## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic. This lesson is intended to help teachers identify and assist students who have difficulties in: Computing perimeters, areas, and arc lengths of sectors using formulas and finding the relationships between arc lengths, and areas of sectors after scaling.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

## Standards

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Construct a tangent line from a point outside a given circle to the circle.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Prove that all circles are similar.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

Cluster: Find arc lengths and areas of sectors of circles

Standard: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Find arc lengths and areas of sectors of circles

Indicator: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Construct a tangent line from a point outside a given circle to the circle.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Prove that all circles are similar.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

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# Tags (12)

- Mathematics
- Geometry and measures
- Area
- CCSS
- Circles
- Common Core Math
- Common Core PD
- Geometry
- ODE Learning
- Perimeters
- Radians
- Sectors

## Comments