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Topic X: Area

COMMON CORE STATE STANDARD(S) &

MATHEMATICAL PRACTICE (MP)

NEXT GENERATION SUNSHINE STATE STANDARD(S)

ESSENTIAL CONTENT OBJECTIVES

MACC.912.G-C.2.5: 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. (MP.2, MP.3, MP.6, MP.7) MACC.912.G-GMD.1.1: Give an informal

argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.. (MP.3, MP.4, MP.5) MACC.912.G-GMD.2.4: Identify the shapes

of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (MP.4, MP.5, MP.7)

MACC.912.G-GPE.2.7: Use coordinates to

compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. (MP.1, MP.2, MP.5, MP.6)

Standard 2: Polygons MA.912.G.2.3: Use properties of congruent

and similar polygons to solve mathematical or real‐world problems. (WI, EOC) MA.912.G.2.5: Explain the derivation and

apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.). (EOC) MA.912.G.2.7: Determine how changes in

dimensions affect the perimeter and area of common geometric figures. Standard 4: Triangles MA.912.G.4.4: Use properties of congruent

and similar triangles to solve problems involving lengths and areas. Standard 6: Circles MA.912.G.6.5: Solve real‐world problems

using measures of circumference, arc length, and areas of circles and sectors. (EOC) Standard 7: Polyhedra and Other Solids MA.912.G.7.5: Explain and use formulas for

lateral area, surface area, and volume of solids. (EOC) MA.912.G.7.6: Identify and use properties

of congruent and similar solids. MA.912.G.7.7: Determine how changes in

dimensions affect the surface area and volume of common geometric solids. (EOC)

A. Area of Polygons 1. Rectangles 2. Parallelograms 3. Triangles 4. Trapezoids 5. Kites 6. Regular Polygons 7. Planar Cross Sections

B. Area of Circles 1. Circle 2. Sector 3. Segment 4. Annulus

C. Total Surface Area 1. Terminology 2. Prisms and Pyramids 3. Cylinders and Cones 4. Sphere and Hemisphere

D. Proportions with Area 1. Area Ratios 2. Surface Area Ratios

E. Applications in the Real World

NGSSS

Derive the formula for the area of a rectangle, parallelogram, triangle, trapezoid, kite, and regular polygons

Apply the area formulas to solve problems

Determine the area of non-regular polygons

Solve area application problems by applying various problem-solving strategies

Define annulus, segment, and sector

Derive the formula for the area of an annulus, segment, and sector of a circle

Calculate the area of an annulus, segment, and sector

Calculate the total surface area of a solid

Determine the effect on the area when one or two dimensions of the polygon are changed

Determine the relationship between areas of similar figures

Derive the formula for surface area of a sphere

Determine the area of the great circle of a sphere

Solve real-world problems involving area.

Key: FI - Fall Interim WI - Winter Interim EOC – Geometry EOC

Pacing Date(s) Traditional 14 days 02/28/14 – 03/19/14

Block 7 days 02/28/14 – 03/19/14

HONORSMIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 2 of 12 Topic X_Third Nine Weeks

INSTRUCTIONAL TOOLS

Core Text Book: Discovering Geometry (4th

Edition)

Benchmark Suggested Lessons Teacher Notes

MA.912.G.2.3 MA.912.G.2.5 MA.912.G.2.7 MA.912.G.4.4 MA.912.G.6.5 MA.912.G.7.5 MA.912.G.7.6 MA.912.G.7.7

8.1 – 8.7

10.7 and 11.5

Notes: Use Geometry EOC Item Specifications as a resource document to help define grade-level content.

Vocabulary: Area, base, height, apothem, sector of a circle, segment of a circle, annulus, surface area, lateral area.

Instructional Strategies: Patty Paper Investigations, Compass/Straightedge Investigations, Geometer’s Sketchpad, Problem Creation, Note Taking

Compute areas of sectors by first considering them as fractional parts of a circle. Then, using proportionality, derive a formula for their area in terms of radius and central angle.

Sectors and segments are often used interchangeably in everyday conversation. Care should be taken to distinguish these two geometric concepts.

Review alternative ways to derive the formula for the area of the circle A = πr2. For example: Cut a cardboard circular disk into 6 congruent sectors and rearrange

the pieces to form a shape that looks like a parallelogram with two scalloped edges. Repeat the process with 12 sectors and note how the edges of the parallelogram look “straighter.” Discuss what would happen in the case as the number of sectors becomes infinitely large. Then calculate the area of a parallelogram with base ½ C and altitude r to derive the formula A = πr

2.

Review vocabulary for names of solids (e.g., right prism, cylinder, cone, sphere, etc.).

Slice various solids to illustrate their cross sections. For example, cross sections of a cube can be triangles, quadrilaterals or hexagons. Rubber bands may also be stretched around a solid to show a cross section.

Graph polygons using coordinates. Explore perimeter and area of a variety of polygons, including convex, concave, and irregularly shaped polygons.

Given a triangle, use slopes to verify that the length and height are perpendicular. Find the area.

Find the area and perimeter of a real-world shape using a coordinate grid and Google Earth. Select a shape (yard, parking lot, school, etc.). Use the tool menu to overlay a coordinate grid. Use coordinates to find the perimeter and area of the shape selected. Determine the scale factor of the picture as related to the actual real-life view. Then find the actual perimeter and area.

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GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 3 of 12 Topic X_Third Nine Weeks

COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION

MACC.K12.MP.1 (back to top)

Make sense of problems and persevere in solving them.

Mathematically proficient students will be able to:

Explain the meaning of a problem and looking for entry points to its solution.

Analyze givens, constraints, relationships, and goals.

Make conjectures about the form and meaning of the solution and plan a solution pathway.

Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.

Monitor and evaluate their progress and change course if necessary.

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.

Check answers to problems using a different method, and continually ask, “Does this make sense?”

Identify correspondences between different approaches.

MACC.K12.MP.2 (back to top)

Reason abstractly and quantitatively.

Mathematically proficient students will be able to:

Make sense of quantities and their relationships in problem situations.

Decontextualize—to abstract a given situation and represent it symbolically.

Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols

Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them.

Know and be flexible using different properties of operations and objects.

MACC.K12.MP.3 (back to top)

Construct viable arguments and critique the reasoning of

others.

Mathematically proficient students will be able to:

Understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Analyze situations by breaking them into cases, and can recognize and use counterexamples.

Justify their conclusions, communicate them to others, and respond to the arguments of others.

Reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Determine domains to which an argument applies.

MACC.K12.MP.4 (back to top)

Model with mathematics.

Mathematically proficient students will be able to:

Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.

Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.

Analyze relationships mathematically to draw conclusions.

Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION

MACC.K12.MP.5 (back to top)

Use appropriate tools strategically.

Mathematically proficient students will be able to:

Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator.

Detect possible errors by strategically using estimation and other mathematical knowledge.

Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems.

Use technological tools to explore and deepen their understanding of concepts

MACC.K12.MP.6

(back to top)

Attend to precision.

Mathematically proficient students will be able to:

Communicate precisely to others.

Use clear definitions in discussion with others and in their own reasoning.

State the meaning of the symbols they choose, including using the equal sign consistently and appropriately.

Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

MACC.K12.MP.7

(back to top)

Look for and make use of structure.

Mathematically proficient students will be able to:

Discern a pattern or structure. Example: In the expression x2 + 9x + 14, students can see the 14 as 2 × 7 and the 9 as 2 + 7.

Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective.

See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. 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.

MACC.K12.MP.8 (back to top)

Look for and express regularity in repeated

reasoning.

Mathematically proficient students will be able to:

Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x-1)(x+1),(x-1)(x2+x+1),and(x-1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series.

Maintain oversight of the process, while attending to the details as they work to solve a problem.

Continually evaluate the reasonableness of their intermediate results.

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NEXT GENERATION SUNSHINE STATE STANDARDS GEOMETRY BODY OF KNOWLEDGE

Standard 2: Polygons

Identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. Find measures of angles, sides, perimeters, and areas of polygons, justifying the methods used. Apply transformations to polygons. Relate geometry to algebra by using coordinate geometry to determine transformations. Use algebraic reasoning to determine congruence, similarity, and symmetry. Create and verify tessellations of the plane using polygons.

BENCHMARK CODE BENCHMARK

MA.912.G.2.3 (back to top)

MC/FR

Use properties of congruent and similar polygons to solve mathematical or real‐world problems.

Remarks/Examples:

Example: Suppose a building is in the shape of a regular hexagon. The architect wants to put walkways as indicated. Show that the triangles formed are equal in size and shape. Cognitive Complexity/Depth of Knowledge Rating: High

Clarification (EOC): Students will use properties of congruent and/or similar polygons to solve problems. (Also assesses: MA.912.G.2.1, MA.912.G.4.1, MA.912.G.4.2, MA.912.G.4.4, and MA.912.G.4.5.)

Content Limit (EOC):

All angle measurements will be in degrees.

Items may require statements and/or justifications to complete formal and informal proofs.

MA.912.G.2.5 (back to top)

MC/FR

Explain the derivation and apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.).

Remarks/Examples:

Example 1: A rectangle of area 360 square yards is ten times as long as it is wide. Find its length and width.

Example 2: Explain the derivation of the formula for the area of a triangle.

Example 3: The design shown is called the Ohio Star. Assuming that it measures 9 inches by 9 inches, calculate the total area of all the orange patches, the total area of all the yellow patches, and the total area of all the green patches. How much fabric of each color will you need to cover an area that measures 72 inches by 90 inches? Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Students will solve problems by using and/or deriving formulas for perimeter and/or area of polygons. (Also assesses MA.912.G.2. 7)

Content Limit (EOC):

Items requiring students to calculate area may require the use of the apothem.

Composite figures may include circles.

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Standard 4: Triangles

Identify and describe various kinds of triangles (right, acute, scalene, isosceles, etc.). Define and construct altitudes, medians, and bisectors, and triangles congruent to given triangles. Prove that triangles are congruent or similar and use properties of these triangles to solve problems involving lengths and areas. Relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. Understand and apply the inequality theorems of triangles.

BENCHMARK CODE BENCHMARK

MA.912.G.4.4 (back to top)

Use properties of congruent and similar triangles to solve problems involving lengths and areas.

Remarks/Examples:

Example: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20 cm2. What is the area of the second

triangle?

Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Assessed with MA.912.G.2.3

Content Limit (EOC): Assessed with MA.912.G.2.3

Standard 2: Polygons

Identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. Find measures of angles, sides, perimeters, and areas of polygons, justifying the methods used. Apply transformations to polygons. Relate geometry to algebra by using coordinate geometry to determine transformations. Use algebraic reasoning to determine congruence, similarity, and symmetry. Create and verify tessellations of the plane using polygons.

MA.912.G.2.7 (back to top)

Determine how changes in dimensions affect the perimeter and area of common geometric figures.

Remarks/Examples:

Example: If the lengths of each side of a trapezoid are tripled, determine the change in its area, and justify your answer. Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Assessed with MA.912.G.2.5 and MA.912.G.7.7

Content Limit (EOC): Assessed with MA.912.G.2.5 and MA.912.G.7.7.

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Standard 6: Circles

Define and understand ideas related to circles (radius, tangent, chord, etc.). Perform constructions, and prove theorems related to circles. Find measures of arcs and angles related to them, as well as measures of circumference and area. Relate geometry to algebra by finding the equation of a circle in the coordinate plane.

MA.912.G.6.5 (back to top)

MC/FR

Solve real‐world problems using measures of circumference, arc length, and areas of circles and sectors.

Remarks/Examples:

Example: Which will give you more: three 6‐inch pizzas or two 8‐inch pizzas? Explain your answer.

Cognitive Complexity/Depth of Knowledge Rating: High

Clarification (EOC): Students will solve problems related to circles. (Also assesses: MA.912.G.6.2 and MA.912.G.6.4)

Content Limit (EOC):

All angle measurements will be in degrees.

Items may require statements and/or justifications to complete formal and informal proofs.

Standard 7: Polyhedra and Other Solids

Describe and make regular and nonregular polyhedra (cube, pyramid, tetrahedron, octahedron, etc.). Explore relationships among the faces, edges, and vertices of polyhedra. Describe sets of points on spheres, using terms such as great circle. Describe symmetries of solids, and understand the properties of congruent and similar solids.

BENCHMARK CODE BENCHMARK

MA.912.G.7.5 (back to top)

MC/FR

Explain and use formulas for lateral area, surface area, and volume of solids.

Remarks/Examples:

Example 1: A gold class ring is dropped into a glass that is a right cylinder with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ring?

Example 2: Given the composite solid consisting of a hemisphere and a cone, calculate the surface area and the volume.

Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Students will explain and/or apply formulas to determine surface area, lateral area, and volume of solids. (Also assesses: MA.912.G.7.4 and MA.912.G.7.6)

Content Limit (EOC):

Solids will be limited to right prisms, right‐circular cylinders, spheres, right pyramids, right‐circular cones, and/or composites of these solids.

Items may not include oblique figures.

Items may ask students to apply knowledge of congruent and similar solids.

Standard 7: Polyhedra and Other Solids

Describe and make regular and nonregular polyhedra (cube, pyramid, tetrahedron, octahedron, etc.). Explore relationships among the faces, edges, and vertices of polyhedra. Describe sets of points on spheres, using terms such as great circle. Describe symmetries of solids, and understand the properties of congruent and similar solids.

HONORSMIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 8 of 12 Topic X_Third Nine Weeks

BENCHMARK CODE BENCHMARK

MA.912.G.7.6 (back to top)

Identify and use properties of congruent and similar solids.

Remarks/Examples:

Example: Explain how the surface area and volume of similar cylinders are related. Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Assessed with MA.912.G.7.5

Content Limit (EOC): Assessed with MA.912.G.7.5

MA.912.G.7.7 (back to top)

MC/FR

Determine how changes in dimensions affect the surface area and volume of common geometric solids.

Remarks/Examples:

Example: Explain how changing the radius or height of a cylinder affects its surface area and volume. Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Students will determine how changes in parameter(s) affect perimeter, area, surface area, or volume, or vice‐versa. Students will

determine how changes to one parameter will change other parameters when the perimeter, area, surface area, or volume is held constant. (Also assesses: MA.912.G.2.7)

Content Limit (EOC):

One or two parameters may be changed, resulting in the change of another parameter.

Three parameters may be changed in one item only if all three are changed by a constant factor.

Solids will be limited to right prisms, right circular cylinders, spheres, right pyramids, right circular cones, and/or composites of these solids.

Items may not include oblique figures.

Items may involve, explicitly and/or implicitly, no more than four parameters.

Changes in dimension may or may not result in similar figures.

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TECHNOLOGY TOOLS

FLORIDA FOCUS http://focus.florida-achieves.com

Sign-in and Password: First 4 letters of Last Name and Last 4 Digits Employee Number. Example: abcd1234

MA.912.G.2.3

MA.912.G.2.5

MA.912.G.6.5

MA.912.G.7.5

MA.912.G.7.7

GIZMO CORRELATION

GIZMO TITLE

Area of Parallelograms - Activity A

Area of Parallelograms - Activity B

Perimeter, Circumference, and Area - Activity B

Circle: Circumference and Area

Perimeter, Circumference, and Area - Activity A

Rectangle: Perimeter and Area

Surface and Lateral Area of Prisms and Cylinders

Surface and Lateral Area of Pyramids and Cones

GEOMETER’S SKETCHPAD ACTIVITIES

UNIT/ TITLE

Area of a Rectangle

Area of a Parallelogram

Areas of Regular Polygons and Circles

Maximum Volume & Area

TOPIC X

VIDEO TITLE

Science of NHL Hockey: Hockey Geometry

Science of NFL Football: Vectors

Science of NHL Hockey: Vectors

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TOPIC X

Micro Apartments Become Large Trend

Hurricane Charley Makes Landfall in Florida

Beetles Threaten British Columbia's Evergreen Trees

Arkansas: Search On for Those Swept Away by "Wall of Water"

Blown Out Oil Well Faces Final Pressure Test

IMAGE

TOPIC X DISCOVERY EDUCATION CORRELATION

VIDEO TITLE

The Area of Parallelograms & Triangles

Reviewing Angles, Arcs, & Sectors

Review: Area

Area and Circles Quick Quiz

MODEL LESSON

Circles: Understanding Structure: Session 2: The Large Hadron Collider: Radian Measure

Circles: Understanding Structure: Session 3: Sector Area of a Pool Cover

Geometry: Three-Dimensional Shapes: Session 1: Digging Deep

Geometry: Three-Dimensional Shapes: Session 3: Stuck In The Middle

Coordinate Geometry and How It's Used: Session 3: Classifying Polygons

MATH EXPLANATION TITLE

Geometry: Area of Parallelograms: Area of Parallelograms

Geometry: Area of Regular Polygons and Circles: Comparing Shapes

Geometry: Area of Triangles, Rhombi, and Trapezoids: Area of a Triangle

Geometry: Area of Triangles, Rhombi, and Trapezoids: Lengths, Perimeter, and Area of Triangles

Geometry: Area of Triangles, Rhombi, and Trapezoids: Area of Trapezoid

Geometry: Area of Triangles, Rhombi, and Trapezoids: Area of a Rhombus

Geometry: Area of Regular Polygons and Circles: Formula for Area

Geometry: Area of Regular Polygons and Circles: Price Per Area

Geometry: Area of Regular Polygons and Circles: Comparing Shapes

Geometry: Area of Regular Polygons and Circles: Apothem, Area, and Perimeter

Geometry: Area of Regular Polygons and Circles: Inscribed In and Circumscribed About

Geometry: Area of Regular Polygons and Circles: Finding Perimeter and Area

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TOPIC X DISCOVERY EDUCATION CORRELATION

Geometry: Area of Regular Polygons and Circles: Graphing and Finding Area

Geometry: Area of Regular Polygons and Circles: Circumference and Area of Circles

MATH OVERVIEW

Geometry: Area of Parallelograms

Geometry: Area of Regular Polygons and Circles

Geometry: Angles and Arcs

Geometry: Area of Regular Polygons and Circles

Geometry: Area of Regular Polygons and Circles

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Date Pacing Guide

Standards Data Driven Standard(s)

Activities Assessment(s) Strategies

Traditional 14 days

Block 7 days

02/28/14 – 03/19/14

(back to top)

Standard 2: Polygons

MA.912.G.2.3

MA.912.G.2.5

MA.912.G.2.7

Standard 4: Triangles

MA.912.G.4.4 Standard 6: Circles

MA.912.G.6.5 Standard 7: Polyhedra and

Other Solids MA.912.G.7.5

MA.912.G.7.6

MA.912.G.7.7