Patterns and the Coordinate Plane

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.

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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Summer Packets

Packet of problems designed to help prepare students for the start of next school year.

Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Essential Understandings

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  • 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

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axis, x-axis, y-axis

coordinate, x-coordinate, y-coordinate

coordinate pair

coordinate plane

ordered pair

corresponding terms

origin

term

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Unit Materials, Representations and Tools

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  • Ruler
  • Inch Grid Paper (It is recommended, even when lessons do not call for it, to have grid paper available for students to use throughout the unit.)

Intellectual Prep

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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 unit assessment.  

Lesson Map

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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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.

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

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Geometry
  • 4.G.A.1

  • 4.G.A.2

  • 4.G.A.3

  • 5.G.B.4

Measurement and Data
  • 2.MD.B.6

Number and Operations—Fractions
  • 3.NF.A.2

Operations and Algebraic Thinking
  • 4.OA.C.5

Future Standards

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Expressions and Equations
  • 6.EE.C.9

Geometry
  • 6.G.A.3

Ratios and Proportional Relationships
  • 6.RP.A.3

The Number System
  • 6.NS.C.6

  • 6.NS.C.8

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.