# MA.5.AR.3.1 Export Print
Given a numerical pattern, identify and write a rule that can describe the pattern as an expression.

### Examples

The given pattern 6,8,10,12… can be describe using the expression 4+2x, where x=1,2,3,4… ; the expression 6+2x, where x=0,1,2,3… or the expression 2x, where x=3,4,5,6….

### Clarifications

Clarification 1: Rules are limited to one or two operations using whole numbers.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Coefficient

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to identify and write an expression that shows the rule for a given pattern. Students have been identifying and generating patterns since grade 3. In grade 5, the expectation extends to students writing a rule as an expression that may have 1 or 2 operations. In grade 6, the focus is on patterns involving ratios (MA.6.AR.3.3).
• The rules for given patterns are limited to one or two operations using whole numbers.
• Vocabulary (e.g., coefficient, terms, variables) should be interwoven into instruction of this benchmark. These terms are introduced in grade 5, but not expected to be mastered until grade 6.
• Students should understand that determining a rule for patterns helps them determine the value of future terms in the pattern (MTR.2.1, MTR.5.1).
• During instruction, teachers can have students compare their rules and justify them using properties of operations.
• For example, have students determine why the rule for the pattern in the benchmark example could be 6 + 2x or 2x + 6 (MTR.5.1, MTR.6.1). Instruction of this benchmark should be paired with MA.5.AR.3.2. The combination of determining rules and completing tables is important for students to begin understanding ratios and functions in the middle grades (MTR.5.1).
• Instruction includes recognizing patterns that arise from geometrical figures with different lengths and their perimeter or area.
• For example, a pattern can arise from the following sequence of rectangles: 1 unit by 1 unit, 1 unit by 2 units, 1 unit by 3 units, 1 unit by 4 units. Students can describe the pattern of the perimeter or of the area.

### Common Misconceptions or Errors

• A common mistake that students make is to determine a rule based on the change in only the first two terms. During instruction, teachers should emphasize that a rule must work for the change in any two terms in a pattern.

### Strategies to Support Tiered Instruction

• Instruction includes opportunities to determine a rule given a numerical expression. After determining the rule, teachers provide guidance to support students as they work to describe the pattern as an expression. Special attention should be given to ensure that the rule is based on changes in all terms within the pattern (not just the first two terms).
• For example, the teacher provides students with the first four terms of a pattern:
3,8,13,18...
• The teacher guides students to notice what pattern they see between the four terms (each number is five greater than the previous number). If students have difficulty, a number line or hundreds chart may be used to support finding the pattern. Students should identify that the rule is to add five. Based on this rule, the teacher guides students to represent the pattern as an expression (e.g., 3 + 5x, where x = 0, 1, 2, 3 ...) having students use the expression to check for accuracy with each of the terms in the pattern and identify the next two terms in the pattern (... 23, 28 ...).
• For example, the teacher provides students with the first four terms of a pattern.
6,10,14,18...
• The teacher guides students to notice what patterns they see between the four terms (each number is five greater than the previous number). A number line or hundreds chart is used to support finding the pattern. Students identify that the rule is to add four. Based on this rule, the teacher guides students to represent the pattern as an expression (e.g., 6 + 4x, where x = 0, 1, 2, 3 ...) having students use the expression to check for accuracy with each of the terms in the pattern and identify the next two terms in the pattern (... 22, 26 ...). The first four terms of a pattern are below.
9,13,17,21,...
• Part A. Write a mathematical description for a rule that matches these terms.
• Part B. Write an expression that describes your rule.
• Part C. Use your answer from Part B to determine the value of the 16th term.

### Instructional Items

Instructional Item 1

Write an expression that can be a rule for the terms shown below.
2,7,12,17,...

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.5.AR.3.AP.1: Given a numerical pattern, identify a one-step rule that can describe the pattern.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

Cool School:

In this lesson, students will take an imaginary trip to my lake house. They will solve a variety of real life word problems. Once at the lake they will experience many activities where they will need number sense. For the assessment, students will record and analyze data in order to determine who the best angler was at "Cool School."

Type: Lesson Plan

Gummy vs. Gum (Number Pattern):

In this lesson from the Beacon Learning Center, students use gummy bears and sticks of gum to discover a number pattern and write an representation that describes it.

This lesson can be adapted for several grade levels and instead of writing an equation, students can write a "rule" or expression based on the pattern or relationship between number of pieces of gum and gummy bears.  Consider scaffolding this activity to meet your grade-level needs.

Type: Lesson Plan

## Perspectives Video: Teaching Idea

The Mathematics of Juggling: Patterns, Rules, and Siteswap Notation:

Sometimes it is difficult to juggle math concepts. See how patterns can help.

Type: Perspectives Video: Teaching Idea

## Tutorial

Number Pattern Example: Growing Sequence:

In this Khan Academy tutorial video a table is used to track a growing sequence of design.

Type: Tutorial

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Tutorial

Number Pattern Example: Growing Sequence:

In this Khan Academy tutorial video a table is used to track a growing sequence of design.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.