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Transcript
LESSON
10.3
Parallel Lines and Perpendicular
Lines
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute,
obtuse), and perpendicular and parallel lines. Identify these in two
d­imensional figures.
Mathematical Practices
MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision.
F C R Coherence:
Standards Across the Grades
Before
Grade 4 After
3.G.A.1
4.G.A.1
5.G.B.3
5.G.B.4
F C R Rigor:
Level 1: Understand Concepts....................Share and Show ( Checked Items)
Level 2: Procedural Skills and Fluency.......On Your Own
Level 3: Applications..................................Think Smarter and Go Deeper
Learning Objective
Identify and draw parallel lines and perpendicular
lines.
Language Objective
Students discuss with a partner which tools they
need to identify and draw parallel lines and
perpendicular lines.
Materials
MathBoard, straightedge
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 547J.
About the Math
Professional Development
Why Teach This?
Interactive Student Edition
In this lesson, students learn to distinguish between
intersecting lines and parallel lines in a plane: intersecting
lines cross at exactly one point, while parallel lines never
share a commonProfessional
point. Parallel
lines are always the same
Development
distance apart. Students also learn that perpendicular lines
are intersecting lines that form four right angles.
About the Math
Personal Math Trainer
Math on the Spot
As students work through the lesson, be sure to relate
the new terms to real-world examples. The language of
geometry is key to describing things around us.
Encourage students to draw and label figures
described. Creating models of parallel, perpendicular,
and intersecting lines will help students later when
distinguishing between parallelograms and rectangles.
Professional Development Videos
561A Chapter 10
Animated Math Models
iTools: Geometry
HMH Mega Math
Daily
Daily Routines
Routines
Common Core
Common Core
Problem of the Day 10.3
In baseball, home plate is a five-sided figure
with five angles. What type of angle is
angle A? right
A
E
B
D
1 ENGAGE
with the Interactive Student Edition
Essential Question
How can you identify and draw parallel lines and perpendicular lines?
Making Connections
Invite students to tell you what they know about parking lots.
What is a parking lot? A parking lot is usually a rectangular shaped area
with lines for people to park their cars in.
Learning Activity
What is the problem the students are trying to solve? Connect the
story to the problem.
C
Vocabulary intersecting lines, parallel lines,
perpendicular lines
• What problem are you trying to solve? Find out what kind of lines
have been formed in the parking lot.
• If the two lines that are side by side were made longer, would
they ever cross? No, the lines always stay the same distance apart
• Interactive Student Edition
• Multimedia eGlossary
• What kind of angles are formed where the line that runs along
the row of cars meets with the lines that the cars are parked
between? right angles
Literacy and Mathematics
Vocabulary Builder
Materials Word Description (see eTeacher Resources)
Word Description Have students use the
Word Description graphic organizer to
define the new vocabulary words in this
lesson and any previous terms in this
chapter with which they have difficulty.
Drawing examples and nonexamples is
important to distinguish between the
different types of lines or angles.
Write the
definition.
• Have students construct a miniature parking lot in the classroom.
Then have students summarize their observations.
• Have students discuss how a parking lot reflects the properties
they know about pairs of parallel sides in rectangles.
Describe using
facts and
characteristics.
parallel
lines
Write or draw an
example.
Choose one or more of the following activities.
How can you identi fy and
draw parallel lines and
perpendicular lines?
Write or draw a
nonexample.
Lesson 10.3 561B
10.3
Unlock the Problem
Parallel Lines and Perpendicular Lines
To introduce the lesson, have students watch
the Real World Video, Maglev Trains. Maglev
is short for “magnetic levitation”. Levitation
means to rise in the air.
Unlock
Unlock the
the Problem
Problem
You can find models of lines in the world around you.
For example, two streets that cross each other model
intersecting lines. Metal rails on a train track that never
cross model parallel lines.
MP6 Attend to precision.
Review the information in the chart.
• How are intersecting lines and
perpendicular lines the same? How are they
different? Possible answer: both types of lines cross
Term and Definition
Try This!
Explain that the phrase “appear to be” is used
because it is not possible to tell whether the
streets on the map are really parallel; if they
were extended, they may intersect.
• Can more than two lines be parallel?
Explain. Yes; possible explanation: if all the lines are
‹__›
F
G
‹__›
‹__›
Line DE is parallel to line
FG.
DE i FG
The symbol i means “is
parallel to.”
‹___›
‹___›
Line LM is perpendicular LM ' NO
to line NO.
The symbol ' means “is
perpendicular to.”
N
M
O
37
th
W
Write perpendicular, parallel, or intersecting.
36
th
W
• W 36th St and Broadway
intersecting
_____
• W 35th St and 7th Ave
intersecting, perpendicular
_____
• W 37th St and W 36th St
parallel
_____
Math
th
Talk
Yes; possible explanation: a ray is a part of a line. If
the lines the rays are part of are parallel, then the
rays are parallel.
35
St
St
St
ay
W
Try This! Tell how the streets appear to be related.
Broadw
Draw on the board two parallel lines.
•Use hand gestures as you tell students that
parallel lines never intersect and never cross
each other.
I
E
L
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Bernd Mellmann/Alamy Images
Restate
Write It
‹__›
Line HI intersects line JK HI and JK intersect at
at point X.
point X
J
D
Perpendicular lines are lines in a
plane that intersect to form four right
angles.
intersect and parallel lines never intersect.
Read It
K
X
Parallel lines are lines in a plane that
are always the same distance apart.
Parallel lines never intersect.
• Can parallel lines ever be perpendicular?
Explain. No; possible explanation: perpendicular lines
▲ Maglev trains use magnets to lift them
above the tracks while moving.
Draw It
H
Intersecting lines are lines in a
plane that cross at exactly one point.
Intersecting lines form four angles.
at one point, forming four angles. Only perpendicular
lines form four angles that are right angles.
Geometry—
4.G.A.1
MATHEMATICAL PRACTICES
MP4, MP6, MP7
Essential Question How can you identify and draw parallel lines and
perpendicular lines?
MATHEMATICAL PRACTICES
ELL Strategy:
Lesson 10.3
Name
Av
e
2 EXPLORE
4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify
DO
NOTinEDIT--Changes
must
be made through “File info”
these
two-dimensional
figures.
CorrectionKey=B
7th
LESSON
MATHEMATICAL PRACTICES 6
Use Math Vocabulary
Can two rays be parallel?
Explain.
Chapter 10 561
extended, it is possible that more than two of the lines
will never touch, so those lines are all parallel.
• Are all intersecting lines perpendicular?
Explain. No; possible explanation: perpendicular lines
are special types of intersecting lines. Some intersecting
lines that form angles are not right angles.
MP4 Model with mathematics. Have
students make a diagram to help explain.
• Two lines are perpendicular to a third
line. How are the first two lines related to
each other? Explain. Possible answer: the lines
4_MNLESE342286_C10L03.indd 561
Reteach 10.3
parallel.
Use Math Talk to help students
recognize that two rays can be
• Suppose two lines are parallel. Can a part
of each of these lines be a ray? Explain.
2
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
1
Lesson 10.3
Reteach
Name Alphabet Soup
Use all 26 capital letters of the alphabet. Place them into as many
“soups” as possible.
Parallel lines are lines in a plane that are always the same
distance apart. Parallel lines or line segments never meet.
Possible answers are given.
In the figure, lines AB and CD, even if extended, will never meet.
The lines are parallel. Write ABiCD.
Lines AD and BC are also parallel. So, ADiBC.
B
D
C
Lesson 10.3
Enrich
Name Parallel Lines and Perpendicular Lines
A
Differentiated
Instruction
Enrich 10.3
3
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
are parallel. My drawing shows that the two lines are
always the same distance apart.
Math
Talk
10/13/14 8:07 PM
1. Letters
Intersecting lines cross at exactly one point. Intersecting lines that
form right angles are perpendicular.
In the figure, lines AD and AB are perpendicular because
they form right angles at vertex A. Write AD'AB.
with parallel line segments
E, F, H, M, N, Z
2. Letters
with perpendicular line
segments
E, F, H, L, T
Lines BC and CD are also perpendicular. So, BC'CD.
Use the figure for 1–3.
1. Name
RV and ST
2. Name
3. Letters
V
two sides that appear to be parallel.
two sides that appear to be perpendicular.
RV and RS or RS and ST
T
with intersecting, but not
perpendicular, line segments
R
two sides that appear to be intersecting,
but not perpendicular.
with no parallel, perpendicular,
or intersecting line segments
S
3. Name
4. Letters
RV and VT or TV and ST
A, K, M, N, V, W,
X, Y, Z
B, C, D, G, I, J, O,
P, Q, R, S, U
Possible answer: yes; the part of each line from a point
on the line extending in one direction is a ray.
561 Chapter 10
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
4_MNLEAN343153_C10R03.indd 9
10-9
Reteach
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
24/02/14 5:15 PM
4_MNLEAN343153_C10E03.indd 10
10-10
Enrich
24/02/14 5:02 PM
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
Activity Draw and label YX ' YZ
___›
___›
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
Activity
Students draw two perpendicular rays to form
right angle ZYX.
• How are rays YX and YZ related? Possible
intersecting at point Y.
Materials ■ straightedge
answer: they both have the same endpoint Y.
___›
STEP 1: Draw and label YX .
• How can you check if two rays are
___›
STEP 2: Then draw and label YZ .
I can use a square corner to check if
Z
Y
• What does the upside-down T symbol
mean? that the rays are perpendicular
• What are all the ways to name the figure
you drew? angle ZYX, angle XYZ, angle Y
perpendicular?
the rays are perpendicular.
___›
X
___›
STEP 3: Make sure YX and YZ intersect at point Y.
3 EXPLAIN
STEP 4: Make sure the rays are perpendicular.
1. Name the figure you drew.
∠ZYX
2. Can you classify the figure? Explain.
__›
Share and Show
__›
∠ZYX is a right angle, since YX and YZ are perpendicular,
MATH
BOARD
___ ___
1. Draw and label QR i ST .
Think: Parallel lines never intersect. Parallel line segments are parts of parallel lines.
Q
R
S
T
Use Math Talk to focus on
students’ understanding of the
symbols for perpendicular and parallel lines.
___
C
D
B
F
CD and BF
3. Name two line segments that appear to be perpendicular.
___
___
CD and BC
Possible answer: the symbol i looks like parallel line
segments, and the symbol ' looks like perpendicular
line segments.
Math
Talk
MATHEMATICAL PRACTICES 4
Use Symbols How could
the symbols ' and i help
you remember which
relationships they describe?
562
© Houghton Mifflin Harcourt Publishing Company
Possible answers are given.
2. Name two line segments that appear to be parallel.
___
Math
Talk
© Houghton Mifflin Harcourt Publishing Company
Use the figure for 2 and 3.
• What do the symbols for parallel and
perpendicular look like? two parallel line
segments; two perpendicular line segments
If
If
4_MNLESE342286_C10L03.indd 562
Quick Check
Quick Check
3
2
1
3
2
1
Rt I
Rt I
a student misses the checked
exercises
10/13/14 9:19 PM
Then
Advanced
AdvancedLearners
Learners
BOARD
The first problem connects to the learning
model. Have students use the MathBoard to
explain their thinking.
Use the checked exercises for Quick Check.
Students should show their answers for the
Quick Check on the MathBoard.
and perpendicular rays form right angles.
Share
Share and
and Show
Show
MATH
Visual / Kinesthetic
Partners
Materials ruler or straightedge
•Have students draw street maps based on a
description given by their partner.
•One student starts by drawing a horizontal line
across the middle of a blank sheet of paper and
labeling it Main Street.
•The partner then describes another street and
its geometric relationship to Main Street, while
the other student draws it. For example, “Oak
Street is above and parallel to Main Street.”
•Have the partner describe five streets that are
either perpendicular, parallel, or intersecting to
another street on the map.
•Have partners switch roles and repeat.
Differentiate Instruction with
• Reteach 10.3
• Personal Math Trainer 4.G.A.1
• RtI Tier 1 Activity (online)
COMMON ERRORS
COMMON ERRORS
Error Students may confuse perpendicular
lines and parallel lines.
Example For Exercise 2, students
name sides CB and BF as parallel.
Springboard to Learning Help students
remember the meaning of parallel by
underlining the pair of l’s in the word, which
look like parallel lines.
Lesson 10.3 562
Name
On Your Own
On
On Your
Your Own
Own
If students complete the checked exercises
correctly, they may continue with the
remaining exercises.
In Exercises 4 and 5, students identify
perpendicular lines and parallel lines in a
geometric figure. Have students note that lines
or sides of figures can appear to be parallel or
perpendicular.
Use the figure for 4–5.
4. Name a pair of lines that are perpendicular.
‹__›
‹___›
6. RS i TU
C
S
T
M
U
‹___›
‹___›
K
L
‹___›
9. JK ' LM
D
E
‹___›
‹___›
‹___›
10. ST intersecting UV at
11. AB i FG
point X
J
L
S
M
SMARTER
V
X
K
Exercise 12 requires students to recognize
that if the lines were extended, they would
eventually intersect.
• Are lines HL and IM parallel? Explain. No;
563 Chapter 10
H
J
Possible drawings are shown.
___›
____›
___
___
7. KL and KM
8. CD ' DE
___ ___
MATHEMATICAL PRACTICES
the perpendicular line will intersect both parallel lines at
right angles since the parallel lines are the same distance
apart.
B
C
Draw and label the figure described.
Problem Solving • Applications
U
T
A
B
F
G
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the figure for 12–13.
12.
© Houghton Mifflin Harcourt Publishing Company
MP7 Look for and make use of structure.
• If a line is perpendicular to one of two
parallel lines, is it perpendicular to the other
line as well? Explain. Yes; possible explanation:
‹__›
DC and FJ
4 ELABORATE
DEEPER
G
F
A
5. Name a pair of lines that appear to be parallel.
R
possible explanation: the lines appear not to be the same
distance apart.
D
‹___›
FJ and HG
‹___›
SMARTER
‹___›
Dan says that HL is parallel to IM. Is Dan
correct? Explain.
No; possible explanation: if you extend the lines, they
G
J
K
H
L
I
M
will intersect.
13.
DEEPER
Name two intersecting line segments that are
not perpendicular.
___
___
Possible answer: HL and LM
Chapter 10 • Lesson 3
563
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
MATHEMATICAL PRACTICES
ANALYZE • LOOK FOR STRUCTURE • PRECISION
Use the house plan at the right for 14–16.
SMARTER
14. What geometric term describes a corner
Kitchen
of the living room?
Master
Bedroom
Living Room
Possible answers: right angle or perpendicular line
segments
Family
Room
Dining
Room
15. Name three parts of the plan that show line segments.
Bedroom
Bathroom
Possible answers: a wall in the master bedroom, the
For Exercise 16, students must find a pair
of line segments that are always the same
distance apart. In the case of these house
plans, opposite walls of rooms and outer
walls of the house appear to be parallel.
bathroom door, a wall in the dining room
16.
SMARTER
Math on the Spot
Video Tutor
Name a pair of line segments that appear
to be parallel.
Use this video to help students model and
solve this type of Think Smarter problem.
Possible answer: two walls in the living room that do not
intersect and are opposite each other
Use the map at the right for 17–19.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
17. Name a street that appears to be parallel to S 17th Street.
t St
stnu
Che
t
tin S
0th
St
9th
MP4 Model with mathematics. In Exercises
17–19, students identify and name
parallel and perpendicular streets on a map.
• Why do you think city streets often are
designed to be parallel or perpendicular?
St
St
St
Vernon St
Use Diagrams Name a street that appears to be
parallel to Vernon Street.
4
7th
MATHEMATICAL
PRACTICE
8th
S1
18.
S1
Street
S1
Aus
S2
Possible answers: S 18th Street, S 19th Street, S 20th
Perry St
Perry Street
Possible answer: it is easier to navigate and to get
from one place to another.
19. Name a street that appears to be perpendicular to S 19th
Street.
Possible answers: Austin Street, Chestnut Street
20.
SMARTER
Choose the labels to make
a true statement.
‹___›
IJ
I
G
B
‹___›
GH
‹__›
A
E
EF
is perpendicular to
‹___›
AB
___
AE
‹___›
C
GH
F
H
J
D
© Houghton Mifflin Harcourt Publishing Company
SMARTER
564
4_MNLESE342286_C10L03.indd 564
3/9/15 3:27 PM
DIFFERENTIATED INSTRUCTION
INDEPENDENT ACTIVITIES
Activities
Picture Perfect!
5 EVALUATE Formative
Assessment
Essential Question
Differentiated Centers Kit
Activities
Concentrate!
Exercise 20 assesses students’ application of
the concept of perpendicular lines. Students
who choose lines AB and CD have chosen
parallel lines instead of perpendicular
lines. These students may have reversed
the meanings of the two terms. Students
who choose line GH may not understand
the concept of parallel or perpendicular
lines since this line is neither parallel nor
perpendicular to any lines shown in the
figure.
Using the Language Objective
Literature
Skateboarding
Takes Shape
Reflect Have students discuss with a partner
which tools they need to answer the Essential
Question.
How can you identify and draw parallel lines
and perpendicular lines? Possible answer: I can
use the definitions. Perpendicular lines are lines in a plane
that intersect to form four right angles. Parallel lines are
lines in a plane that are always the same distance apart;
they never intersect.
Students complete
orange Activity Card
13 by matching
the names of lines,
angles, and twodimensional shapes
to pictures.
Students complete
orange Activity Card
13 by matching
the names of lines,
angles, and twodimensional shapes
to pictures.
Students read
about how plane
figures and
angles are used
in skateparks.
Math Journal
WRITE
Math
Draw and label an example of two parallel
lines that are perpendicular to a third line.
Lesson 10.3 564
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
Practice and Homework
Name
Lesson 10.3
Parallel Lines and Perpendicular Lines
COMMON CORE STANDARD—4.G.A.1
Draw and identify lines and angles, and classify
shapes by properties of their lines and angles.
Use the figure for 1–2.
1. Name a pair of lines that appear to be perpendicular.
Practice and Homework
Think: Perpendicular
lines form right angles.
‹___›
‹___›
AB and EF appear to form right angles.
‹___›
‹__›
AB and EF
Use the Practice and Homework pages to
provide students with more practice of the
concepts and skills presented in this lesson.
Students master their understanding as they
complete practice items and then challenge
their critical thinking skills with Problem
Solving. Use the Write Math section to
determine student’s understanding of content
for this lesson. Encourage students to use their
Math Journals to record their answers.
A
C
E
F
B
2. Name a pair of lines that appear to be parallel.
D
‹___›
AB and CD
‹___›
Possible drawings are shown.
Draw and label the figure described.
‹___›
‹___› ‹___›
‹____›
‹__›
‹___›
3. MN and PQ intersecting at
4. WX || YZ
5. FH ⊥ JK
point R
M
W
X
Q
R
P
N
F
J
Y
Z
K
H
Problem
Problem Solving
Solving
Use the street map for 6–7.
6. Name two streets that intersect but do not appear
Maple
to be perpendicular.
Birch
k
Oa
7. Name two streets that appear to be parallel to each other.
Elm
© Houghton Mifflin Harcourt Publishing Company
Oak and Maple or Oak and Birch
Maple and Birch
8.
Math Draw and label an example of
WRITE
two parallel lines that are perpendicular to a
third line.
Check students’ drawings.
Chapter 10
4_MNLESE342286_C10P03.indd 565
565 Chapter 10
565
10/13/14 8:16 PM
TEST
PREP
Lesson Check (4.G.A.1)
1. Write a capital letter that appears to have
perpendicular line segments?
2. In the figure, which pair of line segments
appear to be parallel?
F
J
Possible answer: T
Continue concepts and skills practice with
Lesson Check. Use Spiral Review to engage
students in previously taught concepts and
to promote content retention. Common Core
standards are correlated to each section.
G
H
FG and JH
Spiral Review (4.NBT.B.5, 4.NBT.B.6 , 4.NF.A.2, 4.G.A.2)
acute angles did he draw?
2 acute angles
5. A school principal ordered 1,000 pencils.
He gave an equal number to each of
7 teachers until he had given out as many as
possible. How many pencils were left?
6 pencils
4. Mike drank more than half the juice in his
glass. What fraction of the juice could Mike
have drunk?
_
Possible answer: 5
8
6. A carton of juice contains 64 ounces.
Ms. Wilson bought 6 cartons of juice. How
many ounces of juice did she buy?
384 ounces
© Houghton Mifflin Harcourt Publishing Company
3. Nolan drew a right triangle. How many
FOR MORE PRACTICE
GO TO THE
566
Personal Math Trainer
Lesson 10.3 566