Download 2.01 Construct congruent segments, congruent angles

The learner will demonstrate an
understanding and use of the
properties and relationships in
geometry, and standard units
of metric and customary
measurement.
2
Notes and textbook
references
Notes and textbook
references
2.01 Construct congruent segments,
congruent angles, bisectors of line segments
and bisectors of angles.
A.
Patty Paper Geometry Paper Patty Geometry is a book by
Michael Serra in which he explains how patty paper can be used to
demonstrate many geometric concepts. Patty paper is a square of waxed
paper used to separate hamburger patties. It is available in many restaurant
supply stores as well as in educational supply stores.
The paper is translucent, so congruent segments and angles may be
constructed by tracing one segment using the paper. Segments and angles
may be proved congruent by placing one over the other so that they
coincide. Bisectors of segments may be formed by folding the paper so that
the segment endpoints coincide. Angle bisectors may be formed by folding
the paper so that the rays of the angles coincide.
B.
Where in the World is Carmen Sandiego? (Blackline
Masters II -1 and II - 2)
Materials needed: Rulers and protractors. A map of Europe is used to locate
Carmen Sandiego. The instructions for finding her involve concepts of
congruence and bisection.
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Notes and textbook
references
2.02 Define and identify interior, exterior,
complementary, and supplementary angles
and pairs of lines including skew lines.
A.
Ins and Outs (Blackline Masters II - 3 and II - 4)
Materials needed: Protractors. In this activity, students are introduced to
the concept of exterior angles of a polygon. Students will explore triangles,
quadrilaterals, pentagons, and other polygons in the attempt to determine the
rule for the sum of the interior angles and the rule for the sum of the exterior
angles.
B.
Protractors in Paris (Blackline Master II - 5) Sixth grade
students are studying Europe and Asia. Here they will use a map of Paris to
explore the intersection of streets and the angles they form. Students should
not use a protractor for this activity but rather rely on their knowledge of the
relationships of angles.
2.03 Define and identify alternate interior,
alternate exterior, corresponding and
vertical angles.
A.
To Be or Not To Be
Develop the definition of alternate
interior angles, alternate exterior
angles, and corresponding angles
through the use of a definition
development card. Cards may be
developed similar to thegiven example for alternate interior angles.
B.
Guess Which Figure
(Blackline Master II - 6) This is an
activity to build geometry
vocabulary. In the game, the leader
will secretly select one of the squares
as the target. The students ask “Yes”
or “No” questions to help determine
which figure is the secret square. An
example question is, “Are A and B
alternate exterior angles?”
In each question, the students must
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Grade 6 Classroom Strategies
These are alternate interior angles.
These are NOT alternate interior angles.
Draw a pair of alternate interior angles.
What are alternate interior angles?
use appropriate geometry terms such as alternate interior, alternate exterior,
supplementary, corresponding, obtuse, acute, right, parallel, transversal.
Notes and textbook
references
This activity is a good warm up or class opener. It takes relatively little
time, but each time the secret figure is changed, or each time the students
ask a different sequence of questions, different vocabulary is exercised.
This is a good way to recall geometry vocabulary some months after it has
been taught in class. Periodic review is a good way to help students recall
geometric terminology before it has been completely forgotten and thus
needs to be retaught.
2.04 Identify and distinguish among similar,
congruent and symmetric figures; name
corresponding parts.
A.
Sorting Activity (Blackline Master II - 7)
Materials needed per group: Two pieces of yarn approximately 30 inches in
length, a set of cards cut apart.
Procedure: Students will arrange the string on their table to resemble the
two overlapping circles of a Venn diagram. Cards will be divided among
the members of the group. The teacher will identify two categories for
sorting. Some possible categories include: contains a right angle, contains
an obtuse angle, contains parallel lines, quadrilaterals, triangles, shows two
similar figures which are not congruent, shows two figures with line
symmetry, shows two congruent figures.
The students then spend time sorting the cards according to the categories.
Some cards may fall in the intersection and some may fall outside both
circles. The point of the activity is for students to discuss with each other
how they determine whether the figures fit into the category or not. The
discussion may clear up some misconceptions that students often have. For
example, some think that all triangles are similar or that all rectangles are
similar.
Alternate procedure: The teacher may put up a large Venn diagram on a
bulletin board. The cards may be thumbtacked to the board and students can
place figures in the circles. From time to time, the teacher can reset the
activity by changing the circle category labels and replacing all the cards
outside the circles.
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Notes and textbook
references
2.05 Locate, give the coordinates of, and
graph plane figures that are the results of
translations or reflections in the first
quadrant.
A.
Slammin’ Sammy (Blackline Masters II - 8 and II - 9)
Students explore transformations on the coordinate grid by letting Sammy
run the bases. His finger, shoulder, back, toe, heel, and fist are the points
used to map his journey around the diamond.
2.06 Investigate and determine the
relationship between the diameter and
circumference of a circle and the value of
pi; calculate the circumference of a circle.
A.
Measuring Activity
Materials needed: Tape measure, a variety of round lids. Have students
measure the circumference and diameter of each lid. Record the results and
calculate the ratio of circumference to diameter. Use these data to introduce
the concept of π as a ratio. If the data are graphed, a nearly linear pattern
should form. This will coordinate with objective 3.05.
B.
Circumference Stumpers (Blackline Masters II - 10 and
II -11) These diagrams show how unusual shapes can be the combination of
circles and other basic geometric shapes. Students can use their knowledge
of circles to find perimeters of these shaded shapes.
C.
Sir Cumference and the First Round Table
Sir Cumference and the First Round Table, by Cindy Neuschwander, is a
tale about why and how King Arthur’s round table became round. It
highlights the characteristics of various shapes and gives meaning to the
names radius, circumference, and diameter.
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Grade 6 Classroom Strategies
Notes and textbook
references
D.
Eyes on Space (Blackline Master II - 12) – Sixth grade
students are studying the solar system and technology of space exploration.
In this activity, students are given information about various large telescopes
in use. They use this information to determine radius and circumference and
to arrange the sizes in order.
2.07 Identify the relationship between areas
of triangles and rectangles with the same
base and height.
A.
Index Card Activity
Materials needed: 3 x 5 index cards and scissors. Students are instructed to
mark a point anywhere along the edge of a 3 x 5 card. They will then use
this point and the corners of the opposite side to form a triangle. Students
should calculate the area of the triangle and the area of the entire card. The
triangle is cut out. Students should use the pieces cut away from the triangle
to verify that the area of the triangle is half the area of the entire card.
B.
Geoboard Activity
Materials needed: Geoboard, rubber bands, grid paper. Students will form a
triangle on the geoboard with rubber bands. They will then form a rectangle
around the triangle so that the heights and bases are the same. Students can
count squares to determine the area of the rectangle and to estimate the area
of the triangle. Each figure should be recorded on the grid paper. Do the
students see a pattern? Even when estimating, do they find the area of the
triangle to be about half the area of the square?
2.08 Use models to develop formulas for
finding areas of triangles, parallelograms
and circles.
Slicing π (Blackline Master II - 13)
Materials needed: Scissors. Students cut the segments of a circle apart and
rearrange them to form a “rectangle” as shown. The base of this
“rectangle” is half the
circumference of the circle, that is
half of 2π x radius. The height of
the “rectangle” is the radius. The
area is π x radius squared.
A.
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Notes and textbook
references
Bean π (Blackline Master II - 14)
Materials needed: Scissors, dried beans. Students fill the circle with a layer
of beans to cover it as completely as possible without overlapping. These
beans are set aside to use in the next step. Then the squares are cut apart.
Students should note that the area of each square is the radius squared. The
squares are placed end to end to form a long rectangle. Now the beans are
used to cover the squares that form the rectangle from one end to the other.
It should be observed that the beans will cover slightly more than three of
the rectangles. This provides evidence that the area of the circle is 3.14
times the radius squared.
B.
Nearly π (Blackline Master II - 15) Students compare the
area of a circle with the area of a polygon of nearly the same size. In the
diagram, if the diameter of the circle is d, then the side length of each small
square is 1/3 the diameter, or 2/3 the radius. The area of each small square is
4r2/9. The area of the polygon is equivalent to the area of seven of the small
squares or 28r2/9. This value is 3.11r2, which is very close to π r2.
C.
D.
Shape Exploration (Blackline Masters II - 16 and II - 17)
Materials needed: scissors, recording paper. Procedure: Students are asked
to cut out the shapes provided and find the area and perimeter of each one.
Then the students should use the shapes in combinations to form
parallelograms and larger triangles. The area of each of these can be found
by adding the areas of the parts. These examples can be used to help
generalize formulas for areas of triangles and parallelograms and even
trapezoids.
2.09 Calculate areas of triangles,
parallelograms and circles.
A.
Pick-Up Area (Blackline Masters II - 18 through II - 24)
Materials needed: Deck of area cards, spinner.
Procedure: The cards are shuffled and placed in a draw pile, face down. On
a student’s turn, he may turn over one card and place it face up in the center
of the table. He then spins the spinner. If he can pick up a card with the
area shown on the spinner, he collects that card and may keep spinning.
When he can no longer pick up a card with the correct area, play passes to
the next player. Cards not picked up remain face up in the play area. Play
continues until all cards have been picked up. The student with the most
cards in his possession is the winner.
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Grade 6 Classroom Strategies
Notes and textbook
references
B.
Area of a Polygon
Materials needed: Scissors, rulers. Have students use rulers to construct
polygons with five, six, and seven sides respectively. Have the students
estimate the area of each polygon by overlaying it with centimeter grid
paper. (Blackline Master II - 17) Then have students dissect each polygon
into triangles and measure the base and height for each one. They can then
calculate the areas of the triangles and add to get the calculated area of each
polygon. They should compare the estimated area to the calculated area and
discuss possible errors in deriving the area in each of these ways.
C.
Finding Area (Blackline Master II - 25) Students use their
knowledge of the area of squares and triangles to find the area of the space
station figure shown.
D.
Mini Review – Area (Blackline Masters II - 26 and II - 27)
This mini review covers most of the area skills from this unit. Allow
students to work in pairs to share strategies and skills.
2.10 Model the concept of volume for
rectangular solids as the product of the area
of the base and the height.
A.
Candy Boxes Present
students with this problem. A certain
candy company makes fudge cut into
pieces that are one cubic inch in volume.
That is, they are one inch deep, one inch
wide, and one inch long. If they plan to
make a package that will hold 36 pieces
of candy, what shape could they give the
box? Is there an advantage over one
shape as compared to the others? If the
top of the box is much more expensive
than the sides and bottom, which shape is best? If the cost of the box is
more expensive than the candy, then which shape is best?
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Notes and textbook
references
B.
Buildings An office building is constructed so that an office
can be put in each 20’ x 20’ space. If the ground floor is 60’ by 80’, how
many offices can fit inside? If the building is going to be five stories tall
how many offices can fit inside? See the extension to this problem in
Objective 2.13.
2.11 Convert measures of length, area,
capacity, weight and time expressed in a
given unit to other units in the same
measurement system.
A.
Metric Match (Blackline Master II - 28) Students should
work in groups to reassemble this puzzle. In each place where three
hexagons come together at a corner, three equivalent metric measurements
can be found. The countries around the perimeter of the puzzle are some of
the first countries to use the metric system. As noted on the master, today
only Liberia, Mayanmar, and the United States do not officially use the
metric system.
Teachers may find many metric materials and information on the
status of the U.S. going metric at the website:
http://lamar.colostate.edu/~hillger/
B.
Equivalent Triangles (Blackline Master II - 29) Students
will work in groups to reassemble the triangle puzzle. Where edges touch,
equivalent measurements match each other. When the puzzle is completed,
the students should have reproduced the shaded shape shown on the page.
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Grade 6 Classroom Strategies
Notes and textbook
references
2.12 Estimate solutions to problems
involving geometry and measurement.
Determine when estimates are sufficient for
the measurement situation.
A.
Apple Statistics Have students explore apple statistics at
the web site shown here: http://www.michiganapples.com/quickfacts.html.
Have students use these statistics to create and solve problems related to
measurement. Can one group create a problem that will stump the others?
B.
Measurement Tasks in the Real World Divide students
into groups and give each group a topic of interest such as automobiles, pet
care, amusement park rides, medical care, aviation, etc. Challenge the
group to think of ways in which measurements must be made in each of
these areas. They should attempt to include measurements of length,
weight, volume, and area. Have the students discuss tools used, precision of
commonly used tools, when estimates might be used, results of
overestimation, and results of underestimation.
C.
Estimation Problem Discussion Cards (Blackline Master
II - 30) Divide students into groups to discuss each of the situations
presented on the cards. After the group has analyzed each situation, have
groups share with each other their ideas on estimation.
D.
Hubble Telescope (Blackline Master II - 31) Sixth grade
students are studying the solar system and technology used to explore space.
This activity on the Hubble Telescope asks students to use their knowledge
of geometry and measurement to become better acquainted with the Hubble.
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Notes and textbook
references
2.13 Analyze problem situations, select
appropriate strategies, and use an organized
approach to solve non-routine and
increasingly complex problems involving
geometry and measurement. Use technology
as appropriate.
A.
Geometry Problem Discussion Cards
(Blackline Master II - 32) Divide the students into groups. Students will
discuss each geometry problem and brainstorm a solution. The teacher may
use the same card for all groups, or allow the groups to work on different
problems. Students should share results with the entire class at the end of
their work time.
B.
Space Ship Storage (Blackline Master II - 33) Students
explore various shapes of rectangles to maximize area.
C.
Comets in the News (Blackline Master II - 34) Students use
reports of comets in the news to get an idea of the size of comets and the
frequency of their appearance. Useful materials for this activity are a state
road map and a foam Hefty plate.
D.
Kepler’s Laws (Blackline Masters II - 35 and II - 36) Sixth
grade students are studying the solar system. Kepler’s Laws relate area and
time in the orbit of a comet or planet. Students will estimate irregular areas
on a grid to solve the problems in this activity.
E.
Building Extension (Blackline Master II - 37) Refer to
activity 2.10 B. Suppose the ground floor could be built in one of four
shapes: 60’ x 80’ , 100’ x 40’ , 110’ x 60’ or 120’ x 50’ . Offices on the
ground floor cost $120 per square feet to build. Offices on the 2nd floor are
only $30 per square foot to build, but as the height goes up from there, the
cost goes up $5 per square foot. Which shape should be used to house at
least 500 offices at the lowest cost?
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Grade 6 Classroom Strategies