5th Grade Mathematics | Patterns and the Coordinate Plane

Unit Summary

In Unit 7, the final unit of the year for Grade 5, students are introduced to the coordinate plane and use it to represent the location of objects in space, as well as to represent patterns and real-world situations.

Students have coordinated numbers and distance before, namely with number lines. Students were introduced to number lines with whole-number intervals in Grade 2 and used them to solve addition and subtraction problems, helping to make the connection between quantity and distance (2.MD.5—6). Then in Grade 3, students made number lines with fractional intervals, using them to understand the idea of equivalence and comparison of fractions, again connecting this to the idea of distance (3.NF.2). For example, two fractions that were at the same point on a number line were equivalent, while a fraction that was further from 0 than another was greater. Then, in Grade 4, students learned to add, subtract, and multiply fractions in simple cases using the number line as a representation, and they extended it to all cases, including in simple cases involving fraction division, throughout Grade 5 (5.NF.1—7). Students’ preparation for this unit is also connected to their extensive pattern work, beginning in Kindergarten with patterns in counting sequences (K.CC.4c) and extending through Grade 4 work with generating and analyzing a number or shape pattern given its rule (4.OA.3).

Thus, students start the unit thinking about the number line as a way to represent distance in one dimension and then see the usefulness of a perpendicular line segment to define distance in a second dimension, allowing any point in two-dimensional space to be located easily and precisely (MP.6). After a lot of practice identifying the coordinates of points as well as plotting points given their coordinates with coordinate grids of various intervals and scales, students begin to draw lines and figures on a coordinate grid, noticing simple patterns in their coordinates. Then, after students have grown comfortable with the coordinate plane as a way to represent two-dimensional space, they represent real-world and mathematical situations, as well as two numerical patterns, by graphing their coordinates. This visual representation allows for a rich interpretation of these contexts (MP.2, MP.4).

This work is an important part of “the progression that leads toward middle-school algebra” (6—7.RP, 6—8.EE, 8.F) (K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics, p. 7). This then deeply informs students’ work in all high school courses. Thus, Grade 5 ends with additional cluster content, but that designation should not diminish its importance this year and for years to come.

Pacing: 14 instructional days (12 lessons, 1 flex day, 1 assessment day)

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Assessment

The following assessments accompany Unit 7.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 7.

Post-Unit

Use the resources below to assess student mastery of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Intellectual Prep for All Units

Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan.
Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind.
Take the Post-Unit Assessment.

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Just as the understanding of number and length can be applied in one dimension using a number line, number and length can be coordinated across two dimensions to understand the location of objects in a space.
The point (2, 3) can be viewed in two different ways: (1) as instructions, “right 2, up 3,” and (2) as the point defined by being a distance 2 from the y-axis and a distance 3 from the x-axis. In these two interpretations the 2 is associated with the x-axis (in the first interpretation) and with the y-axis (in the second interpretation).
Just as relationships can exist between terms in one pattern, relationships can exist between corresponding terms in two patterns. This is the basis for all functional understanding.
Graphing coordinate points that represent a real-world situation or patterns can help illuminate trends and features that may have otherwise been difficult to identify.

Vocabulary

Terms and notation that students learn or use in the unit

axis, x-axis, y-axis

corresponding terms

coordinate pair

coordinate, x-coordinate, y-coordinate

coordinate plane

ordered pair

origin

term

To see all the vocabulary for Unit 7, view our 5th Grade Vocabulary Glossary.

Materials

The materials, representations, and tools teachers and students will need for this unit

Markers or crayons (3 per student)
Graph paper (3 sheets per student) — You can provide students with Inch Grid Paper if you don't have enough printed graph paper.
Optional: Quarter Inch Graph Paper (3 per student) — This is optional in case you don't have graph paper.

Unit Practice

Word Problems and Fluency Activities

Access daily word problem practice and our content-aligned fluency activities created to help students strengthen their application and fluency skills.

Lesson Map

Topic A: Introduction to the Coordinate Plane

1

Construct a coordinate plane and identify the coordinates of given points.

5.G.A.1

2

Construct a coordinate plane and plot points given their coordinates.

5.G.A.1

3

Construct a coordinate plane with non-unit intervals and use it to plot and identify points.

5.G.A.1

4

Determine the interval needed to plot various points and practice plotting and identifying points on coordinate grids with various axes.

5.G.A.1

Topic B: Drawing Figures and Shapes in the Coordinate Plane

5

Plot horizontal and vertical lines on a coordinate plane and find patterns in their coordinates.

5.G.A.1 5.G.A.2

6

Draw parallel lines on a coordinate plane.

5.G.A.1 5.G.A.2

7

Draw perpendicular lines on a coordinate plane.

5.G.A.1 5.G.A.2

8

Draw shapes on a coordinate plane.

5.G.A.1 5.G.A.2

9

Draw symmetric figures on a coordinate plane.

5.G.A.1 5.G.A.2

Topic C: Real-World Problems and Patterns on the Coordinate Plane

10

Solve real-world problems by graphing information represented in a table in the coordinate plane and interpret coordinate values of points in the context of the situation.

5.G.A.2

11

Solve real-world problems by graphing information given as a description of a situation in the coordinate plane and interpret coordinate values of points in the context of the situation.

5.G.A.2 5.OA.B.3

12

Generate two numerical patterns using two given rules, plotting the points, and identify relationships between corresponding terms.

5.OA.B.3

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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).

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 — 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.

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.

Operations and Algebraic Thinking

5.OA.B.3 — Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Operations and Algebraic Thinking

5.OA.B.3 — Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Geometry

4.G.A.1

Geometry

4.G.A.1 — Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.A.3

Geometry

4.G.A.3 — Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
5.G.B.4

Geometry

5.G.B.4 — Classify two-dimensional figures in a hierarchy based on properties.

Measurement and Data

2.MD.B.6

Measurement and Data

2.MD.B.6 — Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Number and Operations—Fractions

3.NF.A.2

Number and Operations—Fractions

3.NF.A.2 — Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Operations and Algebraic Thinking

4.OA.C.5

Operations and Algebraic Thinking

4.OA.C.5 — Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Future Standards

Standards in future grades or units that connect to the content in this unit

Expressions and Equations

6.EE.C.9

Expressions and Equations

6.EE.C.9 — Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Geometry

6.G.A.3

Geometry

6.G.A.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.

Ratios and Proportional Relationships

6.RP.A.3

Ratios and Proportional Relationships

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

The Number System

6.NS.C.6

The Number System

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

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.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.