• MA.5.AR.3 Algebraic Reasoning

    Analyze patterns and relationships between inputs and outputs


    Given a numerical pattern, identify and write a rule that can describe the pattern as an expression. Example: 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 ….
    Benchmark Clarifications:
    Clarification 1: Rules are limited to one or two operations using whole numbers.


    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.
    o 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.

    Instructional Tasks
    Instructional Task 1
    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, …

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