MA.8.AR.2.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.8.NSO.1.2
• MA.8.NSO.1.7
• MA.8.GR.1.1
• MA.8.GR.1.2

Terms from the K-12 Glossary

• Integer
• Real Numbers

Vertical Alignment

Previous Benchmarks

• MA.7.AR.2.2

Next Benchmarks

• MA.912.AR.3.1

Purpose and Instructional Strategies

• This benchmark involves students understanding the concepts of how to square a number and find the square root as well as how to cube a number and find the cube root.
• Students should recognize that squaring a number and taking the square root of a number are inverse operations, therefore, cubing a number and taking the cube root are inverse operations as well. Students should use this understanding to solve equations containing square or cube numbers.
• In finding the square root, instruction involves discussion that there is both a positive and negative solution. Instruction can include relating the lengths of the sides of a square for square root and the length of the side of a cube to cube roots.
• Within this benchmark, it is not the expectation that students are required to isolate the $x$² term or the $x$³ term when solving an equation.

Common Misconceptions or Errors

• Students may incorrectly conclude that squaring a number means to multiply by 2. Likewise, cubing may be mistaken as multiplying by 3. Use length to show doubling and area of a square to show an exponent of 2. Use of two-dimensional and three-dimensional manipulatives (MTR.2.1) may also help to emphasize squares and cubes versus increasing length.
• Students may think that since a negative number has no square root in the real number system, then a negative number has no cube root in the real number system.

Strategies to Support Tiered Instruction

• Instruction includes modeling the differences between doubling and squaring a value using a graphic organizer. Doubling a value would be represented by multiplying a given length by 2 whereas squaring a number would be represented by the area of a square with a given length.
  • For example, students can be given the table below to show how the left column doubles a length whereas the right column squares a length.
    
    
    
• Instruction includes modeling the differences between tripling or cubing a value using a graphic organizer. Tripling a value would be represented by multiplying a given length by 3 whereas cubing a number would be represented by the volume of a cube with a given length.
  • For example, students can be given the table below to show how the left column triples a length whereas the right column cubes a length.
    
    
    
    
• Instruction may include providing students with the opportunity to develop their own note sheet or graphic organizer for the cubes of numbers from -5 to 5.
  
  
  

Instructional Tasks

• Part A. What is the length of one side of the square tile in inches? Is this tile smaller or larger than a one foot by one foot tile?
• Part B. The owner of the house, Kiana, wants to put larger tile in their kitchen to change the look of the kitchen. The new tile is a square with an area of 196 square inches.
  • What is the length of the side of the new tile?
  • How does this larger tile compare to the current tile used in the kitchen?
• Part C. A third tile has a side length of 2$\sqrt{11}$. Kiana is trying to determine which square tile covers the most area. Put the tiles side lengths in order from greatest to least. Justify your thinking.

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