MA.7.GR.1.4

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.7.NSO.2
• MA.7.AR.4.1
• MA.7.GR.2
• MA.7.DP.1.4

Terms from the K-12 Glossary

• Area
• Circle
• Pi (π) 
• Rectangle

Vertical Alignment

Previous Benchmarks

• MA.6.GR.2.2

Next Benchmarks

• MA.912.GR.4.4
• MA.912.GR.4.6
• MA.912.GR.6.4

Purpose and Instructional Strategies

• Students are not expected to memorize the formula for the area of a circle (MTR.5.1).
• Students are expected to know approximations of pi ($\frac{\text{355}}{\text{113}}$, $\frac{\text{22}}{\text{7}}$ or 3.14).
• Instruction includes students exploring circles. Provide students with a circle and have them highlight the circumference. Students will then fold the circle in half, half again, and half once more to allow them to cut it into 8 wedges of equal size. Then arrange the wedges so they alternately point up and down, forming a rectangle, with the highlighted circumference being the bases. Ensure students realize that the length of the rectangle is approximately equal to half the circumference, or π$r$, of the circle and the height of the rectangle is equal to the radius, $r$, of the circle (MTR.4.1).
  • For more accuracy, provide a circle with dashed lined for students to cut the circle into 16 equal-sized wedges.
    
    
    
  • Have students describe the area of a circle and explain if the area of a circle changes if it is cut up and rearranged.
  • Ask questions to elicit student thinking (MTR.4.1) such as:
    • What formula was used to find the area of a circle?
    • How is the formula for the area of a circle related to the formula for the area of a parallelogram?
• Instruction includes using circles on grid paper for students to estimate area before making precise calculations.
• The expectation of this benchmark is not to find the radius or diameter of a circle when given the area.

Common Misconceptions or Errors

• Students may invert the terms radius and diameter. To address this misconception, review parts of a circle including radii, diameters and chords.
  
  
  
• Students may incorrectly believe pi (π) is a variable, rather than a constant that does not change from one circle to the next. Review the development of pi in MA.7.GR.1.3.
• Students may confuse circumference and area. To address this misconception, help students connect area of a circle to area of a rectangle.
• Students may incorrectly double the radius (finding diameter), rather than squaring it, when finding an area. To address this misconception, review exponent rules from MA.6.NSO.3.3 and MA.7.NSO.1.1.

Strategies to Support Tiered Instruction

• Teacher provides opportunities for students to measure the radius and diameter of various circles to explore and discuss the similarities and differences between radius and diameter.
• Instruction includes modeling the area of a given circle in terms of pi before replacing the value of pi with an approximation to determine the estimated area.
• Teacher co-constructs a graphic organizer with students containing color-coded examples of circumference, area, diameter and radius.
• Teacher co-constructs a table to find the constant proportionality between the diameter and circumference of a circle, allowing students to discover the pattern that represents pi (π).
• Instruction includes modeling the differences between doubling and squaring a radius. Doubling a radius would be represented by multiplying the given length by 2, whereas squaring a number would be represented by the area of a square with the given radius.
  • 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 rewriting the area formula for a circle in expanded form before evaluating.
  • For example, the formula for the area of a circle, $A$ = π$r$², can be rewritten as $A$ = (π)($r$)($r$).
• Teacher helps students connect area of a circle to area of a rectangle.

Instructional Tasks

• Part A. What is the area of the large circle?
• Part B. What is the area of the shaded part of the figure?

Instructional Items

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