Developmental Math – An Open Curriculum Instructor Guide

Developmental Math – An Open Curriculum Instructor Guide

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Unit 7: Geometry

Unit Table of Contents Lesson 1: Basic Geometric Concepts and Figures Topic 1: Figures in 1 and 2 Dimensions Learning Objectives

• Identify and define points, lines, line segments, rays and planes. • Classify angles as acute, right, obtuse, or straight. Topic 2: Properties of Angles Learning Objectives • Identify parallel and perpendicular lines. • Find measures of angles. • Identify complementary and supplementary angles.

Topic 3: Triangles Learning Objectives

• Identify equilateral, isosceles, scalene, acute, right, and obtuse triangles. • Identify whether triangles are similar, congruent, or neither. • Identify corresponding sides of congruent and similar triangles. • Find the missing measurements in a pair of similar triangles. • Solve application problems involving similar triangles. Topic 4: The Pythagorean Theorem Learning Objectives • Use the Pythagorean Theorem to find the unknown side of a right triangle. • Solve application problems involving the Pythagorean Theorem.

Lesson 2: Perimeter, Circumference, and Area

Topic 1: Quadrilaterals Learning Objectives

• Identify properties, including angle measurements, of quadrilaterals.

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Unit 7 – Table of Contents and Learning Objectives

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Topic 2: Perimeter and Area Learning Objectives • Find the perimeter of a polygon. • Find the area of a polygon. • Find the area and perimeter of non-standard polygons. Topic 3: Circles

Learning Objectives • Identify properties of circles. • Find the circumference of a circle. • Find the area of a circle. • Find the area and perimeter of composite geometric figures.

Lesson 3: Volume of Geometric Solids

Topic 1: Solids Learning Objectives

• Identify geometric solids. • Find the volume of geometric solids. • Find the volume of a composite geometric solid.


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Unit 7: Geometry

Instructor Notes The Mathematics of Geometry

Geometry is a familiar part of our everyday lives. This unit formalizes the subject with a focus on the definitions of geometric terms and on the special properties of angles and geometric figures. Students will learn to classify angles, triangles, and quadrilaterals as well as how to find the perimeter, area, and volume of shapes and solids. They'll also practice applying the Pythagorean Theorem to solve real world problems. Teaching Tips: Challenges and Approaches

Most students have been studying geometry since they were introduced to the basic shapes of square, rectangle, and circle in kindergarten. The concepts won't trouble them, but some of the calculations and formulas may be a bit tricky, and there are a lot of definitions to learn. Encourage students to make flashcards with definitions, pictures, and equations if needed. Angles One common difficulty is measuring angles correctly. Practice is the best way for students to become proficient with a protractor.

Unit 7 – Instructor Notes


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[From Lesson 1, Topic 1, Topic Text] Students may not be familiar with the terms complementary and supplementary and stumble over which angles have measures that add to 90 degrees and which to 180 degrees. One way to help your students keep this straight is to tell them that complementary comes first alphabetically, and 90 is before 180 numerically. Put the 'firsts' together, and complementary angles add to 90 degrees, while supplementary angles make 180 degrees. Triangles Congruent and similar triangles may also be new ideas to students. Congruent triangles are easy to understand because corresponding sides and angles must be equal in measure. This means that the triangles are exact copies of each other. Explain the use of hash marks to show sides (and also angles) that are congruent, as in:

[From Lesson 1, Topic 3, Topic Text] Similar triangles will be a little more difficult for students to grasp. Explain to them that these triangles have the same shape but one is either stretched or shrunk from the other. Use diagrams to show that the corresponding angles are still equal in measure but the corresponding sides are not. Practice will help illustrate this concept, so give students a number of examples where the measure of one side of a pair of similar triangles needs to be found, as in this example:


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[From Lesson 1, Topic 3, Worked Example 3] Use problems like this to help students discover that the ratio of the two pairs of known corresponding sides is the same. They then will know that the ratio of the third pair of corresponding sides will also have to be the same. Be aware that because the triangle pictured above is a right triangle, some students want to use the Pythagorean Theorem to get the length of the missing side. (This is a good lead-in to the next topic in the lesson which just happens to be the Pythagorean Theorem.) It would then be appropriate to give another pair of similar non-right triangles with a missing side and have the class figure out why the Pythagorean Theorem should not be used. There are many application problems that can be solved using the concepts of similar triangles and the Pythagorean Theorem. For example, the problem below calculates the hypotenuse of a triangle in order to build a ramp:


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[From Lesson 1, Topic 4, Presentation] After solving a few examples, have students come up with examples that use the Pythagorean Theorem and that are pertinent to their own lives. Quadrilaterals It is important for your students to understand the relationships between the different types of quadrilaterals. Using the following diagram, have your students answer true/false questions like “All squares are rectangles” and “All rectangles are squares."


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[From Lesson 2, Topic 1, Topic Text]

This diagram does not include a rhombus or trapezoid. After discussing the properties of each, have your students determine where these shapes would be placed in the diagram. Perimeter and Area Students usually do not have problems understanding the concept of perimeter. The most frequent mistake that is made is that if a rectangle is pictured with only two sides labeled with their lengths (or if an application problem only gives the length and width), they will forget to include the lengths of the two unlabeled sides in the perimeter. Area is generally more difficult to understand and find because there are different formulas for different polygons. Students need to know these formulas as well as how to use them. Finding the area of triangles and trapezoids is more difficult for students because they need to figure out the height. The best way for students to become comfortable with all the formulas for area is with practice. Once students are comfortable finding perimeters and areas of polygons, composite geometric figures should be introduced. Carefully explain that composite figures are two or more “simple” figures combined to produce an interesting shape. The example that follows has a rectangle on top of a trapezoid:



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Have students practice how to view composite figures so that perimeters and areas can be easily calculated. It should be noted that composite figures can also include circles and semi-circles. Circles Students will have studied circles before. After reviewing the definition of diameter and radius, have students practice finding the circumference and area of various circles. You should stress the difference between exact answers and approximations, for example:


Many students think that equals 3.14. This would be a good time to review that is an

irrational number and that 3.14 and are just good approximations for .

Volume Students often confuse volume and area. Point out that area is two-dimensional and volume is three-dimensional. When describing a rectangle prism, explain that it is many rectangles stacked and this is what gives it its third dimension. There are a number of formulas to calculate the volume for the different solids. Be sure to illustrate these with many examples and have students practice identifying which formula needs to be used. Once students understand how to calculate the volume of the different solids, show them a composite solid and calculate its volume. Keep in Mind

This geometry unit can actually be taught at any time in the course as it is not algebra based. For example, minimal algebra is needed to solve the following problem:


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[From Lesson 1, Topic 2, Topic Text] However, once your students have mastered solving algebraic equations, these geometry problems can be made more challenging. A more challenging problem would be to say that two angles are supplementary and one angle measures four times that of the other angle. Additional Resources In all mathematics, the best way to really learn new skills and ideas is repetition. Problem solving is woven into every aspect of this course—each topic includes warm-up, practice, and review problems for students to solve on their own. The presentations, worked examples, and topic texts demonstrate how to tackle even more problems. But practice makes perfect, and some students will benefit from additional work. The site http://www.mathopenref.com/common/indexpage.html has an index of geometry terms. Click on the items to find information and applets to illustrate concepts. Use the http://www.amblesideprimary.com/ambleweb/mentalmaths/protractor.html applet for more practice measuring an angle with a protractor. Practice finding the measures of complementary and supplementary angles at http://www.mathvillage.info/node/41 and http://www.mathvillage.info/node/43. http://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_congruent.htm allows students to work with similar and congruent triangles. The Pythagorean Theorem can be practiced at http://www.shodor.org/interactivate/activities/PythagoreanExplorer/. http://www.mathvillage.info/node/134 has a good review of perimeter and area of different geometric figures. Click on each link for more practice.


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Find help with circumference and area of a circle at http://www.mathvillage.info/node/21 and http://www.mathvillage.info/node/56 respectively. Review the volumes of geometric solids at http://www.mathvillage.info/node/111 and http://www.mathvillage.info/node/112. Summary

After completing this unit students will be familiar with many different geometric terms including those relating to lines, angles, triangles, and quadrilaterals. They will understand how to solve application problems using the Pythagorean Theorem, and will be able to choose an appropriate formula to calculate the perimeter, circumference, area, and volume of various forms.


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Unit 7: Geometry

Instructor Overview Tutor Simulation: Building a Slide

Purpose

This simulation allows students to demonstrate their ability to work with geometry in a real world problem. Students will be asked to apply what they have learned to solve a problem involving:

• Angles • Triangles • Quadrilaterals • Congruency and Similarity • Pythagorean Theorem

Problem Students are presented with the following problem: You will be helping your local city park design a new slide for the playground. The job has been started, but they need your help to figure out some of the angles and measurements so construction can begin. Objectives

Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation:

• Make sure they have completed all other unit material. • Explain the mechanics of tutor simulations:

o Students will be given a problem and then guided through its solution by a video tutor;

o After each answer is chosen, students should wait for tutor feedback before continuing;

o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor.

• Emphasize that this is an exploration, not an exam.

Unit 7 – Tutor Simulation


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Unit 7: Geometry

Instructor Overview Puzzle: Pythagoras' Proof

Objectives

Pythagoras' Proof is a puzzle designed to show students the reasoning behind the Pythagorean Theorem of a2 + b2 = c2. According to the theorem, the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse. A simple and elegant proof of this is to make each side of the triangle one side of a square. If the theorem is correct, the combined areas of the two smallest squares will be the same as the area of the largest square.

In this puzzle, students will see that this proof holds, and the theorem is indeed true.

Figure 1. Pythagoras' Proof reinforces students' grasp of the Pythagorean Theorem by turning

a2 and b2 into shapes that they combine to form c2.

Unit 7 – Puzzle


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Description

This puzzle is a Tetris-style game that asks players to combine two irregular shapes to form a square. Each pair consists of a yellow shape made up of nine small squares and a blue shape formed from 16 squares. The yellow shape represents the a2 in the Pythagorean Theorem and the blue shape the b2. When these two shapes are lined up correctly, they form a green square composed of 25 small squares—this square represents the c2. After sufficient play, students will see that no matter how the squares are oriented relative to one another, a2 and b2 always combine to equal c2.

To play the game, players rotate dropping polyominoes so that they fit together. If the arrangement creates a square, the polyominoes turn green. If it doesn't, the pieces turn red. Players earn points for every square they put together and strikes for every mistake. Game play continues until three strikes occur.

Pythagoras' Proof is suitable for individual play, but it could also be used in a classroom situation to introduce the simple but profound theorem that it illustrates.


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Unit 7: Geometry

Instructor Overview Project: Geometric Designs

Student Instructions

Introduction #

You are working for a stained-glass company and need to analyze and design geometric patterns. You will use your ability to work with geometric shapes to calculate dimensions of shapes including angle measurements, side measurements, and area. In addition, you will need to be creative in order to make your own stained-glass design. #

Task

In this project you will play the part of an artist working for a stained-glass company. When considering different glass designs there are several factors to consider: 1) area of shapes, 2) dimensions of shapes including angle and side measurements, and 3) aesthetic appeal (color, design, shapes utilized.). Working together with your group, you will analyze several designs and finally design your own stained glass window. #

Instructions#

Solve each problem in order and save your work along the way, as you will create a professional report at the conclusion of the project. #

• First problem: Area of Shapes

Most stained-glass windows are a combination of many shapes. The artists come up with sketches and designs. Then the glassmaker needs to determine the side lengths of each individual shape and total area. Look carefully at each glass below and determine the shapes, their side lengths, and area.

Stained-Glass #1

Unit 7 – Project


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Name of Shape Sketch of shape with

side lengths in centimeters

Area of shape

in square centimeters

Total area in square centimeters


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Stained-Glass #2



Area of shape



• Second Problem: Area of Shaded Regions

Along with combining various shapes, stained-glass makers use a variety of different colors to enhance the beauty of their designs. Look carefully at each glass below and determine the areas of each different color in the stained-glass design.


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Stained-Glass #3 Rectangles

Color Area of shape in square centimeters

Orange

Red

Yellow


Stained-Glass #4 Trapezoid


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Aqua-Blue

Purple


• Third Problem: Dimensions and Area of Pattern Blocks

In order to make intricate designs, it is nice to have shapes with known area and dimensions (side lengths and angles). In this problem, you will be determining the area (to the nearest hundredth) and dimensions of each pattern block. The area of the gray rhombus is given. Each side of the square, gray rhombus and equilateral triangle has a length of 2.5 cm. These pattern blocks will be used in the fourth problem to design your own stained glass window.

Pattern Block Number of Equilateral Triangles in Shape

Sketch with dimensions of sides

and angles

Area of Shape

(to the nearest hundredth)


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Equilateral Triangle

1

Blue Rhombus

2

Isosceles Trapezoid

3

Regular Hexagon

6

Square

Does not apply

Gray Rhombus

Does not apply Area = 3.125 square cm


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• Fourth Problem: Make your own Stained-Glass Window

Using the pattern blocks from the template, you need to design your own stained-glass window. You should try to make it appealing in design and color. After you have your design, frame it with a square so it could be used in a window pattern. It does not have to be a “fitted” square meaning that it could be larger than the shape without touching each of the sides of the design. You should color the space between the inside design and the square. Measure the length of the square and determine the angle measurements of the glass needed to fill the square. Make a sketch of the entire design. Fill in the chart below for each shape and determine the total area of the inside shape and the area of the fill between the inside shape and the square.

Pattern Block Number in Design Area of Shape

Total Area


Blue Rhombus

Isosceles Trapezoid

Regular Hexagon

Square


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Gray Rhombus

Area = 3.125 square cm

Total Area of Pattern Blocks (in square cm)

Area Square = ______________ square cm

Area of Fill Needed (Between Inside Shape and Square) = ________________ square cm

Collaboration

Get together with another group to compare your answers to each of the four problems. Discuss how you might analyze the shapes differently or combine your answers to make a more complete and convincing analysis of the situation.

Conclusions

Present your solution in a way that makes it easy for someone beginning geometry to be able to understand your results. Be sure to clearly explain your reasoning at each stage and conclude with recommendations about combining shapes, areas of different shaded shapes, and the design and making of a stained-glass window. You should look at different designs of windows and stained-glass windows to see how your design compares. Explain how your results can transfer to these different situations by scaling your window design.

Instructor Notes

Assignment Procedures This project contains several different types of problems in order to give students practice in working with geometric shapes including calculating dimensions and aesthetic appeal. The first two problems are completely separate from the remaining two problems and are not necessary to complete each other. The third and fourth problems do not depend on the first two problems, but they are connected to each other. Therefore, this project can be easily tailored by assigning only those problems corresponding to those skills you would like to reinforce from the section of material. Problem 1


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All solutions should be the same for each student.



Area of shape


Rectangle

Area = base × height

Area = 8×5

40 square centimeters

Triangle

Area = ½ base × height

Height was found by:

Height of entire shape – height of rectangle

(9.5 – 5) = 4.5 cm

Area =

18 square centimeters

Total Area in square centimeters 58 square centimeters


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Name of Shape

Sketch of shape with


Area of shape


Semi-circle

(Half of a circle)

Area = ½ π r2

square centimeters

Triangle


square centimeters


square centimeters

Problem 2 All of the students’ answers should be the same.


Orange Area = base × height

Area = 4 ×5

Area = 20 square centimeters


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Red Area = base × height

Area = 4 ×1.5


Yellow

Area of Yellow Shape = Total Area – Area of Orange – Area of Red

Area of Yellow = 136 – 20 – 6

Area of Yellow = 110 square centimeters


Area = base × height

Area = 17×8


Each student’s method to obtain the answer may vary. However, all of the students’ answers should be the same.


Aqua-Blue


Area = ½ (13.4) ×4

Area = 26.8 square centimeters

Purple

Area of Purple Shape = Total Area – Area of Aqua Blue

Area of Purple = 62.8 - 26.8

Area of Purple = 36 square centimeters


Area of Trapezoid = ½ (base1 +base2)×height

Area = ½(18 + 13.4) × 4

Area = 62.8 square centimeters

Problem 3 You may want students to cut out the pattern blocks from the template to see how they relate to each other.


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Each student’s method to obtain the answer may vary. However, all of the students’ answers should be the same. All of the shapes (except for longer base on trapezoid, which is 5 cm) measure 2.5 cm in length.

Pattern Block Sketch with dimensions of sides and angles

Area of Shape

(to the nearest hundredth)



square cm

Blue Rhombus

Since there are two equilateral triangles, the area is found by:

square cm

Isosceles Trapezoid

Since there are three equilateral triangles, the area is found by:

square cm

Regular Hexagon

Since there are six equilateral triangles, the area is found by:

square cm


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Square

Area = base × height or

Area = side2

Area = (2.5)(2.5)


Gray Rhombus


Problem 4 Two different examples are provided to give students an idea of what is needed in the stained glass design. Encourage students to be creative in designing their window.

Example #1: Hexagon Window


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Angle measurements for fill: Each of the trapezoids meet at a 120º angle and the external angle of each corner of the trapezoid is 240º. This example shows that the design does not have to be a “fitted” square touching the design on all four sides.

Example #1: Hexagon Window


Total Area


6 square cm

square cm

Blue Rhombus

6 square cm

square cm

Isosceles Trapezoid

6 square cm

square cm

Regular Hexagon

1 square cm

square cm

Square

0 Area = 6.25 square cm

0


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Gray Rhombus


0

Total Area of Pattern Blocks (in square cm) 113.82

Area of Square = (20)(20) = 400 square cm

Area of Fill Needed (Between Inside Shape and Square) = 400 - 113.82 = 286.18 square cm

Example #2: Snowflake Window

Angle measurements for fill: Each of the four corners has the same angle measurements as do the small isosceles triangles in the middle of each of the sides.

Example #2: Snowflake Window


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Total Area


0 square cm

0

Blue Rhombus

12 square cm

square cm

Isosceles Trapezoid

0 square cm

0

Regular Hexagon

0 square cm

0

Square


Area= 12(6.25)

Area = 75 square cm

Gray Rhombus


Area= 12 (3.125)



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Total Area of Pattern Blocks (in square cm) 177.54

Area of Square = (16.4)(16.4) = 268.96 square cm

Area of Fill Needed (Between Inside Shape and Square) = 268.96-177.54 = 91.42 square cm

Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students’ instructions list several websites that provide information on numbering systems, game design, and graphics.

The following are other examples of free Internet resources that can be used to support this project:

http://www.moodle.org

An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students.

http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview

Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work with parents.

http://www.docs.google.com

Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects.

http://why.openoffice.org/

The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose.


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Rubric

Score Content Presentation/Communication

4

• The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution.

• The solution completely addresses all mathematical components presented in the task.

• The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results.

• Mathematically relevant observations and/or connections are made.

• There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.

• Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.

• There is precise and appropriate use of mathematical terminology and notation.

• Your project is professional looking with graphics and effective use of color.

3

• The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution.

• The solution addresses all of the mathematical components presented in the task.

• The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem.

• Most parts of the project are correct with only minor mathematical errors.

• There is a clear explanation. • There is appropriate use of accurate

mathematical representation. • There is effective use of

mathematical terminology and notation.

• Your project is neat with graphics and effective use of color.

2

• The solution is not complete indicating that parts of the problem are not understood.

• The solution addresses some, but not all of the mathematical components presented in the task.

• The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning.

• Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures.

• Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.

• There is some use of appropriate mathematical representation.

• There is some use of mathematical terminology and notation appropriate for the problem.

• Your project contains low quality graphics and colors that do not add interest to the project.

1 • There is no solution, or the solution has no

relationship to the task. • No evidence of a strategy, procedure, or

mathematical reasoning and/or uses a

• There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem.


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strategy that does not help solve the problem.

• The solution addresses none of the mathematical components presented in the task.

• There were so many errors in mathematical procedures that the problem could not be solved.

• There is no use or inappropriate use of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.).

• There is no use, or mostly inappropriate use, of mathematical terminology and notation.

• Your project is missing graphics and uses little to no color.


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Unit 7: Geometry

Common Core Standards Unit#7,#Lesson#1,#Topic#1:##Figures#in#1#and#2#Dimensions#

Grade:#9C12#C#Adopted#2010#STRAND'/'DOMAIN' CC.G.' Geometry#

CATEGORY'/'CLUSTER' G3CO.' Congruence#

STANDARD' '' Experiment#with#transformations#in#the#plane#

EXPECTATION' G3CO.1.' Know#precise#definitions#of#angle,#circle,#perpendicular#line,#parallel#line,#and#line#segment,#based#on#the#undefined#notions#of#point,#line,#distance#along#a#line,#and#distance#around#a#circular#arc.#

##Unit#7,#Lesson#1,#Topic#2:##Properties#of#Angles#





Unit#7,#Lesson#1,#Topic#3:##Triangles#

Grade:#8#C#Adopted#2010#STRAND'/'DOMAIN' CC.8.G.' Geometry#

CATEGORY'/'CLUSTER' '' Understand#congruence#and#similarity#using#physical#models,#transparencies,#or#geometry#software.#

STANDARD' 8.G.1.' Verify#experimentally#the#properties#of#rotations,#reflections,#and#translations:#

EXPECTATION' 8.G.1(a)' Lines#are#taken#to#lines,#and#line#segments#to#line#segments#of#the#same#length.#

EXPECTATION' 8.G.1(b)' Angles#are#taken#to#angles#of#the#same#measure.#

EXPECTATION' 8.G.1(c)' Parallel#lines#are#taken#to#parallel#lines.#

STRAND'/'DOMAIN' CC.8.G.' Geometry#

Unit 7 – Correlation to Common Core Standards

Learning Objectives


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CATEGORY'/'CLUSTER' '' Understand#congruence#and#similarity#using#physical#models,#transparencies,#or#geometry#software.#

STANDARD' 8.G.2.' Understand#that#a#twoCdimensional#figure#is#congruent#to#another#if#the#second#can#be#obtained#from#the#first#by#a#sequence#of#rotations,#reflections,#and#translations;#given#two#congruent#figures,#describe#a#sequence#that#exhibits#the#congruence#between#them.#

STANDARD' 8.G.4.' Understand#that#a#twoCdimensional#figure#is#similar#to#another#if#the#second#can#be#obtained#from#the#first#by#a#sequence#of#rotations,#reflections,#translations,#and#dilations;#given#two#similar#twoCdimensional#figures,#describe#a#sequence#that#exhibits#the#similarity#between#them.#

STANDARD' 8.G.5.' Use#informal#arguments#to#establish#facts#about#the#angle#sum#and#exterior#angle#of#triangles,#about#the#angles#created#when#parallel#lines#are#cut#by#a#transversal,#and#the#angleCangle#criterion#for#similarity#of#triangles.#For#example,#arrange#three#copies#of#the#same#triangle#so#that#the#sum#of#the#three#angles#appears#to#form#a#line,#and#give#an#argument#in#terms#of#transversals#why#this#is#so.#





STRAND'/'DOMAIN' CC.G.' Geometry#


STANDARD' '' Understand#congruence#in#terms#of#rigid#motions#

EXPECTATION' G3CO.7.' Use#the#definition#of#congruence#in#terms#of#rigid#motions#to#show#that#two#triangles#are#congruent#if#and#only#if#corresponding#pairs#of#sides#and#corresponding#pairs#of#angles#are#congruent.#



STANDARD' '' Prove#geometric#theorems#

EXPECTATION' G3CO.10.' Prove#theorems#about#triangles.#Theorems#include:#measures#of#interior#angles#of#a#triangle#sum#to#180#degrees;#base#angles#of#isosceles#triangles#are#congruent;#the#segment#joining#midpoints#of#two#sides#of#a#triangle#is#parallel#to#the#third#side#and#half#the#length;#the#medians#of#a#triangle#meet#at#a#point.#


CATEGORY'/'CLUSTER' G3SRT.' Similarity,#Right#Triangles,#and#Trigonometry#


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STANDARD' '' Understand#similarity#in#terms#of#similarity#transformations#

EXPECTATION' G3SRT.2.' Given#two#figures,#use#the#definition#of#similarity#in#terms#of#similarity#transformations#to#decide#if#they#are#similar;#explain#using#similarity#transformations#the#meaning#of#similarity#for#triangles#as#the#equality#of#all#corresponding#pairs#of#angles#and#the#proportionality#of#all#corresponding#pairs#of#sides.#



STANDARD' '' Prove#theorems#involving#similarity#

EXPECTATION' G3SRT.5.' Use#congruence#and#similarity#criteria#for#triangles#to#solve#problems#and#to#prove#relationships#in#geometric#figures.#

Unit#7,#Lesson#1,#Topic#4:##The#Pythagorean#Theorem#


CATEGORY'/'CLUSTER' '' Understand#and#apply#the#Pythagorean#Theorem.#

STANDARD' 8.G.6.' Explain#a#proof#of#the#Pythagorean#Theorem#and#its#converse.#

STANDARD' 8.G.7.' Apply#the#Pythagorean#Theorem#to#determine#unknown#side#lengths#in#right#triangles#in#realCworld#and#mathematical#problems#in#two#and#three#dimensions.#







STANDARD' '' Define#trigonometric#ratios#and#solve#problems#involving#right#triangles#

EXPECTATION' G3SRT.8.' Use#trigonometric#ratios#and#the#Pythagorean#Theorem#to#solve#right#triangles#in#applied#problems.#

Unit#7,#Lesson#2,#Topic#1:##Quadrilaterals#




EXPECTATION' G3CO.1.' Know#precise#definitions#of#angle,#circle,#perpendicular#line,#parallel#line,#and#line#segment,#based#on#the#undefined#notions#of#


1.36#

point,#line,#distance#along#a#line,#and#distance#around#a#circular#arc.#

Unit#7,#Lesson#2,#Topic#2:##Perimeter#and#Area#





Unit#7,#Lesson#2,#Topic#3:##Circles#





Unit#7,#Lesson#3,#Topic#1:##Solids#


CATEGORY'/'CLUSTER' '' Solve#realCworld#and#mathematical#problems#involving#volume#of#cylinders,#cones,#and#spheres.#

STANDARD' 8.G.9.' Know#the#formulas#for#the#volumes#of#cones,#cylinders,#and#spheres#and#use#them#to#solve#realCworld#and#mathematical#problems.#






CATEGORY'/'CLUSTER' G3GMD.' Geometric#Measurement#and#Dimension#


1.37#

STANDARD' '' Explain#volume#formulas#and#use#them#to#solve#problems#

EXPECTATION' G3GMD.3.' Use#volume#formulas#for#cylinders,#pyramids,#cones#and#spheres#to#solve#problems.#

Developmental Math – An Open Curriculum Instructor Guide

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