Download Unit 6 Geometry: Constructing Triangles and Scale

Unit 6 Geometry: Constructing Triangles and
Scale Drawings
Introduction
In this unit, students will construct triangles from three measures of sides and/or angles, and will
decide whether given conditions make exactly one triangle (up to congruence), more than one
triangle, or no triangle. Students will also solve problems involving scale diagrams.
Materials. Students will use protractors frequently in this unit. If commercial protractors are
unavailable, photocopy BLM Protractors (p. G-58) onto a transparency and cut it into eight
separate protractors. Do this as needed for each student to have one.
In Lesson G7-5, students are asked to use a triangle-building machine. We recommend
preparing enough of them ahead of time so each student and you can have one, instead of
using class time to build them. See BLM Triangle-Building Machine (p. G-59) for instructions
on how to make them. Alternatively, if you have class time available, you could have students
make their own machine. You will need a pack of 100 paper fasteners to do this. These should
be available at your local office supplies store.
Grid paper. We recommend that students have grid paper and that you have a background grid
on your board. If students do not have grid paper, you will need to have lots of grid paper
available (e.g., from BLM 1 cm Grid Paper on p. J-1). If you do not have a background grid on
your board, you will need to project a transparency of a grid onto the board so you can write
over the grid and erase the board without erasing the grid.
Technology: dynamic geometry software. Students are expected to use dynamic geometry
software to draw geometric shapes. Some of the activities in this unit use a program called The
Geometer’s Sketchpad®, and some are instructional—they help you teach students how to use
the program. If you are not familiar with The Geometer’s Sketchpad®, the built-in Help Centre
provides explicit instructions for many constructions. Use phrases such as “How to construct a
line segment of given length” to search the Index. NOTE: If you use a different dynamic
geometry program to complete these activities, the instructions provided may need to be
adjusted.
Fraction notation. We show fractions in two ways in our lesson plans:
Stacked:
1
2
Not stacked: 1/2
If you show your students the non-stacked form, remember to introduce it as new notation.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-1
G7-1
Angles
Pages 152–153
Standards: preparation for 7.G.A.2
Goals:
Students will recognize lines, rays, points, and line segments.
Students will compare angles and will name angles as right, obtuse, or acute.
Vocabulary: acute, angle, arc, arm, degree (°), endpoint, line, line segment, obtuse, point,
ray, right angle, rotation, vertex
Materials:
large part of a small circle and small part of a large circle, made from bristol board
transparency of BLM Quarter-Circle Protractors (p. G-56)
overhead projector
2 thin rays cut from bristol board with arrows at one end
Introduce points, lines, line segments, and rays. Draw the pictures below on the board,
making sure the line segment is 70 cm long and the line is drawn shorter than the line segment:
point
line
line segment
ray
Always be sure to model using a ruler to make lines, line segments, and rays straight, since
students will need to do this themselves.
Point to the point, and SAY: This is called a point. Point to the line segment and SAY: A line
segment is a straight path between two points, called endpoints. You can measure the length of
a line segment. Have a volunteer do so. Then SAY: This line segment has lots of points. Show
some of them on the board:
Now point to the line. SAY: A line extends in a straight path forever in two directions. It has no
endpoints. The arrows at both ends show that you can extend the line in both directions. Point
out that it looks shorter than the line segment, but it is actually much longer because you can
extend it as much as you want in both directions. You cannot measure the length of a line
because it goes on in both directions. Point to the ray and SAY: A ray has one starting point and
G-2
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
goes on forever in one direction, like a ray of sunlight starting at the sun goes on forever. You
cannot measure a ray either.
Exercises: Name each object as a point, a line, a line segment, or a ray.
a)
b)
c)
d)
e)
Answers: a) point, b) line, c) ray, d) ray, e) line segment
Draw on the board:
Point to the picture on the left and ASK: Do these objects meet when extended as much as
possible? (yes) Show the meeting by extending the line. Then point to the picture on the right
and ask the same question. (no) SAY: Only the line can be extended. Show the extending and
how, this time, there is no meeting point. In the Exercises below, students can signal thumbs up
for yes and thumbs down for no when you take up the answers.
Exercises:
1. Do the rays, lines, or line segments meet when extended where possible?
a)
b)
c)
d)
e)
f)
g)
h)
Answers: a) yes, b) no, c) yes, d) no, e) no, f) yes, g) no, h) yes
2. Is the given point on the object (the line, the line segment, or the ray)?
a)
b)
c)
d)
Answers: a) yes, b) yes, c) no, d) no
Introduce angles. Tell students that an angle is the space between two rays that have the
same endpoint. Draw on the board:
Tell students that the endpoint is called the vertex and the two rays are called the arms of the
angle. The vertex is easy to see, so they don’t need to draw the dot to show it.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-3
The size of an angle is the amount of rotation between the arms. Tell students that the size
of an angle is the amount of rotation between the two arms. Cut out two thin rays from bristol
board and tape one of them to the board. Show how you can make a small or large angle using
the other ray by rotating it away from the first ray either a small or large amount. Have students
stand up facing the front and rotate in place until they see various objects in the classroom,
such as a clock, a bookshelf, a computer, etc. After each rotation, ask students if the rotation
was a greater or lesser amount of rotation than the previous object.
Draw the following pictures on the board without the arcs to illustrate what you mean by smaller
and larger angles:
Explain that angles are drawn with an arc (a part of a circle around the vertex of an angle) to
show how much one arm turns to get to the other. Add the arcs to the picture.
Using the amount of space between the arms to compare sizes. Draw on the board:
Tell students that you want to use the definition of an angle as the space between the arms as a
way to compare angles to say which one is larger. Color the space between the arms in both
pictures and ASK: Which angle has more space between the arms? (the one on the right)
SAY: So the one on the right is a larger angle. But you need to be careful because sometimes
the same angle can be drawn with shorter arms, which make it look as though there is less
space between the arms. Draw the same angles as above on the board, but this time draw the
one on the right with shorter arms, as shown below:
Tell students that if you want to compare angles by coloring the space between the arms, you
need to make sure you draw the arms the same length. Instead of coloring the space, another
way to compare the space between the arms is to check which angle fits onto the other. Show
students two angles: a large angle cut from a small circle and a small angle cut from a large
circle. Demonstrate how the smaller angle fits onto the larger angle when you place their
vertices together as shown below:
G-4
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Exercises:
1. Which angle is larger?
a)
A
B
b)
c)
A
B
A
B
Answers: a) A, b) B, c) A
2. Which picture in Exercise 1 is deceptive?
Answer: The picture for part b) is deceptive because the smaller angle has longer arms drawn.
The size of an angle is measured in degrees. Tell students that they can measure the size of
an angle. SAY: The unit of measurement for an angle is a degree. Write on the board:
90 degrees = 90°
Point at the degree symbol and SAY: We write this symbol for degrees. Show students a piece
of paper and point out the horizontal and vertical sides. SAY: The amount of rotation needed to
go from a horizontal line to a vertical line is 90 degrees. The angle between a horizontal line and
a vertical line is called a right angle.
Project BLM Quarter-Circle Protractors on the board. Draw several angles starting at the 0°
mark (10°, 70°, 60°, Bonus: 35°, 75°) and have students individually record the angles shown.
Then continue with more angles that do not start at the 0 mark. (50° to 90°, 20° to 30°, 60° to
80°, Bonus: 55° to 90°, 75° to 85°)
Introduce acute and obtuse angles. Tell students that the angles they have seen so far were
all less than 90° because they all fit into a right angle. Smaller angles have smaller degree
measures. SAY: Angles that measure less than 90° are called acute angles and angles that
measure more than 90° are called obtuse angles. You can compare an angle to a square corner
to decide if it is acute or obtuse. Draw several angles on the board (see examples below):
Have volunteers compare the angles to a square corner (e.g., from a sheet of paper) to decide
whether the angle will measure more or less than 90°. Then have the volunteers say whether
each angle is acute or obtuse. (From left to right, the angles are obtuse, acute, acute, and
obtuse.)
Ask a volunteer to look up the word “acute” in the dictionary: it means “sharp.” Point out that a
small angle is sharper than a large angle; it would hurt more if you walked into an edge that had
an acute angle than if you walked into an edge that had an obtuse angle.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-5
Have students draw a mix of several acute and obtuse angles in their notebook. Emphasize the
importance of using a ruler to make the rays straight, so that their partner can easily compare
their angle to a square corner. Have them switch notebooks with a partner and label their
partner’s angles as obtuse or acute.
Exercises: Does the degree measure represent an acute angle, an obtuse angle, or a right angle?
a) 75°
b) 83°
c) 100°
d) 90°
e) 12°
f) 94°
Answers: a) acute, b) acute, c) obtuse, d) right, e) acute, f) obtuse
Activity
Teach students to draw lines, line segments, and rays using the line tool of The Geometer’s
Sketchpad®. Then ask them to draw a line and an independent point. Ask them to move the
point so it looks as though it is on the line. Then have them modify the line. Does the point stay
on the line? (no) Now show students how to construct a line through two given points and a
point on the line, so that modifying the points keep the line and the points together.
(end of activity)
Extensions
1. What is the angle between the hands of a clock?
a) at 3:00
b) at 1:00
c) at 2:00
Answers: a) 90°, b) 30°, c) 60°, d) 120°
d) at 4:00
(MP.1) 2. Do the objects below intersect (meet)? Check all possibilities of the objects being
lines, line segments, and rays in both directions.
G-6
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-2
Measuring and Drawing Angles
Pages 154–156
Standards: preparation for 7.G.A.2
Goals:
Students will use protractors to measure and draw angles.
Prior Knowledge Required:
Can recognize qualitatively when one angle is larger than another
Vocabulary: acute, angle, arc, arm, base line, degree (°), endpoint, line, line segment, obtuse,
origin, point, protractor, ray, vertex
Materials:
protractors
transparency of BLM Simple Protractors (p. G-57)
transparency of BLM Protractors (p. G-58)
BLM Protractors (p. G-58)
overhead projector
Introduce protractors. Give each student a protractor and SAY: This is a protractor. Have
students examine their protractors and compare them to rulers. Remind students that when they
measure with a ruler, they have to line up one end of the object with the zero mark. Have
students find the zero mark on their protractors (There are two of them, one at each end!) Point
out that a protractor has two scales, both with the same unit, but in opposite directions. A ruler
can have two scales, too, but the two scales would use different units. Explain that having two
identical scales going in different directions allows you to measure the angles from both sides,
but this also means that you need to decide which scale you will use each time.
Project BLM Simple Protractors onto the board. Explain that the protractors on the BLM are
simplified pictures of a protractor, without all the tiny markings in between the larger angles.
Draw the angles below on the board:
Pointing at the first picture, ASK: Is this angle an acute angle or an obtuse angle? (acute) Circle
the numbers 30 and 150 that the arm of the angle passes through. ASK: Which one is the
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-7
answer? (30°) How do you know? (the angle is acute, so the angle measure has to be less
than 90°) Repeat for the second picture. (this time, the angle is obtuse, so the correct choice is
150°, not 30°) Point out that there is another way to check that you are using the correct scale.
The correct scale starts with zero on the arm of the angle.
Exercises: Measure the angle.
a)
b)
c)
d)
Answers: a) 60°, 120°, 120°, 60°
Now project BLM Protractors on the board and ask similar questions.
Exercises: Measure the angle.
a)
b)
c)
d)
Answers: a) 130°, b) 55°, c) 155°, d) 45°
G-8
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Placing protractors on angles. Point out the base line and the origin on a large protractor or
on a picture of a protractor on the board, as shown below:
Have students find the base line and the origin on their protractors. Demonstrate how to place a
protractor correctly, so that the base line lines up with one arm of the angle and the origin is at
the vertex. Point out that this is similar to placing a ruler with the zero at the beginning of the
object you are measuring. Provide students with BLM Protractors for the following Exercises.
Exercises: Draw the angle on BLM Protractors two ways, starting at either zero.
a) 50°
b) 70°
c) 15°
d) 165°
Selected answer:
d)
Have students draw an acute angle in their notebooks, then ask them to place their protractors
correctly. Circulate in the classroom to check that all students have done so. Then have
students measure the angle they drew. Repeat with an obtuse angle. Have students exchange
notebooks with a partner and measure each other’s angles to check their work.
(MP.3) Draw an angle on the overhead projector and demonstrate measuring incorrectly. Have
students tell you what you are doing wrong. For example, place the vertex at several different
incorrect locations, such as exactly on the bottom of the protractor or along the small central arc
of the protractor. You could also place the vertex correctly, but have one arm outside the range
of the protractor while the other is aligned with 0 or have neither arm aligned with the zero.
Extending the arms of an angle to meet the angle measures on a protractor. Draw a small
angle on the overhead projector and use a protractor to demonstrate how it is difficult to read
the measurements because the arms do not reach the scale. SAY: Sometimes you have to
extend the arms so that you can read where they meet the scale. Do so, then have a volunteer
read the angle measure.
Exercises: Copy the angle onto grid paper. Then extend the arms and measure the angle.
a)
b)
c)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-9
Answers: a) 53°, b) 45°, c) 34°
Drawing angles. Model drawing angles step by step. To draw a 60° angle:
Step 1: Use a ruler to draw a ray with the arm long enough to pass through the zero mark.
Place the protractor so that the origin is at the vertex of the angle, and the zero mark lies along
the arm you drew.
Step 2: Make a mark at 60°. Be sure to use the same scale that shows zero on the arm of the
angle.
Step 3: Using a ruler, join the endpoint of the ray to the mark. Add an arrow to complete the
second ray.
Exercises: Draw the angle.
a) One arm is horizontal, pointing right, and the angle opens upward and measures …
i) 40°
ii) 72°
iii) 154°
b) One arm is horizontal, pointing left, and the angle opens upward and measures 120°.
c) One arm is vertical, pointing down, and the angle opens right and measures …
i) 70°
ii) 96°
Answers:
a) i)
ii)
ii)
b)
G-10
c) i)
ii)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Activities 1–2
1. Teach students to draw and measure angles using The Geometer’s Sketchpad®. Then ask
them to try moving different points (on the arms of the angle or its vertex) so that the size of the
angle becomes, say, 50°. ASK: Is it easy or hard to do? (hard) When you move the line
segments, does the angle change? (yes) When you move the vertex or other point on the arms,
does the angle change? (yes) Show students how to draw an angle of fixed measure by using
menu options. ASK: Will moving the endpoints change the size of the angle now? (no) Show
students how to draw angles equal to a given angle.
2. Have students draw polygons in The Geometer’s Sketchpad® and measure the size of the
angles and the length of the sides of these polygons. Have students check that the angle
measures they obtain make sense. For example, if they click on three vertices of a quadrilateral
then use menu options to measure the angle, it might produce different angles depending on
the order in which the vertices were selected. Also, the software sometimes measures angles in
the wrong direction, producing an answer more than 180°.
(end of activities)
Extensions
(MP.1) 1. Have students find the angle between the clock hands at …
a) 3:30
b) 2:30
c) 4:30
d) 12:12
e) 12:24
f) 1:36
g) 3:48
Hints: An hour is 60 minutes and a whole circle is 360°. What angle does the minute hand cover
every minute? (6°) How long does it take the hour hand to cover that many degrees? (12
minutes) How do you know? (because the hour hand covers only one twelfth of the full circle in
an hour, moving 12 times slower than the minute hand) If the time is 12:12, where do the hour
hand and the minute hand point? (the hour hand points one fifth of the way from 12 to 1, the
minute hand points 2/5 of the way from 2 to 3) What angle does each hand make with a vertical
line? (hour hand: 6°; minute hand: 72°) What is the angle between the hands? (66°)
Answers: a) 75°, b) 105°, c) 45°, d) 66°, e) 132°, f) 168°, g) 174°
2. Some scientists think that moths travel at a 30° angle to the Sun when they leave home at
sunrise. Note that the Sun is far away, so all the rays it sends to us seem parallel.
a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the
Sun in the evening?
b) A moth sees the light from a candle flame and thinks it’s the Sun. The candle is very near to
us and the rays it sends to us go out in all directions. Where does the moth end up? Draw the
moth’s path.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-11
Answers: a) 30°, b) The moth spirals toward the candle flame.
G-12
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-3
Constructing Triangles from Angles and Sides
Pages 157–158
Standards: 7.G.A.2
Goals:
Students will learn how to recognize congruent shapes.
Students will understand that two given angles with a side length between them make exactly
one triangle (up to congruence), when the sum of the two angles is less than 180°.
Prior Knowledge Required:
Can use a protractor to measure angles
Can draw angles
Knows that parallelograms have opposite sides equal
Vocabulary: angle, congruent, degree (°), line, line segment, parallel, protractor, ray
Materials:
protractors
grid paper or BLM 1 cm Grid Paper (p. J-1)
small triangles cut from bristol board, a different one for each student
blank paper
scissors
Some pairs of angles can be used to make a triangle and others cannot be. Draw on the
board:
a)
b)
c)
d)
e)
For each picture, ASK: Are the rays getting closer together, farther apart, or neither? (a) closer
together, b) farther apart, c) closer together, d) farther apart, e) neither) ASK: If you were to
extend the rays, would they meet? (a) yes, b) no, c) yes, d) no, e) no) Have a volunteer extend
the rays in part a) so they meet. Point out that the volunteer drew a triangle. Tell students that
this class is about constructing triangles. If you know two of the angles you can sometimes
extend the rays to make a triangle and sometimes you cannot.
(MP.8) Determining a condition on the angles for the rays to make a triangle. Draw on the
board:
60°
Tell all students to draw a 60° angle in their notebook with one arm being a horizontal line
segment and the other arm being a ray on the left side of the first line segment, pointing upward.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-13
When students finish, tell one third of the class to draw a ray on the right side of the line
segment that will make the rays get closer together, another third to make them get farther
apart, and the rest to try to make them stay the same distance apart. Then have everyone
measure the angle they drew. Write on the board:
Angles that make
the rays get closer
Angles that make
the rays get farther apart
Angles that make the rays
stay the same distance apart
Fill in the chart as a class. ASK: What angles make the rays get closer together? (angles less
than 120°) What angles make the rays get farther apart? (angles greater than 120°) What
angles make the rays stay the same distance apart? (exactly 120°) Tell students that if they
were trying to make the rays stay the same distance apart, they should have drawn an angle
very close to 120°. It won’t be exact because exact measurements are not possible, but they
should be close. To summarize, SAY: When the second angle is 120°, the rays stay the same
distance apart—they are parallel and always go in the same direction—they do not make a
triangle. When the second angle is less than 120°, the rays get closer together so they make a
triangle. When the second angle is larger than 120°, the rays get farther apart so they do not
make a triangle.
When two lines are parallel, the angles add to 180°. Provide students with grid paper or BLM
1 cm Grid Paper if they do not already have grid paper. Remind students that parallelograms
have opposite sides equal and if there are two parallel sides that are equal in length, students
can make a parallelogram. Show students how they can draw, for example, two horizontal sides
of length 3 and make parallel lines to join them. Start by drawing two line segments that go right
1 square and up 3 squares, as shown below:
SAY: If both line segments go right the same number of squares and up the same number of
squares, then they are parallel. Have students draw in their notebooks the parallel lines with a
horizontal side of length 3 between them. Tell students that they have created two angles, as
shown below:
Exercise: Use a protractor to measure the two angles you just drew.
Answer: 72° and 108°
G-14
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Bonus: Create other pairs of parallel line segments with a horizontal line segment between
them. Measure the angles created in this way.
(MP.8) When students finish, ASK: What do the two angles add to? (180°) Point students’
attention to the 60° and 120° angles that also created parallel lines. SAY: When two lines are
parallel, the angles they make in this way add to 180°. If the lines weren’t parallel and the
angles added to less than 180°, the rays would be getting closer and could create a triangle if
extended. If the angles added to more than 180°, the rays would be getting farther apart.
(MP.8) Exercises: Add the angles to predict whether you can create a triangle satisfying these
conditions. Then check your prediction.
a) two angles are 70° and the side between them has length 5 cm
b) two angles are 90° and the side between them has length 4 cm
c) a 40° angle and a 50° angle and the side between them is 3 inches long
d) a 70° angle and a 120° angle and the side between them is 7 cm long
Answers: a) yes, 70 + 70 < 180; b) no, 90 + 90 = 180; c) yes, 40 + 50 < 180;
d) no, 70 + 120 > 180
Congruent shapes. SAY: Two shapes are congruent when they are the same size and the
same shape. Tell students that you’re interested in congruent shapes because if two people
each draw a triangle that satisfies certain conditions, you want to know if the two triangles have
to be congruent or if there are two non-congruent triangles that satisfy the conditions.
SAY: Color, pattern, and orientation don’t affect congruence, only size and shape do.
Exercises: Are the two shapes congruent?
a)
b)
c)
d)
f)
e)
Answers: a) no, b) yes, c) no, d) yes, e) yes, f) yes
SAY: When two shapes are congruent you can fit one exactly on top of the other with no spaces
or overlaps. You might need to flip it or rotate it, but you will be able to fit it onto the other one
exactly.
Activity
Give each student a unique, fairly small, triangle cut out from bristol board. Have students trace
the triangle several times on a sheet of paper, being sure to flip and/or rotate it. Ask them to
also draw some non-congruent triangles and to try to make it difficult to see which ones are
congruent. Then have students exchange sheets and triangles with a partner. The partner will
then predict which shapes are congruent to the triangle and check their answers by rotating or
flipping to see if the triangle fits exactly.
(end of activity)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-15
Using a grid on the board (e.g., projected from BLM 1 cm Grid Paper), draw several shapes on
the grid as shown below:
Ask volunteers to draw shapes congruent to the shapes on the board, but ask them to orient the
shapes differently by rotating and/or flipping them. Draw the first two shapes below on the
board:
Ask students to explain why the shapes are congruent. Add the third shape and ask them to
explain why this shape is not congruent to the other two. (sample answer: it has a row of
squares that is three squares long and the other two don’t)
Do two given angles and the side length between them make exactly one triangle (up to
congruence), more than one triangle, or no triangle? Write on the board:
a 60° angle, a 70° angle, and the side between them is 2 inches long
ASK: Will it be possible to create a triangle satisfying these conditions? (yes) How do you
know? (60 + 70 = 130, which is less than 180°) Tell students that you know you can create at
least one triangle, but now you want to know how many different triangles you can create. Show
the four ways of starting the triangle below on the board:
A.
B.
60°
70°
2 in
C.
70°
60°
70°
D.
60°
2 in
2 in
70°
2 in
60°
Give each student a letter (A, B, C, or D) and have them draw the triangle according to their
letter on a blank sheet of paper, then cut out the triangle.
Bonus: Draw a triangle that is different from any of the four shown, but still satisfies the
conditions.
G-16
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
When students are finished, have them cut out their triangles and get into groups of four
students, each with a different letter than themselves. Have them check for congruence by
rotating or flipping when necessary. (all triangles are congruent) If some students made extra
triangles, they can check those for congruence, too. Tell students that all triangles satisfying the
given conditions are congruent.
Exploring congruent shapes. Draw on the board:
SAY: You can add a square to this shape in 10 different places (point to them), but some of the
ways will be congruent. Show the first five ways below on the board:
Point to the first two shapes and ASK: Are these two shapes congruent? (no) Point to the third
shape and ASK: Is this congruent to either of the first two shapes? (yes, to the first shape) How
do you know? (you can flip the first shape to get the third shape) Is the fourth shape congruent
to any previous shape? (no) How do you know? (it has a row of squares that is four squares
long) How about the fifth shape? (yes, it is congruent to the fourth shape) Cross out the shapes
that are the same as a previous shape, as shown below:
SAY: So we have three non-congruent shapes so far.
Exercise: Continue going around the shape adding squares. Did you find any new shapes or
are they all congruent to a previous shape?
Answer: They are all congruent to a previous shape.
Draw on the board:
a)
b)
c)
d)
e)
f)
For each shape, ASK: How many places can you add a square to this shape by going around in
order? (8, 10, 12, 14, 16, 18)
(MP.1) Exercises: Copy the shape into your notebook as many times as you need to. How
many non-congruent ways can you add a square to the shape?
a)
b)
c)
d)
e)
f)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-17
(MP.8) Bonus: How many non-congruent ways can you add a square to a rectangle made of 3
rows of 19 squares?
Sample solution: e)
Answers: a) 3; b) 3; c) 2; d) 4; e) 5; f) 5; Bonus: 12, because except for the 3 by 3 array, the
pattern would be 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, and so on.
Extensions
(MP.3) 1. The pattern in the last Bonus question of the lesson “should” be 3, 3, 4, 4, 5, 5, ….
Why does it change when there are 3 rows in each square?
Answer: When there are 3 rows of 3 squares, adding a square to the side does the same thing
as adding a square to the top or bottom. In any of the other cases, adding a square to the side
is always different from adding a square to the top.
(MP.1) 2. How many non-congruent shapes can you make by removing one square from the
3 by 3 array?
Answer: 3 shapes, as shown below:
(MP.1) 3. How many non-congruent shapes can you make by removing 2 squares from the
3 by 3 array?
Answer: 8 shapes. Encourage students to proceed systematically in looking for the answer. For
instance, they might start by finding all the shapes they can make after they have removed a
corner square:
Then they could try removing a middle square on the outside of the figure:
G-18
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
(Notice the last two shapes have been crossed out because they are already on the list for the
previous figure.) Finally, they could try removing the middle square (but all of the shapes that
can be made after removing the middle square are already listed).
WORKBOOK 6:1 PAGE 163-164
4. Investigate to find what the relationship is between the angle measures a and b in the picture
below that makes the rays to get closer together and hence makes a triangle.
a
b
Answer: b > a
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-19
G7-4
Drawing Triangles on Grid Paper
Pages 159–160
Standards: 7.G.A.2
Goals:
Students will draw triangles on grid paper to satisfy given conditions.
Students will understand that some conditions are consistent with more than one non-congruent
triangle.
Prior Knowledge Required:
Understands that congruent shapes are the same shape and size
Can draw a 90° angle on grid paper
Can measure angles
Vocabulary: angle, congruent, degree (°), protractor, ray, unique triangle
Materials:
grid paper or BLM 1 cm Grid Paper (p. J-1)
transparency of BLM 1 cm Grid Paper (p. J-1)
Review drawing triangles with a protractor and ruler. Remind students that in the last lesson
they drew triangles from given conditions—in particular, two given angles and a side between
them. ASK: What do you have to know about the angles to know that it’s possible to create the
triangle? (that they add to less than 180°) If the conditions do make a triangle, how many
different triangles do they make? (only 1; all possible triangles are congruent) What tools did
you use to draw the triangles? (a protractor and a ruler)
Drawing right triangles with grid paper instead of a protractor and ruler. Tell students that
they will draw triangles from given conditions, but instead of using a protractor and a ruler, they
will use grid paper. Explain that some angles are really easy to draw using grid paper. ASK: How
can you use grid paper to draw a 90° angle? (the horizontal and vertical lines make a 90° angle
with each other)
Tell students that in the Exercises below, all lengths are in units equal to the length of one grid
square.
Exercises: Draw a triangle according to the given conditions.
a) two sides of length 3 and the angle between them is 90°
b) a side of length 2, a side of length 5, and the angle between them is 90°
Have students cut out their triangles from the Exercises above and check with a partner that
their triangles from each part are congruent.
G-20
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Drawing 45° angles on grid paper. Have students draw a 90° angle on grid paper, then fold it
in half. Have students predict what the degree measure will be (45°) and explain their prediction.
(45 is half of 90) Have students use a protractor to check that their prediction is correct. Then
have students look at the fold marks they made and ASK: How can you use the grid lines to
draw a 45° angle on grid paper? (go up or down one unit for every unit you go right)
NOTE: For the Exercises below, you may need to get students started by drawing a vertical or
horizontal ray for them.
Exercises: On grid paper, draw as many ways of making a 45° angle as you can.
Bonus: Draw 135° angles.
Sample answers:
Drawing triangles that have two 45° angles.
Exercises:
a) Draw a triangle with two 45° angles. The side between the angles is …
i) 2 units long
ii) 5 units long
iii) 7 units long
b) Then cut out the triangles and make sure they are congruent to a partner’s triangles.
Bonus: Draw the first triangle from part a), but not on grid paper. Make sure it is congruent to
the triangle you drew on grid paper.
(MP.8) The third angle in a triangle with two 45° angles is always 90°. Then have students
use a protractor to measure the third angle. ASK: What do you notice? (the third angle is always
90°) Tell students that when two angles in a triangle are 45°, the other angle is always 90°.
NOTE: In the next lesson students will learn that the sum of the angles in a triangle is always
180°. For now, students only need this special case.
Non-congruent triangles with the same angles in both triangles and one same side
length. Draw on the board:
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-21
Have students list some of the things these two triangles have in common. Prompt them to think
about angles and side lengths. (sample answers: both have two 45° angles and one 90° angle,
both have at least one side of length 4)
SAY: These two triangles have a lot in common. ASK: Are they congruent? (no) How do you
know? (they are not the same size) If some students say they are not the same shape, tell
students that actually they are the same shape, and that would be easier to see if you could
rotate one of them. ASK: Which triangle has the 90° angle between two 4 cm sides? (the
second one) Which triangle has the 90° angle opposite the 4 cm side? (the first one) SAY: If you
know two angles and a side length, or even three angles and a side length, you can still have
two non-congruent triangles satisfying the conditions. If there is exactly one triangle (up to
congruence) that satisfies the conditions, then you can say that the triangle is the unique
triangle satisfying the conditions.
Exercise: On grid paper, draw two non-congruent triangles with two 45° angles and a side of
length 3.
Answer: Both triangles will have a 90° angle. One triangle will have the sides adjacent to the
90° angle of length 3, and the other triangle will have the side opposite the right angle of
length 3.
Extension
How many non-congruent triangles can be made to satisfy the conditions: a unique triangle, no
triangle, or more than one triangle?
a) a 90° angle, a 45° angle, and the side between them is 5 cm long
b) two 45° angles and the side between them is 2 inches long
c) three 45° angles
d) two 45° angles and a 90° angle
e) two 45° angles and a side 2 inches long
f) two 90° angles and the side between them is 4 inches long
Answers: a) a unique triangle, b) a unique triangle, c) no triangle, d) more than one triangle,
e) more than one triangle, f) no triangle
G-22
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-5
Constructing Triangles from Three Sides or
Three Angles
Pages 161–162
Standards: 7.G.A.2
Goals:
Students will determine the conditions for which three side measures or three angle measures
make exactly one triangle, more than one triangle, or no triangle.
Prior Knowledge Required:
Can determine whether two triangles are congruent by trying to fit one on top of the other
Can measure angles
Vocabulary: angle, congruent, degree (°), point, protractor, unique triangle, vertex
Materials:
a triangle-building machine for each student (plus one for yourself), made using BLM TriangleBuilding Machine (p. G-59) and two paper fasteners
paper clips
rulers
Constructing triangles from three side lengths. Provide each student with a triangle-building
machine made from BLM Triangle-Building Machine and two paper fasteners. Also give each
student a paper clip. Tell students that when they use their triangle-building machine to make
triangles, the longest side will always be 10 units long. They could change the machine to make
any number of units up to 10, but in this class, they will always use 10 as the longest side.
Show students how to make a triangle with any given side lengths. For example, to make a
triangle with side lengths 5, 7, and 10, fold one of the strips under at the 5 and the other strip
under at the 7, and then rotate them until the 5 dot and the 7 dot meet. Tell students to make
sure they keep all the strips flat along the table. Show students how they can do it wrong if they
stretch the strips by lifting one end off the table. That makes the dots meet at a point farther
from the other side than they should. Students can hold the third vertex of the triangle together
with a paper clip, then check with a partner to see if their triangles are congruent. Remind
students that when two triangles are congruent, one will fit exactly on top of the other without
gaps or overlaps, but they might need to rotate or flip the triangle to make it fit exactly. Make
sure students do at least one of the following Exercises. Students who finish early can do more.
Exercises: Use the triangle-building machine to construct the triangle. Check with a partner that
your triangles are congruent.
a) 3, 9, 10
b) 6, 8, 10
c) 8, 8, 10
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-23
If some students find that their triangles are not congruent, emphasize the importance of
building the machine correctly and of not stretching any sides. Point out also that if some
students made the 3 and 9 sides from part a) on opposite sides of the 10, then one partner will
need to flip their triangle to fit onto the other one. Be sure that students have done it correctly
and have found their triangle to be congruent to those of everyone else who has made the
triangle with the same side lengths.
(MP.8) Not all side lengths make a triangle. Tell students that not all possible side lengths will
work. Demonstrate with sides of length 2, 3, and 10 (see diagram below). SAY: The sides of
length 2 and 3 are too far from each other to complete the triangle.
2 cm
3 cm
10 cm
Tell students that you would like to investigate when three side lengths make a triangle. In the
Exercises below, students will need their triangle-building machine.
Exercises: Complete the table.
Sum of the Two Do the Sides Make
Three Sides
Shortest Sides
a Triangle?
a)
4, 5, 10
b)
4, 9, 10
c)
7, 7, 10
d)
2, 4, 10
3, 7, 10
Bonus:
Answers: a) 9, no; b) 14, yes; c) 14, yes; d) 6, no; Bonus: 10, no
(MP.3) SAY: When the shorter sides put together are too short compared to the longest side,
they can’t fit together to make the third point of the triangle. Have volunteers try to articulate the
precise rule. Some students might say that the shorter sides put together can’t be shorter than
the longest side, others might say that the shorter sides put together have to be longer than the
longest side. Point out that there is a slight difference between the two answers if they both
come up. Then tell students you want to know what happens when the longest side is exactly
equal to the shorter two sides put together. Demonstrate by trying to make a 3-7-10 triangle.
Point out how the sides of length 3 and 7 land onto the side of length 10 exactly (you will need
to turn the triangle machine face down to see this) and so, while the two points do meet at a
third point, the “triangle” has no height, so it’s not a triangle!
(MP.6) Use this to emphasize the importance of students being precise while they work. If they
made their triangle incorrectly or stretched the sides by lifting one end off the table, they would
make an incorrect triangle.
Exercises: Without constructing a triangle, decide whether the side lengths make a triangle.
a) 3 in, 5 in, 8 in
b) 2 mm, 5 mm, 6 mm
c) 18 mi, 23 mi, 42 mi
d) 2 cm, 3 cm, 4 cm
e) 2 cm, 5 cm, 8 cm
f) 17 in, 3 in, 20 in
(MP.2) Bonus: 4 cm, 18 mm, 23 mm (Hint: Be careful with the units.)
G-24
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Answers: a) no; b) yes; c) no; d) yes; e) no; f) no; Bonus: yes, because 4 cm = 40 mm, and
18 + 23 > 40
When do three angle measures make a triangle? SAY: We know when three side lengths
make a triangle. Now I want to know when three angle measures make a triangle. Have
students draw triangles using a ruler to make sure that the sides are straight, but without paying
attention to the side lengths. Have them measure the angles in their triangles. When students
finish, point out that the sum of the angles in a triangle is always exactly 180° and it is only
measurement error that will make them get something slightly different. Then remind students
that, in the previous lesson, they learned that when two angles in a triangle are 45°, the third
angle is 90°. SAY: That makes sense because 45 + 45 + 90 = 180.
SAY: Let’s assume we have three angles that add to 180°, so we know that they make a
triangle. Write on the board:
40°, 60°, 80°
SAY: I want to know if they will make a unique triangle or if there will be more than one noncongruent triangle. Have students make predictions, then suggest ways of checking the
prediction. SAY: You don’t know the length of the side between the 40° and 60° angle, so try
making one up and see if it works. Write on the board:
A. a 40° angle, a 60° angle, and a 5 cm side between them
B. a 40° angle, a 60° angle, and a 6 cm side between them
C. a 40° angle, a 60° angle, and a 7 cm side between them
Point to A and ASK: Do these conditions make a unique triangle? (yes) Repeat for B and C.
(both yes) Point to each in turn again, and ASK: What will the third angle be? (80°) How do you
know? (because the three angles have to add to 180°) Have students use a ruler and protractor
to draw all three triangles.
SAY: So all of these triangles have angles 40°, 60°, and 80°, so you can actually make many
triangles having angles 40°, 60°, and 80°. None of these triangles are congruent because if you
put the 40° angle from A onto the 40° angle from B, the 60° angles won’t match up. In B, it is
farther from the 40° angle than it is in A. The triangles are all the same shape, but they are all
different sizes.
Extensions
(MP.1) 1. a) A triangle with perimeter 8 has whole-number side lengths. What are the sides of
the triangle?
b) A triangle with perimeter 13 has whole number side lengths. Its shortest two sides are equal.
What are the sides of the triangle?
Answers: a) 2, 3, 3; b) 4, 4, 5
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-25
(MP.7) 2. Can the quadrilateral below exist? Hint: Divide the shape into a triangle and square.
8
7.5
4
4
Answer: Dividing the shape into a triangle and a square makes a right triangle with side lengths
4, 3.5, and 8, but 4 + 3.5 < 8, so the triangle, and hence the quadrilateral, does not exist.
G-26
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-6
Constructing Triangles from Three Measures
Pages 163–164
Standards: 7.G.A.2
Goals:
Students will be given three measures and decide whether the three measures determine a
unique triangle, no triangle, or more than one triangle.
Prior Knowledge Required:
Can use a protractor and a ruler to draw triangles
Can determine whether two triangles are congruent
Can measure angles
Knows that the sum of the angles in a triangle is 180°
Knows what acute, right, and obtuse angles are
Vocabulary: acute, angle, congruent, degree (°), endpoint, obtuse, point, ray, right angle,
unique triangle, vertex
Materials:
protractors
rulers
Review the conditions for three given angles to make a triangle. Tell students that in this
unit they have been learning what kinds of conditions on triangles make a unique triangle, more
than one triangle, or no triangle. ASK: When do three angles make a triangle? (when they add
to 180°) Review the terms “right angle” (90°), “acute angle” (less than 90°), and “obtuse angle”
(greater than 90°). Then have students signal their answers to the questions below.
ASK: Can there be a triangle with…
a) two right angles?
b) two obtuse angles?
c) three acute angles?
d) exactly two acute angles?
e) a right angle and an obtuse angle?
f) an acute angle and an obtuse angle?
(MP.3) Have volunteers explain their reasoning for any “no” answers or draw a sample for any
“yes” answers. (a) no, two right angles already add to 180°; b) no, they add to more than 180°;
c) yes, for example an equilateral triangle; d) yes, any triangle with a right angle or an obtuse
angle will have exactly two acute angles; e) no, they add to more than 180°; f) yes, any triangle
with an obtuse angle will also have an acute angle)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-27
ASK: Do three angles that add to 180° make a unique triangle or more than one triangle? (more
than one) Ask a volunteer to sketch on the board two non-congruent triangles having angles
90°, 45°, and 45°. (see sample answers below)
Review the conditions for three given side lengths to make a triangle. ASK: When do three
side lengths make a triangle? (when the shortest two sides add to more than the longest side)
What if the shortest two sides add to the same as the longest side? Do they make a triangle?
(no) Why not? Remind students that the two shorter sides will meet at a third point, but that third
point is on the longest side, so the “triangle” has no height and is not actually a triangle.
Another way to frame the side lengths rule for triangles. Remind students that the perimeter
of a shape is the distance around the shape. Draw on the board:
5 cm
4 cm
3 cm
ASK: What is the perimeter of this triangle? (12 cm) How did you get that? (add the three side
lengths) Tell students that they can always get the perimeter of a triangle by adding its three
side lengths. Write on the board:
Longest Side
(cm)
8
8
8
8
8
Perimeter
(cm)
14
15
16
17
18
Sum of the Two
Shortest Sides
Did it Make a
Triangle?
(MP.3) Point to the first row and ASK: What is the sum of the two shortest sides of the triangle?
(6 cm) PROMPT: The longest side is 8 cm and the sum of all three sides is 14 cm. Write “6” in
the third column. ASK: If the two shortest sides add to 6, and the longest side is 8, do the sides
make a triangle? (no) Complete the first row, then have volunteers help you complete the table.
(7, no; 8, no; 9, yes; 10, yes) ASK: How can you explain the rule in terms of longest side and
perimeter? (when the perimeter is more than double the longest side, you can make a triangle)
Exercises: Can you make a triangle with …
a) longest side 10 cm and perimeter 18 cm?
b) longest side 5 inches and perimeter 12 inches?
c) longest side 7 mi and perimeter 18 mi?
d) longest side 1.83 miles and perimeter 3.4 miles?
Bonus: longest side 18 ft and perimeter 11 yd?
Answers: a) no; b) yes; c) yes; d) no; Bonus: no, because 11 yards = 33 feet, and 33 < 18 × 2
G-28
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Review the conditions for two angles and a side. SAY: If two angles are given and the length
of the side between them is given, then you can make a unique triangle satisfying the
conditions. Have students construct a triangle with a 40° angle, a 50° angle, and a 4 cm side
between them. When students finish, write on the board:
• a 40° angle
• a 50° angle
• a 4 cm side
ASK: Does the triangle you drew satisfy all these conditions? (yes) What will the other angle
always be? (90°) How do you know? (because the three angles have to add to 180°) Add this
condition to the list on the board:
• a 40° angle
• a 50° angle
• a 4 cm side
• a 90° angle
SAY: The 4 cm side doesn’t have to be between the 40° and 50° angles; it can also be between
the 40° and 90° angles or between the 50° and 90° angles. So there are actually three different
cases. Show the three different cases on the board:
4 cm
4 cm
40°
50°
4 cm
40°
50°
40°
50°
Provide each student with a protractor and a ruler.
Exercises: Draw all three possible triangles you can make with a 40° angle, a 60° angle, and a
4 cm side.
Bonus: How many non-congruent triangles can you make with …
a) a 40° angle and a 70° angle and a 4 cm side?
b) two 60° angles and a 4 cm side?
Answers: There are three non-congruent triangles depending on where the 4 cm side is;
Bonus: a) there are only two different triangles because the other angle is 70°, and the triangle
with the 4 cm side between one 70° angle and the 40° angle is congruent to the triangle with the
4 cm side between the other 70° angle and the 40° angle; b) there is only one possible triangle
because all angles are 60°, so no matter which two 60° angle the 4 cm side is between, all the
triangles are congruent.
When students finish the Exercises above, point out that when they are told the side length
between two given angles, then the triangle is unique. But if they are just given a side length
without being told where it is, there can be more than one triangle.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-29
Two sides and an angle. SAY: There are four ways you can be given three pieces of
information about a triangle. Write on the board:
• three given side lengths
• three given angle measures
• two angles and a side
• two sides and an angle
SAY: We’ve looked at the first three so far, but we haven’t considered the last one yet. Write on
the board:
• a 3 cm side
• a 5 cm side
• an 80° angle
(MP.3) SAY: Let’s consider all possible cases. ASK: Where can the 80° angle be relative to the
sides? (between the two sides; beside the 3 cm side, but not the 5 cm side; beside the 5 cm
side, but not the 3 cm side)
Demonstrate beginning each case as shown below. NOTE: In the first picture, both sides have
given lengths. The next two diagrams have one side without a given length, so that side
requires a ray to show that the length is infinite.
5 cm
80°
3 cm
80°
80°
3 cm
5 cm
Point out that in the second case, students need to try to find a point on the ray that is 5 cm from
the other endpoint of the 3 cm side. They can do this by using a ruler and rotating it around the
endpoint of the 3 cm side. ASK: At what point on the ray is the distance 5 cm?
In the third case, they need to try to find a point on the ray that is 3 cm from the other endpoint
of the 5 cm side.
Tell students that some cases might not be possible and they have to decide which ones are
possible. The best way to do that is to try to construct all three triangles.
G-30
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Exercises:
1. Try to construct all three triangles. Which ones are possible?
Answer: Only the first two are possible; in the third case, the ray is always farther than 3 cm
from the opposite vertex.
2. Draw two non-congruent triangles with a 3 inch side, a 5 inch side, and a 90° angle.
Answers: The two possibilities are for the 90° angle to be between the 3 inch side and the 5
inch side, and for the 90° angle to be adjacent to the 3 inch side but not the 5 inch side.
Extensions
(MP.3) 1. Sara says a triangle can have longest side 5 cm and perimeter 16 cm. Explain why
she’s wrong.
Answer: If the perimeter is 16 cm, then the sum of the remaining sides is 11 cm. Therefore, the
5 cm side is not the longest.
(MP.3) 2. Find four different triangles that have a 3 cm side, a 5 cm side, and a 30° angle.
Answer: There are three different cases to consider.
Case 1: the 30° angle is between the two sides.
5 cm
30°
3 cm
Case 2: the 30° angle is adjacent to the 5 cm side, but not to the 3 cm side.
3 cm
3 cm
30°
5 cm
This case has two triangles.
Case 3: the 30° angle is adjacent to the 3 cm side, but not to the 5 cm side.
5 cm
30°
3 cm
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-31
3. How many non-congruent triangles can you make with a 60° angle and two 3 cm sides?
Answer: Only one—any triangle you can make with these conditions will be an equilateral
triangle with all sides equal to 3 cm.
NOTE: In Grade 8, students will learn that when two sides are equal in a triangle, the angles
opposite those sides are equal as well. They will then be able to prove their discovery from
Extension 3.
G-32
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-7
Counterexamples
Pages 165–167
Standards: 7.G.A.2
Goals:
Students will determine whether statements are true or false.
Students will use counterexamples to disprove false statements and reasoning to prove true
statements.
Prior Knowledge Required:
Knows that congruent triangles can fit exactly on top of one another without overlaps or gaps
Knows that the sum of the angles in a triangle is 180°
Knows that two non-congruent triangles can have all the same angles
Vocabulary: angle, area, congruent, counterexample, degree (°), perimeter
Materials:
BLM Sudoku—Introduction (pp. G-60–61, see Extension 1)
BLM Sudoku—Another Strategy (p. G-62, see Extension 1)
BLM Sudoku—Advanced (p. G-63, see Extension 1)
Anno’s Hat Tricks by Akihiro Nozaki and Mitsumasa Anno (see Extension 8)
Introduce the term “counterexample.” Draw and write on the board:
All circles are shaded.
Have a volunteer identify which circle shows that the statement isn’t true. (the third one) Tell
students that an example that proves a statement false is called a counterexample to the
statement.
NOTE: For the Exercises below, draw the triangles so that A and D are isosceles, B is right
scalene, and C is equilateral but rotated.
Exercises: Which shape is the counterexample to each statement?
a) All triangles are striped.
b) All triangles have a horizontal side.
Bonus: All triangles have at least two equal sides.
Answers: a) D, b) C, Bonus: B
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-33
Recognizing when a statement does not apply to all examples. Draw and write on the
board:
All circles are shaded.
ASK: What is this statement about? (circles) Underline all the circles. Emphasize that the
statement refers only to the circles—it doesn’t matter whether any of the other shapes are
shaded. ASK: Are all circles shaded? (no) Have a student circle the counterexample. (E) Erase
the underlining and the circling and repeat with new statements (see below), underlining the
relevant shapes first. Emphasize in each case that the sentence is only about the shapes you
underline; the shapes that are not underlined don’t matter. Write the following statements on the
board, one at a time:
• All squares are large.
• All large squares are shaded.
• All shaded circles are small.
Have volunteers name the counterexample for each statement. (D, F, and A)
(MP.3) Exercises: Use the same picture to name a counterexample for the statement.
a) All shaded shapes are circles.
b) All unshaded shapes are small.
c) All shaded shapes are large.
d) All unshaded shapes are squares.
e) All large shapes are squares.
f ) All large shapes are shaded.
g) All small shapes are unshaded.
h) All small unshaded shapes are squares.
Answers: a) B, b) F, c) C, d) E, e) A, f ) F, g) C, h) E
As students do the Exercises above, encourage them to first write down the shapes that the
statement is talking about. For example, the statement in part a) is about the shaded shapes A,
B, and C. This is where students should look for a counterexample. For students who need
extra help, you can draw the shapes in their notebooks for them, and they can underline the
shapes each question is referring to (and erase the underlining before doing each new
question).
False statements about words. Write on the board:
All words start with the letter b.
G-34
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Ask if each of the words below is a counterexample to the statement and have students explain
why or why not:
• bat (no, because it does start with b)
• cat (yes, it is a word that does not start with b)
• boat (no, because it does start with b)
• bxcv (no, because it does start with b OR no, it’s not a word and the statement only talks
about words, so something that is not a word cannot be a counterexample)
• xcvb (no, because it’s not a word,)
(MP.3) Exercises: Circle the counterexample to the statement.
a) All nouns have an “e.”
red
brown
truck
bike
b) All even numbers have a digit 2.
23
32
34
43
c) All numbers divisible by 5 have ones digit 5.
35
40
52
55
Answers: a) truck, b) 34, c) 40
Review the word “vowel” if necessary. The letters a, e, i, o, u, and sometimes y are vowels.
Exercises:
1. To which of these statements is “Bob” a counterexample?
A. All names have two vowels.
B. All names have three letters.
C. All names have four letters.
D. All boys’ names start with D.
E. All names are boys’ names.
F. All names read the same backward as they do forward.
Answers: A, C, and D.
2. Find a counterexample to the three statements in Exercise 1 for which “Bob” is not a
counterexample. Try to find one example that works as a counterexample to all three
statements at the same time.
Sample answer: Sara.
(MP.3) Bonus: Explain why there cannot be a counterexample to all six statements in Exercise 1.
Answer: To be a counterexample to D, the name would have to be a boy’s name. On the other
hand, to be a counterexample to E, the name would have to not be a boy’s name. So, there
cannot be a counterexample that disproves both D and E at the same time. Therefore, there
cannot be a counterexample that disproves all six statements at the same time.
If a hint is necessary for the Bonus above, point students’ attention to statements D and E.
Explain that to reject D, you need a boys’ name that doesn’t start with D and, for E, you need a
name that is not a boys’ name.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-35
Proving a statement true by checking all examples. Draw and write on the board:
All squares are shaded.
Demonstrate checking all the squares to see if they are shaded. They are, so the statement is
true. Repeat with the following statement: “All triangles have a horizontal side” and have
volunteers check all triangles. (again, the statement is true) Repeat with “All squares have a
horizontal side.” (this statement is false; I is a counterexample) Point out that to show a
statement is true, students need to check all examples. To show a statement is false, they need
to identify just one—any—counterexample.
Have students use the same shapes to complete the following Exercises.
Exercises: Decide whether the statement is true or false.
a) All striped shapes are large.
b) All triangles are large.
c) All large circles are shaded.
d) All small squares have a horizontal side.
e) All small shapes have a horizontal side.
Bonus: All large shaded triangles are equilateral.
Answers: a) true; b) false; D; c) false, H or K; d) true; e) false; J; Bonus: false, E
Using reasoning to prove a statement true. Write on the board:
Whenever it is raining, there are clouds.
ASK: Is this statement true or false? (true) Do you have to check for clouds every time it rains to
know that the statement is true? (no) How do you know without checking that it is true? (rain can
only come from clouds) Tell students that there is often a reason why a statement is true. When
there is, students don’t have to check all examples to prove it. Review the words “even” and
“odd” as they apply to numbers. (even numbers are multiples of 2; odd numbers are not even)
Then write the following two statements on the board:
All even numbers have an even digit.
All even numbers have an odd digit.
Tell students that one of the statements is true and the other is false. ASK: Which statement is
true? (all even numbers have an even digit) How do you know it’s true? (because the ones digit
G-36
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
is always even for any even number) Explain that if you had to check all even numbers one by
one, you would be checking forever! But because you know the reason this statement is true,
you don’t have to check every example. Have a volunteer name a counterexample to the
second statement. Again, point out that students don’t need to check every even number—one
counterexample is enough to prove it’s false.
(MP.3) Exercises: Either explain why the statement is true or find a counterexample.
a) All 3-digit numbers less than 200 have a digit 1.
b) All 3-digit numbers more than 200 have a digit 1.
c) All 3-digit numbers less than 900 have a digit 9.
d) All 3-digit numbers more than 900 have a digit 9.
(MP.1) Bonus: What is the smallest number x that will make the statement below true? Explain
your reasoning.
All 3-digit numbers more than x have a digit 9.
Answers:
a) true, because all 3-digit numbers less than 200 are in the 100s, so their hundreds digit is 1
b) false, sample counterexample: 202
c) false, sample counterexample: 100
d) true, because all 3-digit numbers more than 900 are in the 900s and have hundreds digit 9
Bonus: 888, because 889 has ones digit 9, and any number in the 890s has tens digit 9, and
any number in the 900s has hundreds digit 9.
Determining whether geometric statements are true or false. Write on the board:
If two shapes are congruent, they have the same perimeter.
(MP.3) ASK: Is this statement true? (yes) How do you know? (if the two shapes are congruent,
one fits exactly on top of the other, so all their side lengths are the same and their perimeters
are the same, too) Now tell students you are going to turn the sentence around. Write on the
board:
If two shapes have the same perimeter, they are congruent.
ASK: Is this statement true? Take guesses. Then, draw a 3 by 3 square on grid paper and
challenge students to find a shape that fits inside the 3 by 3 square that has the same
perimeter. Take various answers. Here are some sample answers:
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-37
SAY: All of these shapes have the same perimeter, but none of them are congruent, so any two
of them make a counterexample to the statement. Point to any shape and ASK: Is this shape by
itself a counterexample? (no) Why not? (you need two shapes with the same perimeter to be a
counterexample)
Exercises: Is the statement true? If not, find a counterexample.
a) All squares are congruent.
b) All right triangles have two sides equal.
c) All rectangles with the same area are congruent.
d) All squares with the same area are congruent.
e) All triangles with the perimeter 12 are congruent.
f) Any two triangles having a side of length 5 cm, a 30° angle, and a 60° angle are congruent.
g) All triangles with perimeter 3 cm have all sides equal.
Sample answers:
a)
b)
c)
3
6
2
1
d) true; e) a triangle with sides 3 cm, 4 cm, and 5 cm, and a triangle with sides 4 cm, 4 cm, and
4 cm; f) a triangle with the 5 cm side between the two given angles and a triangle with the 5 cm
side opposite the 30° angle; g) a triangle with sides 0.8 cm, 1 cm, and 1.2 cm has perimeter
3 cm but does not have all sides equal.
Extensions
1. Have students complete BLM Sudoku—Introduction and BLM Sudoku—Another
Strategy. These BLMs introduce students to sudoku puzzles, which require substantial logical
thinking to solve. Students can complete BLM Sudoku—Advanced for an added challenge.
2. Is the statement true? If so, explain your reasoning. If not, give a counterexample.
a) All solids expand when they melt.
b) All ice cubes are colder than 40°F.
Answers: a) false, ice is a counterexample; b) true, because all ice cubes have temperature at
most 32°F which is the freezing point of water
(MP.1) 3. How many numbers do you have to check to show that the following statement is true?
All numbers less than one thousand, when written out in words, do not contain the letter “a.”
Solution: Number words to check: zero to twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety,
one hundred. That’s it! Every other number less than one thousand is written as a combination
of these words, and so also will not have the letter “a” in it. Examples: three hundred forty-two;
one hundred seventeen.
(MP.3) 4. Is the statement true?
a) If a shape has all sides equal, it has all angles equal.
b) If a triangle has all sides equal, it has all angles equal.
G-38
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Answer: a) no; sample counterexamples:
b) yes, check by drawing two or three triangles that have all sides equal then measuring their
angles. NOTE: Students will be able to prove this formally with the tools they learn in Grade 8.
(MP.3) 5. Is the statement true?
a) All triangles are not equilateral.
b) All shapes are not striped.
c) No circles are large.
d) No circles are shaded.
e) No squares are not unshaded.
f) No triangles are not shaded.
Answers: a) false, B; b) true; c) true; d) false, D; e) true; f) false, B
(MP.3) 6. Which statement is true? Explain how you know the statement is true. Write a
counterexample for the other statement. Hint: What are the ones digits of multiples of 2, 5, or 10?
A. All numbers that are multiples of both 2 and 5 are multiples of 10.
B. All numbers that are multiples of both 4 and 6 are multiples of 24.
Answer: Statement A is true because the multiples of 2 are the numbers with ones digit 0, 2, 4,
6, or 8; the multiples of 5 are the numbers with ones digit 0 and 5, so numbers that are both
multiples of 2 and 5 have ones digit 0, but these are exactly the multiples of 10. One
counterexample to statement B is 12, for example, because 12 is a multiple of both 4 and 6, but
is not a multiple of 24.
(MP.3) 7. Some students might enjoy seeing how far they can get in the book Anno’s Hat Tricks
by Akihiro Nozaki and Mitsumasa Anno. The book walks students step-by-step through a
difficult logic problem.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-39
G7-8
Scale Drawings and Similar Shapes
Pages 168–169
Standards: 7.G.A.1, 7.RP.A.2
Goals:
Students will recognize scale drawings by using the ratios of corresponding sides.
Students will understand that when one shape is a scale drawing of another, the two shapes are
similar.
Prior Knowledge Required:
Can recognize equivalent ratios
Vocabulary: congruent, ratio, scale drawing, similar
Materials:
grid paper or BLM 1 cm Grid Paper (p. J-1)
The Geometer’s Sketchpad® or equivalent software (optional)
Introduce scale drawings. Give students 1 cm grid paper or BLM 1 cm Grid Paper, and have
them draw a right triangle with horizontal side length 3 units and vertical side length 4 units. For
more accurate results, you could instead have students do so using The Geometer’s
Sketchpad® or equivalent software, instead of grid paper. Then have them draw two more
triangles, one with horizontal and vertical sides twice as long and the other with horizontal and
vertical sides three times as long. When students are finished, ask them to predict whether the
third side will also be two or three times as long as the original third side. Then have students
use a centimeter ruler or geometry software to measure the third side of each triangle.
ASK: What do you notice? (the third side is also two or three times as long as the third side of
the original triangle) SAY: If you stretch both the horizontal length and the vertical length by the
same multiple, then all sides in the shape get stretched by the same multiple.
Exercises: I stretched the shape so it is twice as tall and twice as wide. What are the new side
lengths?
a)
b)
c)
Answers:
a)
6
22
4
2
b)
2
2
6
c)
6
6
4
4
2
8
6
8
8
G-40
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Tell students that when you stretch all side lengths by the same multiple, it is called a scale
drawing. Maps and floor plans are examples of scale drawings.
Recognizing scale drawings. Draw on the board:
A.
B.
Point to the original and scale drawings in A. ASK: What is the length of the original rectangle?
(3; students can signal their answer by holding up fingers) What is the length of the new
rectangle? (6) What is the ratio of the lengths? (3 : 6 or 1 : 2) Repeat for the widths. (1 : 2)
ASK: Are the ratios the same? (yes) SAY: That’s how you know it’s a scale drawing: if the ratio
of the lengths is the same as the ratio of the widths, then it’s a scale drawing. ASK: What is the
ratio of lengths for B? (2 : 5) And for the widths? (1 : 3) So is B a scale drawing? (no)
Exercises:
1. Which figures are scale drawings of the original? For those that are scale drawings, what is
the scale?
Original
A.
B.
C.
D.
Answer: B has scale 1 : 2, D has scale 1 : 3.
2. Draw a scale drawing on grid paper by making all sides twice as long:
a)
b)
Answers:
a)
b)
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-41
Introduce similar shapes. Remind students that two shapes are congruent if they are the
same size and shape. SAY: Two shapes are similar if they are the same shape but different
sizes. When you make a scale drawing, you are changing the size without changing the shape,
so the two shapes are similar. Draw on the board:
_____
3
2
10
Tell students that the two shapes are similar. ASK: How many times taller is the second shape
than the first? (5) SAY: 10 is 5 times 2. So the second rectangle must also be 5 times longer.
ASK: If the first rectangle’s length is 3, what is the second rectangle’s length? (15) Point out that
it doesn’t matter what the units are, as long as the units are the same for both.
Exercises: Find the missing side length in the similar shapes. Include the units.
a)
b)
7 ft
28 ft
3 in
9 in
5 in
15 ft
_____
_____
c)
1 mm
6 mm
30 mm
_____
d)
______
3 mm
2 mm
5 mm
5 mm
6 mm
7 mm
______
______
______
G-42
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Bonus: Find all the missing sides in the similar shapes.
e)
f)
2.5 in
1.5 in
5 mm
2 in
13 mm
2 cm
12 mm
6 in
g)
3m
2m
3.2 m
4m
10 m
Answers: a) 15 in; b) 60 ft; c) 5 mm; d) in clockwise order from the given length: 6 mm, 9 mm,
15 mm, 21 mm, 15 mm; Bonus: e) 6 in, 4.5 in, 7.5 in; f) 2 cm, 5.2 cm, 4.8 cm; g) smaller shape
sides: 3.2 m, 2 m, 3 m, 5 m, 6.2 m, 3 m; larger shape sides, in the same order: 6.4 m, 4 m, 6 m,
10 m, 12.4 m, 6 m
Extensions
1. Find the missing side length.
4 cm
2 cm
a)
3 cm
b)
5 cm
c)
2 cm
3 cm
_____
7 cm
e)
6 cm
2 cm
_____
8 cm
f)
2 cm
x cm
3 cm
3 cm
_____
2 cm
3 cm
3 cm
_____
d)
2 cm
_____
_____
Answers: a) 6 cm, b) 7.5 cm, c) 9 cm, d) 10.5 cm, e) 12 cm, f) 3x/2
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-43
2. Find the missing length in terms of x.
_____
a)
b)
2 cm
2 cm
x cm
3 cm
x cm
3 cm
c)
x cm
2 cm
3 cm
_____
_____
Answers: a) 2x/3, b) 3x/2, c) 6/x
3. Is the statement true? If not, find a counterexample.
a) All squares are similar.
b) All rectangles are similar.
c) All rectangles with the same area are similar.
d) If two similar shapes have the same perimeter, they are congruent.
e) If two similar shapes have the same area, they are congruent.
f) Two similar triangles have all the same angles.
Sample answers: a) true; b) false, sample counterexample: a 1 by 2 rectangles is not similar to
a 1 by 3 rectangle; c) false, sample counterexample: a 4 by 6 rectangle and a 3 by 8 rectangle
have the same area but are not similar; d) true; e) true; f) true
G-44
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-9
Scale Drawings and Ratio Tables
Pages 170–172
Standards: 7.G.A.1, 7.RP.A.1, 7.RP.A.2
Goals:
Students will use ratio tables to make scale drawings.
Students will discover that the ratios “length : width” in similar rectangles are the same.
Prior Knowledge Required:
Can complete a ratio table when given enough information
Can multiply a fraction by a whole number
Can create equivalent ratios
Vocabulary: equivalent ratio, multiplier, ratio, ratio table, scale drawing, similar
Making a ratio table from a scale drawing. Draw on the board:
3 cm
5 cm
10 cm
6 cm
4 cm
8 cm
SAY: If you draw a scale drawing with every side twice as long, then you can write that as a
ratio. Write on the board:
(original length) : (scale drawing) = 1 : 2
Original Length (cm)
3
4
5
Scale Drawing (cm)
Have volunteers fill in the second column. (6, 8, 10) Remind students that a table is a ratio table
if the rows are equivalent ratios. ASK: Is this a ratio table? (yes) What ratio are all the rows
equivalent to? (1 : 2) SAY: When you have a scale drawing, you can always make a ratio table
by recording the lengths of the original lines and the lengths of the lines in the scale drawing.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-45
Exercises: Complete the ratio table for the scale drawing.
a) (original length) : (scale drawing) = 1 : 3
b) (original length) : (scale drawing) = 1 : 4
5
3
2
3
Original
Scale Drawing
Original
2
3
3
5
Answers: a) width 6 by length 9, b) width 12 by length 20
Scale Drawing
Making a scale drawing from a ratio table. SAY: A rectangular soccer field is 56 feet by
21 feet. I want to make a scale drawing. Write on the board:
actual : scale drawing = 7 : 2
SAY: I start by making a ratio table. Since every row is a ratio equivalent to the scale ratio, I can
add a row for the scale ratio and still have a ratio table. Draw on the board:
Scale
Length
Width
Actual
(square feet)
7
56
21
Scale Drawing
(grid squares)
2
SAY: I put the scale we’re using for the drawing right into the table. I know it’s a ratio table and
the first row is completed. ASK: How can I complete the second row? (the second row is 8 times
the first row) PROMPT: What do you multiply 7 by to get 56? Remind students that in a ratio
table, all rows are equivalent. SAY: You can use this to finish the table. Write on the board:
7:2
= 56 :
7:2
= 21 :
SAY: Since you multiply 7 by 8 to get 56, then you have to multiply 2 by 8 to get the length in
the scale drawing. Write “16” as the missing number in the ratio and in the second row of the
table. ASK: What do you multiply the first row by to get the third row? (3) So what is the missing
term in the third row? (3 × 2 = 6) Write “6” in the ratio and in the table. SAY: So now we know
what our scale drawing would look like on grid paper. It would be a 16 by 6 rectangle.
G-46
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Exercises: Complete the table then draw the scale drawing on grid paper.
a)
b)
Actual
Scale Drawing
Actual
(square feet) (grid squares)
(square feet)
Scale
3
2
Scale
4
Length
12
Length
12
Width
9
Width
8
Answers: a) length by width = 8 by 6, b) length by width = 9 by 6
Scale Drawing
(grid squares)
3
Review writing unit ratios from given ratios. SAY: The scale can always be written as 1 to
something, but that something might not be a whole number. Refer to parts a) and b) above,
and ASK: What would the scale be if written in this way? (a) 1 : 2/3, b) 1 : 3/4)
Exercises: Write the unit ratio.
a) (original length) : (scaled length) = 5 : 2 = 1 : _____
b) (original length) : (scaled length) = 3 : 7 = 1 : _____
Answers: a) 2/5, b) 7/3
The unit ratio as multiplier. SAY: You can think of the unit ratio as the multiplier. It tells you
what to multiply the original lengths by to get the new lengths. Write on the board:
12
original : new = 3 : 2
9
ASK: What is the original to new ratio written as a unit ratio? (1 : 2/3) SAY: So you multiply the
length and the width by 2/3 to get the new length and width. Demonstrate with the length, 12,
and have a volunteer demonstrate with the width, 9. (see solutions below)
12 ×
2 12 × 2
=
3
3
24
=
3
=8
2 9´2
9´ =
3
3
18
=
3
=6
Exercises: Write the ratio (original length) : (new length) as a unit ratio, then find the length and
width of the scale drawings.
a) length = 12 cm, width = 9 cm, (original) : (new) = 3 : 4
b) length = 14 cm, width = 10 cm, (original) : (new) = 2 : 5
Answers: a) length = 16, width = 12; b) length = 35, width = 25
The length-to-width ratio in a scale drawing is the same as in the original drawing. Have
volunteers write on the board their answers to the previous Exercises:
a) length = 16, width = 12
b) length = 35, width = 25
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-47
ASK: What is the length-to-width ratio in the scaled drawing in part a)? (16 : 12) Write the
answer on the board, then have a volunteer reduce the ratio to lowest terms:
length : width = 16 : 12 = 4 : 3
ASK: What is the length-to-width ratio in the original object in part a)? Write the answer on the
board, then have a volunteer reduce the ratio to lowest terms:
length : width = 12 : 9 = 4 : 3
ASK: Are the length-to-width ratios the same in the scale drawing as in the original drawing?
(yes) Summarize by drawing the table below:
Length
12
16
Original drawing
Scale drawing
Width
9
12
SAY: Both of the rows are equivalent to the ratio 4 : 3, so this is a ratio table.
Exercise: For part b), fill in the table. Check that the length-to-width ratio in the scale drawing is
the same as in the original rectangle.
Length Width
Original drawing
Scale drawing
Answer: original drawing has length : width = 14 : 10 = 7 : 5, scale drawing has length : width
= 35 : 25 = 7 : 5.
Two ways of finding the missing number. Draw on the board:
4
3
2
?
SAY: The second rectangle is a scale drawing of the first, so the two rectangles are similar.
There are two ways to find the missing number. You can use the length-to-width ratios in both
rectangles. Write the first table below on the board. SAY: Or, you can make the ratios of
corresponding sides equal. The side of length 4 is a scale drawing of the side of length 3 and
the missing side is the side of length 2 drawn to the same scale. Write the second table below
on the board:
Length
3
4
G-48
Width
2
?
Original
3
2
Scale
4
?
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Have volunteers show the two ways of getting the missing number. (see answers below)
From the first table: 2 ×
4 8
=
3 3
From the second table: 4 ×
2 8
=
3 3
Point out that when you have a ratio table, the rows are equivalent ratios. But so are the
columns, because the number you multiply the first row by to get the second row is the unit rate
for the columns. The length-to-width table is really the same as the original-to-scale table—the
only difference is that rows become columns and columns become rows.
Exercises: Find the missing number two ways. Make sure you get the same answer.
a)
b)
6
5
3
8
?
3
5
?
Answers: a) 6 × 3/5 = 18/5 or 3 × 6/5 = 18/5, b) 8 × 5/3 = 40/3 = 13 1/3 or 5 × 8/3 = 40/3 =
13 1/3
Comparing scale drawings with and without units. SAY: When the scale has no units, the
numbers tell you which shape is larger. Point to the shapes in part a) above. SAY: Here, the
scale is 5 : 6. The fact that 6 is larger than 5 tells you that the second shape is larger than the
first. Write on the board:
2 feet : 3 grid squares
(MP.6) SAY: Here, the number part of the scaled length is larger than the number part of the
original length, but the units are much smaller, so the scale drawing is still smaller than the
original shape.
Extensions
1. Which rectangle is longer?
2 feet for 5 grid squares
5 ft for 2 grid squares
Answer: The first rectangle is 4 feet long and the second rectangle is 5 feet long, so the second
rectangle is longer.
2. The scale in a scale drawing can be expressed as a unit ratio without units by writing both
terms in the same unit. Example: 3 inches to 5 feet = 3 inches to 60 inches = 1 to 20.
Change the scale to a unit ratio without units.
a) 5 cm : 2 km
b) 3 in : 4 ft
c) 2 cm : 0.3 km
Answers: a) 5 cm : 200,000 cm = 1 : 40,000; b) 3 in : 48 in = 1 : 16;
c) 2 cm to 30,000 cm = 1 : 15,000; d) 4 in : 60 in = 1 : 15
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
d) 4 in : 5 ft
G-49
3. Three drawings of the same object are done to different scales.
A. 2 cm : 5 km
B. 1 in : 5 mi
C. 1 mm : 32 m
Which drawing is largest? Which is smallest? Hint: Use the ideas from Extension 1. Also,
1 mile = 1,760 yards.
Solution: The unit ratios are:
A: 2 cm : 500,000 cm = 1 : 250,000;
B: 1 in : 316,800 in, = 1 : 316,800;
C: 1 mm = 32,000 mm = 1 : 32,000, so C is largest and B is smallest. (If this seems opposite to
what one would expect because 32,000 is the smallest number, notice that a 1 : 1 drawing
would be largest of all.)
4. A rectangle has length 6 cm and width 3 cm. Change the scale to a unit ratio, then determine
the length and width of the scale drawing.
1 2
:
b) scale is 0.2 : 1.1
c) scale is 0.6 : 3
a) scale is
2 3
Answers: a) 1 : 4/3, length = 8 cm, width = 4 cm; b) 1 : 5.5, length = 33 cm, width = 16.5 cm;
c) 1 : 5, length = 30 cm, width = 15 cm
5. The side lengths (in millimeters) of a trapezoid are shown below:
30
25
15
50
A scale drawing is done to the scale of
1
original : new = 1 : .
5
a) Write the new side lengths.
1
of 15 = 15 ÷ 5
3
5
b) Write the ratio of the side lengths in the actual drawing, then in the scale drawing.
Are the ratios equivalent? (answers are in italics)
Actual
Scale Drawing Equivalent?
G-50
:
15 : 50
3 : 10
yes
:
30 : 50
6 : 10
yes
:
15 : 25
3:5
yes
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G7-10
Scale Drawings in the Real World
Pages 173–175
Standards: 7.G.A.1, 7.RP.A.1, 7.RP.A.2
Goals:
Students will use scale drawings to determine actual distances.
Given a scale drawing to a given scale, students will draw another scale drawing to a different
scale.
Prior Knowledge Required:
Can make a ratio table for a scale drawing
Can determine actual measurements from a scale drawing
Can make a scale drawing from actual measurements
Vocabulary: ratio, ratio table, scale drawing
Materials:
transparency of grid paper or BLM 1 cm Grid Paper (p. J-1)
BLM Floor Plan (p. G-64)
grid paper or BLM 1 cm Grid Paper (p. J-1)
Finding real-life distances from a map. Draw on the board:
City B
3 cm
City A
4 cm
6 cm
1 cm : 30 km
City C
Tell students that a map always shows the scale. SAY: On this map, every 1 cm represents
30 km in real life. ASK: How far apart are City A and City B in real life? (90 km) Write on the
board:
1 cm : 30 km
×3
3 cm : ? km
×3
SAY: 3 cm represents 3 times as much as 1 cm. So if 1 cm represents 30 km, then 3 cm
represents 90 km.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-51
(MP.4) Exercises: Use the map to answer the question.
a) How far apart are City B and City C in real life?
b) How far apart are City A and City C in real life?
c) Jack estimates that gas costs 9¢ per km. How much will the gas cost to go from City A to City B?
d) If Jack drives 80 km per hour, how long will it take him to travel from City B to City C?
Answers: a) 120 km, b) 180 km, c) $8.10, d) 1.5 hours
(MP.4) Using floor plans to find real-life distances. Tell students that another thing that
people draw scale drawings for is when they are moving and need a floor plan of their new
place. Provide students with BLM Floor Plan.
Exercises: Answer the following questions about the floor plan.
a) How big is the entire apartment?
b) How long is the kitchen countertop (not including the sink and kitchen appliances)?
c) How wide is the entrance door?
d) How long is the closet near the entrance?
e) How wide is the balcony sliding door?
Answers: a) 24 ft by 33 ft, b) 63 in or 5 ft 3 in, c) 36 in or 3 ft, d) 72 in or 6 ft, e) 90 in or 7 1/2 ft
Activity
(MP.4) Tell students that when people move into a new apartment, they usually have a floor
plan to work with and some furniture that they already own, so it’s just a matter of determining
how to place the furniture.
Using the same scale as on BLM Floor Plan (1 cm = 18 inches), have students cut out the
following furniture pieces from 1 cm grid paper or BLM 1 cm Grid Paper:
• kitchen table: 45 in wide by 72 in long
• couch: 72 in wide by 36 in deep
• bed: 81 in long by 54 in wide
• TV stand: 36 in wide by 12 in deep
• desk: 54 in wide by 27 in deep
• bookshelf: 27 in wide by 18 in deep
Students then design a room by using the floor plan on the BLM. Tell students that there are
some practical considerations to keep in mind when they design the floor plan. For example,
some objects should be near each other, doors need to open, and people need enough space
to walk from one place to the other. Then have students answer the following questions about
their design.
a) How far from the TV is the couch?
b) How far is the bed from the closet?
c) How far is the kitchen table from the kitchen sink?
d) How far is the desk from the bookshelf?
e) Does anything block any doors from opening?
f) Is there enough space to walk between items?
(end of activity)
G-52
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Making scale diagrams of the same object to different scales. Write on the board:
12 ft wide and 15 ft long
1 grid square to 3 ft
Tell students that the scale “1 grid square to 3 feet” means that the length of 1 grid square
represents 3 feet in real life. Then SAY: A room is 12 feet wide and 15 feet long. ASK: If I use
the scale 1 grid square for every 3 feet to draw the room, how many grid squares wide will the
room be? (4) How many grid squares long will it be? (5) SAY: If 1 grid square is 3 feet, then 4
grid squares is 12 feet and 5 grid squares is 15 feet. Have a volunteer draw the room to scale
on the board (have BLM 1 cm Grid Paper already projected onto the board):
Tell students you want to do another drawing of the same room, but this time using the scale 2
grid squares to 3 feet. You can use a ratio table to determine the length and width in grid
squares on the scale drawing. Draw on the board:
Grid Squares
2
Feet
3
12
15
Have volunteers fill in the ratio table, then have another volunteer draw the scale drawing. (8
units wide by 10 units long)
Exercises: A room is 15 feet wide by 25 feet long. Draw the scale drawing.
a) 1 grid square : 5 feet
b) 2 grid squares : 5 feet
Bonus: 3 grid squares : 4 feet
Answers: a) 3 by 5, b) 6 by 10, Bonus: 11 1/4 by 18 3/4
(MP.5) ASK: Why is 1 grid square to 5 feet a good choice of scale? (because both dimensions
are divisible by 5 so the scale drawing will have whole-number side lengths) Explain that the
scale 3 grid squares to 4 feet is not as good of a choice because it requires having fractional
lengths for the dimensions.
Making scale diagrams from other scale diagrams of the same object. SAY: Sometimes
you’re given a scale drawing and you want another scale drawing. The first step is to find the
size of the actual object using the scale drawing you’re given. The next step is to use the size of
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-53
the actual object to draw the second scale drawing. Write on the board:
2 grid squares to 3 feet
SAY: The scale drawing is 3 grid squares by 5 grid squares and you know the scale, so you can
fill in part of the ratio table to get the other part. This time it’s the scale drawing that you know
and you have to find the actual size. Draw on the board:
Grid Squares
2
3
5
Feet
3
Ask volunteers to fill in the ratio table. (3 × 3/2 = 9/2 feet wide and 3 × 5/2 = 15/2 feet long, so
4 1/2 feet by 7 1/2 feet)
Exercises: Find the actual dimensions from the scale drawing.
a) 2 grid squares : 5 feet
b) 3 grid squares : 2 m
c) 2 grid squares : 7 feet
Answers: a) 10 feet by 15 feet; b) each grid square represents 2/3 of a meter, so the rectangle
is 2 m by 10/3 = 3 1/3 m; c) each grid square represents 7/2 feet, so the rectangle is 7/2 × 5 by
7/2 × 7/2, or 35/2 by 49/4, or 17 1/2 ft by 12 1/4 ft
SAY: Once you have the actual dimensions, you can draw a scale drawing to any scale.
Exercises: Find the actual dimensions, then draw the scale drawing with scale 5 grid squares
to 2 miles.
a) 3 grid squares to 4 miles
b) 2 grid squares to 3 miles
G-54
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
Answers: a) actual dimensions: 4 miles by 8 miles, new scale drawing: 10 grid squares by 20
grid squares; b) actual dimensions: 6 miles by 9 miles, new scale drawing: 15 grid squares by
22.5 grid squares
For students who are struggling with the Bonus problem on AP Book 7.1 p. 174, suggest that
they draw C as a 1 by 2 rectangle, then use the information to draw B then to draw A. Then
students only need to compare the actual drawings of A and C to answer the question.
(MP.4) Word problems practice.
a) An ostrich, when drawn to the scale 2 in : 5 ft, is 4.2 inches tall. How tall is the ostrich in real
life?
b) Two cities are 2 1/4 inches apart on a map with scale 2 in : 3 miles. How far apart are the
cities in real life?
c) Two cities are 8 cm apart on a map with scale 1 : 500,000. How far apart would the cities be
on a map with scale 1 : 200,000?
d) A room drawn to the scale 2 cm : 3 ft is 8 cm wide by 10 cm long. What would the dimensions
be on a scale drawing with scale 1 in : 4 ft?
Answers: a) 10.5 ft, b) 3 3/8 miles, c) 20 cm apart, d) 3 in by 3 3/4 in
(MP.1, MP.3) When students finish, discuss the reasonableness of their answers to the word
problems practice questions. For example, in part b) 2 1/4 inches is only slightly over 2 inches,
so the actual distance should be only slightly over 3 miles. In part c), one might be tempted to
think that the 1 : 500 drawing should be larger than the 1 : 200 drawing. In fact, it is the
opposite. The 1 : 500 drawing is much further from the size of the actual object than the 1 : 200
drawing, so the 1 : 200 drawing is the larger one.
Extensions
(MP.3) 1. Two scale drawings are drawn of the same object. Which scale drawing is larger?
Explain how you know.
a) 1 cm : 300 km and 1 cm : 400 km
b) 1 cm : 3 ft or 2 cm : 5 ft
Answers: a) the 1 cm : 300 km drawing will be closer to the actual size than the 1 cm : 400 km
drawing, so the first scale drawing will be larger; b) 2 cm : 5 ft = 1 cm : 2.5 ft, which will be a
larger scale drawing than 1 cm to 3 ft.
(MP.3) 2. When Room A is drawn to a scale 3 cm : 2 ft, it is identical to Room B drawn to a
scale 2 cm : 3 ft. Which room is larger? How do you know?
Answer: Room A is drawn to the scale 1 cm : 2/3 ft and Room B is drawn to the scale 1 cm to
1 1/2 ft or 3/2 ft. The drawings are the same size, and each centimeter represents more in
Room B than in Room A, so Room B is larger.
Teacher’s Guide for AP Book 7.1 — Unit 6 Geometry
G-55