Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Table of Contents and
Learning Objectives
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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1.1#
Developmental Math – An Open Curriculum
Instructor Guide
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.
1.2#
Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Instructor Notes
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.
1.3#
Developmental Math – An Open Curriculum
Instructor Guide
[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:
1.4#
Developmental Math – An Open Curriculum
Instructor Guide
[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:
1.5#
Developmental Math – An Open Curriculum
Instructor Guide
[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."
1.6#
Developmental Math – An Open Curriculum
Instructor Guide
[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:
[From Lesson 2, Topic 2, Topic Text]
1.7#
Developmental Math – An Open Curriculum
Instructor Guide
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 semicircles.
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:
[From Lesson 2, Topic 3, Topic Text]
Many students think that
equals 3.14. This would be a good time to review that
irrational number and that 3.14 and
are just good approximations for
is an
.
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:
1.8#
Developmental Math – An Open Curriculum
Instructor Guide
[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.
1.9#
Developmental Math – An Open Curriculum
Instructor Guide
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.
1.10#
Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Tutor Simulation
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.
1.11#
Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Puzzle
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
2
2
2
a and b into shapes that they combine to form c .
1.12#
Developmental Math – An Open Curriculum
Instructor Guide
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.
1.13#
Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Project
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
1.14#
Developmental Math – An Open Curriculum
Instructor Guide
Name of Shape
Sketch of shape with
Area of shape
side lengths in centimeters
in square centimeters
Total area in square centimeters
1.15#
Developmental Math – An Open Curriculum
Instructor Guide
Stained-Glass #2
Name of Shape
Sketch of shape with
Area of shape
side lengths in centimeters
in square centimeters
Total area in square centimeters
•
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.
1.16#
Developmental Math – An Open Curriculum
Instructor Guide
Stained-Glass #3 Rectangles
Color
Area of shape in square centimeters
Orange
Red
Yellow
Total area in square
centimeters
Stained-Glass #4 Trapezoid
1.17#
Developmental Math – An Open Curriculum
Instructor Guide
Color
Area of shape in square centimeters
AquaBlue
Purple
Total area in square
centimeters
•
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
1.18#
Sketch with
dimensions of sides
and angles
Area of Shape
(to the
nearest
hundredth)
Developmental Math – An Open Curriculum
Instructor Guide
Equilateral Triangle
1
Blue Rhombus
2
Isosceles Trapezoid
3
Regular Hexagon
6
Square
Does not apply
Gray Rhombus
Does not apply
1.19#
Area = 3.125
square cm
Developmental Math – An Open Curriculum
Instructor Guide
•
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
Equilateral Triangle
Blue Rhombus
Isosceles Trapezoid
Regular Hexagon
Square
1.20#
Area of
Shape
Total Area
Developmental Math – An Open Curriculum
Instructor Guide
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
1.21#
Developmental Math – An Open Curriculum
Instructor Guide
All solutions should be the same for each student.
Name of Shape
Sketch of shape with
Area of shape
side lengths in centimeters
in square centimeters
Area = base × height
Rectangle
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
1.22#
58 square centimeters
Developmental Math – An Open Curriculum
Instructor Guide
Name of
Shape
Sketch of shape with
Area of shape
side lengths in centimeters
in square centimeters
Area = ½ π r2
Semi-circle
(Half of a
circle)
square
centimeters
Area = ½ base × height
Triangle
square
centimeters
Total area in square centimeters
square
centimeters
Problem 2
All of the students’ answers should be the same.
Color
Area of shape in square centimeters
Orange
Area = base × height
Area = 4 ×5
Area = 20 square centimeters
1.23#
Developmental Math – An Open Curriculum
Instructor Guide
Red
Area = base × height
Area = 4 ×1.5
Area = 6 square centimeters
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
Total area in square
centimeters
Area = base × height
Area = 17×8
Area = 136 square centimeters
Each student’s method to obtain the answer may vary. However, all of the students’ answers
should be the same.
Color
Area of shape in square centimeters
AquaBlue
Area = ½ base × height
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
Total area in 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.
1.24#
Developmental Math – An Open Curriculum
Instructor Guide
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
Equilateral Triangle
Area of Shape
(to the nearest hundredth)
Area = ½ base × height
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
1.25#
Developmental Math – An Open Curriculum
Instructor Guide
Square
Area = base × height or
Area = side2
Area = (2.5)(2.5)
Area = 6.25 square cm
Gray Rhombus
Area = 3.125 square cm
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
1.26#
Developmental Math – An Open Curriculum
Instructor Guide
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
Pattern Block
Equilateral Triangle
Number in Design
Area of
Shape
Total Area
6
square cm
square cm
Blue Rhombus
6
square cm
square
cm
Isosceles Trapezoid
6
square cm
square
cm
Regular Hexagon
Square
1
0
1.27#
square cm
square cm
Area = 6.25
square cm
0
Developmental Math – An Open Curriculum
Instructor Guide
Gray Rhombus
0
Area = 3.125
square cm
Total Area of Pattern Blocks (in square cm)
0
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
1.28#
Developmental Math – An Open Curriculum
Instructor Guide
Pattern Block
Equilateral Triangle
Number in Design
Area of
Shape
0
Total Area
0
square cm
Blue Rhombus
12
square cm
square
cm
Isosceles Trapezoid
0
0
square cm
Regular Hexagon
0
0
square cm
Square
Gray Rhombus
12
12
1.29#
Area = 6.25
square cm
Area= 12(6.25)
Area = 3.125
square cm
Area= 12 (3.125)
Area = 75 square
cm
Area = 37.5 square
cm
Developmental Math – An Open Curriculum
Instructor Guide
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.
1.30#
Developmental Math – An Open Curriculum
Instructor Guide
Rubric
Score
Content
•
•
4
•
•
•
•
3
•
•
•
•
2
•
•
•
1
•
Presentation/Communication
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.
•
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.
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.
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
•
•
1.31#
•
•
•
•
•
•
•
•
•
•
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.
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.
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.
There is no explanation of the
solution, the explanation cannot be
understood or it is unrelated to the
problem.
Developmental Math – An Open Curriculum
Instructor Guide
•
•
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.
1.32#
•
•
•
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.
Developmental Math – An Open Curriculum
Instructor Guide
Unit 7 – Correlation to Common Core
Standards
Learning Objectives
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#
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#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)'
EXPECTATION'
8.G.1(b)'
Lines#are#taken#to#lines,#and#line#segments#to#line#segments#of#the#
same#length.#
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#
1.33#
Developmental Math – An Open Curriculum
Instructor Guide
CATEGORY'/'CLUSTER'
''
Understand#congruence#and#similarity#using#physical#models,#
transparencies,#or#geometry#software.#
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.#
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.#
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.#
STANDARD'
8.G.2.'
STANDARD'
8.G.4.'
STANDARD'
8.G.5.'
STRAND'/'DOMAIN'
CC.G.'
Geometry#
CATEGORY'/'CLUSTER'
G3CO.'
Congruence#
STANDARD'
''
Experiment#with#transformations#in#the#plane#
EXPECTATION'
G3CO.1.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3CO.'
Congruence#
STANDARD'
''
Understand#congruence#in#terms#of#rigid#motions#
EXPECTATION'
G3CO.7.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3CO.'
Congruence#
STANDARD'
''
Prove#geometric#theorems#
EXPECTATION'
G3CO.10.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3SRT.'
Similarity,#Right#Triangles,#and#Trigonometry#
Grade:#9C12#C#Adopted#2010#
1.34#
Developmental Math – An Open Curriculum
Instructor Guide
STANDARD'
''
Understand#similarity#in#terms#of#similarity#transformations#
EXPECTATION'
G3SRT.2.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3SRT.'
Similarity,#Right#Triangles,#and#Trigonometry#
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#
Grade:#8#C#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.G.'
Geometry#
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.#
STRAND'/'DOMAIN'
CC.G.'
Geometry#
CATEGORY'/'CLUSTER'
G3CO.'
Congruence#
STANDARD'
''
Experiment#with#transformations#in#the#plane#
EXPECTATION'
G3CO.1.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3SRT.'
Similarity,#Right#Triangles,#and#Trigonometry#
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.#
Grade:#9C12#C#Adopted#2010#
Unit#7,#Lesson#2,#Topic#1:##Quadrilaterals#
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#
1.35#
Developmental Math – An Open Curriculum
Instructor Guide
point,#line,#distance#along#a#line,#and#distance#around#a#circular#
arc.#
Unit#7,#Lesson#2,#Topic#2:##Perimeter#and#Area#
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#2,#Topic#3:##Circles#
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#3,#Topic#1:##Solids#
Grade:#8#C#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.G.'
Geometry#
CATEGORY'/'CLUSTER'
''
STANDARD'
8.G.9.'
Solve#realCworld#and#mathematical#problems#involving#volume#of#
cylinders,#cones,#and#spheres.#
Know#the#formulas#for#the#volumes#of#cones,#cylinders,#and#
spheres#and#use#them#to#solve#realCworld#and#mathematical#
problems.#
Grade:#9C12#C#Adopted#2010#
STRAND'/'DOMAIN'
CC.G.'
Geometry#
CATEGORY'/'CLUSTER'
G3CO.'
Congruence#
STANDARD'
''
Experiment#with#transformations#in#the#plane#
EXPECTATION'
G3CO.1.'
STRAND'/'DOMAIN'
CC.G.'
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.#
Geometry#
CATEGORY'/'CLUSTER'
G3GMD.'
Geometric#Measurement#and#Dimension#
1.36#
Developmental Math – An Open Curriculum
Instructor Guide
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.#
1.37#