Download EXPLORING GEOMETRIC FIGURES Grade 10 6

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EXPLORING GEOMETRIC
FIGURES
Grade 10
6-Day Lesson Plan
Tangrams
Geoboards
Equation Grapher
Green Globs
AlphaShapes
Protractor
Compass
Created by Sandra Metzler
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OVERALL OBJECTIVES
1- Students will increase their knowledge of geometric shapes such as
triangles, polygons, circles, and quadrilaterals.
2- Students will be able to classify triangles, polygons and
quadrilaterals.
3- Students will find the measures of angles within geometric shapes
and also outside these shapes.
4- Students will become confident using a compass and a protractor.
5- Students will become confident using computer technology such as
equation grapher.
6- Students will be able to recognize parallel and perpendicular lines by
their slopes.
7- Students will be able to measure central angles and arcs of circles.
8- Students will be able to display and read data in a circle graph.
9- Students will make connections between geometric figures and
everyday life.
10- Students will experience math hands-on and work cooperatively.
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LEARNING STANDARDS
ELA Standard 1: Students will read, write, listen and speak for information
and understanding.
ELA Standard 4: Students will read, write, listen and speak for social
interaction.
MST Standard 1: Students will use mathematical analysis and scientific
inquiry to pose questions, seek answers, and develop solutions.
MST Standard 2: Students will access, generate, process, and transfer
information using appropriate technologies.
MST Standard 3: Students will understand mathematics and become
mathematically confident by communicating and reasoning mathematically,
by applying mathematics in real-world settings, and by solving problems
through the integrated study of geometry and algebra.
MST Standard 5: Students will apply technological knowledge and skills to
design, construct, use, and evaluate products and systems to satisfy human
and environmental needs.
MST Standard 6: Students will understand the relationships and common
themes that connect mathematics, science, and technology to address real
-life problems and make informed decisions.
MST Standard 7: Students will apply the knowledge and thinking skills of
mathematics, science, and technology to address real-life problems and
make informed decisions.
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RESOURCES AND TOOLS NEEDED:
1- Textbook:
Bass, Laurie E., et al. New York Math A/B an Integrated
Approach Volume 2. New Jersey: Prentice-Hall, Inc., 2001
pages 66-101.
2- Class set of Geoboards and rubber bands
3- Class set of Tangrams
4- Computers
5- Equation Grapher computer program
6- Green Globs computer program
7- Class set of AlphaShapes
8- Protractors
9- Compasses
10- Scissors
11- Construction paper
12- Graph paper
13- Paper
14- Pencils or pens
15- Overhead
16- Overhead markers
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OVERVIEW
My goal for the next six days is to involve the students in hands on learning of different
geometric shapes and their relationships with one another. I hope that after this
instructional period, the students will walk away with an enhanced learning of
measurement and classification of the different geometric shapes.
Day 1:
A. Triangles
1. Construction of triangles
2. Measurement of angles
3. Triangle Angle-Sum Theorem
4. Exterior Angle Theorem
Day 2:
A. Triangles continued
1. Classification of triangles
2. Use of tangrams for triangle exploration
Day 3:
B. Polygons
1. Properties of polygons
2. Use of geoboards to explore polygons and their relationship to
triangles
3. Polygon Interior Angle-Sum Theorem
4. Polygon Exterior Angle-Sum Theorem
5. Exploring and classifying polygons using AlphaShapes
Day 4 and Day 5:
C. Quadrilaterals
1. Classifying quadrilaterals
2. Identifying slopes of different lines and how it relates to the
classification of quadrilaterals
3. Practice with Equation Grapher
4. Fun practice with Green Globs
Day 6:
D. Circles
1. Definition of a circle and its parts
2. Practice constructing circles and parts using compass and
protractor
3. Graphing circles from coordinates and locating the center
4. Constructing and reading a circle graph
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DAILY LESSON PLANS
DAY 1
Topic: Triangles
Learner Objectives:
Students will be able to find the measure of interior and exterior angles of
a triangle using the theorems presented.
Students will be comfortable using a protractor.
Instructional Presentation:
•The teacher will give a brief overview of the chapter to the students.
•The teacher will ask leading questions about the angles of a triangle to
activate the learner’s prior knowledge. For example, one question may
be “if we know we have a right triangle what is one of the angle
measures of that triangle?”
•The teacher will pass out a protractor warm up worksheet to each
student.(See below)
•The teacher will instruct the students to get out their rulers, scissors,
protractors, and a pencil.
•The teacher will model and instruct the students on how to do the
protractor sheet.
•The teacher will give the students time to complete the three problems.
•After sufficient time the teacher will model the steps of how to do each
problem.
•The teacher will then pass out a piece of construction paper to
each student.
•The teacher will instruct the students to draw a triangle that has sides of
length 9 cm and 4 cm and a 70-degree angle between these two
sides.(see below)
•The teacher will also instruct the students to find the measurement of the
other two angles.
•While the students are doing this activity, the teacher will assist those
students who need help or if needed the teacher will model how to do this
on the overhead.
•Once everyone has completed the triangle, the students will label each
angle with its measurement and then cut out their triangle.
•The angles will then be torn off the triangle.
•The students will be instructed to put the three angles adjacent to one
another and note their findings.
•The teacher will then lead in a discussion about the sum of the angles of
a triangle.
•The Triangle Angle-Sum theorem will then be introduced formally from
page 68 in their math textbooks.
•The teacher will then go through example 3 on the bottom of page 68.
•The teacher will then ask the students to try number 5 on page 69.(see
below)
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DAY 1cont.
•The teacher will then ask for a volunteers to do the each part of the
problem on the chalkboard.
•If the students are not grasping the Triangle Angle-Sum Theorem, then
more examples will be done.
•Moving on to exterior angles, the students will be instructed to perform a
similar activity as we did earlier with the construction paper and
constructing a triangle.
•The teacher will again pass out a piece of construction paper to each
student.
•While doing so the teacher will discuss exterior angles and remote
interior angles.
•For this activity, the students will draw a triangle, where one side there is
an extension, so the triangle will not be cut out for this exercise. The
students will use the example in the book on page 69 as a guide.
•The students will cut out the two remote interior angles and place them
on the extension line which is the exterior angle of the triangle.
•The teacher will then ask the students what have we learned by doing
this and a discussion will follow.
•The teacher will then formally introduce the Exterior Angle Theorem
which is described on page 70.
•The students will then be asked to try problem 7 on page 70.
•Volunteers will again be asked to show their work to each of the
problems on the chalkboard.
•A discussion will follow about the corollary to the theorem and
problems 8 and 9 on page 70.
•The teacher will then assign homework that consists of problems 10-19
and problem 21 A,B,C on page 72 of their textbook. Time permitting, the
students will be able to start it in class and the teacher can assist if
needed.
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PROTRACTOR WARMUP
1. Using your protractor, draw a right triangle that has a side
that measures 8.1 cm and another side that measures 5.3
cm.
2. Using your protractor, construct a triangle that has sides of
length 4.3 cm and 5.2 cm and a 29-degree angle between
the two sides.
3. Create your own triangle. Label the angle measurements
and the lengths of the sides. Then find the area of the triangle
you created.
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DAY 2
Topic: Triangles and Tangrams
Learner Objectives:
Students will be able to classify types of triangles by using a combination
of prior knowledge and congruency of sides and angles.
Students will work cooperatively and investigate triangles by using
tangrams.
Instructional Presentation:
•The teacher will review the theorems discussed last class.
•The teacher will ask the students if there was any questions from the
homework assigned and will address them.
•The teacher will collect the homework for evaluation later.(see below)
•Discussion will then commence with classifying triangles. In order to
activate prior knowledge the students will be asked how we determine if a
triangle is scalene, obtuse, acute, or equilateral.
•Discussion will continue with page 71 of their math textbooks. The Math
A Prep questions and problem 11 will be discussed as a group.
•The students will then be asked to do problems 1- 9 on page 71.
•After sufficient time is given to the students to complete the problems,
the teacher will go through each problem on the overhead projector with
the students’ involvement.
•The students will then be grouped into pairs and each given a tanagram
set. The students will perform the activity given to them to investigate
more about triangles.(see below)
•Problem number 23 on page 72 is assigned for extra credit and is
due by the following Monday. It must be typed, double spaced and be no
longer than one page.
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Names: ____________________________________________
TANGRAM FUN
With your partner try the following exercises.
1. Make a triangle using two of the pieces from the set.
What classification does this triangle belong?
_____________________________________
Are there other two-piece combinations you can use to
make a different triangle? _________________
If so, are they still the same classification or different?
__________________________________________
2. Make a triangle using three of the pieces form the set.
What classification does this triangle belong?
Is there another three-piece combination you could use to make a
different triangle? ______________________
If so, is the new triangle in the same classification or different?
________________________________________
3. Take any one of the triangles and trace the angles one by one
next to each other. What can you conclude from this about the
sum of the triangles angles?
_______________________________________________
4. Make a conjecture: Is it possible to make a triangle out of all 7
pieces? _______
Now investigate and try doing so.
Was your conjecture true or false? _______
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MORE TANGRAM FUN
SEE IF YOU CAN MAKE THESE FIGURES USING ALL 7 OF YOUR TANGRAMS.
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Names:_____SAMPLE STUDENT WORK
TANGRAM FUN
With your partner try the following exercises.
1. Make a triangle using two of the pieces from the set.
What classification does this triangle belong?
__Isosceles__________________________
Are there other two-piece combinations you can use to
make a different triangle?_______yes____
If so, are they still the same classification or different?
____Isosceles__________________________
2. Make a triangle using three of the pieces form the set.
What classification does this triangle belong?
Isosceles
Is there another three piece combination you could use to make a
different triangle?__none found__________
If so, is the new triangle in the same classification or different?
________________________________________
3. Take any one of the triangles and trace the angles one by
one next to each other. What can you conclude from this
about the sum of the triangles angles?
The sum of the angles is equal to 180 or a straight line.
4. Make a conjecture: Is it possible to make a triangle out of all 7
pieces?__no___
Now investigate and try doing so.
Was your conjecture true or false?_false_
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MORE TANGRAM FUN
SEE IF YOU CAN MAKE THESE FIGURES USING ALL 7 TANGRAM PIECES.
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DAY 3:
Topic: Polygons
Learner Objectives:
Students will be able to classify polygons.
Students will also be able to find the sum of the measures of the interior and
exterior angles of the polygons.
Instructional Presentation:
•The teacher will give a brief overview of the chapter on polygons which starts on
page 76 to 81 in their math textbook.
•To activate the student’s prior knowledge, the teacher will lead in a discussion
about properties of polygons and examples. The students will be asked to
participate in sharing what they know with the class.
•The teacher will hand out geoboards and rubber bands to each student along
with
a worksheet on polygons.(see below)
•The students will work on the worksheet in pairs and use the geoboards to
construct the polygons and the diagonals they need to complete it.
•A discussion about their findings will follow and the teacher will ask about what
conjectures can be made about a larger polygon, for example one with 20
sides.
•The teacher will then formally introduce the Polygon Interior Angle-Sum
Theorem that is on page 77 of their math textbooks.
•The teacher will then do some examples.
•The teacher will then discuss the sum of exterior angles of polygons and
introduce the Polygon Exterior Angle-Sum Theorem that is located on page 78
of their textbooks.
•The teacher will then go over examples 2 and 3 on pages 78 and 79.
•The teacher will then hand out AlphaShapes and the AlphaShapes worksheet to
the students.(see below)
•The students will complete the worksheet in groups of two or three and hand it
in
at the end of the class period. The teacher will assist as needed.
•The teacher will assign page 80 #10,12,14,16,20,22,24,26.
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Name:______________________________________________
POLYGONS
Using your geoboard and rubber bands, construct the following polygons and then add the
diagonals using rubber bands to see how many triangles are formed. Compute the sum of
interior angles.
POLYGON
TRIANGLE
QUADRILATERAL
PENTAGON
HEXAGON
HEPTAGON
OCTAGON
NONAGON
DECAGON
NUMBER OF
SIDES
NUMBER OF
TRIANGLES
FORMED
SUM OF
INTERIOR
ANGLES
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POLYGONS
Using your geoboard and rubber bands, construct the following polygons and then add the
diagonals using rubber bands to see how many triangles are formed. Compute the sum of
interior angles.
POLYGON
NUMBER OF
SIDES
NUMBER OF
TRIANGLES
FORMED
1
SUM OF
INTERIOR
ANGLES
180
TRIANGLE
3
QUADRILATERAL 4
2
360
PENTAGON
5
3
540
HEXAGON
6
4
720
HEPTAGON
7
5
900
OCTAGON
8
6
1080
NONAGON
9
7
1260
DECAGON
10
8
1440
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DAY 4 and DAY 5
Topic: Quadrilaterals
Learner Objectives:
Students will be able to classify special types of quadrilaterals.
Students will be able to graph quadrilaterals and explore the relevance of
slope, parallel and perpendicular lines.
Instructional Presentation:
•The teacher will go over the homework (page 80) from the last class and
then collect it. (see below)
•The teacher will give a brief overview of lesson on quadrilaterals, slope,
perpendicular and parallel lines.
•The teacher will pass out the AlphaShapes and ask students to find each
of the special quadrilaterals that are described on page 91 of their
textbooks.
•The teacher will then pass out the worksheet on classifying quadrilaterals.(see
below)
•A discussion will follow on concave and convex.
•The teacher will hand out a worksheet on finding the slopes of lines. The
teacher will
model the first example and the students will attempt the rest on their own. After
a
reasonable amount of time, the worksheet will be reviewed and discussed as a
class.(see below)
•The rest of the remaining class time, the students will explore graphing lines
with
different slopes using their computers and equation grapher. The teacher will
walk the students through some examples from the book, such as page 86 2022.
(see below)
•The students can then play Green Globs in the time remaining. (see below)
•The teacher will assign homework and pass out the necessary graph paper for
the
students to complete the assignment on. (see below)
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SAMPLE STUDENT WORK
Homework page 80
10. regular pentagon interior angles
(5-2) 180 = 540
540/5 = 108 degrees
exterior angles= 180 – 108 = 72 degrees
12. regular 18-gon interior angles=
(18-2) 180= 2880
2880/18 = 160 degrees
exterior angles= 180 – 160 = 20 degrees
14. Find the number of sides when given the exterior angle of a regular
polygon.
Exterior angle = 72
180 – 72 = 108
108 * 5 = 540 it works so the
regular polygon is a pentagon
16. Find the number of sides when given the exterior angle of a regular
polygon.
Exterior angle = 18
180 – 18 = 162
162 * 10 = 1620 it works so the
regular polygon is a decagon
Find the missing variables.
20. 7 sides (7-2) 180 = 900
116 +129 + 130 + 135 + 125 + 120 = 755
22. 5 sides (5-2) 180 = 540
117 + 100 + 105 + 115 = 437
24. 4 sides (4-2) 180 = 360
h + h + 2h + 2h = 360
6h = 360
h = 60
900
-755
Y= 145
540
-437
X= 103
angles are 60, 60, 120, 120
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26. exterior angles are = 360
108,131
z + (z – 13) + (z + 10) = 360
3z-3 = 360
z = 121
x = 180 – 121
x = 59
exterior angles are 121,
y = 180 – 131
y = 49
w = 180 - 108
w = 72
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Name:
Classifying Quadrilaterals Worksheet
Using your Quadrilateral Alphashapes determine ways to sort your quadrilaterals into subsets.
SUBSET #1
What is the common feature that all these quadrilaterals have in common?
How many belong? And what letters are they?
SUBSET #2
What is the common feature that all these quadrilaterals have in common?
How many belong? And what letters are they?
SUBSET #3
What is the common feature that all these quadrilaterals have in common?
How many belong? And what letters are they?
Now trace one of the quadrilaterals from subset #1, #2, and #3 here and draw all the diagonals.
What happened?
Now trace AlphaShapes E and T and draw all the diagonals. What happened?
How are these quadrilaterals different from the ones you traced above?
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Name: SAMPLE STUDENT WORK
Classifying Quadrilaterals Worksheet
Using your Quadrilateral AlphaShapes determine ways to sort your quadrilaterals into subsets.
SUBSET #1
What is the common feature that all these quadrilaterals have in common?
TWO PAIRS OF PARALLEL SIDES
How many belong? And what letters are they?
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Q,W,K,O,U
SUBSET #2
What is the common feature that all these quadrilaterals have in common?
ONE PAIR OF PARALLEL SIDES
How many belong? And what letters are they?
2
C,S
SUBSET #3
What is the common feature that all these quadrilaterals have in common?
NO PARALLEL SIDES
How many belong? And what letters are they?
1
G
Now trace one of the quadrilaterals from subset #1, #2, and #3 here and draw all the diagonals.
What happened?
ONLY TWO DIAGONALS WERE DRAWN IN EACH FIGURE MAKING FOUR
TRIANGLES.
Now trace AlphaShape E and T and draw all the diagonals. What happened?
How are these quadrilaterals different from the ones you traced above?
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THESE HAVE DIAGONALS THAT RUN OUTSIDE OF THE QUADRILATERALS.
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Name:
Slope Worksheet
Directions: Find the slope of each of the following graphs.
1.
2.
3.
5.
4.
6.
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7.
8.
9.
10.
11.
12.
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Slope Worksheet
Directions: Find the slope of each of the following graphs.
1. M=1
3. M= UNDEFINED
5. M= -1/2
2. M=0
4.M= -2
6. BOTH LINES
UNDEFINED AND
PARALLEL
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7.
M= 3 FOR BOTH
SO THEY ARE PARALLEL.
8. M= -2 for blue; M = _ red
lines are perpendicular
11. M= 3
8. M= 0 for blue graph; undefined for red
Lines are perpendicular.
9. M= -1 for blue; M= 1 for red
lines are perpendicular
12. M = - 1/2
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SAMPLE STUDENT WORK
PAGE 86 #20
Which of the lines is not parallel to the line y = -2/3x + 8?
a. 2x + 3y = 1
b. 4x = 3 – 6y
c. x = -1.5y
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d. 24 = 2x – 3y
e. 9y = -6x -2
SAMPLE STUDENT WORK
PAGE 86 # 21
a.
b.
c.
d.
e.
sketch a vertical line containing the point (-5,2)
write an equation for the line
On the same graph, sketch horizontal line containing (-5,2)
Write an equation for the line
What is the relationship between the two lines?
Sample student work
Page 86 # 22
a.
b.
c.
d.
e.
Sketch line w perpendicular to y= 5, and containing (1,4)
Write an equation for line w.
On the same graph, sketch line r parallel to y = 5, and containing (1,4).
Write an equation for line r.
What is the relationship between line r and line w?
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SAMPLE STUDENT WORK
GREEN GLOBS
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Name:_______________________________________________
Homework Assignment: Graphing
On the provided graph paper, construct the following geometric shapes on a 10 X 10
grid and answer the related questions right on the graph paper next to the figure
constructed. Label each shape with the appropriate number.
1. Plot the following points on the graph paper: A(-2,1), B(0,3), C(2,1), D(0,-1).
What kind of quadrilateral is this and why?
2. Plot the following points on the graph paper: A(1,2), B(3,3), C(5,2), D(3,1).
What kind of quadrilateral is this and why?
3. Plot the following points on the graph paper: W(-4,-3),X(-1,3),Y(4,3),Z(-1,3).
What kind of quadrilateral is this and why?
4. Plot the following points on the graph paper: J(-2,2), K(2,2), L(5,-1), M(-5,-1).
What kind of quadrilateral is this and why?
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Answer Key to homework graphing assignment
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DAY 6
Topic: Circles
Learner Objectives:
Students will be able to recognize and define a circle and it’s parts: radius,
diameter,
central angle, major and minor arcs and adjacent arcs.
Students will be able to find the measures of central angles and arcs of circles.
Students will be comfortable using a compass to draw circles and arcs.
Students will be able to find the center of a circle and compute the radius.
Students will be able to recognize the relationships that the different arcs have to
the
circle and to one another.
Instructional Presentation:
•The quadrilateral graphing homework will be collected for grading.
•The teacher will then ask the students if they have any questions about any of
the
material covered thus far.
•The teacher will give a brief overview of the next section: circles. A discussion
will
begin on how to define a circle and who can come up with a good definition that
has no counterexample.
•The students will then get out their compasses and protractors and begin to
practice
drawing circles with them, labeling the radius, diameter, and arcs. The teacher
will
model this on the overhead with them.
•The teacher will explain (while modeling) minor arcs, major arcs, adjacent arcs
and
semicircles and how to find the measure of each.
•The teacher will then go through example 3 on page 98 of the textbook which
involves finding the measures of arcs.
•The students will then try to do #5 and #6 of example 3 on their own. After
sufficient
time the teacher will go through each problem with them.
•The teacher will then pass out graph paper and the students will practice finding
the
coordinates of the center and the radius of different circles given two points on
the
circle, construct an arc, and construct a circle graph using data from a table;
problems 22,23,24,37, and 50 on pages 99-101. (see below)
•The teacher will assist as needed and may model an example on the overhead
to
help get the students started.
•The teacher will collect the student’s work for grading at the end of the class
period.
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•The teacher will then assign problems 1-19 odds, 25-35 odds, 41-45 odds for
homework.
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SAMPLE STUDENT WORK PAGE 99 #22-24
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STUDENT SAMPLE WORK PAGE 99, # 37 and PAGE 101 #5
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