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Last Updated: July 23, 2002
Functioning in the Real World Chapter 8.2 Lessons 1 and 2
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience:
Self-contained Basic Geometry class of 4 sophomore and junior students
who are hard of hearing.
Description of Lessons:
Using TI-83 calculators and Geometer Sketchpad the students
will practice the use of the formulas in section 2 of chapter 8 regarding
distance of a line, midpoint of a line, and equation of a circle. The students will work in pairs for both of
these lessons and integrate their knowledge of graphing and Geometer Sketchpad
to complete assignments. Students will
use the TI-83 tutorial provided by Math Teacher Link to familiarize themselves
with the variety of functions on the graphing calculators. The students will use the calculators while
solving the problems related to the above equations.
Functioning in the Real World Chapter 8.2 Lesson 1
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience: Self-contained Basic Geometry class of 4 sophomore and junior students who are hard of hearing.
Prerequisites:
Students need a basic understanding of graphing. Completion of TI 83 tutorial is needed to
use calculators.
Class time: 50 minutes
Materials:
graph paper, TI 83 calculators, Chapter 8 Notes, worksheet homework
paper, and copies of Functioning in the Real World to complete homework
problems.
Procedure: Divide the students into pairs
that will work together during this lesson.
My students worked as partners to complete the TI 83 tutorial.
1.
Give each student a piece of graph paper and have him fold
it in fourths. Instruct the students to
draw an x and y axis in each of the 4 sections. Each student should draw a right triangle in 2 of his sections
then trade papers with his partner and draw 2 more right triangles in the
remaining sections.
2.
Review the order of coordinates (x,y). Ask the students to place the coordinates of
2 of the triangles near the vertices.
Trade papers with the partner and write the coordinates of the 2
remaining triangles on this paper.
Direct the partners to check the work on both papers to see that they
are in agreement on the coordinates.
3.
Direct the students to count the number of squares on each
of the legs so the length will be known.
Ask if they can determine the length of the hypotenuse by counting
squares on the paper. Ask for reasons
why they can or why they cannot.
4.
Hand out notes/worksheet for Chapter 8. Have the partner teams read the top 2
paragraphs. Guide the entire class through the practice exercise involving
triangle ABC. Hand out the calculators
and allow them to enter the same formula on the calculator to solve.
5.
When the students have completed the first triangle
instruct the pairs to go on to triangle DEF.
Follow the same procedure and use the calculator to find the distance of
DF. When each set of partners has
finished they will share their answer with another set of partners. If the 2 groups answers are the same one
member of the group will share their answer with the teacher. If the 2 pairs answers are not the same the
foursome should try to find what is different and how to correct it, then share
a common answer with the teacher.
6.
Before continuing with the notes have students select one
of the triangles they drew on graph paper to start. Ask if they know how to find the middle point or midpoint of each
side. Wait for responses. If no one suggests folding direct them to
fold each side in half by placing vertex to vertex. Write the approximate coordinate of the midpoint at the
fold. Trade papers with their partner
and have the partner check for midpoint accuracy. Ask if this is a way that can always be used for finding
midpoints. Discuss how it would be
impossible at times to fold due to size, location and medium (i.e., a computer
screen).
7.
Have the students return to the note/worksheet pages. Ask if anyone remembers how to find the
average of 2 things. Have one person
from each pair write the formula for averaging 2 numbers on the board. Correct any errors.
8.
Go through the given example for AB together. Ask if there are any questions. Direct the students to work with their
partner to find the midpoints of the other sides. Give 5 to 10 minutes to complete. As pairs are working check progress.
9.
Students should add these 2 new formulas to their Geometry
notebooks and begin their homework.
Students may continue working in pairs until class is finished. Remainder of homework should be completed
individually and returned tomorrow.
The homework is from pages 523-524, numbers 1 to 8. Have the students copy the problems from the
book and include the formula for each answer for full credit. 8 formulas and 8 answers equal 16 total points.
Functioning in the Real World Chapter 8.2 Lesson 1
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience: Self-contained Basic Geometry class of 4 sophomore and junior students who are hard of hearing.
Prerequisites:
Students need a basic understanding of graphing. Completion of TI 83 tutorial is needed to
use calculators.
Class time: 50 minutes
Materials:
graph paper, TI 83 calculators, Chapter 8 Notes, worksheet homework
paper, and copies of Functioning in the Real World to complete homework
problems.
Procedure: Divide the students into pairs
that will work together during this lesson.
My students worked as partners to complete the TI 83 tutorial.
10.
Give each student a piece of graph paper and have him fold
it in fourths. Instruct the students to
draw an x and y axis in each of the 4 sections. Each student should draw a right triangle in 2 of his sections
then trade papers with his partner and draw 2 more right triangles in the
remaining sections.
11.
Review the order of coordinates (x,y). Ask the students to place the coordinates of
2 of the triangles near the vertices.
Trade papers with the partner and write the coordinates of the 2
remaining triangles on this paper.
Direct the partners to check the work on both papers to see that they
are in agreement on the coordinates.
12.
Direct the students to count the number of squares on each
of the legs so the length will be known.
Ask if they can determine the length of the hypotenuse by counting
squares on the paper. Ask for reasons
why they can or why they cannot.
13.
Hand out notes/worksheet for Chapter 8. Have the partner teams read the top 2
paragraphs. Guide the entire class through the practice exercise involving
triangle ABC. Hand out the calculators
and allow them to enter the same formula on the calculator to solve.
14.
When the students have completed the first triangle
instruct the pairs to go on to triangle DEF.
Follow the same procedure and use the calculator to find the distance of
DF. When each set of partners has
finished they will share their answer with another set of partners. If the 2 groups answers are the same one
member of the group will share their answer with the teacher. If the 2 pairs answers are not the same the
foursome should try to find what is different and how to correct it, then share
a common answer with the teacher.
15.
Before continuing with the notes have students select one
of the triangles they drew on graph paper to start. Ask if they know how to find the middle point or midpoint of each
side. Wait for responses. If no one suggests folding direct them to
fold each side in half by placing vertex to vertex. Write the approximate coordinate of the midpoint at the fold. Trade papers with their partner and have the
partner check for midpoint accuracy.
Ask if this is a way that can always be used for finding midpoints. Discuss how it would be impossible at times
to fold due to size, location and medium (i.e., a computer screen).
16.
Have the students return to the note/worksheet pages. Ask if anyone remembers how to find the
average of 2 things. Have one person
from each pair write the formula for averaging 2 numbers on the board. Correct any errors.
17.
Go through the given example for AB together. Ask if there are any questions. Direct the students to work with their
partner to find the midpoints of the other sides. Give 5 to 10 minutes to complete. As pairs are working check progress.
18.
Students should add these 2 new formulas to their Geometry
notebooks and begin their homework.
Students may continue working in pairs until class is finished. Remainder of homework should be completed
individually and returned tomorrow.
The homework is from pages 523-524, numbers 1 to 8. Have the students copy the problems from the
book and include the formula for each answer for full credit. 8 formulas and 8 answers equal 16 total
points.
Functioning in the Real World Chapter 8.2 Lesson 2
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience: Self-contained Basic Geometry class of 4 sophomore and junior students who are hard of hearing.
Prerequisites:
Students need a basic understanding of graphing and Geometer Sketchpad
program. Completion of TI 83 tutorial
is needed to use calculators. Lesson 1
of this chapter will have been completed the previous day.
Class time: 50 minutes
Materials:
TI 83 calculators, Geometer Sketchpad in the computer lab, white board
and marker or black board and chalk, Chapter 8/2 worksheet, and copies of Functioning
in the Real World to complete homework problems.
Procedure: Divide the students into pairs
that are different from the previous day and have each pair log on to a
computer.
1.
Direct the students to open Geometer Sketchpad, select
graph, and show grid.
2. Assign one partner the responsibility of constructing 1 circle in any quadrant. Next, the other partner should construct a different sized circle using a different quadrant on the same grid.
3. Instruct the students to work together with their partners to construct a point on the circles, find the coordinates, select the centers and find the coordinates. Each person should attempt this on one circle. While the students are working write the equation of the circle with radius r centered at (x0, y0) with the example from pg. 521(A circle w/ radius of 7 centered at C (6,1). Handout the worksheets and calculators to each student.
4. Introduce the formula using the example from the book. Demonstrate the use of the formula. As a class work on the example that gives a point on the circle and the center. Direct the students to fill in the blanks as we go through this.
5. Following the fill in the blank part of the worksheet the partners should use the formula and coordinates of their circles to find the equations of the circles. After they have solved for r or radius, they should select the circle they are working on and use the measure menu to find the exact radius. Compare the answers from the formulas and the measure command. When they have completed both equations, they should explain to the teacher their answers. If these are completed correctly, the pair should work on problems 11 and 12 on page 524 of Functioning in the Real World.
6. In the last 10 minutes of class the pairs will take turns using the white board and marker to explain how they wrote the formula and obtained the answer for 11 or 12. Each pair is responsible for one of them.
7. Direct the students to add this circle equation to their Geometry notebooks.
8. The homework for this lesson is to use the computer lab during study hall or study lunch to construct 2 more circles in the same way the partners did. In a text box use the formula to solve for radius and use the measure menu to find the exact measurement of the radius. 5 points will be given for each construction/text box. 1 point will be given for the constructed circle, coordinates of point and center, measured radius, formula and radius obtained through equation. For extra credit students can use the computer lab during study hall to construct up to 3 more circles with the radii and coordinates of points on the circles and centers in order to use the formula from today. 2 points of extra credit will be given for each sketch and equation.
Geometry Notes from Chapter 8
Section 2
Lesson 1
Functioning in the Real World
Reminders: (x,y)
The x is the horizontal axis (goes across), and the y is the vertical
axis (goes up and down). Always find
the x point first before going up or down to locate the y point.
Sometimes it is very
difficult to use the grids on a graph to figure out distance or midpoint
because the line does not lie horizontally or vertically. Use the graph below and triangles ABC and
DEF to practice using a formula that will give the distance of a line.
To find the midpoint of a
line segment you can find the average of the x coordinates and the y
coordinates to give the coordinate of the midpoint.
Using triangle ABC find the
midpoint of side AB, BC and AC by filling in this formula ½ (x1 + x0) and ½ (y1 + y0). For segment AB that would equal
½(-4+-3) = ½ (-7) = -3.5
and ½(0+0) = ½(0) = 0. The coordinates of the midpoint of AB are (-3.5,
0). Use the space below to find the
midpoints of BC, CA, DE, DF and EF.
Plot the midpoints on the graph above and write the coordinates in
parenthesis.
BC
CA
Notes from Chapter 8
continued
DE
DF
EF
Add the distance formula
and the midpoint formula to your Geometry Notebook.
Homework: Page 523-524: 1-8. You may use this paper to complete your work.
1.
2.
3.
4.
5.
6.
7.
8.
Worksheet for 8.2
Lesson 2
Functioning in the Real World
Equation of the circle with radius r centered at (x0,y0)
(x – x0) squared + (y – y0) squared – r squared
Example: A circle has
a radius of 7 and is centered at C (6,1)
(x- 6)squared + (y – 1)squared = 7 squared = 49
Try this: A circle
is centered at (3, -2) and passes through the point (-2, 1). Find the equation of the circle. (A small sketch may help)
( - )___ + ( - ) ___ = r ___
( )___ +
( )___ = r___
______+ ______= r____
_______= r_____
_______= r
Using the information from the circles you and your partner
have constructed use the above form to write the equations and solve.
Circle 1 Circle 2
Use the back of this paper to do problems 11 and 12 on page
524.
With your partner, be ready to explain how you did these
problems.
Lesson Evaluation for Functioning in the Real World Chapter 8.2 Lesson 1
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience: Self-contained Basic Geometry class of 4 sophomore and junior students who are hard of hearing.
Prerequisites:
Students need a basic understanding of graphing. Completion of TI 83 tutorial is needed to
use calculators.
Class time: 50 minutes
Materials:
graph paper, TI 83 calculators, Chapter 8 Notes, worksheet homework
paper, and copies of Functioning in the Real World to complete homework
problems.
As in any lesson involving
technology there are good and bad points.
My students were thrilled to have the opportunity to learn about the
graphing calculators, however it was hard to keep them on track going through
the tutorial as they were searching for games and other functions on the
keypad. It was a good idea to have them
only be in pairs as I could keep a better eye on their progress and off-task
behavior. Only 1 of my students has a
TI-82, and he had no idea what most of the keys were for other than the regular
calculator keypad.
Steps 1,
2, and 3 of the procedure included a much-needed review of graphing, quadrants
and coordinates, as my students have had no graph paper work since before
Christmas. I had to draw examples on
the board as they went through these steps.
Step 6 of the
procedure was not as obvious as I thought it might be. Actually even after I gave thinking time and
suggested they think of something we could do with the graph paper, they still
were unable to come up with folding for midpoint. It was also difficult for them to see through the paper to fold
vertex to vertex and we couldn’t cut them out because the coordinates were on
the outside of the triangles. Next time
I would instruct them and show them on the board to write the coordinates just
inside the vertices. We then could cut
out the triangles for easier folding.
All of the
students remembered how to find averages and had no difficulty with steps 7 and
8.
During the
“homework time” of the lesson the students were exploring with the
calculators. Because this was the first
opportunity to use them I did not redirect as often as I might normally. However, the math department policy is that
the graphing calculators can only be used in class. So the students had to use another study hall or study lunch to
complete the homework so I could be with them.
If I had a larger class this might not have been possible. My students’ study hall is the same period,
and I have a small class that period so it was not an interruption to have them
come to my room to work on the homework.
A teacher with large classes would most likely not have this luxury. Possibly a study/resource center would have
graphing calculators that students could use.
We are opening a resource center next year in our building, and I may
suggest that a classroom set of calculators be part of the supplies ordered.
Overall, my
students enjoyed the break from our book.
They worked well in partners for the most part. This class is difficult to split into pairs,
as 2 of the 4 students are ADHD as well as hard of hearing. Another of the students is a very dependent
learner and prefers to ask me questions as he moves through procedures. I had to continually direct students back to
their partners for advice. The homework
grades were higher than the grades the students have been receiving in class,
but it also involved fewer problems than we have been doing lately. Any change in routine is definitely a
positive for this group!
Lesson evaluation for Functioning in the Real World Chapter 8.2 Lesson 2
Jill McCue
Streamwood High School, 701 W. Schaumburg Rd.
Streamwood, IL 60107
Student Audience: Self-contained Basic Geometry class of 4 sophomore and junior students who are hard of hearing.
Prerequisites:
Students need a basic understanding of graphing and Geometer Sketchpad
program. Completion of TI 83 tutorial
is needed to use calculators. Lesson 1
of this chapter will have been completed the previous day.
Class time: 50 minutes
Materials:
TI 83 calculators, Geometer Sketchpad in the computer lab, white board
and marker or black board and chalk, Chapter 8/2 worksheet, and copies of Functioning
in the Real World to complete homework problems.
The best thing
about this lesson was the ease at which the boys used Geometer Sketchpad. We have not used this program as a class
since February, but the students had no trouble completing the steps using the
program. The most difficult part of
this lesson for the students was the use of positives and negatives in the
quadrants when filling in the formula.
This provided very good practice for the use of the negative key as
opposed to the subtraction key on the calculators.
Steps 1
through 4 of the procedure went very well and took less time than I
anticipated. However, when the students
worked in their pairs on step 5, filling in the x and y coordinates correctly
was a little tricky. Problems 11 and 12
from the book were solved easily after I gave a hint to quickly sketch a
circle, point and center if they were having difficulty with filling in the
formula. This step was especially
enjoyed because they got to use the whiteboard in the computer lab. Dry erase markers were more fun than
Geometer Sketchpad and TI-83 calculators as none of my classrooms have a white
board!
The students
used their study hall period to complete the homework, and all of them had
enough time to complete some extra credit.
Because I was not in the lab during the study hall time, I do not know
how independently the boys worked on this assignment, however, they did have
different circle sizes and it did not appear as if any copying and pasting
transpired.
I feel
confident that the students understand how to enter the numbers in the formulas
of the last 2 days. I am less confident
that they know the formulas without looking at them. This will be assessed on the final exam at the end of this
month. The motivation to use the
calculators and computer lab continues to be high, and I received 2 sets of
completed homework papers from all 4 students.
This is not the usual in this class, and the students had less work time
on the assignment due the next day than they normally do. This, once again, shows me that the use of
technology and altering the routine of classes is a benefit to students and
teachers.