordinate

plane. (SMP 3) o How do you know if a point is to the right or the

left of the y-axis? How do you know if a point is above or below the x-axis?

o Describe the location of an ordered pair with zero as its x or y coordinate.

o What conjectures can you make about the signs of coordinates in each quadrant?

o Describe how to reflect a point over the x-axis, y-axis, or both axes.


Section 6.5 (Activities 1 and 2; Examples 1, 2, and 4)

Extension 6.5 (Examples 1 and 2)


Plotting Points in the Coordinate Plane Task

Graphing Points on a Coordinate Plane Task


Desmos: Coordinate Plane Activity

Desmos: The (Awesome) Coordinate Plane Activity

Coordinate Plane: Reflecting Points Video

Reflecting Points on the Coordinate Plane Technology Activity

Khan Academy: Coordinate plane

Desmos: Battle Boats

Application:

Port Royal A Sunken City Task






1 day

I will check my understanding of graphing on the coordinate plane by participating in the FAL.

OPTION 2: FORMATIVE ASSESSMENT LESSON

Reflecting Points Over Coordinate Axes (SMP 1, 2, 3, 5, 6, 7, 8)

2 days

I will solve real-world and mathematical problems involving the coordinate plane by…

6.G.3

Drawing polygons in the coordinate plane given coordinates for the vertices.

Finding distance between ordered pairs on a graph with the same first coordinate or the same second coordinate.

Using of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Answering questions such as… o Why is the length of a segment always positive? o Explain how you find the distance between two

points. o When finding distance between ordered pairs on

a graph with the same first coordinate or the same second coordinate, which do you prefer: counting the distance or adding the absolute values? Explain your reasoning.


Section 4.4 (Activities 1, 2, and 3; Examples 2 and 3)

Section 6.5 (Example 3; Exercises 25-34, 53)


Desmos: Zombie Apocalypse

Area of a Quadrilateral on a Coordinate Plane Task


Distance Between Points Task

Khan Academy: Polygons on the coordinate plane

Desmos: Rectangle Ordered Pairs Card Sort

Application:

Facial Recognition Task






1 day

I will prepare for the unit assessment on integers and the coordinate plane by…

Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit.

Ch. 6 Study Help (p. 266)

Ch. 6 Review (p. 285 – 287)

Ch. 11 Study Help (p. 494)

Ch. 11 Review (p. 509 – 511)

Application:

Temperature Task

Hotel Elevator Task

Bottling Task

1 day Unit Assessment

Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit.

At this point, all standards addressed in the Grade 6 SBAC Interim Assessment Block – The Number System and Grade 6 SBAC Interim Assessment Block – Geometry have been covered.

This block may now be administered.