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5�1 Angles
145
160230–232
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1. Measure each angle to the nearest degree. Write the measure next tothe angle.
C
B
A
D
G
E
F
H
K
I
J
L
2. a. Which angle above is an acute angle?
b. A right angle? c. An obtuse angle?
d. Which angles above are reflex angles?
3. a. Measure each angle in triangle ADB at the right.
b. Find the sum of the 3 angle measures. °
c. Use Problem 3b to calculate the sum of the interior angle measures in quadrangle ABCD.
°
A
D C
B
Try This
4. Find the measure of �KLM. Then draw an angle that is 60% of the measure of �KLM on the reverse side of this paper. Label it as �NOP.
L M
K
LESSON
5�1
Name Date Time
146
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opyright © W
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Making an Angle Measurer
LESSON
5�1
Name Date Time
Constructing a Hexagon
147
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Follow the directions below to draw hexagon ABCDEF. Line segment AF at thebottom of the page is one of the sides of the hexagon. The completed drawingshould be a convex hexagon. Use your Geometry Template and a pencil with asharp point.
1. Draw a 130� angle with its vertex at point A.One side of the angle is A�F�. Draw a point on the other side that is 5 centimeters from point A.Label this point B.
2. Draw a 115� angle with its vertex at point B. One side of the angle is A�B�.Draw a point on the other side that is 1�
12� inches from point B. Label it point C.
3. Draw a 145� angle with its vertex at point C. One side of the angle is B�C�.Draw a point on the other side that is 6.5 centimeters from point C. Label itpoint D.
4. Draw a 90� angle with its vertex at point D. One side of the angle is C�D�. Draw a point on the other side that is 2�
34� inches from point D.
Label it point E. Then draw E�F� to complete the hexagon.
5. What is the measure of �E?
of �F ?
6. What is the length of E�F� to the nearest �
18� inch?
7. What is the sum of the measures of the angles of your hexagon?
The closer this sum is to 720°, the more accurate your drawing is.
B
A F
130°5 cm
A F
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5�2 Angle RelationshipsC
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148
163 233
Name Date Time
Find the following angle measures. Do not use a protractor.
1. 2. 3.
45°c at
50° 15°x60° y
m�y � m�x � m�c �
m�a �
m�t �
4. m�q � m�r � m�s �
5. m�a � m�b �
m�c � m�d �
m�e � m�f �
m�g � m�h �
m�i �
6. m�w � m�a �
m�t � m�c �
m�h �
70°
60°
s
q
r
c
hg
i80°
60°40°b f
dea
105°
75°
at
ch
w
Practice
7. �34� of 16 � 8. �
35� of 50 � 9. �
13� of 330 �
LESSON
5�2
Name Date Time
Angle Measures: Triangles and Quadrangles
149
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Cut out any or all of the triangles and quadrangles on this page.
B C
A
tear
tear
tear
B C
A
tear
tear tear
B
C
A
tear
tear
tear
AB
C
tear
tear
tear
D E
G F
tear tear
tear tear
D
E
G F
tear
tear
tear
tear
D E
G F
tear tear
teartear
D E
G F
tear
tear
tear
tear
LESSON
5�2
Name Date Time
Finding Sums of Angle Measures
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1. Cut out one of the triangles on Math Masters, page 149. Carefully cut or tear off each angle. Use point P at the right to position the angles so they touch but do not overlap. The shaded regions should form a semicircle. Use tape or glue to hold the angles in place.
2. Notice that the combined shaded regions form an angle. How many degrees does this angle measure?
3. Compare your results with those of other students. What do your triangles seem to have in common?
4. Complete the following statement. The sum of the measures of the angles of any triangle is
°.
A
BC
P
5. Cut out one of the quadrangles on Math Masters, page 149. Carefully cut or tear off each angle. Use point Q at the right to position the angles so they touch but do not overlap. The shaded regions should form a circle. Use tape or glue to hold the angles in place.
6. Notice that the combined shaded regions form a figure. How many degrees are in this figure?
7. Compare your results with those of other students. What do your quadrangles seem to have in common?
8. Complete the following statement. The sum of the measures of the angles of any quadrangle is .
°
E
F GDQ
LESSON
5�2
Name Date Time
Applying Angle Relationships
151
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1. Extend each side of the regular pentagon in both directions to form a star. The first extension has been done for you. Then use what you know aboutangle relationships to find and label the measures of each interior angle inyour completed star.
108�
2. Describe the angle relationships you used to determine the measures of the interior angles.
LESSON
5�3
Name Date Time
Graphing Votes
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Percent of Degree Measure of SectorCandidate
Votes Received (to nearest degree)
Connie 28% 0.28 º 360� � 100.8� � 101�
Josh º 360� �
Manuel º 360� �
Total 100% 360�
LESSON
5�3
Name Date Time
Pet Survey
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Fraction Decimal PercentDegree MeasureKind Number of Total Equivalent of Total
of Sectorof Pet of Pets Number (to nearest Numberof Pets thousandth) of Pets
Dog 8 �284� 0.333 33�
13�% º 360� �
Cat 6 º 360� �
Guinea pig3 º 360� �or hamster
Bird 3 º 360� �
Other 4 º 360� �
120��13
�
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5�3 Circle GraphsC
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154
Name Date Time
1. The table below shows a breakdown, by age group, of adults who listen toclassical music.
a. Calculate the degree measure of each sector to the nearest degree.
b. Use a protractor to make a circle graph. Do not use the Percent Circle. Write a title for the graph.
AgePercent of DegreeListeners Measure
18–24 11%
25–34 18%
35–44 24%
45–54 20%
55–64 11%
65� 16%
Source: USA Today, Snapshot
2. On average, about 8 million adults listen to classical music on the radio each day.
a. Estimate how many adults between the ages of 35 and 44 listen to classical music on the radio each day. About
(unit)b. Estimate how many adults at least
45 years old listen to classical music on the radio each day. About
(unit)
Practice
Order each set of numbers from least to greatest.
3. 7, 0.07, �7, 0.7, 0
4. 0.25, 0.75, 0.2, �45�, �
44�, 0.06, 0.18, �1
10�
49 147
LESSON
5�3
Name Date Time
Fractions of 360�
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Shade each fractional part. Then record the number of degrees in each shaded region.
1. Shade �12� of the circle. 2. Shade �
14� of the circle.
�12� of 360� �
°�14� of 360� �
°
3. Shade �16� of the circle. 4. Shade �
13� of the circle.
�16� of 360� �
°�13� of 360� �
°
5. Shade �112� of the circle. 6. Shade �
34� of the circle.
�112� of 360� �
°�34� of 360� �
°
0360°
350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
50
402010
330
300
270
240
210180
150
12090
60
300360°
350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
50
402010
330
300
270
240
210180
150
12090
60
30
0360°
350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
50
402010
330
300
270
240
210180
150
12090
60
300360°
350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
50
402010
330
300
270
240
210180
150
12090
60
30
0360°
350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
5040
2010
330
300
270
240
210180
150
12090
60300
360°350340
320
310
290
280
260
250
230
220
200 190 170 160
140
130
110100
8070
50
402010
330
300
270
240
210180
150
12090
60
30
LESSON
5�3
Name Date Time
Sums of Angle Measures in Polygons
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A diagonal is a line segment that connects two vertices of a polygon and is not a side. You can draw diagonals from one vertex to separate polygons into triangles.
1. Draw diagonals from the given vertex to separate each polygon into triangles.Then complete the table.
2. a. Study your completed table. Use any patterns you notice to write a formulato find the sum of the angle measures in any polygon (n-gon).
Formula
b. Use the formula to find the sums of the angle measures in a
heptagon. °
nonagon. °
dodecagon. °1,8001,260900
(n � 2) º 180
Number of Number ofSum of Angle MeasuresPolygon
Sides (n) Triangles
Example: Quadrangle
4 2 2 º 180� � 360�
Pentagon
5 3 º ��
Hexagon
6 4 º ��
Octagon
8 6 º ��
Decagon
10 8 º ��8
6
4
3
diagonal
180� 540
180�
180�
180�
720
1,080
1,440
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5�4 More Polygons on a Coordinate Grid
157
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For each polygon described below, some vertices are plotted on the grid.Either one vertex or two vertices are missing.
� Plot and label the missing vertex or vertices on the grid. (There may be more than one place you can plot a point.)
� Write an ordered number pair for each vertex you plot.� Draw the polygon.
1. Right triangle ABC Vertex C: ( , )
2. Parallelogram DEFG Vertex F: ( , ) Vertex G: ( , )
3. Scalene triangle HIJ Vertex J: ( , )
4. Kite KLMN Vertex M: ( , ) 5. Square PQRS Vertex Q: ( , )
–1–2–3–4–5–7–8 –6 1 2 3 4 5 6 7 8
1
2
4
3
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
0
–9
–10
9
10
9 10x
–9–10
D
E
H
I
L
K
N
P
S
R
A
B
y
LESSON
5�4
Name Date Time
X and O—Tic-Tac-Toe
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Materials � 4 each of number cards 0–10(from the Everything Math Deck, if available)
� Coordinate Grid (Math Masters, p. 417)
Players 2
Object of the game To get 4 Xs or Os in a row, column, or diagonal on thecoordinate grid.
Directions
1. Shuffle the cards and place the deck facedown on the playing surface.
2. In each round:
� Player 1 draws 2 cards from the deck and uses the cards in any order toform an ordered pair. The player marks this ordered pair on the grid with anX and places the 2 cards in the discard pile.
� Player 2 draws the next 2 cards from the deck and follows the sameprocedure, except that he or she uses an O to mark the ordered pair.
� Players take turns drawing cards to form ordered pairs and marking the ordered pairs on the coordinate grid. If the 2 possible points that the player can make have already been marked, the player loses his or her turn.
3. The winner is the first player to get 4 Xs or 4 Os in a row, column, or diagonal.
LESSON
5�4
Name Date Time
Plotting Triangles and Quadrangles
159
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Plot each of the described triangles and quadrangles on Math Masters, page 417.Record the coordinates of each vertex in the table below.
Description Coordinates of Each Vertex
Example: Square with side measuring 3 units (6,�6); (6,�3); (9,�3); (9,�6)
1. A rhombus that is not a square, which has at least one vertex with a negative x-coordinate and a positive y-coordinate
2. An isosceles triangle that has an area of 2 square units
3. A rectangle that has a perimeter of 16 units
4. A right scalene triangle that has each vertex in a different quadrant
5. A kite that has one vertex at the origin
6. A parallelogram that has an area of 18 square units and one side on the y-axis
7. An obtuse scalene triangle that has at least two vertices with a negative x-coordinate and a negative y-coordinate
8. Write your own description.
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5�5 Transforming PatternsC
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160
180 181
Name Date Time
A pattern can be translated, reflected, or rotated to create many different designs. Consider the pattern at the right.
The following examples show how the pattern can be transformed to create different designs:
1. Translate the pattern at the right across 2 grid squares. Then translate the resulting pattern (the given pattern and its translation) down 2 grid squares.
2. Rotate the given pattern clockwise 90�
around point X. Repeat 2 more times.
3. Reflect the given pattern over line JK.Reflect the resulting pattern (the given pattern and its reflection) over line LM.
A
B
C D
Translations Rotations Reflections
4. 26 � 5. 35 � 6. 70 � 7. 43 �
Practice
X
J
K
L M
LESSON
5�5
Name Date Time
Degrees and Directions of Rotation
161
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When a figure is rotated, it is turned a certain number of degrees around a particular point. A figure can be rotated clockwise or counterclockwise.
Position a trapezoid pattern blockon the center point of the anglemeasurer as shown at the right. Then rotate the pattern block as indicated and trace it in its new position.
Example: Rotate 90� clockwise.
For each problem below, rotate and then trace the pattern block in its new position.
1. Rotate 90� counterclockwise. 2. Rotate 270� clockwise.
12
6
11
5
10
4
1
7
2
839
degrees
12
6
11
5
10
4
1
7
2
839
degrees
12
6
11
5
10
4
1
7
2
839
degrees
LESSON
5�5
Name Date Time
Scaling Transformations
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Some scaling transformations produce a figure that is the same shape as the originalfigure but not necessarily the same size. Enlargements and reductions are types of scaling transformations.
Enlargement: Follow the steps to draw a triangle D�E�F� with angles that are congruent totriangle DEF and sides that are twice as long as triangle DEF.
Step 1 Draw rays from P through each vertex. The first ray P���D has been drawnfor you.
Step 2 Measure the distance from point P to vertex D. Then locate the point onP���D that is 2 times that distance. Label it D�.
Step 3 Use the same method from Step 2 to locate point F� on P���F andpoint E � on P���E.
Step 4 Connect points D�, E�, and F�.
Reduction: Change Steps 2 and 3 to draw a triangle D E F with angles that are congruent to triangle DEF and sides that are half as long as triangle DEF.
P
D
F
E
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5�6 Congruent Figures and Copying
163
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Column 1 below shows paths with the Start points marked. Complete each path in Column 2 so that it is congruent to the path in Column 1. Use the Start points marked in Column 2. In Problems 2 and 3, the copy will not be in the same position as the original path.
(Hint: If you have trouble, try tracing the path in Column 1 and then slide, flip,or rotate it so that its starting point matches the starting point in Column 2.)
Example: These two paths are congruent, but they are not in the same position.
Start
Start
Start
Start
Start
Start
Start
Start
1.
Column 1 Column 2
2.
3.
LESSON
5�6
Name Date Time
Quadrangles and Congruence
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16151413
12119
5
4321
Cut out the 16 quadrangles and the 6 set labels.
All Right AnglesA
All Sides CongruentA
Adjacent Sides CongruentC
All Pairs Opposite Sides CongruentB
At Least 2 Acute AnglesB
At Least 1 Right AngleC
6 7 8
10
LESSON
5�6
Name Date Time
Quadrangles and Congruence continued
165
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Cut
out
the
quad
rang
les
and
set l
abel
s fro
m M
ath
Mas
ters
,pag
e 16
4. P
lace
the
two
A, B
, or C
set
labe
ls in
the
plac
ehol
ders
abo
ve th
e rin
gs a
nd s
ort t
he q
uadr
angl
es a
ccor
ding
ly.
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5�7 Angle RelationshipsC
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Write the measures of the angles indicated in Problems 1–6. Do not use a protractor.
1.
m�r �
m�s �
m�t �
3.
m�a �
m�b �
m�c �
5. Angles x and y have the same measure.
m�x �
m�y �
m�z �
xy z
60°a c
b
2. �JKL is a straight angle.
m�NKO �
4. Angles a and t have the same measure.
m�a �
m�c �
m�t �
6.
m�p �p
36°
a
tc
21°
135°
MJ
K
L O
N
15°
93°
62°
133°rs t
Practice
7. 0.09 º 0.03 � 8. 0.15 º 0.8 �
9. 0.07 º 0.07 � 10. 0.75 º 0.3 �
LESSON
5�7
Name Date Time
Circle Constructions
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Use the directions and the pictures below to make one or both of the constructions.You may need to make several versions of the construction before you are satisfiedwith your work. Cut out your best constructions and tape or glue them to anothersheet of paper.
Construction #1
Step 1 Draw a small point that will be the center of thecircle. Press the compass anchor firmly onthe center of the circle.
Step 2 Hold the compass at the top and rotate thepencil around the anchor. The pencil must go allthe way around to make a circle. You may find iteasier to firmly hold the compass in place andcarefully turn the paper under the compass.
Step 3 Without changing the opening of thecompass, draw another circle that passesthrough the center of the first circle.Mark the center of the second circle.
Step 4 Repeat Step 3 to draw a third circle thatpasses through the center of each of thefirst two circles.
Construction #2
Follow the steps to make the three circles fromConstruction #1. Then draw more circles to createthe construction shown at the right.
LESSON
5�7
Name Date Time
Octagon Construction
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Use the directions and the pictures to construct an octagon.
Step 1 Draw a circle with a compass. Label thecenter of the circle A. Then draw adiameter through A. Label the two pointswhere the diameter intersects the circleS and T.
Step 2 Use the length of ST� to set the compass opening.Then place the anchor of your compass on S anddraw an arc below the circle and another arc above the circle. Without changing the compassopening, place the compass anchor on T and draw another set of arcs above and below thecircle. Label the points where the arcs intersectas D and E. Draw a line through D and E.Label the points where DE��� intersects the circleas M and N.
Step 3 Set the compass opening so that it is equal tothe length of SM�. Then place the compass anchoron S and draw an arc; reposition the anchoron M and draw another arc. Label the point where the arcs intersect as G. Draw a linethrough G and A. Label the points where GA���
intersects the circle as H and I.
Step 4 Repeat Step 3, using the length of MT� to setthe compass opening. Label the points ofintersection as X, Y, and Z.
Step 5 Connect the points on the circle to form an octagon.
AS T
A
G
H
D
M
N
E
S
I
T
A
G
H
D
M XY
N
E
Z
S
I
T
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5�8 Isometry Transformations on a Grid
169
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1. Graph and label the following points on the coordinate grid. Connect the points to form quadrangle ABCD.
A: (�2,1) B: (�6,2)C: (�8,4) D: (�5,7)
2. Translate each vertex ofABCD (in Problem 1) 0 units to the left or rightand 8 units down. Plot and connect the new points. Label them A�, B�, C�, and D�.
Record the coordinates ofthe image.
3. Reflect quadrangle ABCD across the y-axis. Plot and connect the new points.Label them A , B , C , and D . Record the coordinates of the image.
4. Rotate quadrangle A B C D 90° clockwise around point (0,0). Plot and connect the newpoints. Label them A���, B ���, C���, and D���. Record the coordinates of the rotated image.
–1–2–3–4–5–7–8 –6 1 2 3 4 5 6 7 8x
y
1
2
4
3
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
0
Try This
Practice
5. 300 � 0.001 � 6. 143 � 10�3 � 7. 35.9 � �1,0100� �
LESSON
5�8
Name Date Time
Inscribing a Circle in a Triangle
170
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Follow the steps to inscribe a circle in triangle ABC.
Step 1 Construct the bisectors of �A and �B. Then label the intersection of the angle bisectors as P.
Step 2 Construct a line segment through point P, perpendicular to AB�. Label the point at which the line segment intersects AB� as Q.
Step 3 Center the compass anchor on P and the pencil point on Q. Draw a circle through Q. The circle will be tangent to all three sides of the triangle.
B
CA
LESSON
5�8
Name Date Time
Constructing Perpendicular Bisectors
171
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1. Draw a large triangle. Construct perpendicular bisectors for each side of the triangle. What observations can you make?
2. Use a protractor and a ruler to draw a large square.Draw a diagonal. Then construct the perpendicularbisector of the diagonal. What observations can you make?
1. Use a ruler and a straightedge to draw 2 parallel lines. Then draw anotherline that crosses both parallel lines.
2. Measure the 8 angles in your figure. Write each measure inside the angle.
3. What patterns do you notice in your angle measures?
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5�9 Parallel Lines and a TransversalC
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172
163230 231
Name Date Time
Practice
Remember: 1,000 milliliters (mL) � 1 liter (L)
4. 500 mL � L 5. 2.5 L � mL
6. 1,300 mL � L 7. 0.95 L � mL
8. 3,250 mL � L 9. 0.045 L � mL
LESSON
5�9
Name Date Time
Angle Relationships and Algebra
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Apply your knowledge of angle relationships to find the missing values and angle measures. Do not use a protractor.
1.
x �°
2.
Lines l and m are parallel.
y �°
m�p �°
m�r �°
m�q �°
m�s �°
3.
4. Turn this page over. Using only a straightedge and a compass, design aproblem that uses angle relationships. Create an answer key for your problem.Then ask a classmate to solve your problem.
150�
2x � 10�
100° y � 40°
2y ° n
sp q
r
l
m
50° 70°
70°
a
x � 30° c 2x � 70°
x �°
m�a �°
m�c �°
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5�10 Parallelogram ProblemsC
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All of the figures on this page are parallelograms. Do not use a ruler or a protractor to solve Problems 1, 2, or 3.
1. a. The measure of �X �°. Explain how you know.
b. The measure of �Y �°. Explain how you know.
2. Alexi said that the only way to find the length of sides CO and OA is to measure them with a ruler. Explain why he is incorrect.
3. What is the measure of �MAR? . Explain how you know.
4. Draw a parallelogram in which all sides have the same length and all angles have the same measure.
What is another name for this parallelogram?
5. Draw a parallelogram in which all sides have the same length and no angle measures 90°.
What is another name for this parallelogram?
130°
YX
ZW
C O
T A10 cm
30 c
m
70°40°
M
K R
A
LESSON
5�10
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Using Quadrangles to Classify Quadrangles
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Read each statement. Then decide if the statement is always, sometimes, or never true. If you write sometimes, identify a case for which the statement is true.
1. A square is a rectangle.
2. A rhombus is a trapezoid.
3. A square is a parallelogram.
4. A rhombus is a parallelogram.
5. A kite is a parallelogram.
6. A rhombus is a rectangle.
7. A square is a rhombus.
8. A trapezoid is a parallelogram.
Fill in the blank using always, sometimes, or never. If you write sometimes, identify a case for which the statement is true.
9. A rectangle has consecutive sides that are congruent.
10. The diagonals of a rhombus are congruent.
Example: A rectangle is a square. sometimes
A rectangle is a square when its 4 sides are the same length.
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5�11 Unit 6: Family LetterC
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Number Systems and Algebra ConceptsIn Fourth and Fifth Grade Everyday Mathematics, your child worked with addition andsubtraction of positive and negative numbers. In this unit, students use multiplicationpatterns to help them establish the rules for multiplying and dividing with positive and negative numbers. They also develop and use an algorithm for the division of fractions.
In the rest of the unit, your child will explore beginning algebra concepts. First, theclass reviews how to determine whether a number sentence is true or false. Thisinvolves understanding what to do with numbers that are grouped within parenthesesand knowing in what order to calculate if the groupings of numbers are not madeexplicit by parentheses.
Students then solve simple equations by trial and error to reinforce what it means tosolve an equation—to replace a variable with a number that will make the numbersentence true.
Next, they solve pan-balance problems, first introduced inFifth Grade Everyday Mathematics, to develop a moresystematic approach to solving equations. For example, tofind out how many marbles weigh as much as 1 orange inthe top balance at the right, you can first remove 1 orangefrom each pan and then remove half the remaining orangesfrom the left side and half the marbles from the right side.The pans will still balance.
Students learn that each step in the solution of a pan-balance problem can be represented by an equation,thus leading to the solution of the original equation. Youmight ask your child to demonstrate how pan-balanceproblems work.
Finally, your child will learn how to solve inequalities—number sentences comparing two quantitiesthat are not equal.
Please keep this Family Letter for reference as your child works through Unit 6.
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VocabularyImportant terms in Unit 6:
cover-up method An informal method forfinding the solution of an open sentence by coveringup a part of the sentence containing a variable.
Division of Fractions Property A property ofdividing that says division by a fraction is the same asmultiplication by the reciprocal of the fraction. Anothername for this property is the “invert and multiplyrule.” For example:
5 � 8 � 5 � �18
� � �58
�
15 � �35
� � 15 � �53
� � �735� � 25
�12
� � �35
� � �12
� � �53
� � �56
�
In symbols: For a and nonzero b, c, and d,
�ba
� � �dc
� � �ba
� � �dc
�
If b � 1, then �ba
� � a and the property is applied asin the first two examples above.
equivalent equations Equations with the samesolution. For example, 2 � x � 4 and 6 � x � 8 areequivalent equations with solution 2.
inequality A number sentence with a relationsymbol other than �, such as , �, �, �, �, or �.
integer A number in the set {..., �4, �3, �2, �1,0, 1, 2, 3, 4, ...}. A whole number or its opposite,where 0 is its own opposite.
Multiplication Property of �1 A property of multiplication that says multiplying any number by �1 gives the opposite of the number. For example,�1 � 5 � �5 and –1 � �3 � �(�3) � 3. Somecalculators apply this property with a [�/�] key thattoggles between a positive and negative value in the display.
open sentence A number sentence with one ormore variables. An open sentence is neither true norfalse. For example, 9 � __ � 15, ? �24 � 10, and 7 � x � y are open sentences.
opposite of a number n A number that is thesame distance from zero on the number line as n,
but on the opposite side of zero. In symbols, theopposite of a number n is –n, and, in EverydayMathematics, OPP(n). If n is a negative number, –nis a positive number. For example, the opposite of –5�5. The sum of a number n and its opposite is
zero; n � –n � 0.
order of operations Rules that tell the order in which operations in an expression should becarried out. The conventional order of operations is:
1. Do the operations inside grouping symbols. Work from the innermost set of grouping symbols outward. Inside grouping symbols, follow Rules 2–4.
2. Calculate all the expressions with exponents.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
For example: 52 � (3 � 4–2)/5 � 52 � (12–2)/5� 52 � 10/5� 25 � 10/5� 25 � 2� 27
reciprocals Two numbers whose product is 1. For example, 5 and �
15
�, �35
� and �53
�, and 0. 2 and 5 areall pairs of multiplicative inverses.
trial-and-error method A method for finding the solution of an equation by trying asequence of test numbers.
0 3 541�1�2�3 2�4�5
Unit 6: Family Letter cont.STUDY LINK
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Do-Anytime ActivitiesTo work with your child on concepts taught in this unit, try these interesting andengaging activities:
1. If your child helps with dinner, ask him or her to identify uses of positive and negativenumbers in the kitchen. For example, negative numbers might be used to describethe temperature in the freezer. Positive numbers are used to measure liquid anddry ingredients. For a quick game, you might imagine a vertical number line with thecountertop as 0; everything above is referenced by a positive number, and everythingbelow is referenced by a negative number. Give your child directions for getting outitems by using phrases such as this: “the �2 mixing bowl”; that is, the bowl on thesecond shelf below the counter.
2. If your child needs extra practice adding and subtracting positive and negativenumbers, ask him or her to bring home the directions for the Credits/Debits Game.Play a few rounds for review.
3. After your child has completed Lesson 6, ask him or her to explain to you what thefollowing memory device means: Please Excuse My Dear Aunt Sally. It represents therule for the order of operations: parentheses, exponents, multiplication, division,addition, subtraction. Your family might enjoy inventing another memory device thatuses the same initial letters; for example, Please Excuse My Devious Annoying Sibling;Perhaps Everything Might Drop Again Soon, and so on.
In Unit 6, your child will work on his or her understanding of algebra concepts by playing games like theones described below.
Algebra Election See Student Reference Book, pages 304 and 305.Two teams of two players will need 32 Algebra Election cards, an Electoral Vote map, 1 six-sided die, 4 pennies or other small counters, and a calculator. This game provides practice with solving equations.
Credits/Debits Game (Advanced Version) See Student Reference Book, page 308.Two players use a complete deck of number cards and a recording sheet to play the advanced versionof the Credits/Debits Game. This game provides practice with adding and subtracting positive and negative integers.
Top-It See Student Reference Book, pages 337 and 338.Top-It with Positive and Negative Numbers provides practice finding sums and differences of positiveand negative numbers. One or more players need 4 each of number cards 0–9 and a calculator to playthis Top-It game.
Building Skills Through Games
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As You Help Your Child with HomeworkAs your child brings assignments home, you might want to go over the instructions together, clarifying them asnecessary. The answers listed below will guide you through some of the Unit 6 Study Links.
Study Link 6�1
2. ✓ 3. ✓ 5. ✓ 7. �119�
9. �276� 11. �
34
� 13. 12�12
� lb 14. 38�14
� in.
15. 67�12
� in.3 16. 81 17. �2 18. �67
Study Link 6�2
1. �45
� 3. 1 5. 1 7. �958�
9. 10 10. 14 11. 17 12. 13.56
13. 589.36 14. 13
Study Link 6�3
1. a. 46 � (�19) � 27 c. �5 � 6.8 � 1.8
2. a. �29 c. �2�15
� e. �3�14
� g. �18.2
3. a. (�2) c. 2�14
� e. �3.7 g. ��176�
4. 2 5. 11 6. 8 7. �6
Study Link 6�4
1. �60 3. �6 5. �5 7. �6
9. �1,150 11. �54 13. �2 15. ��59
�
17. �2 19. a. 36 b. 77
Study Link 6�6
1. 21 3. �2312� 5. 72 7. 1
9. 28 11. 3 12. 23 13. 6, 1
14. 2, 1 15. 4, 4
Study Link 6�7
1. a. 17 � 27; 3 � 15 � 100; (5 � 4) � 20 � 20;12 � 12
b. Sample answer: A number sentence mustcontain a relation symbol. 56�8 does notinclude one.
2. a. true b. false c. false d. true
3. a. (28 � 6) � 9 � 31 b. 20 � (40 � 9) � 11
c. (36/6) / 2 � 12 d. 4 � (8 � 4) � 16
4. a. 60 � 14 � 50; false b. 90 � 3 � 30; true
c. 21 � 7 � 40; true d. �36� �12
� � 10; true
5. 0.92 6. 3.51 7. 251.515
Study Link 6�8
1. a. b � 19 b. n � 24 c. y � 3 d. m � �15
�
2. a. �6x
� � 10; x � 60 b. 200 � 7 � n; n � 193
c. b � 48 � 2,928; b � 61
3. Sample answers:
a. (3 � 11) � (12 � 9) b. 2 � 18 � 14
4. 54 5. 3.6 6. 121
Study Link 6�9
1. 1 2. 1�12
� 3. 5 4. 1
6. Answers vary. 7. 10 8. �14
� 9. �23
�
10. �12
�
Study Link 6�10
1. k � 4 � 5; 3k � 12 � 15; 20k � 12 � 15 � 17k
2. Multiply by 2; M 2 3. Add 5m; A 5mSubtract 3q; S 3q Divide by 2; D 2Add 5; A 5 Subtract 6; S 6
Study Link 6�11
1. k � 12 3. x � 1 5. r � 2
Study Link 6�12
1. a. 15 � 3 � 7 b. x � 5 � 75
c. �99
� � 13 � 14
2. a. 200 � (4 � 5) � 10
b. 16 � 22 � (5 � 3) � 12
3. a. 46 b. 18 c. 0 d. 8
4. a. x � �1 b. y � 6.5
5. a. Sample answers: �3, �2�12
�, �2
6. $0.25; $0.21 7. 1; 1.28 8. 800; 781
Unit 6: Family Letter cont.STUDY LINK
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