Download Between-ness, Distances, Midpoints and Bisectors

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2:1:18:Geometric Concepts:Between-ness, Distances, Midpoints and Bisectors
TITLE OF LESSON
Geometry Unit 1 Lesson 18 – Geometric Concepts: Between-ness, Distances, Midpoints and Bisectors
Prove it! What’s on the outside? What’s on the inside? Of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
Introductory lesson necessary for:
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding
parts of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties
of quadrilaterals, and the properties of circles.
MATERIALS
None
LESSON OBJECTIVES
•
To relate angle types to color, sound, movement etc.
•
To introduce the concepts of betweenness, distance, midpoints, and angle bisectors
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Stamp/initial complete homework assignment. Pass back graded work and have
students place in the appropriate sections of their binders.
2) Agenda – Have students copy the agenda.
3) Present Homework – (15 minutes) Each student will demonstrate or explain how to express various types
of angles as music, movements, words, color, feelings or numbers from yesterday’s homework. Watch the
clock so you don’t spend too much time here. Students can choose any one of the homework answers they
wish to demonstrate.
ACTIVITIES – INDIVIDUAL AND GROUP
1.
Discussion: Between-ness – (5 minutes) Ask the question: What does it mean for a point to be between two
other points? (You want the definition to imply ultimately that all three points are on the same line). If X and Y
(Use student names from your class) are in line (line segment or ray) for a movie and their friend Z comes along
and stands in between them, can you easily state where Z is in the line (in relation to X and Y)?
2.
Lecture/Discussion – (10 minutes) Draw a ruler on the board. |--|--|--|--|--|--|--|--|--|--|--| Label the ruler 0 through
12. Ask the question: Is 2 between 0 and 1? (No, of course not.) Ask the question: Is 1.5 between 1 and 2? (Yes)
Is 1 ½ between 1 and 2? (Yes, that is the same question except we used a fraction instead of a number. Now,
let’s transfer this to the angles we’ve been working with. What does it mean to draw a ray between the two sides
of an angle? Have someone draw this on the board.
Refer to the same ruler drawn on the board. Ask the question: What is the distance between a point drawn at 2
and a point drawn at 5? (3) What is the distance between a point drawn at 0 and a point at 1? (1) What is the
distance between a point at 0 and a point at 2? (2) If we labeled the point at 0 A and the point at 1 B and the
point at 2 C and the point at 3 D etc. what is the distance between points A and B? (1) What is the distance
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© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:18:Geometric Concepts:Between-ness, Distances, Midpoints and Bisectors
between the points A and G? (6). We can then say that the length of line segment AG is 6 and the length of line
segment AB is 1, the length of line segment AC is 2 etc. Continue with questions like these. Ask the question: If
we say that two angles that are equal in measurement are congruent, does it make sense to call two line
segments that have equal lengths congruent? (Yes, because congruent means that they are equal in measurement
and the way we measure line segments is by distance.) Have the students write the following definitions in the
Terms and Definitions section of their binders. Line segments are said to be congruent if they have the same
measure. Angles are said to be congruent if they have the same measure.
3.
Lecture: Midpoint – (10 minutes) Write the following definition on the board: Point M is the midpoint of line
segment AB if M is between A and B and line segment AM is congruent to line segment MB. Have students copy
this into their binders. Ask the question: What does this mean? Can any three students demonstrate this idea by
standing in the room? Write the following definition on the board: If M is the midpoint of line segment AB then
M bisects AB. Have the students copy this definition into their binders. Ask a number of questions using the
number line that has been drawn on the board. What is the midpoint of the number line? (6) What is the
midpoint of the line segment that starts at 2 and goes to 4? (3) What is the midpoint of 2 and 6? (4) What is the
midpoint of 2 and 3? (2.5) If students are having trouble with this, mark out a tape ruler on the ground and have
them walk through the questions.
4.
Lecture – (10 minutes) Write the following definition on the board: An angle bisector is a ray that divides an
angle into two equal angles. Have the students copy the definition into their binders. Ask for a volunteer to
demonstrate this concept with a diagram on the board. They will need to draw an angle, then draw the ray and
label the two angles created by the bisector. The two angles should be equal to each other in measurement.
The diagram should look something like the above diagram. Using the following list of angle sizes (30°, 60°,
90°, 120°, 150°, 160°, 180° (and other angle sizes to insure that each student has a unique angle) have each
student repeat the exercise on the board. Each bisector will divide the angle into two congruent angles. For
instance the 30° angle will be divided into 2 15° angles.
5.
Homework Review – Explain the homework assignment. Field questions.
HOMEWORK
Draw a number line that starts at 0 and goes to 10. Write down all the line segments that start and end at a
whole number that are of length 2. How many are there? Write down all the line segments that are of length 3.
How many are there? Write down all the line segments that are of length 4. How many are there? Write down
all the line segments that are of length 6. How many are there? Do you see a pattern here? What do you think
the pattern is? Bonus: Identify the midpoints of each line segment you’ve noted.
2
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:18:Geometric Concepts:Between-ness, Distances, Midpoints and Bisectors
GROUP ROLES
None
DOCUMENTATION FOR PORTFOLIO
None
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© 2001 ESubjects Inc. All rights reserved.