Standard #: MA.6.AR.1.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.6.NSO.1.2
• MA.6.NSO.1.3
• MA.6.AR.2.1
• MA.6.DP.1.6

 

Terms from the K-12 Glossary

• Number Line

 

Vertical Alignment

Previous Benchmarks

• MA.5.NSO.1.4
• MA.5.AR.2.1
• MA.5.AR.2.4

Next Benchmarks

• MA.7.AR.2.1

 

Purpose and Instructional Strategies

In grade 5, students plotted, ordered and compared multi-digit decimal numbers up to the thousandths using the greater than, less than, or equal to symbols. Students translated written real-world and mathematical situations into numerical expressions. In grade 6, students plot, order and compare on both sides of zero on the number line with all forms of rational numbers. The algebraic inequalities include “is less than or equal to” (≤) and “is greater than or equal to” (≥). In grade 7, students will write and solve one-step inequalities in one variable and represent the solutions both algebraically and graphically.

• Instruction emphasizes the understanding of defining an algebraic inequality both numerically and graphically. Students should explore how “is greater than or equal to” and “is strictly greater than” are similar and different as well as “is less than or equal to” and “is strictly less than.” Students should use appropriate language when describing the algebraic inequality.
• The expectation of this benchmark includes translating from an algebraic inequality into a real-world written description.
• As students identify and write the inequality relationships it is important to ask students about other values on the number line (MTR.6.1, MTR.7.1).
  • For example, when representing the inequality $x$ > 1 on the number line, students should compare numbers to 1 in order to determine whether to shade to the left or to the right of 1.
• Instruction includes inequalities where the variable is on the left and right side of the inequality symbol. This will create flexibility in their thinking to be able to apply in future applications of inequalities. Students should form equivalent statements to demonstrate their understanding of how inequalities are related. Students should use context to determine which situation is most appropriate for forming the inequality (MTR.2.1).
  • For example, $a$ > 8 can be written as 8 < $a$.
• Students should understand the solution set and its graphical representation on a number line. This includes showcasing inclusive (closed circle) or non-inclusive (open circle) as well as shading to represent the solution set.
• A number line is a useful tool for modeling inequality situations.
  • For example, students can model on a vertical number line having to be at least 40 inches tall to ride a roller coaster.
• Instruction includes the understanding how inequality relationships can be represented in real-world contexts from the beginning. It can be talked about in terms of character lives in a video game, or amount of money to purchase apps or popular items. Allow students opportunities to identify their own inequality relationships as written descriptions and share with classmates to have them create the algebraic inequality and represent it on the number line (MTR.1.1, MTR.4.1, MTR.5.1, MTR.7.1).

 

Common Misconceptions or Errors

• Students may incorrectly think that the variable must always be written on the left-hand side of the inequality symbol. It is important that students be able to read and understand the context regardless of the side of the inequality that the variable is placed.
• Students may incorrectly shade the solution set of an inequality on a number line. Using verbal descriptions acknowledging the relationship between the number and the variable can support students in identifying the solution set. Students should test numbers on each side of the value to determine if they are true statements.
  • For example, if the relationship is $x$ > 3, have students replace the $x$ with −5 and 5 to see which one creates a true statement.
• Students may not understand the difference between inclusive and non-inclusive solutions sets on a number line. Students can benefit from reasoning within a real-world context.

 

Strategies to Support Tiered Instruction

• Teacher provides inequalities with variables on the left or right side of the symbol and has students write two verbal comparisons for each statement.
• For example, $x$ >15 can be read as “$x$ is greater than 15” or as “15 is less than $x$.”
• For example, $k$ ≤ 6 can be read as “$k$ is less than or equal to 6” or as “6 is greater than or equal to $y$.”
• Instruction includes the use of substitution to test various numbers to determine which set of numbers will make true statements and then to draw a ray in the appropriate direction. The same strategy holds true for reasoning with inclusive and non-inclusive solution sets and the difference in the solid or open point at the indicated value.
• Teacher provides students with flash cards to practice and reinforce academic vocabulary.
• Instruction includes creating a list of common terms from contextual situations that may be used to describe situations requiring an inequality symbol and add to as an interactive anchor chart as additional situations arise.
  • Examples include the table below.
    
    
• Teacher uses verbal descriptions to acknowledge the relationship between the number and the variable to support students in identifying the solution set. Students should test numbers on each side of the value to determine if they are true statements.
  • For example, if the relationship is $x$ > 3, have students replace the $x$ with −5 and 5 to see which one creates a true statement.

 

Instructional Tasks

Instructional Task 1 (MTR.4.1) 

• The graph below describes the altitudes, measured in feet, at which civilian aircraft must provide everyone with supplemental oxygen, according to the U.S. Federal Aviation Regulation.

• Determine an inequality to represent the situation. How can this be written in another way?

 

Instructional Task 2 (MTR.4.1) 

According to historical records, the highest price for regular gas in Florida over the last ten years was just under $4.06. Tammy states that this is an inclusive relationship. Christine says it is a non-inclusive relationship. Who is correct and why?

 

Instructional Items

Instructional Item 1

Graph x > 3 on a number line.