LESSON
Name
61
Teacher Note:
page 394
• For additional practice, students
may complete Fraction Activity 61.
• Remaining Fraction
• Two-Step Equations
New Concept
• Remaining Fraction • If we know the size of one portion of a whole, we can figure
out the size of the other portion.
3 is shaded.
__
8
• Two−Step
Equations
5 is not shaded.
__
8
• To solve a two−step equation:
1. Find the answer to one side
of the equation.
2. Find the letter (variable)
• A raised dot shows multiplication.
4 · 6 means “4 times 6.”
2n = 7 + 5
2n = 12
n = 12 ÷ 2
n=6
© 2008 Saxon
Example
3m = 4 · 6
1. Find the answer to one side.
3m = 24
2. Divide to find the variable.
m=8
Saxon Math Intermediate 4
397
Adaptations Lesson 61
Lesson Practice
Use fraction pieces for help.
a. What fraction of this rectangle is not shaded?
b. Three fifths of the race was over.
What fraction of the race was left?
Find each missing number:
c. 2n = 2 + 8
d. 2 + n = 2 · 8
2n =
2+n=
n=
n=
page 396
Written Practice
1. Find the radius.
2.
lunches bought
apple slices per lunch
24 in.
total
The
d
the
r
,
=
.
© 2008 Saxon
so 24 ÷
is twice
×
Saxon Math Intermediate 4
398
=n
Adaptations Lesson 61
page 397
Written Practice, continued
3. Two nickels
4. Multiply the loop.
Divide by the outside number.
a. fraction
cups
_____
?
= __
days 00 1
b. decimal
39
___
_____
)
a.
b.
5.
6. What fraction of this rectangle is not
shaded?
28 to Ariana’s house
to restaurant
back to Ariana’s
back to Maya’s
total
7. Use the multiplication table.
8. What fraction is left (not shaded)?
Which of these numbers is not a
factor of 10?
9.
A 2
B 5
2
__
3
3
__
4
C 10
D 20
10. 5 0 70
© 2008 Saxon
3 8 40
+
Use fraction pieces.
Use work area.
Saxon Math Intermediate 4
399
Adaptations Lesson 61
page 397
Written Practice, continued
12.
11. Use fraction pieces.
4 cm
100% answers
60% true
8 cm
a. perimeter
false
b. area
More true answers or more false
answers?
a.
b.
13.
14. 5n = 12 + 18
$62.59
+ $17.47
15. 1000 − (110 × 9)
5n =
110
n =
×
÷
9
1000
−
n=
3.675
16.
17.
×
18.
703
× 6
.
© 2008 Saxon
−
$6.70
4
Saxon Math Intermediate 4
400
Adaptations Lesson 61
Written Practice, continued
19.
page 398
8 _____ R
21. 7 ) 3 9
8 _____ R
20. 5 ) 3 9
$346
×
9
23. 16 ÷ 3 =
24. 26 ÷ 6 =
8 _____ R
22. 4 ) 3 9
___
25.
36
÷ √ 36
=
÷
=
26. Point A represents what number on this number line?
A
© 2008 Saxon
–5
27. a. 745
b. 132
0
5
754
99
Use work area.
Saxon Math Intermediate 4
401
Adaptations Lesson 61
page 398
Written Practice, continued
28. a. fraction not shaded
b. decimal not shaded
c. percent not shaded
a. 100
b.
0
.
c.
___
29. AB is parallel to
.
is perpendicular to
.
is perpendicular to
.
Angle
is a right angle.
Angle
is a right angle.
A
C
Angle
is obtuse.
Angle
is acute.
B
D
Use work area.
postcards
−
© 2008 Saxon
30.
stamps
elapsed years
Saxon Math Intermediate 4
402
Adaptations Lesson 61
LESSON
Name
62
Teacher Note:
page 399
• Multiplying Three or More
Factors
• Exponents
• Refer students to “Exponents”
on page 21 in the Student
Reference Guide.
New Concept
• Multiplying Three
or More Factors
• To find the product of three numbers:
1. Multiply any two of the numbers.
2. Multiply that answer by the third number.
• It does not matter which numbers are multiplied first.
• Exponents
Math Language
Exponents are
numbers that show
how many times
another number
(the base) is to be
used as a factor.
base
52
3 × 4 × 5 = 00
3 × 4 × 5 = 00
12 × 5 = 60
3 × 20 = 60
• If the exponent is 2, we say “squared.”
squared.”
• If the exponent is 3, we say “cubed.”
cubed.”
2
5 is read “five
3
5 is read “five
• Exponents can be used in formulas:
area of a square
exponent
A = s2
2
area = (length of side)
52 means 5 × 5.
© 2008 Saxon
52 = 25
Lesson Practice
Simplify:
a. 2 × 3 × 4 =
b. 3 × 4 × 10 =
c. 82 =
d. 33 =
Saxon Math Intermediate 4
403
Adaptations Lesson 62
Lesson Practice, continued
e. 102 – 62 =
f. 32 – 23 =
–
=
–
=
g. Rewrite this expression using an exponent: 4 × 4 × 4 =
h. Formula for area of a square:
2
5 in.
Area of this square =
×
Written Practice
1. missing factor
2h =
=
page 402
Divide.
peacocks
2. 00$1.98 paper
00$ .98 pen
00$ .98 pen
00$ .
tax
total
$
+$
+$
r
.
+$
=
$
My estimate is close to the actual sum of
Saxon Math Intermediate 4
404
$
.
Adaptations Lesson 62
© 2008 Saxon
I will round each number to see if my answer is
page 402
Written Practice, continued
3. Use your student clock.
time class starts:
Count minutes back.
4. Finish the table. How many miles
in 4 hours?
60 = __
miles ___
?
2
hours 1
Hours
Miles
1
60
miles ___
00 = __
?
hours 00 3
2
3
miles ___
00 = __
?
00
4
hours
4
5.
Fraction of race over:
_2
3
© 2008 Saxon
Fraction of race left:
7. 9 × 11 = 100 − y
6. $8. 8 7
$2. 9 1
+
=
=y
y=
Saxon Math Intermediate 4
405
Adaptations Lesson 62
page 403
Written Practice, continued
8.
1
__
2
2
__
4
9. Which are prime numbers?
A prime number has exactly two factors
(itself and 1).
A 7
B 8
C 9
D 10
Use fraction pieces.
Use work area.
11.
10. According to this calendar, July 4,
2014, is what day of the week?
18
6
3
JULY 2014
S M T W T F S
1 2
6 7 8 9
13 14 15 16
20 21 22 23
27 28 29 30
×
3 4 5
10 11 12
17 18 19
24 25 26
31
_____
)
×
_____
)
Use work area.
12. Associative Property of
13. 43 =
14.
476,385
+ 259,518
16.
17.
$4.17
×
8
Multiplication
15.
$20.00
– $17.84
c
– 19,434
45,579
c=
Saxon Math Intermediate 4
406
Adaptations Lesson 62
© 2008 Saxon
5×6×7=
page 403
Written Practice, continued
608
× 4
20. 8 _____ R
4)2 9
21. 8 _____ R
8)6 5
22. 8 _____ R
5)2 9
23. 65 ÷ 7 =
24. 29 ÷ 5 =
25. 65 ÷ 9 =
26.
18.
19.
$470
×
7
100% students
40% boys
girls
27. a. perimeter
6 in.
28. What type of angle is each angle of a
square?
© 2008 Saxon
b. area
A acute
B right
C obtuse
D straight
a.
b.
Saxon Math Intermediate 4
407
Adaptations Lesson 62
page 404
Written Practice, continued
29.
Colors of Candles in One Package
Number of Candles
10
8
6
4
2
Red
Orange Yellow
Green
White
a. How many red candles?
b. How many more green than orange candles?
a.
b.
30. Show 12_1, 1.25, and 14_3.
2
© 2008 Saxon
1
Use work area.
Saxon Math Intermediate 4
408
Adaptations Lesson 62
LESSON
Name
63
Teacher Note:
page 405
• Refer students to “Common
Polygons” on page 14 in the
Student Reference Guide.
• Polygons
New Concept
Polygons are closed, flat shapes made from line segments.
Math Language
Three-sided polygons are triangles.
A vertex is a corner
of a polygon (plural
is vertices).
Regular polygons
have:
(Regular triangle)
Four-sided polygons are quadrilaterals.
• sides the same
length.
• angles the same
size.
(Regular quadrilateral, or square)
Five-sided polygons are pentagons.
(Regular pentagon)
Six-sided polygons are hexagons.
(Regular hexagon)
Eight-sided polygons are octagons.
(Regular octagon)
© 2008 Saxon
Ten-sided polygons are decagons.
(Regular decagon)
Saxon Math Intermediate 4
409
Adaptations Lesson 63
Lesson Practice
Draw an example of each polygon.
a. triangle
b. quadrilateral
c. pentagon
d. hexagon
e. octagon
f. decagon
Name each polygon shown and describe its angles:
g.
h.
angles
i.
angles
j.
angles
k.
and
l.
m.
Written Practice
3 = __
0
ft __
1. ___
yd 1 ?
angles
more vertices
page 408
_____
)
2.
30
3
×
_____
)
_____
)
© 2008 Saxon
×
10
Use work area.
Saxon Math Intermediate 4
410
Adaptations Lesson 63
Written Practice, continued
3. 6 quarters
3 dimes
14 pennies
page 408
$00.0000
$00.0000
$00.0000
4. Find the sum of all even numbers
greater than 10 but less than 20
+
5. $ 7.15
$ 5.94
6. Use containers to solve.
1 gallon =
+
cups
Use work area.
7.
7
6
© 2008 Saxon
8.
9. product of 4 and 3
sum of 4 and 3
–
pieces of cornbread eaten: 7
fraction of cornbread left:
Use work area.
Saxon Math Intermediate 4
411
Adaptations Lesson 63
page 409
Written Practice, continued
10. 92
11.
a. name
__
√9
b. perimeter
cm 1
2
3
a.
b.
berries
13. ________
minutes
12. Multiply the loop.
Divide by the outside number.
berries
________
minutes
00 = __
?
___
8
1
00 = __
?
___
1
5
__
)
___
14.
$40.00
−
15. 5 × n = 15 + √2 5
16. Associative Property of
Multiplication
d
$ 2.43
=
6 × 4 × 10 =
n=
d=
18.
3.5
+ 0.0
Saxon Math Intermediate 4
412
19.
1.95
−
.00
Adaptations Lesson 63
© 2008 Saxon
17. 53 =
n=
Written Practice, continued
page 409
20. $1.00 − ($0.36 + $0.57) =
00$0.36
00
00$1.00
00
00$0.57
00 $0.00
22.
×
21.
23. 8 _____ R
6)3 4
$7.60
7
÷
8_____ R
)
=n
n=
The remainder
© 2008 Saxon
24. 8 _____ R
8)6 2
63
26. ___
7 =
25. 8 _____ R
5)2 4
27.
349
× 8
m
represents
players. If only 4
are used, then those players would not have a ride.
minivans will be needed to fit all of the
p
.
Use work area.
Saxon Math Intermediate 4
413
Adaptations Lesson 63
Written Practice, continued
page 409
28. Use the multiplication table.
Which of these numbers is a multiple of 10?
A 3
1,
B 5
,
C 15
D 40
, 10
a. fraction shaded
29.
b. decimal not shaded
c. percent not shaded
a.
b.
c.
30. a. type of polygons:
b.
angles
© 2008 Saxon
line segments
line segments
Use work area.
Saxon Math Intermediate 4
414
Adaptations Lesson 63
LESSON
Name
64
Teacher Notes:
page 411
• Introduce Hint #35 “Short
Division.”
• Division with Two-Digit
Answers, Part 1
• For additional practice, students
may complete Targeted Practice 64.
New Concept
• Short division rules:
1.
2.
3.
4.
Any number “left over” goes in front of the next digit.
There must be a digit above each digit.
Use zero as a placeholder.
Any final number “left over” becomes the remainder.
Example
Solve 78 ÷ 3 using short division.
2
____
3)7 8
1. 3 divides into 7 two times.
2
____
3 ) 7 18
3. Put the leftover 1 in front of the 8.
26
____
3 ) 7 18
2. Multiply: 2 × 3 = 6. Subtract 6 from 7.
4. 3 divides into 18 six times.
5. Multiply: 6 × 3 = 18. Subtract 18 from 18.
There is no remainder.
Example
0 7 8
______
)
3 2 3 24
0 7 8 R1
______
)
3 2 3 25
Tests for Divisibility
© 2008 Saxon
A number is able to be divided by:
2
if the last digit is even.
5
if the last digit is 0 or 5.
Already
know
10 if the last digit is 0.
3
Saxon Math Intermediate 4
if the sum of the digits can be divided by 3.
415
New
Adaptations Lesson 64
New Concept, continued
Example
Use the divisibility rules to divide 87 by 3.
8 + 7 = 15
87
Since 15 is a multiple of 3, 87 can be divided by 3 with no
remainder.
Lesson Practice
Use short division.
b. 4 ) 5
1
_____
d. 3 ) 7
_____
_____
_____
a. 3 ) 5
c. 5 ) 7
2
_____
e. 4 ) 9
2
_____
f. 2 ) 7
6
g. Find the missing factor: 3n = 45
5
_____
)
4
n=
h. Which of these numbers can be divided by 3 with no remainder?
A 75
The sum of
B 76
C 77
and
D 79
is 12, which is a multiple of 3.
i. There are 6 students per row.
29 students fill the first available seat.
What is a reasonable estimate of the number of rows that will be filled?
, and 30 seats ÷ 6 students =
rows.
© 2008 Saxon
29 is close to
Saxon Math Intermediate 4
416
Adaptations Lesson 64
page 415
Written Practice
1. twenty-seven million, eight hundred seventy-eight thousand, four hundred
square feet
,
,
2. number in each group × number of groups
×
3.
=
has
m
more
will hold
m=
4. Multiply the loop.
days
0
7 = __
______ __
weeks 1 ?
5.
_____
sides of a stop sign
stop signs
)
6.
inch
1
2
3
© 2008 Saxon
7. 406,912
expanded form:
words:
Use work area.
Saxon Math Intermediate 4
417
Adaptations Lesson 64
page 415
Written Practice, continued
8. Multiply the loop.
9. The students washed 8 cars in 90 minutes.
Estimate the number of minutes spent washing
each car.
1 = __
4
ft
__
___
in. 0 ?
com
I can use
numbers.
is close to 88.
88 ÷ 8 =
minutes
Use work area.
10.
3
__
6
1
__
2
11. a. 614
609
b. 88
106
Use fraction pieces.
Use work area.
Use work area.
last week
12.
this week
112 total
–
r
I will round to the nearest 10 to see if my answer is
+
=
.
papers
Use work area.
$32.47
+ $67.54
14.
51,036
− 7,648
15.
53.6
2.9
97.4
8.8
+ 436.1
© 2008 Saxon
13.
Use work area.
Saxon Math Intermediate 4
418
Adaptations Lesson 64
page 416
Written Practice, continued
16. short division
18. short division
17. short division
5n = 75
_____
3)8
_____
4)9
4
2
____
)123
n=
19. short division
_____
6)5 8
22.
21.
20. Carry on your fingers.
257
× 5
R
24. 4n = 36
23. short division
$334
×
9
_____
2)3
$7.09
×
3
_____
)
6
n=
© 2008 Saxon
25.
42
+
23
+
Saxon Math Intermediate 4
=
26.
3.5 − (2.4 − 1.3) =
−
=
419
=
Adaptations Lesson 64
page 416
Written Practice, continued
27. See
page 416.
List ways to pair two bills:
$5
and $10
$5
and $
$10
and
and
and
$20
and
Use work area.
fraction of game over: 4_3
28.
fraction of game remaining:
Use work area.
29. a. fraction shaded
b. decimal shaded
c. percent not shaded
.
%
Use work area.
30. prime = 2 factors
What are the next two prime numbers?
A 4 and 5
Saxon Math Intermediate 4
B 5 and 6
C 5 and 7
420
D 7 and 9
Adaptations Lesson 64
© 2008 Saxon
The first two prime numbers are 2 and 3.
LESSON
Name
65
Teacher Notes:
page 417
• Review Hint #35 “Short Division.”
• Refer students to “Division” on
page 8 in the Student Reference
Guide.
• Division with Two-Digit
Answers, Part 2
• For additional practice, students
may complete Targeted Practice 65.
New Concept
• The numbers in a division problem are called the divisor,
dividend, and quotient.
dividend quotient
dividend
divisor
78 = 26
3 =
78 ÷ 3 = 26
quotient
divisor
divisor
26
___
)
3 78
quotient
dividend
Tests for Divisibility
A number is able to be divided by:
2
if the last digit is even.
5
if the last digit is 0 or 5.
Already
know
© 2008 Saxon
10 if the last digit is 0.
3
if the sum of the digits can be divided by 3.
9
if the sum of the digits can be divided by 9.
New
• Use short division:
0 7
8 • Any number “left over” goes in front of the
_________
3)2 3 24
next digit.
• Place a digit above each digit.
• Use zero as a placeholder.
Saxon Math Intermediate 4
421
Adaptations Lesson 65
Lesson Practice
In the division fact 32 ÷ 8 = 4,
a. What number is the divisor?
b. What number is the dividend?
c. What number is the quotient?
Use short division.
d.
g.
j.
e.
0
_________
3)1
4 4
h.
0
_________
5)2
2 5
f.
0
_________
4)1
4 4
i.
0
_________
7)4
5 4
0
_________
6)1
4 4
0
_________
8)2
0 0
Which of these numbers can be divided by 9 without a remainder?
A 288
B 377
C 466
D 555
How do you know?
The sum of the digits
is a multiple of 9.
, and
,
equals
, which
k. Find the missing factor in this equation:
5m = 125
l.
_____
)
m =
An oil change business sold 157 quarts of motor oil.
About how many gallons is that?
Round.
1 5 7 quarts
quarts
157 quarts is close to
There are
quarts.
quarts in each gallon.
÷
A reasonable estimate is
Saxon Math Intermediate 4
© 2008 Saxon
quarts
0 = __
4
_______ __
galons ? 1
422
=
.
Adaptations Lesson 65
page 421
Written Practice
1. missing factor
72 end
2.
Divide.
____
)
3.
years
_______
months
0
1 = __
___
12 ?
begin
minutes long
Use work area.
4. 7 half gallons = how many gallons?
half
gallons __
2 = __
7
___________
1 ?
gallons
___
)
36 = __
miles ___
?
5. ______
1
hours 4
R
_____
)
and
The quotient is between
Madeline purchases
.
or
gallons of juice each month.
Use work area.
6. Finish the table. How many miles in 5 days?
7.
© 2008 Saxon
Find the pattern.
Days
Miles
1
20
2
40
a. fraction not
shaded
b. perimeter
3
a.
Use work area.
Saxon Math Intermediate 4
423
b.
Adaptations Lesson 65
page 422
Written Practice, continued
8. see
page 422.
Average Annual Precipitation
14
Amount of Rain
(in inches)
12
10
8
6
4
2
Phoenix
Reno
Boise
Albuquerque
City
Use work area.
9. 8:05 to 10:05 is how many hours?
hr
10:05 to 10:25 is how many minutes?
10. What is the diameter?
min
18 in.
A
B
cm 1
2
Saxon Math Intermediate 4
3
4
5
© 2008 Saxon
11. How long is segment BC?
C
6
7
424
8
9
10
11
Adaptations Lesson 65
page 422
Written Practice, continued
13. 27 ÷ 32
12. Which of these words names the
answer to a division problem?
A product
B dividend
C divisor
D quotient
__
27 ÷ √ 9
Use work area.
14.
$97.56
+ $ 8.49
18. short division
378 ÷ 7 =
© 2008 Saxon
22. short division
_________
4)1
3 6
Saxon Math Intermediate 4
15.
19.
$60.00
− $54.78
16.
840
× 3
20. 4 × 564 =
23. short division
_________
2)1
3 2
425
37.64
29.45
3.01
+ 75.38
17. short division
168 ÷ 3 =
21.
304
× 6
24. short division
_________
6)1
9 2
Adaptations Lesson 65
page 423
Written Practice, continued
25. 7n = 50 + 34
and
First, add
Then divide
. The sum is
by
. The answer is
.
.
n=
26. Associative Property of Multiplication
27. See
page 423.
12 × 7 × 10 =
29. See page 16 in the Student Reference Guide
Draw a quadrilateral.
A quadrilateral has how many
vertices?
28. three quarters
a. fraction
b. percent
a.
b.
Use work area.
30. a. Which side is parallel to side CB?
A
D
B
C
a.
b.
Saxon Math Intermediate 4
426
Adaptations Lesson 65
© 2008 Saxon
b. Which angle appears to be obtuse?
LESSON
Name
66
Teacher Note:
page 424
• Use Lesson Activity 29 for the
activity.
• Similar and Congruent
Figures
New Concept
• Congruent figures match exactly (the same shape and the
same size).
X
C
B
A
Z
Y
• Triangle ABC and triangle XYZ are congruent.
∠A corresponds to ∠X.
___
___
AB corresponds to XY.
• Similar figures have the same shape but not necessarily
the same size.
Similar
not similar
or congruent
A
B
C
D
© 2008 Saxon
Congruent
Saxon Math Intermediate 4
427
Adaptations Lesson 66
New Concept, continued
Activity
page 426
Determining Similarity and Congruence
Use Lesson Activity 29 to complete this activity.
1. Is the first shape similar to the bike sign?
Is the shape congruent to the bike sign?
Cut out the shape on the left and place it on top of the bike sign to
check your answers.
2. Is the triangle similar to the yield sign?
Is the triangle congruent to the yield sign?
Cut out the triangle and place it on top of the yield sign to check your
answers.
3. How can you tell if the octagon on the left is congruent to the stop sign?
The octagon is
c
They are the same
to the stop sign.
s
and the same
s
Are these shapes similar?
, congruent shapes are also
s
.
A
B
C
© 2008 Saxon
Lesson Practice
D
a. Which are similar (same shape)?
b. Which are congruent (match exactly)?
Saxon Math Intermediate 4
428
Adaptations Lesson 66
page 426
Written Practice
2. There are 141 beads.
Every third bead is red.
How many beads are red?
earned each hour
hours per day
missing factor
_____
earned each day
) 789
100% square
25% shaded
4. North
20 km
3.
Divide.
not shaded
15 km
1.
South
5. Time now:
Count minutes forward.
Count hours forward.
6. a. perimeter:
units
6 units
3 units
© 2008 Saxon
area:
square units
b. Draw a rectangle with 18 square units arranged in two rows.
Use work area.
Saxon Math Intermediate 4
429
Adaptations Lesson 66
page 427
Written Practice, continued
30 = __
?
mi ___
7. ___
0
gal 1
8.
I multiplied
by
miles
cheered wildly:
gallons.
The answer is
miles.
_2
7
stood quietly:
Use work area.
9. Make 3-digit numbers using 4, 2, and 7. Label even or odd.
2
7
odd
2
4
7
different numbers
10.
1
__
2
2
__
5
11. n + 2 = 3 × 12
© 2008 Saxon
=
n=
Use work area.
Saxon Math Intermediate 4
430
n=
Adaptations Lesson 66
page 427
Written Practice, continued
___
13. √ 81 + 82 + 32 =
12. 6.42 − (3.3 − 1.5) =
3.3
1.5
6.42
+ 82 +
0.42
14.
15.
$10.00
43,016
=
16. Associative Property of
Multiplication
−$
17.
−
.
$4.86
×
7
18.
307
×
8
24 × 3 × 10 =
19.
$460
×
9
20. short division
_________
2)1
© 2008 Saxon
21. short division
22. short division
23. short division
230 ÷ 5 =
_________
6)2
6 4
Saxon Math Intermediate 4
5 2
24. short division
91 ÷ 7 =
_____
4)5
6
431
Adaptations Lesson 66
page 428
Written Practice, continued
26. a. 8¢
25. short division
135 ÷ 3 =
$
b. $11. 8 9
.
$
.
Use work area.
3
__
27. a. 210
in words:
b. 2.3 in words:
Use work area.
28. a. Which two triangles are congruent?
congruent
exactly the same
A
B
b. Triangles
C
29. Draw a pentagon.
A pentagon has how
many vertices?
D
and
are
congruent because they have the same
s
and the same
s
.
© 2008 Saxon
Use work area.
30. Are all squares similar? Why or why not?
All squares
equal sides and
similar because every square has
right angles.
Use work area.
Saxon Math Intermediate 4
432
Adaptations Lesson 66
LESSON
Name
67
Teacher Note:
page 429
• For additional practice, students
may complete Targeted Practice 67.
• Multiplying by Multiples
of 10
New Concept
• To multiply a whole number by 10, attach a zero after the
number.
25 × 10 = 250
Multiples of 10
37 × 10 = 370
10, 20, 30, 40, 50, 60, 70…
• Use offset multiplication to multiply a whole or decimal
number by a multiple of 10.
1.
2.
3.
4.
Write the multiple of 10 as the bottom number.
Let the zero “hang out” (offset to the right).
Copy the zero into the answer.
Multiply.
Example
Example
34
× 20
34
Zero “hangs out” to the right. × 20
0
34
× 20
680
• When multiplying a money amount by a multiple of 10, put
two decimal places in the answer.
$1.43
20
$28.60
© 2008 Saxon
×
Saxon Math Intermediate 4
433
Adaptations Lesson 67
Lesson Practice
Multiply.
a. 75 × 10 =
b. 10 × 32 =
c. 10 × 53¢ =
d.
26
× 20
$1.64
×
30
f.
45
× 50
e.
g. Write 12 × 30 as a product of 10 and two other factors. Then multiply.
See Example 1 on page 430.
×
× 10 =
page 431
Written Practice
_____
1. missing factor
Divide.
)
2. a. perimeter:
8 units
3 units
b. Sketch a rectangle that is four units wide with the same area as the
rectangle above.
Area = length × width
perimeter:
Use work area.
Saxon Math Intermediate 4
434
Adaptations Lesson 67
© 2008 Saxon
area:
page 431
Written Practice, continued
3. The server placed a full pitcher of
water on the table.
4. Which of these numbers is not a
factor of 12?
Use the multiplication table.
Which of the following is a reasonable
estimate of the amount of water in
the pitcher?
5.
1
A 2 gallons
B 2 quarts
C 2 cups
D 2 ounces
Starting time
11
10
11
10
2
3
7
6
,
D 3
,
,
12
6. 3,097,600 sq. yd
words:
2
3
4
8
5
,
C 4
12 1
9
4
8
B 5
,
Stopping time
12 1
9
A 6
7
6
5
The starting time was before dawn.
The stopping time was in the afternoon.
What was the difference in the two times?
Use work area.
© 2008 Saxon
7.
a. fraction not shaded
8. last Saturday of the month
b. shaded part:
more or less than
50%?
a.
b.
Saxon Math Intermediate 4
,
435
Adaptations Lesson 67
page 432
Written Practice, continued
9. See
10. mixed number
page 432.
The exact product of 5 × 12 is
.
7
8
Round 12 to the nearest ten. Then multiply.
5×
=
The estimate is
than the
exact product.
Use work area.
11. See
Use work area.
page 433.
A better
e
would be to round 68 to the nearest
instead of to the nearest hundred:
4×
=
Use work area.
12.
23
2×3
13.
$6.25
14.
3.6
.
.
17.
409
× 70
$0.00
$0.00
15.
$30.25
−
b
$13.06
16.
149,384
− 98,765
b=
Saxon Math Intermediate 4
436
Adaptations Lesson 67
© 2008 Saxon
Use work area.
page 433
Written Practice, continued
18.
$3.46
$0.79
19.
20. 10 × 39¢ =
×
×
_____
6 )9
23. short division
22. short division
21. short division
0
24. short division
6
25. short division
234 ÷ 3 =
_________
8 )4
5 6
26. fraction and decimal
shaded
© 2008 Saxon
95 ÷ 5 =
_____
4 )9
Saxon Math Intermediate 4
437
Adaptations Lesson 67
Written Practice, continued
page 433
27. a. Which two figures are congruent?
congruent
A
exactly the same
B
C
D
b. If one of them is flipped, they are the same
the same
.
a.
28. How much money is 4_1 of a dollar?
100¢ ÷ 4
and
and
b.
Use work area.
29. Draw a hexagon.
A hexagon has how many vertices?
30. a. The first temperature of the morning was recorded at
.
What was that temperature?
b. Was the noon temperature warmer or colder than the 10 a.m. temperature?
© 2008 Saxon
How many degrees warmer or colder was the noon temperature?
Use work area.
Saxon Math Intermediate 4
438
Adaptations Lesson 67
LESSON
Name
68
Teacher Notes:
page 435
• Review Hint #35 “Short Division.”
• For additional practice, students
may complete Targeted Practice 68.
• Division with Two-Digit
Answers and a Remainder
New Concept
• Use short division:
1.
2.
3.
4.
Any number “left over” goes in front of the next digit.
Place a digit above each digit.
Use zero as a placeholder.
Any final number “left over” becomes the remainder.
To check a division answer that has a remainder:
1. Multiply the quotient by the divisor.
2. Add the remainder.
3. The answer should match the dividend.
Example
02 7 R 2
______
5 ) 1 3 37
Check:
135
+ 2
137
27
× 5
135
dividend
Lesson Practice
© 2008 Saxon
Divide. Use short division.
0
__________
R
0
__________
R
a. 3 ) 1 3 4
d. 8 ) 2 5 9
Saxon Math Intermediate 4
0
__________
b. 7 ) 2 4 0
______
e. 4 ) 9 5
R
439
R
______
c. 5 ) 8 8
R
0
__________
f. 6 ) 3 2 5
R
Adaptations Lesson 68
Lesson Practice, continued
g. Shou divided 235 by 4 and got 58 R 3 for her answer.
Describe how to check Shou’s calculation.
To check her calculation, I would
Then I would
a
m
58 by
.
3 to the product. The answer should be
.
h. 175 birds live in the 9-acre marsh.
What is a reasonable estimate of the number of birds in each acre?
Explain why your estimate is reasonable.
175 is close to 180. The number
is a factor of both 18 and 180.
A reasonable estimate is 180 ÷
, which equals
birds per acre.
page 437
Written Practice
1. Finish the table. How many miles in
3 hours?
2. perimeter and area
Find the pattern.
6 units
4 units
Hours
Miles
1
6
12
© 2008 Saxon
3
3. twelve and fourteen hundredths seconds
.
Saxon Math Intermediate 4
440
Adaptations Lesson 68
Written Practice, continued
page 437
4. octagon
5. Offset.
47 × 30 =
pentagon
total
6. Offset.
7. Offset.
60 × 39 =
85 × 40 =
a. fraction of the course ran: 5_3
8.
fraction of the course walked:
b. ran: more or less than 50%?
Use work area.
9.
6
7
8
10. Show 0, 1, 2, 13_2, and 2 3_1.
11
3
© 2008 Saxon
2
3
3
Use work area.
11. four thousand, three hundred ninety-two meters
,
Saxon Math Intermediate 4
441
Adaptations Lesson 68
Written Practice, continued
page 438
_____
)
?
35 = __
knots
___
12.________
1
minutes 0
13. 6810
9030
14.
$12.15
$
$ 5.95
$
+
15. $20 − ($8.95 + 75¢) =
$8.95
$0.75
$20.00
$
.00
00
estimate:
actual sum:
23.64
16.
17.
43¢
× 8
18.
$3.05
×
5
19.
×
$2.63
7
20. Change to multiplication and solve:
64 + 64 + 64 + 64 + 64
×
Saxon Math Intermediate 4
442
=
Adaptations Lesson 68
© 2008 Saxon
.
page 438
Written Practice, continued
21. short division
______
5) 9 6
22. short division
R
6
R
__________
7) 1 5 6
__________
3) 2 4 6
25. short division
24. short division
216
____
23. short division
26. short division
195 ÷ 8 =
=
_____
)
4r = 156
r=
27. Use an inch ruler to find the lengths of segments AB, BC, and AC.
A
B
C
___
___
___
AB
BC
AC
28. a. Which word makes the following sentence not true?
All squares are
A polygons
© 2008 Saxon
b. Not all
B rectangles
s
.
C similar
D congruent
are
they are not all the same
s
because
.
Use work area.
29. 2 quarts
_1
2
gallon
See page 1 in the Student Reference Guide.
Use work area.
Saxon Math Intermediate 4
443
Adaptations Lesson 68
page 439
Written Practice, continued
Land Tunnels in the United States
Length (in feet)
30.
4800
4700
4600
4500
4400
4300
4200
4100
4000
3900
3800
3700
3600
3500
3400
3300
3200
Dingess, WV
Cumberland Gap, KY/TN
Tunnel Name and State
Hanging Lake, CO
a. Tunnels from shortest to longest:
shortest
longest
b. How many feet longer is the Hanging Lake Tunnel than the Dingess Tunnel?
Hanging Lake
Dingess
feet longer
miles
feet
2
1 = __
_____
5280
?
2 miles =
Dingess
Cumberland Gap
Hanging Lake
total feet
feet
The combined length of the
t
is
than 2 miles.
Use work area.
Saxon Math Intermediate 4
444
Adaptations Lesson 68
© 2008 Saxon
c. 1 mile = 5280 feet
Are the combined lengths of the tunnels more than or less than 2 miles?
LESSON
Name
69
Teacher Notes:
page 440
• Introduce Hint #36 “Reading Metric
Rulers.”
• Millimeters
• Review “Equivalence Table for
Units” on page 1 in the Student
Reference Guide.
• Use rulers to illustrate the
examples.
New Concept
• 1 centimeter = 10 millimeters
Millimeter
mm 10
20
30
40
50
60
cm 1
2
3
4
5
6
Centimeter
• The length of the segment below is 35 mm. It is also 3.5
centimeters.
mm 10
20
30
40
50
60
• Metric measures are always written as decimal numbers
instead of fractions.
• To convert:
• from centimeters to millimeters
• from millimeters to centimeters
© 2008 Saxon
Activity
Multiply by 10.
Divide by 10.
page 442
Measuring with Metric Units
• Use your text book to complete this activity.
Saxon Math Intermediate 4
445
Adaptations Lesson 69
Lesson Practice
a. The thickness of a dime is about 1 mm.
How many dimes would it take to form a stack that is about 1 cm high?
b. Write the length of the segment below twice,
once in millimeters and once in centimeters.
Millimeter
Centimeter
mm 10
20
30
40
50
60
cm 1
2
3
4
5
6
c. Each side of this square is 1 cm long.
What is the perimeter of this square in millimeters?
1 cm
d. The diameter of a penny is about 19 mm. How many centimeters is that?
e. Write a decimal subtraction problem that shows how to find the length of
segment XY.
W
cm 1
X
Y
2
3
3.2 cm
4
5
−
6
© 2008 Saxon
f. Write 3.4 cm as a fraction.
Saxon Math Intermediate 4
446
Adaptations Lesson 69
Written Practice
1.
page 443
students per bus
2.
384 Celeste
Will
buses
more
total students
4.
3. How many millimeters?
A
B
0
C
1
2
point A
decimal
fraction
decimal
fraction
decimal
fraction
point B
5.2 cm
point C
Use work area.
6.
© 2008 Saxon
5. Shade one sixth.
mm 10
20
30
40
50
60
cm 1
2
3
4
5
6
a. centimeters
b. millimeters
a.
Use work area.
Saxon Math Intermediate 4
447
b.
Adaptations Lesson 69
page 444
Written Practice,
Practice, continued
7.
8. Find the perimeter of the square
in feet.
100% students
25% completed on Thursday
yd
___
completed on Friday
ft
1 = 00
___
__
?
3
1 yd
1 yd
1 yd
1 yd
9. 412
495
379
+
h
I rounded each number to the nearest
+
A reasonable
10. A
3.5 cm
+
e
and then added.
=
is
B
.
?
C
© 2008 Saxon
11.6 cm
11.6 cm
.
Saxon Math Intermediate 4
448
Adaptations Lesson 69
page 445
Written Practice, continued
125 = ___
?
mi ____
11. a. ___
5
00
hr
12. prime = 2 factors
The first three prime numbers are
2, 3, and 5.
?
21 = ___
mi ___
b. ___
00
hr 1
Next three prime numbers:
,
and
,
.
a.
b.
13. $7.95
$8.95
Use work area.
14. short division
15. short division
250 ÷ 6 =
100 ÷ 9 =
+
17. short division
18. 4w = 60
19. 9 × $4.63 =
16.
20.
36.2
4.7
15.9
148.4
30.5
+ 6.0
$0.290
×
80
256
____ =
8
© 2008 Saxon
w=
21.
$10.00
− $ 1.73
22.
36,428
− 27,338
23.
78
× 60
24. short division
_________
4)3
Saxon Math Intermediate 4
449
2 8
Adaptations Lesson 69
page 445
Written Practice, continued
25. short division
_________
7)3
27. a + 5 = 25 + 25
26. short division
=
_________
5)3
R
2 0
a=
7 5
a=
28. 4.7 − (3.6 − 1.7) =
3.6
29.
3 cm
20 mm
4.7
2 cm
30 mm
1.7
First
.
s
a. perimeter in millimeters
1.7
b. area in square centimeters
from 3.6.
The difference is
.
Then subtract
from 4.7.
The answer is
.
a.
30. Each angle of this triangle is
A acute
Saxon Math Intermediate 4
B right
.
C obtuse
450
D straight
Adaptations Lesson 69
© 2008 Saxon
b.
LESSON
Name
70
Teacher Note:
page 446
• Introduce Hint #37 “Fraction of a
Group, Part 1.”
• Word Problems About a
Fraction of a Group
New Concept
Example
One fourth of the team’s 32 points were scored by Thi.
Thi scored how many points?
The whole rectangle stands for 32 points.
Thi scored 4_1 of the points, so we divide the rectangle into
4 equal parts.
32 points
1
4
3
4
scored
by Thi
8 points
not scored
by Thi
8 points
8 points
8 points
__________
4 ) 32 points
8 points
Thi scored 8 points.
• Here are some shortcuts:
_1
2
of a number
Divide by 2.
• To find
_1
3
of a number
Divide by 3.
• To find
_1
4
of a number
Divide by 4.
© 2008 Saxon
• To find
Saxon Math Intermediate 4
451
Adaptations Lesson 70
Lesson Practice
a. What is 3_1 of 60?
b. What is 2_1 of 60?
___
) 60
60
___
) 60
60
1
2
1
3
c. What is 4_1 of 60?
d. What is 5_1 of 60?
___
) 60
60
1
4
___
) 60
60
1
5
e. One half of the 32 children were boys.
How many boys were there?
_1
2
of
=
32 children
1
2
1
2
___
boys
) 32
were
boys.
were
girls.
___ children
___ children
f. One third of the 24 coins were quarters.
How many quarters were there?
of
=
___
quarters
) 24
2 were not
3 quarters.
___ coins
___ coins
___ coins
© 2008 Saxon
_1
3
24 coins
1 were
3 quarters.
Saxon Math Intermediate 4
452
Adaptations Lesson 70
page 448
Written Practice
1.
150 seats
128 filled
2. 12.14 seconds
Angela
Marion
empty
faster
3.
4.
42,000,000
price of lunch
lunches per week
greater
5. perimeter and area
spent during week
6.
30 M
T
5 units
W
4 units
Th
F
in all
I can use compatible
n
to check my answer.
© 2008 Saxon
30 ×
is about 100.
45 is close to
a.
100 +
+
b.
Saxon Math Intermediate 4
, and 26 is near 25.
=
Use work area.
453
Adaptations Lesson 70
Written Practice, continued
7. a.
page 449
___
) 74
_1
of 74
2
1 =
b. __
2
74 seeds
%
1 sprouted.
2
1 did not
2 sprout.
___ seeds
___ seeds
a.
b.
8.
Show all the ways to arrange these bills in a row:
$1
,
$10
$1
,
,
,
,
,
,
,
,
,
,
,
Use work area.
9.
___
_1
of 60
6
) 60
60
1
6
10. Finish the table. How many miles in 3 hours?
Hours
Miles
1
65
hr
65 = ___
?
___
1
00
© 2008 Saxon
2
mi
___
Saxon Math Intermediate 4
454
Adaptations Lesson 70
page 449
Written Practice, continued
_____
)
?
11. ___
248 = __
mi ____
1
0
hr
12.
a. diameter in centimeters
cm 1
2
b. radius in millimeters
a.
b.
13. A
2.7 cm
B
4.8 cm
C
?
14. 00$8.00
00$ .
00$ ... ..
4.8 cm
4.8 cm
5.36
15.
16.
$100.00
− $ 59.47
17.
37,102
− 18,590
.
.
___
© 2008 Saxon
18. √ 49
×
23
=
19. Offset.
20. Offset.
00$1.638
×
Saxon Math Intermediate 4
=
60 × 39 =
0
455
Adaptations Lesson 70
page 450
Written Practice, continued
21.
22. short division
$2.56
23. short division
_________
_____
3)8 9
9)2
R
25. short division
24. short division
26. short division
243 ÷ 7 =
90
___ =
6
3 4
5m = 355
m=
27. 7 + n = 28
28. Write twelve and three tenths as a:
• mixed number
• decimal number
n=
Use work area.
29. Which of these numbers is a factor of both 12 and 20?
B 4
C 5
D 6
Factors of 12:
1
,
,
,
,
,
12
Factors of 20:
1
,
,
,
,
,
20
Use work area.
30. Draw a triangle that has one right angle.
Use work area.
Saxon Math Intermediate 4
456
Adaptations Lesson 70
© 2008 Saxon
A 3
I NVE S TIGATION
7
Name
page 451
Focus on
• Collecting Data with
Surveys
• A survey gathers specific information about a group.
• The group in a survey is called a population.
• A sample is part of a population.
• How survey questions are asked can affect the results of
a survey.
Which of these school
lunches is your favorite?
tuna
turkey
pizza
chicken
Multiple Choice:
Answers limited to
the choices given
Which lunch from
the school menu
is your favorite?
Open-Ended:
Open to many
answers
1. Write two questions to find out students’ favorite drink to have
with lunch.
Make one question multiple-choice.
For the other, leave the question open (do not list options).
© 2008 Saxon
Question 1:
Saxon Math Intermediate 4
457
Adaptations Investigation 7
7
INVE STIGATION
continued
Question 2:
• Survey questions should be written without bias.
Bias makes one choice sound better than another choice.
2. Describe the bias in the following question:
Which drink do you prefer with lunch:
cool, sweet lemonade or milk that has been out of the
refrigerator for an hour?
The words
c
question to favor
and
s
l
bias the
.
3. Cross off words to remove the bias:
Which drink do you prefer with lunch:
cool, sweet lemonade or milk that has been out of the
refrigerator for an hour?
• When using a sample to find information about a larger
population, the sample must be very similar to the population.
4. You will survey students in your class. (Your class is the sample.)
© 2008 Saxon
Which of the following larger populations will your survey apply
to most?
(State why or why not each larger population is like your
sample.)
Saxon Math Intermediate 4
458
Adaptations Investigation 7
7
INVE STIGATION
continued
A all the students in the school
The
d
between ages might be too great.
B all the school children your age in your community
This
p
because the
is most similar to the sample
a
and region are the same.
C all the children your age in the country
r
The differences in
might be too great.
D all the parents of students in the class
Parents might make different choices than their
.
© 2008 Saxon
c
Saxon Math Intermediate 4
459
Adaptations Investigation 7
7
INVE STIGATION
continued
We can track survey answers with a tally sheet.
On the tally sheet, we make tally marks:
Make four vertical marks.
Cross them with a fifth diagonal mark.
This counts as five.
Question: Which of these school lunches is
your favorite?
tuna, turkey, pizza, chicken
Answer: tuna
turkey
pizza
chicken
5. According to the tally marks, how many students named tuna as
their favorite lunch?
6. Make a tally sheet to show favorite drinks to have with lunch.
Write a multiple-choice question.
List the possible answers.
One choice may be “no opinion.”
Question:
Tally:
© 2008 Saxon
Answer Choices:
Saxon Math Intermediate 4
460
Adaptations Investigation 7
INVE STIGATION
7
continued
Duplicating data occurs when information is counted more
than once.
Omitting data occurs when information is left out.
7. Brad found that 11 students in his class had dogs.
Dena found that 9 students in the same class had cats.
Brad and Dena concluded that 20 students in the class had
dogs or cats for pets.
Is this conclusion correct?
Why or why not?
Their
c
might not be correct.
Some students may have both a
a
and
for pets.
Dena and Brad may have counted these students
times.
© 2008 Saxon
t
Saxon Math Intermediate 4
461
Adaptations Investigation 7
7
INVE STIGATION
continued
8. Brad and Dena also concluded that 20 students in the class
have pets.
Is this conclusion correct?
Why or why not?
Their
c
might not be correct.
Brad and Dena did not count some students who may have
other
, such as birds or fish.
Brad and Dena may have counted some students
t
times-those who may have both a
c
Activity
and a
d
for pets.
page 453
Class Survey
© 2008 Saxon
• This activity is optional.
Saxon Math Intermediate 4
462
Adaptations Investigation 7