Lesson 1.4 Parallelograms

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Objective
To model the classification of quadrangles based on
their properties.
1
materials
Teaching the Lesson
Key Activities
Students review the meanings of parallel lines, line segments, and rays. They compare
various parallelograms and quadrangles.
Key Concepts and Skills
• Develop a definition for parallel and intersecting line segments, lines, and rays.
[Geometry Goal 1]
• Develop a definition for perpendicular line segments. [Geometry Goal 1]
• Describe characteristics of parallelograms. [Geometry Goal 2]
Math Journal 1, pp. 10 and 11
Student Reference Book, p. 100 (optional)
Study Link 1 3
Teaching Master (Math Masters, p. 14)
+, – Fact Triangles
Geometry Template or straightedge
See Advance Preparation
• Classify quadrangles based on side and angle properties. [Geometry Goal 2]
Key Vocabulary parallel lines • intersect • parallel line segments • parallel rays
• perpendicular line segments
Ongoing Assessment: Recognizing Student Achievement Use journal page 10.
[Geometry Goal 1]
2
materials
Ongoing Learning & Practice
Students play Subtraction Top-It to practice subtraction facts.
Students practice and maintain skills through Math Boxes and Study Link activities.
Math Journal 1, p. 9
Student Reference Book, pp. 263 and 264
Study Link (Math Masters, p. 15)
Game Master (Math Masters, p. 506)
number cards 1–10 (4 of each); six-sided
or polyhedral dice (optional)
3
materials
Differentiation Options
READINESS
Students explore parallel
line segments with rubber
bands and geoboards.
ENRICHMENT
Students solve a puzzle
involving properties of
parallelograms.
ENRICHMENT
Students play Sz’kwa.
Student Reference Book, p. 310
Teaching Masters (Math Masters, pp. 16
and 17)
Game Master (Math Masters, p. 505)
geoboards; rubber bands; straws;
straightedge; 40 counters (20 each of
2 different colors)
Additional Information
Advance Preparation For Part 1, place copies of Math Masters, page 14 near the
Math Message.
Technology
Assessment Management System
Journal page 10, Problems 2 and 3
See the iTLG.
Lesson 1 4
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Getting Started
Mental Math and
Reflexes
In Lesson 1-1, students were asked to
sort their , Fact Triangles into two
piles: OK and Try Again. Have students
practice the facts in the Try Again pile and
transfer appropriate facts to the OK pile.
When they are finished, have students
fasten the piles with paper clips and store
them until the next practice session.
NOTE Some students may benefit from
doing the Readiness activity before you
begin Part 1 of the lesson. See the
Readiness activity in Part 3 for details.
Math Message
Study Link 1 3
Follow-Up
Take a Properties of
Polygons sheet (Math Masters,
page 14) and follow the directions.
Have students compare
answers with a partner. Ask
volunteers to share how the rectangles
and trapezoids they drew in Problems 1
and 2 are similar and different.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Masters, p. 14)
Invite volunteers to identify what the shapes have in common
and indicate which shapes have that property. Students indicate
thumbs-up if they agree. Have partners share the polygons they
drew for Problem 3.
Explain that geometric shapes can be classified by their
properties. For example, any polygon with four sides is a
quadrangle. As illustrated on Math Masters, page 14, some
quadrangles are squares, some are trapezoids, and so on.
Teaching Master
Name
LESSON
1 4
䉬
Date
Time
Math Message: Properties of Polygons
All of these have something in common.
None of these has it.
1.
Which of these has it? Circle them.
2.
What property do the circled polygons have in common?
Sample answer: All of the polygons have 4
sides. All of the polygons are quadrangles.
3.
Use your straightedge to draw a polygon that has this property.
Sample answer:
Math Masters, p. 14
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Unit 1 Naming and Constructing Geometric Figures
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Developing Definitions of
WHOLE-CLASS
ACTIVITY
Parallel Lines, Line Segments,
and Rays
Tell students that the number of sides is an obvious property of a
shape, but there are many other properties that are less obvious.
This lesson involves one of those properties.
Links to the Future
In Unit 11 of Fourth Grade Everyday
Mathematics, students apply their
understanding of the term parallel as they
describe the relationships between the faces
and edges of geometric solids.
1. Draw two parallel lines on the board, and ask students
what these lines are called. Remind them that a line goes
on without end in both directions.
Parallel lines
●
When you look at a long stretch of straight railroad tracks,
the tracks appear to meet far in the distance. Do they
actually meet? no Do two parallel lines ever meet or cross? no
NOTE To be parallel, lines must be in the
same plane. Two lines that do not meet and
are not in the same plane are called skew
lines. Lay a pencil on a table and stand
another pencil upright a few inches away.
The two pencils suggest skew lines.
Parallel lines are lines on a flat surface that never meet or
cross; they do not intersect.
●
What would happen if railroad tracks were not parallel?
Sample answer: The train’s wheels could not stay on the
tracks.
2. Draw two parallel line segments on the board. Two line
segments in the same plane are parallel if they do not
intersect and they will never intersect no matter how far they
are extended. Parallel line segments are parts of lines that are
parallel. Have students demonstrate parallel line segments
with their arms—either by holding them straight up or by
lining up their forearms, elbow to finger, with a small distance
between them.
Parallel line segments
3. Draw two parallel rays on the board. Two rays in the same
plane are parallel if they do not intersect, and they will never
intersect no matter how far they are extended. Parallel rays
are parts of parallel lines.
Parallel rays
ELL
Adjusting the Activity
To help students remember the definition of parallel, point out that the
three l’s in the word parallel are, in fact, parallel.
Some students may be interested in the mathematical symbols used to indicate
parallel lines or line segments. For example, instead of writing “Line segment AB
is parallel to line segment CD,” students can write AB
D
.
C
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Lesson 1 4
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Student Page
Date
Time
LESSON
Exploring Parallelograms
Parallelograms
14
䉬
1. Circle the pairs of line segments below that are parallel. Check some of
your answers by extending each pair of segments to see if the two segments
in the pair meet or cross.
a.
PARTNER
ACTIVITY
(Math Journal 1, pp. 10 and 11)
94–100
b.
c.
d.
e.
f.
Have students identify the parallel line segments in Problem 1 on
journal page 10. Ask why the other pairs of line segments are not
parallel. If the line segments in Problems 1e and 1f are extended,
they will meet or cross. The line segments in Problem 1b intersect.
Line segments (or lines) that intersect and form right angles, like
those in Problem 1b, are called perpendicular line segments
(or lines).
Use your Geometry Template or straightedge to draw the following quadrangles:
夹
Sample answers:
2. Draw a quadrangle that has 2 pairs of parallel sides.
Have students do Problems 2 and 3 on their own before completing
journal page 11 with a partner.
This is called a
parallelogram .
夹
3. Draw a quadrangle that has only 1 pair of parallel sides.
Ongoing Assessment:
Recognizing Student Achievement
This is called a
trapezoid
.
Journal page 10
Problems 2
and 3
Use journal page 10, Problems 2 and 3 to assess students’ understanding of
parallel line segments. Students are making adequate progress if they are able
to draw appropriate quadrangles. Some students may be able to draw more than
one example.
10
Math Journal 1, p. 10
[Geometry Goal 1]
Ask students to help you list relationships, similarities, and
differences among various quadrangles. For example:
NOTE The term rhombus comes from
Greek by way of Latin. The plural is either
rhombuses or rhombi.
Parallelograms are quadrangles with two pairs of parallel sides.
Squares, rectangles, and rhombuses are parallelograms, but
trapezoids and kites are not.
All squares are rectangles, but not all rectangles are squares.
All four sides of a square or rhombus are the same length.
Squares have right angles; rhombuses can have right angles
but usually do not.
Student Page
Date
Time
LESSON
Parallelograms
1 4
䉬
All squares are rhombuses, but not all rhombuses are squares.
(Rhombuses are usually thought of as “slanted” or diamond
shaped.) A rhombus that is not a square is also not a rectangle.
continued
For Problems 4 and 5, circle the best answer(s). Some items have more than
1 correct answer, so you may need to circle more than 1 answer.
4. A parallelogram is a quadrangle that
has 2 pairs of parallel sides.
Which are parallelograms?
5.
A rhombus is a parallelogram in which
all sides are the same length.
Which are always rhombuses?
A.
squares
A.
squares
B.
rectangles
B.
rectangles
C.
rhombuses
C.
trapezoids
D.
trapezoids
D.
kites
Rhombuses
Try This
A rectangle is a parallelogram that has all right angles. Which of the following
are rectangles? Write always, sometimes, or never to complete each sentence.
Explain your answers.
The key difference between a kite and a rhombus is that all the
sides of a rhombus are equal, but a kite has two adjacent sides
of one length and two adjacent sides of another length.
always rectangles. Explain. A square is a
rectangle with all sides the same length.
7. Rhombuses are sometimes rectangles. Explain. A rhombus is
a rectangle (square) if it has all right angles.
never rectangles. Explain. A trapezoid has
8. Trapezoids are
only 1 pair of parallel sides.
never a parallelogram. Explain. A kite does not
9. A kite is
6. Squares are
have 2 pairs of parallel sides.
Kite
11
Math Journal 1, p. 11
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Unit 1 Naming and Constructing Geometric Figures
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Student Page
Date
Adjusting the Activity
Time
LESSON
Math Boxes
14
䉬
1. Subtract mentally.
Have students open their Student Reference Book to page 100 to see
a visual organizer displaying relationships among quadrangles.
3
b.
K I N E S T H E T I C
T A C T I L E
V I S U A L
d. 15 ⫺ 7 ⫽
e. 13 ⫺ 8 ⫽
9
f.
Draw point C on it.
Sample answers:
B
CA
⫽8⫺5
c. 7 ⫺ 4 ⫽
A U D I T O R Y
2. Draw and label line AB.
6
a. 10 ⫺ 4 ⫽
3
8
5
What are two other names for line AB?
AC, BC, BA,
CA, CB
⫽ 17 ⫺ 8
3. Complete.
Playing Subtraction Top-It
PARTNER
ACTIVITY
Mya sold
boxes.
9
10
Number of Boxes
2 Ongoing Learning & Practice
10
Ana sold
boxes.
8
A. 10 ⫹ 35
6
B. 136 ⫺ 51
4
2
0
C. 200 ⫼ 4
Luz
Ana Mya
Pei
D. 4 ⫻ 15
Students
7
Pei sold
boxes.
box for the number 50? Circle the best
answer.
Cookie Sale
8
Luz sold
boxes.
91
4. Which of these can go in a name-collection
76
149
5. Subtract mentally or with a
paper-and-pencil algorithm.
(Student Reference Book, pp. 263 and 264; Math Masters, p. 506)
a. 76 ⫺ 41 ⫽
b. 52 ⫺ 38 ⫽
Students play Subtraction Top-It to develop automaticity with
subtraction facts. Consider having students record several rounds
of play on Math Masters, page 506.
35
14
12–15
9
Math Journal 1, p. 9
Adjusting the Activity
Use these game variations as appropriate:
Use a regular six-sided die and a polyhedral die with numbers 1–20. Roll both
dice, and subtract the smaller number from the larger one.
Use two polyhedral dice with numbers 1–20. Roll both dice, and subtract the
smaller number from the larger one.
Use only the number cards 1–9. Turn over four cards, form two 2-digit
numbers, and find the difference.
A U D I T O R Y
K I N E S T H E T I C
Math Boxes 1 4
T A C T I L E
V I S U A L
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 9)
Study Link Master
Name
Date
STUDY LINK
Classifying Quadrangles
1 4
䉬
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 1-2. The skill in Problem 5
previews Unit 2 content.
Writing/Reasoning Have students write a response to the
following: For Problem 3, some students wrote that
1
Mya sold 42 boxes of cookies. Explain the mistake they
might have made when reading the graph. Sample answer: They
counted the number of squares. They did not look at the scale to
see that one square represents two boxes of cookies.
1.
A parallelogram is a quadrangle
(quadrilateral) that has 2 pairs
of parallel sides.
99 100
Draw a parallelogram.
Sample answer:
2.
Answer yes or no. Explain your answer.
yes
It has 2 pairs of parallel sides.
yes
b. Is a square a parallelogram?
It has 2 pairs of parallel sides.
yes
c. Is a square a rhombus?
It is a parallelogram with equal sides.
no
d. Is a trapezoid a parallelogram?
It has just 1 pair of parallel sides.
a.
3.
Study Link 1 4
Time
Is a rectangle a parallelogram?
Draw a quadrangle that has at least
1 right angle.
Sample answer:
INDEPENDENT
ACTIVITY
4.
kite .
Sample answer:
(Math Masters, p. 15)
Home Connection Students answer questions and
draw figures to demonstrate their understanding of
the classifications of different quadrangles.
Draw a quadrangle that has 2 pairs
of equal sides but is NOT a parallelogram.
This is called a
Practice
5.
8.
6
6.
⫽ 140 ⫺ 80
9.
12 ⫺ 6 ⫽
60
16 ⫺ 7 ⫽
35
9
⫽ 93 ⫺ 58
7.
10.
210 ⫺ 150 ⫽
123 ⫺ 76 ⫽
60
47
Math Masters, p. 15
Lesson 1 4
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Teaching Master
Name
LESSON
1 4
䉬
Date
Time
Parallel Line Segments
3 Differentiation Options
94
All of these are parallel line segments. Make each pair on your geoboard.
1.
READINESS
None of these are parallel line segments. Make each pair on your geoboard.
2.
Exploring Parallel Line
PARTNER
ACTIVITY
5–15 Min
Segments with Geoboards
(Math Masters, p. 16)
Some of these are parallel line segments. Make each pair on your geoboard.
Circle the parallel line segments.
3.
How would you describe parallel line segments to a friend?
4.
To explore the concept of parallel line segments using a concrete
model, have students make line segments on a geoboard. Ask
them to share their answers to Problem 4. Some students might
use gestures to support their words.
Sample answer: If 2 parallel line segments
in the same plane were to go on forever,
they would never meet or cross.
Solving a Straw-Squares Puzzle
Practice making other parallel line segments on your geoboard.
5.
ENRICHMENT
PARTNER
ACTIVITY
15–30 Min
(Math Masters, p. 17)
Math Masters, p. 16
To apply students’ understanding of the properties of
parallelograms, have them solve a puzzle that requires altering
a rectangular arrangement of straws to create two squares.
ENRICHMENT
Playing Sz’kwa
PARTNER
ACTIVITY
15–30 Min
(Student Reference Book, p. 310; Math Masters, p. 505)
Name
LESSON
1 4
䉬
1.
Date
Time
Straw-Squares Puzzle
Gather 17 straws of the same length.
Arrange them as shown to the right.
To apply students’ understanding of intersecting line segments,
have them play Sz’kwa. Students take turns placing markers on
the Sz’kwa game mat (Math Masters, page 505) at any
intersection that is not already covered by a marker. The goal is to
capture the most markers.
The arrangement of straws forms a rectangle. The object of this puzzle
is to remove straws from the arrangement so that only 2 squares remain.
2.
䉬
You must remove exactly 6 straws from the arrangement.
䉬
You may not move any of the other straws.
Record your work on the picture above by marking an X on the straws
you removed. Trace over the remaining straws that form the 2 squares.
yright © Wright Group/McGraw-Hill
Math Masters, page 17
Sz’kwa Game Mat from Math Masters, page 505
40
Unit 1 Naming and Constructing Geometric Figures