### Clarifications

*Clarification 1*: Instruction focuses on the conceptual understanding that for a given angle measure the length of the intercepted arc is proportional to the radius, and for a given radius the length of the intercepted arc is proportional is the angle measure.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Circle
- Radius

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students solved problems involving the circumference and area of a circle. In Algebra 1, students rearranged formulas to highlight a quantity of interest and solved linear equations in one variable. In Geometry, students use their knowledge of circumference and area to relate arc measure to arc length and to area of sectors of circles. In later courses, students will determine the value of trigonometric functions for real numbers by identifying angle measures in the unit circle, and will convert between radians and degrees.- Instruction includes the student understanding that arcs can be measured in both degrees (arc measure) and in units of length (arc length). For expectations of Geometry, students will only need to work in degrees when discussing arc measure as students will work with radians in later courses.
- Instruction includes the understanding that two or more circles with a common center,
called concentric circles, will have the same arc measure but different corresponding arc
lengths.
- For example, given various concentric circles, students can draw a central angle and extend its side to the length of the radius of the largest circle. Students should notice that the measure of the central angle remains the same and that the larger the circle; the longer the intercepted arc; and that the length of the intercept arc depends proportionally on the radius in the same way that the circumference depends proportionally on the radius.

- When determining an arc length or an area of a sector given the arc measure, instruction
includes the connection to proportional relationships (as was done in grade 7).
- For example, if the arc measure is 57°, and students are asked to find the area of the sector, they can determine the area of the entire circle and multiply by $\frac{\text{57}}{\text{360}}$. Students should realize that areas of sectors are fractional portions of the area of the entire circle.
- For example, if the arc measure is 57°, and students are asked to find the arc length, they can determine the circumference of the entire circle and multiply by $\frac{\text{57}}{\text{360}}$. Students should realize that arc lengths are fractional portions of the circumference of the entire circle

### Common Misconceptions or Errors

- Students may confuse arc measure and arc length, and may try to measure arcs with linear units rather than degrees.

### Instructional Tasks

*Instructional Task 1 (MTR.7.1)*

- De’Veon must create an animal using geometric shapes for his Geometry class. He has
decided to use construction paper scraps from his mom’s crafting box to create a bird, like
the one shown below. The head is made from a sector with radius 1.5 centimeters and
central angle measuring 130°. The body is a semicircle with radius 1.9 centimeters.
- Part A. What fraction of the whole circle is the head?
- Part B. How much glitter string will he need to outline the part of the bird’s head that is not touching the beak or neck?
- Part C. What is the total area of light blue construction paper used to create the bird (i.e., the area of the head and the body)?

### Instructional Items

*Instructional Item 1*

- The North Rose Window in the Rouen Cathedral in France has a diameter of 23 feet. The
stained glass design is equally spaced about the center of the circle. What is the area of the
sector bounded by arc
*GJ*?

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

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Problem-Solving Tasks

## Text Resource

## MFAS Formative Assessments

Students are asked to explain why the length of an arc intercepted by an angle is proportional to the radius and then explain how that proportionality leads to a definition of the radian measure of an angle.

Students are asked to write a formula to find the area of a sector of a circle and then explain and justify that formula.

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

## Student Resources

## Problem-Solving Tasks

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Type: Problem-Solving Task