Download Complete Project - Bemidji State University

Angles &
Lines
Grades 7 & 8
Summer 2009
Jess Stuewe
Detroit Lakes Middle School, Grades 7-8
[email protected]
Steve Bergerson
Pine River-Backus Schools, Grades 6-12
[email protected]
Executive Summary:
Unit Goal: Students will learn to classify, construct, and measure angles and
polygons using tools such as compasses and protractors as well as hands-on
manipulatives such as patty paper.
Teaching Strategies: Students will construct, measure, classify, and
analyze angles using compasses, protractors, rulers and patty paper.
Students will use their knowledge to classify polygons using angle measures
and identify missing angles of parallel and perpendicular lines bisected by a
transversal. The cumulating project for this unit is a “cubical portrait” in
which students will use measurements and constructions to create a portrait
using only straight lines.
2007 Minnesota Academic Standards for Mathematics:
(Published for public use on Mn Dept. of Ed website):
http://education.state.mn.us
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
7.3.2.1
Analyze the effect of change of scale, translations and reflections on the
7.3.2.2
attributes of two-dimensional figures.
7.3.2.3
7.3.2.4
Geometry & Measurement Strand Grade 8
8.3.2.1
Solve problems involving parallel and perpendicular lines on a coordinate
8.3.2.2
system.
8.3.2.3
Page 2 Table of Contents
Sample MCA Questions .......................................................................................................... 5 Lesson 1: Constructing Angles............................................................................................. 6 Lesson 2: Measuring Angles .............................................................................................. 11 Lesson 3: Classifying Polygons......................................................................................... 14 Lesson 4: What’s My Angle?............................................................................................... 19 Lesson 5: Stretching and Shrinking................................................................................ 21 Review Activities for Extension/Remediation ........................................................... 26 Lines and Angles Vocabulary:........................................................................................... 27 Cubical Portrait ..................................................................................................................... 30 Slopes........................................................................................................................................ 37 Slope Investigation with Sketchpad ............................................................................... 43 Parallel & Perpindicular Exploration ............................................................................ 44 Perpendicular Exploration................................................................................................ 46 Roof Bracket Handout......................................................................................................... 47 Parallel Lines Exploration ................................................................................................. 49 Lines & Angles Pre/Post Test ........................................................................................... 54 Page 3 Lesson
Day
0
1 Day
1
1-3 Days
2
1 Day
3
2-4 Days
4
1 Day
5
2 Days
6
1-2 Days
7
1 Day
8
1 Day
9
1 Day
10
1-2 Days
11
1 week
12
Activity
Description
Pre-Test
Constructing Angles
Measuring Angles
Classifying Polygons
What’s My Angle?
Stretching and
Shrinking
Slopes
Slope Investigation with
Sketchpad.
Parallel & Perpendicular
Exploration
Perpendicular
Exploration
Parallel Lines
Exploration
Cubical Portrait Project
Construct angles with a compass and straight
edge (ruler).
Measure angles using protractors.
Find angle measures of polygons using patty
paper.
Determine if the size of a hand affects the
angle measures between the fingers.
Draw similar shapes on graph paper, using a
focal point and scale factor.
Exploring different slopes.
Using technology to explore slopes of lines.
Looking at parallel and perpendicular lines in a
drawing of a garage.
Building roof brackets for a company.
Solving puzzles with intersecting parallel lines
and transversals.
Create a scale portrait of a face-shot picture
using only straight lines. Label angles and
measures and polygons drawn on rough draft.
Post-Test
Page 4 Sample MCA Questions
Geometry & Measurement Strand Grade 7
Geometry & Measurement Strand Grade 8
MCA Practice Problems from Minnesota Department of Education
(Published for public use on their website: http://education.state.mn.us)
Page 5 Lesson 1: Constructing Angles
Modified from “Hamilton’s Math to Build On” © 1993
http://mathforum.org/~sarah/hamilton/ham.cotents.html
Objective: Students will use compasses and rulers to construct angles of
90°, 45°, 60°, 30°, and 15°.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
Geometry & Measurement Strand Grade 8
8.3.2.1
Solve problems involving parallel and perpendicular lines on a coordinate
8.3.2.2
system.
8.3.2.3
Materials:
Paper
Pencils
Compasses
Rulers
Launch:
Students will come to class with some knowledge of angles.
Ask students:
“What do you know about angles?”
“How can we measure an angle?”
“When do we use angle measures? Why should we know how?”
“What kinds of professions use measures of angles?”
This lesson teaches students how to construct angles of 90°, 45°, 60°, 30°,
and 15° using a compass and ruler.
Tell students:
“Today we are going to construct angles in class using a compass and ruler. If
I use a ruler to draw a straight line, what would I have to do to create a 90°
or right angle?”
Page 6 Explore:
Each student should work independently, sharing ideas/strategies in pairs or
small groups to create a 90° angle. When groups have discovered that a
perpendicular line will create a 90°angle, discuss as a class how we can be
sure that they created exactly a 90° angle.
Summarize:
Work through the construction of a 90° with a compass and ruler:
1)
Draw a straight line of any length and mark a point somewhere on
the line.
2)
Place the metal (pointed) end of the compass on the marked point
and turn the compass to mark two points equidistant from the
point.
3)
4)
5)
Set your compass a little wider than in step 2. Set the compass
point on an endpoint (one of the intersections drawn in step 2) and
mark an arc above the center point.
Keep the same compass setting and repeat for the other end point.
Draw a straight line from the point where the arcs cross to the
center point of the line. These two straight lines are perpendicular
to each other and form a 90°.
Page 7 Launch:
If we can construct a 90° using these tools, it makes sense that we should
be able to construct other angles too, but how?
Explore:
Each student should work independently, sharing ideas/strategies in pairs or
small groups to create other angles. Discuss the strategies students use to
construct angles. Students may need a hint to “divide angles” to construct
smaller angles.
Work through the constructions as a class, pausing to check for
understanding after each construction. Each student should work
independently, sharing ideas/strategies/results in pairs or small groups as
the class progresses through each construction.
45°
1)
2)
Work from the construction of 90°. Set the compass at the vertex
of the 90°. Use the compass to mark a point on all three lines that
are equidistant from the vertex.
Set the compass point at each of the three points and mark arcs
between the lines.
Page 8 3)
60°
1)
Draw a straight line. Mark a point on the line close to the center.
Set the compass point on the center point and draw a semi-circle
with the compass.
2)
Keep the compass setting the same. Place the compass point where
the semi-circle and the straight line meet. Mark a small arc across
the semi-circle.
3)
Draw a line from the center point to the arc intersection from
step 2. This will mark 60°
30°
1)
Use a ruler to draw a straight line from the vertex through the
point where the arcs cross. These lines equally divide 90°, so our
constructions now show 45°
Keep the compass the same. Place the compass point at the arc
intersection. Mark another arc above the arc (but between the
straight line and the 60° angle). Repeat the same action from the
intersection of the straight line and the semi-circle.
Page 9 2)
Draw a straight line from the center point through the point where
the arcs cross. This forms two-30° angles.
15°
To mark 15°, bisect the 30° angles using the same process.
Summarize:
Students share constructions with each other while checking for
correctness. When all groups have constructions for each angle (90°, 45°,
60°, 30°, and 15°), ask for a representative from each group to share one
construction with the class. Conclude that using a ruler and a compass, we
can construct angles 90°, 45°, 60°, 30°, and 15°.
Extend:
How many 90°, 45°, 60°, 30°, and 15° are there in a circle?
Construct all of the above angles using the ruler and compass.
Page 10 Lesson 2: Measuring Angles
Objective: Students will use a protractor to find and construct measures of
angles.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
Geometry & Measurement Strand Grade 8
8.3.2.1
Solve problems involving parallel and perpendicular lines on a coordinate
8.3.2.2
system.
8.3.2.3
Materials:
Internet (for launch and extension)
Worksheets with angles for students to measure
Protractors
Pencils
Launch:
Project the “Banana Hunt” game on a SmartBoard and let students
manipulate the program before class starts. Play a few rounds as a class
when all of the students are in attendance and ready to begin class.
http://www.classbrain.com/artgames/publish/banana_angles_hunt.shtml
Ask students:
“We can estimate the measures of these angles, but is there a way to be
SURE that our measures are correct? How do we measure angles?”
Page 11 Introduce today’s lesson: “Today we are going to use protractors to measure
angles.” Distribute the “What is My Angle” worksheet.
In the previous lesson, we used compasses and rulers to construct angles.
These angles are already constructed for us. Use the protractors to find
the measures of the angles.
Explore:
Allow students to use the protractors on their own in small groups to find
the measures of angles. Assign the rolls of 1) Group Leader to keep group on
task 2) Administrative Assistant to record the groups’ measurements and 3)
Media Correspondent to share the groups’ results. (If students become
frustrated with how to use a protractor, hint that the center-point on the
protractor should lie on the vertex of the angle.)
Summarize:
Project the worksheet on the front board (SmartBoard or otherwise). Ask a
student representative from each group to share their results.
Project the webpage:
http://www.amblesideprimary.com/ambleweb/mentalmaths/protractor.html
Choose “Make and Measure” from the teacher controls.
Use the interactive arrows to draw an angle. Using the interactive
protractor, summarize the groups’ findings by reviewing how to read a
protractor.
Extend:
Now that students are prepared to use a protractor to read measures of
angles, ask students to construct various angles using a protractor and a
pencil.
Page 12 What is My Angle?
http://www.mathleague.com/help/geometry/angles.htm#whatisanangle
1)
Use your protractors to find the degree measure of each
of the following angles:
2)
Describe how you used your protractor to find the
measures of the angles:
3)
Use your protractor to construct angles of 180°, 200°,
80°, and 20°. (Hint: start with a straight line).
Page 13 Lesson 3: Classifying Polygons
Part 1: “Polygon Capture”
Adapted from Illuminations “Polygon Capture”
(Originally appearing in the October 1998 edition of
Mathematics Teaching in the Middle School)
http://illuminations.nctm.org/LessonDetail.aspx?id=L270
Part 2: “Polygon Sum Conjectures”
Adapted from Patty Paper Geometry by Michael Serra, Key
Curriculum Press
Objective: Students will classify polygons by examining relationships among
geometric properties. In part 2, students will classify polygons by exploring
the measure of interior angles.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
Part 1: Polygon Capture
Materials:
Polygon Capture Game Rules
Polygon Capture Game Cards
Polygon Capture Game Polygons
Launch:
As students walk into the classroom, ask them to write their responses to
the posed question on the front board:
“What can you tell me about…”
Triangles?
Quadrilaterals?
Shapes with more than 4
sides?
Page 14 As the class settles into their seats, ask them to look over the list. Go over
the lists students have generated. Ask if there are any other attributes
that should be added to the lists.
Review the vocabulary used in this game, such as, parallel, perpendicular,
polygon, acute, and obtuse and ask students where such words might be
included on the list.
When the class is satisfied with the list, partner students.
Distribute copies of Polygon Capture Game Rules, Polygon Capture Game
Cards, and Polygon Capture Game Polygons (each pair will need one copy of
each). Before introducing the game, have students cut out the polygons and
the cards. They should also mark the backs of the cards to designate each
as an “angle” or “side” card.
When all partners are ready, go over the rules of the game by reading the
rules on the Polygon Capture Game Rules sheet. (It may be helpful to play
one round of the game together, teacher vs. class).
Explore:
Students play “Polygon Capture,” using angle and side properties to “capture”
the polygons.
Summarize:
Ask students to share their strategies for capturing the polygons. Was
there a strategy that seemed to be better than others? Why?
Return to the list that students generated at the beginning of class. Did
they discover any other attributes/properties of polygons that were not on
the list? If so, add those properties to the list.
Extend:
1) For a simpler version of this game, students turn over only one card
over and work with one property at a time.
2) Remove the steal or wild card.
3) Use the Polygon Capture Game Polygons to sort before playing the
game.
4) Add polygons such as kites and concave polygons.
5) Add angle cards such as “opposite angles have equal measure” or
“number of vertices is a prime number.”
Page 15 Part 2: Polygon Sum Conjectures
Materials:
Warm-up Sheet (1/2 sheet per student)
Patty Paper
Open Investigation 4.1 Worksheet
(Polygon Sum Conjectures, p. 63 out of the student workbook)
Launch:
As students walk into the room, hand each student a warm-up sheet to work
on until all students have settled into their seats. Before starting the lesson,
ask students what they noticed about the statements and their converses.
Discuss some of their converse statements and results.
In geometry, as in real life, just because something appears to be true or
untrue, we cannot assume that the reverse situation (or converse) will give
us the same result (true or untrue). In this investigation, students will start
by looking at the sum of the angle measures of a triangle. Then they will
explore other polygons (quadrilaterals, pentagons, and hexagons).
Distribute Patty Paper and 4.1 Worksheet (see p. 63 of student workbook).
(It may be helpful to do step 1 and step 2 on the worksheet as a class).
Step 1: Draw a large scalene acute triangle on a patty paper. Label its angles
a, b, and c.
Step 2: On another patty paper, draw or fold a line and draw a point on the
line. Place the point over the vertex of angle “a”. Line it up with on of the
rays of the angle with the line. Trace angle a onto the second patty paper.
Now trace angle “b” on the paper so that it shares one side with angle “a”
and is on the same vertex. Repeat with angle “c” so that angle “b” shares one
side with angle “a” and the other side with angle “c.
Ask students to discuss their observations and write a conjecture (summary
statement) about what they noticed. (Sum of all three angles is 180° =
Triangle Sum Conjecture).
Explore:
Students continue as above using quadrilaterals instead of triangles, using
patty paper to discover the sum conjectures for quadrilaterals, pentagons,
and hexagons. Circulate around the classroom and listen to student
discussion. Regroup if there is confusion.
Page 16 Summarize:
Complete the table from the worksheet on the board:
Number of sides on the
3
4
5
6
polygon
Sum of the angle
measures of the polygon
7
8
n
Ask students to describe patterns they notice. As a class, find a formula or
“rule” for “n” sides of a polygon.
Review sum of angles conjectures:
Triangles: If a polygon has 3 sides, then the interior angle measure is 180°.
Quadrilaterals: If a polygon has 4 sides, then the interior angle measure is
360°.
Pentagons: If a polygon has 5 sides, then the interior angle measure is 540°.
N-gon: the interior angle measure of an n-gon is 180(n – 2).
End class by returning to the conjecture/converse idea from the Launch. If
an object is 180°, does that mean it is a triangle? When is it not? What if we
determine that the object is a polygon? Does that make the conjecture
true?
Extend:
1) Play the Polygon Capture game again, this time adding the polygon sum
conjectures to the “angles” cards. Ask students to predict how adding
this property might change the game, or if it will not change the game.
2) For more investigation with interior angles of polygons, try the
“Adding It All Up” lesson by NCTM’s “Illuminations.” In this lesson,
students draw different kinds of polygons and explore the interior
angles of convex and concave polygons.
(http://illuminations.nctm.org/LessonDetail.aspx?id=L765)
3) Continue classification of polygons using lines of symmetry, diagonals,
central angles, exterior angles etc.
Page 17 Warm-Up:
Read each statement below. Determine whether it is true or false. Then write the converse
(the reverse situation) of the statement and determine whether the converse is true or
false. (The first converse is done for you).
Statement: If today is Friday, then tomorrow is part of the weekend.
True or False?
Converse: If today is part of the weekend, then tomorrow is Friday.
Statement: If tomorrow is Thursday, then yesterday was Tuesday.
True or False?
True or False?
Converse:
Statement: If 9 is an odd number, then 9 is divisible by 2.
True or False?
True or False?
Converse:
True or False?
Statement: If you grew up in Minnesota, then you have seen snow.
True or False?
Converse:
True or False?
Warm-Up:
Read each statement below. Determine whether it is true or false. Then write the converse
(the reverse situation) of the statement and determine whether the converse is true or
false. (The first converse is done for you).
Statement: If today is Friday, then tomorrow is part of the weekend.
True or False?
Converse: If today is part of the weekend, then tomorrow is Friday.
Statement: If tomorrow is Thursday, then yesterday was Tuesday.
True or False?
True or False?
Converse:
Statement: If 9 is an odd number, then 9 is divisible by 2.
True or False?
True or False?
Converse:
True or False?
Statement: If you grew up in Minnesota, then you have seen snow.
True or False?
Converse:
True or False?
Page 18 Lesson 4: What’s My Angle?
Adapted from “Figure This Math Challenges for Families”
Challenge #10-“What’s My Angle?”
http://www.figurethis.org/challenges/c10/challenge.htm
Objective:
Students will measure the angles between their fingers to determine that
the size of their hand does not affect the measure of the angles between
their fingers. This will be a launch activity for exploring stretches
(similitude/dilations) of shapes in the next lesson.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
7.3.2.1
Analyze the effect of change of scale, translations and reflections on the
7.3.2.2
attributes of two-dimensional figures.
7.3.2.3
7.3.2.4
Materials:
Paper and pencil
Protractor
Launch:
Project the question for the students from the Figure This! Webpage.
Ask students to look at their hands and the hands of their classmates. Ask
them “Do people with smaller hands have smaller angles between their
fingers?”
Ask students to estimate the angles between their fingers and share their
guesses with their partners.
Page 19 Explore:
After estimating, ask students to trace their hands on a piece of paper and
label the angles with an estimated measure (they may change their estimates
from earlier if they want to). Be sure to emphasize that their fingers should
be spread wide.
How to the angle measures compare between the fingers? Ask students to
compare their sketches and estimates with at least five other classmates.
Ask students now to measure the angles using a protractor and modify their
measures as needed. When finished with measures, compare sketches and
measures with at least 5 other classmates (different classmates than
earlier).
One possible answer: 45°, 20°, 20°, and 90° between fingers (pinky to
thumb).
Summarize:
Ask students to share their findings. Although measurements may vary (here
is a good time to discuss measurement errors and what could have affected
the accuracy of this investigation), the conclusion is that the lengths of the
fingers do not affect the angle measures.
Extend:
1)
Suppose a bicycle wheel turns around exactly once. This is a 360°
revolution. How far would the bicycle wheel have moved on the
ground?
2)
Explore angle measures in sporting events:
a. Billiards (pool)
b. http://illuminations.nctm.org/ActivityDetail.aspx?ID=28
c. Golf
http://www.linkslearning.org/Teachers/1_Math/6_Learning_Re
sources/2_SuperMath/content/games/golf.htm
d. Soccer
http://illuminations.nctm.org/ActivityDetail.aspx?ID=158
3)
Explore other angles in nature:
a. Tree branches
b. Veins in a leaf
4)
Michael Jordan’s hand activity.
http://www.carlton.k12.mn.us/hs/Staff/glende/webquest.htm
Page 20 Lesson 5: Stretching and Shrinking
Objective: Students will use graph paper to construct similitudes (also
known as dilations or stretching and shrinking) with whole number and
fraction scale factors.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
7.3.2.1
Analyze the effect of change of scale, translations and reflections on the
7.3.2.2
attributes of two-dimensional figures.
7.3.2.3
7.3.2.4
Geometry & Measurement Strand Grade 8
8.3.2.1
Solve problems involving parallel and perpendicular lines on a coordinate
8.3.2.2
system.
8.3.2.3
Materials:
Graph paper
Protractors
Pencils
Launch:
Discuss the following scenario with your students using think-pair-share
(students think about the problem, then pair up with a partner to discuss
their ideas before sharing with the class).
“Imagine you are an architect. You have just been commissioned to design
the new stadium for the Minnesota Vikings. You are on a tight budget and
need to present your ideas to a panel of committee members before you
begin building. You need a plan. How are you going to present your ideas to
the committee?”
Page 21 After discussing and concluding that a drawing of the design (to scale) is the
best and cheapest plan to present to the committee, distribute graph paper
and introduce the first problem:
Stretch the “L” to three times its size.
Explore:
Let students use their graph paper and try the problem first. They might
comment that they know the perimeter will be three times as long, or that
the area will be larger (3 x 3 = 9 times larger!).
Give the following hints as needed:
1) Label the coordinates of the points
2) Scale factor means the size is changing by a factor
3) The shapes will be similar. That means they are the same shape,
but not the same size.
After a few minutes, have students partner and see if they agree with each
other. Encourage students to justify their answers and be able to explain to
Page 22 the class. Discus student answers then show the correct answer before
trying another shape.
Allow students to work in partners through the following problems.
Encourage students to justify their answers and be able to explain to the
class.
Try these similitudes (otherwise known as dilations or stretches):
Stretch the letter M to 2 times its original size.
Page 23 Stretch the letter T to ½ its original size.
Ask students to share their observations. What happened if we needed to
construct a new shape with a scale factor of ½? Do the lines from the focal
point still line up through the corresponding points?
Summarize:
After students have shared their strategies in partners, small groups, and
as a class, discuss which strategies made the most sense and why.
Conclude that we know:
1)
The new coordinates of the shape will be the product of original
coordinates and the scale factor.
2)
The shapes are similar/proportionate (they are the same shape,
but different size)
3)
If we draw a straight line from the focal point through a point on
the original shape, we should find the corresponding new point on
the line.
4)
The new perimeter will be the scale factor times the original
perimeter.
5)
The new area will be the scale factor squared times the original
area (discuss why---shapes are 2 dimensions and we are taking
both dimensions times the scale factor. Therefore we need to
square the scale factor first-then multiply it by the original area
to get the new area).
Page 24 Extend:
1) Continue work with similitudes, but change the focal point to a
different point on the coordinate plane (move it off of the origin).
2) Continue work with similitude, using standard geometric notation.
3) Calculate the scale factor of similar polygons, given all of the
dimensions and the picture.
4) http://www.successlink.org/GTI/lesson_unit-viewer.asp?lid=7867
5) For another way to dilate polygons, check out this interactive
webpage: http://www.mathopenref.com/dilate.html
6) Watch a “BrainPop” movie on scale drawings:
7) http://www.brainpop.com/artsandmusic/artconcepts/scaledrawing/pr
eview.weml
8) Try a lesson on similitudes using Geometer’s Sketchpad to construct
varying types of similar polygons:
http://www.lessonplanspage.com/printables/PMathCISimilarPolygonsU
singGeometersSketchpad7.htm
Page 25 Review Activities for
Extension/Remediation
1)
2)
Java Games from Quia.com
(http://www.quia.com/jg/65822.html)
a. Matching
b. Flashcards
c. Concentration
d. Word Search
Angles and Angle Terms from Mathleague.com
(http://www.mathleague.com/help/geometry/angles.htm)
a. Vocabulary and examples (good reference!)
Page 26 Lines and Angles Vocabulary:
Write your own definition of the vocabulary term and draw a
picture to match the definition.
Vocab Word
Definition
Angle
Degrees
Acute Angles
Obtuse Angles
Right Angles
Complementary
Angles
Supplementary
Angles
Page 27 Picture
Vertical
Angles
Alternate
Interior
Angles
Alternate
Exterior
Angles
Corresponding
Angles
Angle Bisector
Perpendicular
Lines
Parallel Lines
Transversal
Quadrilateral
Page 28 Pentagon
Hexagon
Heptagon
Octagon
Polygon
Other words I learned in this unit:
Page 29 Cubical Portrait
Final Project
Objective: Students will use what they have learned in this unit to create a
“cubical portrait.” Students start with a “head-shot” photo no larger than
3”x3” (preferably not on photo paper or magazine paper-if the picture is one
of these, make a copy, then proceed). Drawing only straight lines using a
ruler, compass, and protractor, students will enlarge the picture x2, x3, or
x4 times onto a poster-sized paper.
Minnesota State Mathematics Standards: Geometry and Measurement
Geometry & Measurement Strand Grade 7
7.3.1.1
Use reasoning with proportions and ratios to determine measurements, justify
7.3.1.2
formulas and solve real-world and mathematical problems involving circles
7.3.1.3
and related geometric figures
7.3.2.1
Analyze the effect of change of scale, translations and reflections on the
7.3.2.2
attributes of two-dimensional figures.
7.3.2.3
7.3.2.4
Geometry & Measurement Strand Grade 8
8.3.2.1
Solve problems involving parallel and perpendicular lines on a coordinate
8.3.2.2
system.
8.3.2.3
Materials:
Magazines (with photographs of celebrities)
Cubical Portrait Checklist
Project Plan
Rulers
Compass
Protractor
Pencils
Legal-sized or larger poster paper
Page 30 Day 1:
This is the culminating project for the students to show what they know
about lines and angles measures. Read the objective to students and display
student work from previous years.
Distribute the “Cubical Portrait Checklist” and the “Project Plan.” Fill in
today’s project goals for the week so students can stay on track and not fall
behind (this is a tedious project and SHOULD NOT be assigned over a
weekend. Students will have lots of questions).
Today’s Project Goal:
Students need to obtain 2 copies of a picture of a person they would like to
draw (one picture will be used for the rough draft, the other for the “Guess
Who” box on the final copy). The photos needs to be no larger than 3”x3”.
Pictures can be printed off of a website or photocopied in the office.
Things for students to consider while choosing a photograph:
1)
Is the photograph 3” x 3” or smaller?
2)
Does the face fill up most of the photo space (or can the photo be
cropped and copied to make it a good face-shot)?
3)
Is there some facial shading that I can detail?
4)
Is it an interesting photo (consider angles, accessories, facial
features, etc.)?
**Note: Daily project goals may vary by the time on task of the class, time
allotted for the project, and daily interruptions. Make adjustments as
needed.
Day 2:
Students will need to have their photograph approved by the teacher in
order to proceed.
Once the photo has been approved, students start working on drawing lines
on one of their pictures (remind students to save the other copy for their
final copy for display in the hallway). Change every feature from soft,
curved lines into sharp, straight lines.
Discuss features that are essential to the face: eyes, nose, ears, and mouth.
Ask students, “What kinds of lines can you see?” (Example: the bridge of the
nose can be drawn with two parallel lines, the eyes might be trapezoids, the
mouth might be constructed using two triangles drawn on top of an inverted
Page 31 trapezoid). Draw some of the examples on the board for students to
consider.
Today’s Project Goal:
Using rulers, draw straight lines on facial features of the picture. Be sure to
include all necessary facial features: eyes, nose, mouth, chin, ears, jaw line
and outline of hair.
Day 3:
Once the most important facial features are converted into straight lines,
turn the focus onto shading. Ask students, “What shading does the face
have that makes our features more interesting?” Eyebrows, shading on the
sides of the nose, cheekbones, dimples, and wrinkles are the focus for today.
Model how to shade facial features using polygons. Outline the area of the
shading with straight lines to create regular or irregular polygons to make
the drawing more interesting. Also consider the hair. If hair is straight,
encourage students to add parallel lines to emphasis that characteristic. If
it is curly, add polygons in the hair to represent curls. Be creative!
Use the “Cubical Portrait Checklist” as a guideline. Make sure you include all
required lines, angles, and polygons (but feel free to add more than is
required).
Today’s Project Goal:
Focus on facial details. Add polygons for shading to make the drawing more
interesting.
Day 4:
Now it is time for students to start measuring and transferring the drawing
onto the final copy. This will take a couple of days. Review what they learned
in previous lessons: that even though a shape is dilated (by a stretch or
shrink), the angle measures stay the same. This will help transfer the
picture because they can check angles as they go.
Ask students “What happened to the polygons when we dilated it by 2?”
(The angles stayed the same, but all of the side lengths doubled and the
area increased by 2 squared). The same will happen with their project. So,
Page 32 today we have two project goals: 1) choose a scale factor and 2) begin
transferring lines to the final draft.
Before transferring lines, choose a starting point. Discuss this with your
students. What is a good starting point-at the top, on a corner, in the center
of the photo? (There isn’t a right or wrong answer---but students should
have a plan before they begin.)
Model how to start the project with your students. After a place is chosen,
work on stretching lines by the scale factor. It is a good idea to keep track
of measures as you go (numbering lines on the rough draft has helped
students in previous years). Remember to check angles as you go!
Today’s Project Goals:
1) Choose dilation (scale factor) of 1:2, 1:3, or 1:4. Measure the length
and width of your picture and multiply both measures by the scale
factor to make sure your picture will fit on your final copy paper.
2) Choose a starting place to begin measuring from.
Day 5:
Measure and transfer! Today is a workday. It is a good idea to keep track of
all measurements either on your rough draft or on a separate paper.
Today’s Project Goals:
1)
Dilate all lines on the rough draft by the scale factor
2)
Transfer dilated lines onto the rough draft, making sure to check
angle measures as you go.
***Today’s project goals may take a few days.
Day 6 & 7:
Finish up! Ask students to go back to the “Cubical Portrait Checklist” and
label their rough draft with all of the requirements (it is helpful to use color
or highlighter in order to see labels on the photograph---by now the rough
draft will look pretty messy). Students should color the final copy to make it
Page 33 interesting. Remind students to include the “Guess Who?” box to the bottom
corner of the portrait for display before grading.
As students finish, meet with each student for grading. With each student,
go through the checklist and assign completion points. Students should
choose 5 lines and 5 angles to measure with the teacher.
Today’s Project Goals:
1)
Make sure rough draft is labeled with all requirements on the
checklist
2)
Color your final copy
3)
Glue the “Guess Who?” box to the bottom corner of your portrait
4)
Grade your project with your teacher
Page 34 Project Plan:
What should I expect to have done each day?
Day
Today’s Goal (What task should I work on?)
1
2
3
4
5
6
7
Page 35 Done?
Cubical Portrait Checklist
_____ Identify a set of parallel lines in blue (1).
_____ Identify a set of perpendicular lines in green (1).
_____ Identify and name 4 different angles in yellow (4).
_____ Identify and name 5 different polygons in orange (5).
_____ Measure all facial features on your picture (2).
_____ Use a 1:2, 1:3, or 1:4 scale to enlarge your picture (1).
_____ Convert the new dimensions correctly (5).
_____ Measure & label the angles around the face correctly (5).
_____ Name & picture under the “Guess Who?” box on final (1).
_____ Total Points Earned (out of 25)
_____ %
_____ Grade
Page 36 Rough Draft (Student 1)
Page 37 Final Draft (Student 1)
Page 38 Rough Draft (Student 2)
Page 39 Final Draft (Student 2)
Page 40 Final Draft (Student 3)
Page 41 Slopes
1 Class Period
MN Standard/Objective: 8.3.2.1 Understand and apply the relationships between the
slopes of parallel lines and between the slopes of perpendicular lines. Dynamic graphing
software may be used to examine these relationships
Launch:
I am allergic to downhill skiing. When I was in middle school, my
friend told me he was going to take me skiing and teach me how to ski. We started
out on something called the “Bunny Hill”. I skied down that hill about 4 times, when
my buddy got bored and convinced me to try a different hill. The hill he took me on
next had a sign at the top with a picture of a black diamond. I pushed off of the
top of the hill and went straight down (I didn’t know how to turn or stop yet). I was
passing all the other skiers and am sure I exceeded the speed of sound. I survived
the ordeal, but that was the last time I skied.
Explore:
Why was the bunny hill so much easier to ski than the black diamond?
Talk about slopes, steepness, and how they might be measured.
Have the students open Geometers Sketchpad, “define coordinate system” and
“snap points”. Snapping points will make it easier to identify slopes. Have the
students generate a line. Let them play with the line, making it into a bunny hill and
a black diamond hill.
Have them “Measure the Slope” in sketchpad and explore what happens with the
slope as they manipulate the line.
They may want to add one or two additional lines.
Share:
worksheet.
Have the students share and explain their findings from the
Summarize: There are four types of slopes; positive, negative, zero, and
undefined.
Slopes can be written as a decimal or a fraction. (rise/run)
The larger the number, the steeper the slope.
A vertical line is undefined because you can’t divide by zero.
If Sketchpad is unavailable, use the following virtual manipulative:
http://nlvm.usu.edu/en/nav/frames_asid_303_g_3_t_3.html?open=activities&from=apple
ts/controller/query/query.htm?qt=slope&lang=en
Page 42 Slope Investigation with Sketchpad
Instructions:
• Open Geometers Sketch Pad and click on Mark Coordinate System in
the Graph menu. Next make sure that Snap Points is selected in the
Graph menu.
• Choose the line tool and create a line on the page.
• Select the line and Measure the Slope of the line in the Measure
menu.
• Observe what happens as you manipulate the line.
When a line goes up from left to right, what do you notice about the slopes?
When a line goes up from right to left, what do you notice about the slopes?
What about the slope of a horizontal line?
What do you notice about the slope of a vertical line?
What might the slope of a “Bunny Hill” be? Write the slopes as a decimal and
as a fraction.
What might the slope of a “Black Diamond” hill be? Write the slopes as a
decimal and as a fraction.
Look at some lines with slopes written as a fraction. What significance does
the numerator and denominator have in relationship to the line?
Page 43 Parallel & Perpendicular Exploration
1-2 Class Periods
MN Standard/Objective: 8.3.2.1 Understand and apply the relationships between the
slopes of parallel lines and between the slopes of perpendicular lines. Dynamic graphing
software may be used to examine these relationships
Launch:
My friend drew a plan of a garage he will be building. I told him that
it looks like the building inspector will not approve his plans because the walls, doors
and roof are not straight and aligned with each other. He insists that his drawing is
perfect. It is up to you to determine whether his drawing is “perfectly square”.
Explore:
Have the students open Geometers Sketchpad, “define coordinate system” and
“snap points”. Snapping points will make it easier to identify slopes. Have the
students generate a line. Show them also how to construct a parallel line and a
perpendicular line. Allow them to manipulate the lines and make observations about
the slopes of the lines.
Hand out the garage drawing (made with Sketchpad) and let the students work in
groups to determine whether the building is square or not.
Share:
Let the students present their findings about whether the slopes of the building
are parallel, perpendicular, or skew. As the students share, make a list of
properties of parallel lines, perpendicular lines, and skew lines on the board. Discuss
the attributes of these lines and move toward formal rules.
Summarize:
•
•
•
Parallel lines always have the same slope and never intersect each
other.
Perpendicular lines have slopes that are negative reciprocals of
each other and intersect at 90° angles.
Skew lines have different slopes and will always intersect at
exactly one point.
Page 44 25
20
15
10
5
-10
10
-5
-10
-15
-20
-25
-30
-35
Page 45 20
30
Perpendicular Exploration
1 Class Period
MN Standard/Objective: 8.3.2.1
Understand and apply the relationships
between the slopes of parallel lines and between the slopes of perpendicular lines.
Dynamic graphing software may be used to examine these relationships
Launch:
A roofing company is concerned about the safety of their
workers as they work high up on various rooftops. In order to enhance the
safety of their workers, they are requiring that foot braces be placed on
the rooftops.
The braces must be manufactured to join the roof surface at a 90° angle.
Your job is to determine the slope of the braces for various roofing jobs
that are coming up.
Explore:
Let the students work in groups on the “Roof Bracket Handout.
Share:
Let the students present their findings about perpendicular slopes, and then
have the students share the results in their tables. Discuss different
strategies to find the slope of a line given two points.
Summarize:
Students should understand that the slope of a perpendicular line is the
negative reciprical of the slope of the other line.
Also, students should be moving toward understanding that given two points:
a!c
c!a
(a,b) and (c,d), the slope can be found by b ! d or d ! b .
Page 46 Roof Bracket Handout
Example: Given the roof slope (thick
line), what should the slope of the
bracket (thin line) be?
The roof slope is 1/-1 or -1.
The slope of the bracket should be
1.
Figure the slopes for each roof below and for the brackets needed.
Page 47 What is the slope of a line passing through the following points? What is the slope of the
perpendicular lines? Use graph paper if necessary.
Points on Line
Slope of Line
Slope of Perpendicular
Line
(3,1) and (6, 10)
(-2, -7) and (1, 5)
(-5, 3) and (3,3)
(7, 2) and (7, -9)
(4, 5) and (-5, 4)
(6.2, 1) and (8.1, 5)
(251, -1000) and (-400, 54)
(a, b) and (c, d)
Can you find a rule to determine the slope of a line without having to
draw it on a piece of paper?
Page 48 Parallel Lines Exploration
1 Class Period
MN Standard/Objective: 8.3.2.1 Understand and apply the relationships between the
slopes of parallel lines and between the slopes of perpendicular lines. Dynamic graphing
software may be used to examine these relationships
Launch:
A city planner drew out the street layout for a new development.
Before he could finish his design, he won the lottery and moved to Tahiti. In order
to get the city workers to quit leaning on their shovels and start constructing the
streets, you need to determine the angles for all of the intersections which are not
labeled on the following street maps.
Explore:
Let the students work in groups on the Angle Puzzle Handouts.
Share:
Have the students post their solutions around the room. Share the strategies they
used to come up with their solutions.
Summarize:
Parallel lines cut by a transversal have interesting angles:
• Alternate Interior
• Alternate Exterior
• Vertical
• Corresponding
Triangles have interior angle sum of 180°
Quadrilaterals have interior angle sum of 360°
Extension:
Have the students use Geometers Sketchpad to design their own “street
angle puzzles”. Make sure that they set the Precision in the Preferences
Menu to Units if they want to keep the angle measures whole numbers.
Page 49 Page 50 Find all the missing
angles in this drawing
using the three angles
given.
70°
43°
91°
Page 51 43°
43°
137°
137°
Find all the missing
angles in this drawing
using the three angles
given.
43°
67°
43°
70°
91°
43°
89°
89°
137°
137°
43°
43°
67°
70°
43°
67°
70°
67°
70°
89°
91°
134°
46°
91°
91°
134°
134°
46°
46°
134°
Find all the missing
angles in this drawing
using the three angles
given.
Page 52 43°
60°
86°
Find all the missing
angles in this drawing
using the three angles
given.
Page 53 60°
120°
60°
120°
60°
60°
43°
43°
60°
60°
120°
60°
34°
94°
86°
86°
120°
94°
94°
86°
86°
94°
77°
43°
43°
77°
43°
137°
60°
137°
43°
146°
146°
34°
Lines & Angles Pre/Post Test
Give the slopes of the following line segments
Line A
What is the slope of a line perpendicular to Line A?
What is the slope of a line perpendicular to Line B?
Are lines L and M parallel? Explain how you know.
Page 54 Line B
Given that lines L & M are parallel and
measures of the following angles:
lines N & O are parallel, find the
Angle a
Angle b
Angle c
Angle d
Angle e
Angle f
Angle g
What are the slopes of the lines passing through the following points?
(-7, 4) and (3, 3)
(4, 5) and (-3, 5)
(2, 2) and (2, 4)
(-7, 2) and (-9, 5)
(a, b) and (c, d)
Page 55 Duplicate this angle in
the space below.
Find the measurement of the
following angles in degrees.
Page 56 A.
B.
D.
E.
C.
Fill out the following table for the polygons above.
(Hint: some shapes can have more than one name.)
Polygon
Name(s)
Sum of interior angles
A.
B.
C.
D.
E.
Page 57