California Mathematics Content Standards
REFRESHER
LESSON
5.MR 2.0, 2.3 Use a variety of methods, such as words,
numbers, symbols, charts, graphs, tables, diagrams,
and models, to explain mathematical reasoning.
4.MG 3.0, 3.5 Know the definitions of a right angle, an
acute angle, and an obtuse angle. Understand that
90°, 180°, 270°, and 360° are associated, respectively,
with 4_1, 2_1, 4_3, and full turns.
17
• Measuring Turns
Refresher Concept
Every hour the minute hand of a clock completes one full turn in a
clockwise direction. How many degrees does the minute hand turn
in an hour? 360°
Turns can be measured in degrees. A full turn is a 360° turn. So the
minute hand turns 360° in one hour.
If you turn 360°, you will end up facing the same direction you were
facing before you turned. A half turn is half of 360°, which is 180°. If
you turn 180°, you will end up facing opposite the direction you were
facing before you turned.
If you are facing north and turn 90°, you will end up facing either east
or west, depending on the direction in which you turned. To avoid
confusion, we often specify the direction of a turn as well as the
measure of the turn. Sometimes the direction is described as being
to the right or to the left. Other times it is described as clockwise
or counterclockwise. Clockwise means “moving in the direction of the
hands of a clock.” Counterclockwise means “moving in the opposite
direction of the hands of a clock.”
Example 1
Leila was traveling north. At the light she turned 90° to the left and
traveled one block to the next intersection. At the intersection she
turned 90° to the left. What direction was Leila then traveling?
A picture may help us answer the question.
Leila was traveling north when she turned
90° to the left. After that first turn Leila
was traveling west. When she turned
90° to the left a second time, she began
traveling to the south. Notice that the
two turns in the same direction (left) total
180°. So we would expect that after the
two turns Leila was heading in the direction
opposite to her starting direction.
Saxon Math Intermediate 6
© Harcourt Achieve Inc. and Stephen Hake. All rights reserved.
N
E
W
S
90í
90í
35
Example 2
Andy and Barney were both facing north. Andy made a quarter turn
(90°) clockwise to face east, while Barney turned counterclockwise
until he faced east. How many degrees did Barney turn?
We will draw the two turns. Andy made a
quarter turn clockwise. We see that Barney
made a three-quarter turn counterclockwise.
We can calculate the number of degrees in
three quarters of a turn by finding 34 of 360°.
Andy
N
E
3
⫻ 360° ⫽ 270°
4
Barney
N
Another way to find the number of degrees is
to recognize that each quarter turn is 90°. So
three quarters is three times 90°.
E
3 × 90° = 270°
Barney turned 270° counterclockwise.
Example 3
As Elizabeth ran each lap around the park, she made six turns to
the left (and no turns to the right). What was the average number of
degrees of each turn?
We are not given the measure of any of the
turns, but we do know that Elizabeth made six
turns to the left to get completely around the
park. That is, after six turns she once again
faced the same direction she faced before the
first turn. So after six turns she had turned a
total of 360°. We find the average number of
degrees in each turn by dividing 360° by 6.
2
1
3
6
4
5
360° ÷ 6 = 60°
Each of Elizabeth’s turns averaged 60°.
Refresher Practice
c.
a.
b. Kiara made one full turn counterclockwise. Mary made two full turns
clockwise. How many degrees did Mary turn? 2 × 360° = 720°
1
2
5
c.
3
4
360í
â 72í
5
36
Jose was heading south on his bike. When he
reached Sycamore, he turned 90° to the right. Then at Highland
he turned 90° to the right, and at Elkins he turned 90° to the right
again. Assuming each street was straight, in which direction was
Jose heading on Elkins? east
Analyze
Model
David ran three laps around the park. On each lap
he made five turns to the left and no turns to the right. What was
the average number of degrees in each of David’s turns? Draw a
picture of the problem.
© Harcourt Achieve Inc. and Stephen Hake. All rights reserved.
Saxon Math Intermediate 6