Objective
To guide students in comparing predicted and actual
results from an experiment with equally likely outcomes.
1
materials
Teaching the Lesson
Key Activities
Students color a 10-by-10 grid. They determine the chance that a centimeter cube, dropped
onto the grid, will land on a particular color. They perform the experiment and compare the
actual results with their predictions.
Key Concepts and Skills
•
•
•
•
•
Rename fractions as percents. [Number and Numeration Goal 5]
Use basic probability terms to describe the likelihood of events. [Data and Chance Goal 3]
Conduct a cube-drop experiment. [Data and Chance Goal 4]
Use fractions and percents to predict the outcomes of an experiment. [Data and Chance Goal 4]
Compare predicted outcomes and actual results. [Data and Chance Goal 4]
ⵧ Math Journal 2, pp. 213– 215
ⵧ Study Link 7 11
䉬
ⵧ Teaching Master (Math Masters,
p. 238)
ⵧ Teaching Aid Master (Math Masters,
p. 388 or 389)
ⵧ colored pencils, markers, or
crayons (yellow, red, green, blue)
ⵧ 1 cm cube per partnership
Ongoing Assessment: Informing Instruction See page 635.
ⵧ slate
ⵧ shoe box or copier-paper box
(optional)
Ongoing Assessment: Recognizing Student Achievement Use a Math Log or Exit Slip.
See Advance Preparation
[Data and Chance Goal 4]
2
materials
Ongoing Learning & Practice
Students solve place-value problems.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Differentiation Options
READINESS
Students use
base-10 blocks to
model fractions
and their
percent
equivalents.
ENRICHMENT
Students combine
the results of
1,000 actual cube
drops and compare
them with the
expected results.
EXTRA PRACTICE
Students solve
probability problems.
ELL SUPPORT
Students add
predicted and
actual to their
Math Word Banks.
Additional Information
Advance Preparation For the cube-drop experiment, gather shoe box or copier-paper box
bottoms or tops to contain the bouncing cubes during the experiment.
632
ⵧ Math Journal 2, pp. 216 and 217
ⵧ Study Link Master (Math Masters,
p. 239)
Unit 7 Fractions and Their Uses; Chance and Probability
ⵧ Teaching Masters (Math Masters,
pp. 240–242)
ⵧ Teaching Aid Master (Math Masters,
p. 388 or 389)
ⵧ 5-Minute Math, pp. 42–47
ⵧ Differentiation Handbook
ⵧ base-10 blocks (1 flat, 10 longs,
cubes)
Technology
Assessment Management System
Math Log or Exit Slip
See the iTLG.
Getting Started
Mental Math and Reflexes
Math Message
Write fractions with denominators of 10 or 100 on the board, and have
students write the equivalent decimals on their slates. Then write decimals on
the board and tell students to write the equivalent fractions and mixed numbers.
Do not insist that they write fractions in simplest form. Suggestions:
Complete journal page 213.
2
0.2
10
50
0.50
100
24
0.24 100
60
0.60 100
8
0.08
100
72
0.072
1,000
3
0.03 100
29
0.029 1,000
Study Link 7 11
Follow-Up
䉬
6
7 1,000 7.006
1
32 100 32.01
9
1.9 11
0
5
6.5 61
0
Have students share
their spinner designs
and descriptions
in small groups.
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 213)
Discuss the answers to Problems 1 and 2, and have students
explain their reasoning. Ask whether students would be surprised
if their predictions were not fulfilled exactly. Discuss why the
actual results for 24 spins and 12 rolls might not match the
predicted outcomes. The predictions are based on what is likely
to happen; the actual outcomes will probably differ from
the predictions.
Explain that this lesson involves an experiment in which students
will compare their predicted outcomes with actual results.
Student Page
䉴 Predicting the Result of
Date
INDEPENDENT
ACTIVITY
an Experiment
(Math Journal 2, p. 214; Math Masters, p. 238)
Time
LESSON
7 12
䉬
Expected Spinner Results
1. If this spinner is spun 24 times, how many times do you
expect it to land on each color?
Color
Expected Number
in 24 Spins
8
8
4
4
red
Direct students to color the grid on Math Masters, page 238
according to the directions given.
Students can color the squares using any pattern they choose as
long as they end up with the specified number of squares of each
color. (Students will actually color only 50 of the squares.)
Have students read about the cube-drop experiment on journal
page 214. Ask:
●
If a cube is dropped onto the 10-by-10 grid, on which color is
it most likely to land? White, because there are more white
squares than squares of any other color
82–86
a. Fill in the table.
blue
yellow
green
Total
yellow
red
red
blue
blue
green
24
b. Explain how you determined the expected number of times the
spinner would land on each color.
2
1
Sample answer: Red and blue each cover 6 or 3
1
of the circle. 3 of 24 spins is 8 spins. Green and
1
1
yellow each cover 6 of the circle. 6 of 24 spins is
4 spins.
Try This
2. If a six-sided die is rolled 12 times, how many times would you expect to roll
a. an odd number?
6
b. a number less than 4?
c. a 6?
6
2
4
6
a triangular number?
6
a prime number?
d. a square number?
●
On which color is it least likely to land? yellow
e.
f.
Have students complete the page on their own.
213
Math Journal 2, p. 213
Lesson 7 12
䉬
633
Adjusting the Activity
Bring the class together to share predicted outcomes. Encourage
statements such as the following:
Have students describe how
dropping a cube onto a colored grid and
spinning a spinner are similar. Look for
answers such as the following:
䉯 “Only 1 square out of 100 is yellow, so I should hit yellow
about once out of every 100 drops. The chance of the cube
landing on yellow is 1 out of 100.”
䉯 For both the grid and the spinner, the
chance of landing on a specific color is
the fraction of times you expect to land
on that color. For example, if 4 out of
every 100 squares on a grid or sections
on a spinner are colored red, then the
4
chance of landing on red is 100 , or
4 out of 100, or 4%.
AUDITORY
䉬
䉬
KINESTHETIC
TACTILE
䉬
䉯 “White is easy. Half of the squares are white, so I would hit
white half of the time. If I dropped the cube 100 times, it
would hit white 50 times. If I dropped the cube 50 times, it
would hit white 25 times.”
VISUAL
䉯 “For green, it is 10 out of 100. So I expect 10 greens if I toss
100 times. If I toss 500 times, I should get 5 times as many—
that’s 50 greens.”
䉴 Performing a Cube-Drop
PARTNER
ACTIVITY
Experiment
(Math Journal 2, p. 215; Math Masters, p. 238)
Review the directions on journal page 215. Have partners take
turns performing the experiment:
1. One partner drops a cube 50 times onto his or her 10-by-10
grid. The other partner records the results in the first
partner’s journal.
2. Partners switch roles.
3. Students count the number of drops for each color and
complete the “My Results for 50 Cube Drops” table on
journal page 215. (See sample table below.)
Color
Student Page
Date
LESSON
7 12
䉬
Time
My Results for 50 Cube Drops
Number of Drops
Percent
yellow
1
2%
red
2
4%
green
3
6%
A Cube-Drop Experiment
Getting Ready
80–86
1. Follow the directions for coloring the grid on Math Masters, page 238. You may
color the squares in any way. The colors can even form a pattern or a picture.
blue
16
32%
centimeter cube about 2 feet above the grid. Without aiming, you will let it drop
onto the grid. You will then record the color of the square on which the cube
finally lands.
white
28
56%
䉬 If the cube does not land on the grid, the drop does not count.
Total
50
100%
2. For this experiment, you are going to place your grid on the floor and hold a
䉬 If the cube lands on more than one color, record the color that is covered
by most of the cube. If you cannot tell, the toss does not count.
Making a Prediction
NOTE Have students kneel when dropping the cube, or place something under
white
yellow
3. On which color is the cube most likely to land?
4. On which color is it least likely to land?
the grid to cushion the cube drop and reduce the bounce.
5. Suppose you were to drop the cube 100 times. How many times would you
expect it to land on each color? Record your predictions below.
Predicted Results of 100 Cube Drops
Color
Number of
Squares
Predicted Results
Fraction
Percent
1
100
yellow
1
red
4
green
10
blue
35
white
50
4
100
10
100
35
100
50
100
Total
100
1
1
4
%
10
%
35
%
50
%
%
100%
214
Math Journal 2, p. 214
634
Unit 7 Fractions and Their Uses; Chance and Probability
Student Page
Date
Ongoing Assessment: Informing Instruction
Watch for students who note that converting fractions to percents in this activity
is similar to calculating percent scores on the multiplication facts tests.
䉴 Comparing Actual and
WHOLE-CLASS
ACTIVITY
Expected Results
Time
LESSON
A Cube-Drop Experiment
7 12
䉬
continued
Doing the Experiment
You and your partner will each drop a centimeter cube onto your own colored grid.
6. One partner drops the cube. The other partner records the color in the grid below
by writing a letter in one of the squares. Drop the cube a total of 50 times.
Write
y for yellow,
r for red,
g for green,
b for blue, and
w for white.
w w w b w
r
g
b
y w b w w b
r
w b w
w g w w b w b
b
b
b
b w
w b w w w b w b
b w
Sample answer: w w b
g w w w w w w
7. Then trade roles. Do another 50 drops, and record the results in the other
partner’s journal.
(Math Journal 2, pp. 214 and 215)
My Results for 50 Cube Drops
Sample answers:
Bring students together to compare their actual results with their
predictions. Individual students’ results probably show a wide
range. For example, some students may have hit a white square as
many as 35 out of 50 tosses (70% of the time), while others may
have done so only 15 times (30% of the time). Students should
notice that although some individual results may be very close to
the expected results, others may be far off.
Number of
Drops
Color
Write it in the “Number of
Drops” column.
green
Check that the total is 50.
white
1
2
3
16
28
Total
50
yellow
8. Count the number for each color.
red
blue
Percent
2%
4%
6%
32%
56%
100%
9. When you have finished, fill in the percent column in the table.
Example: If your cube landed on blue 15 times out of 50 drops, this is the same as
30 times out of 100 drops, or 30% of the time.
215
Math Journal 2, p. 215
Ongoing Assessment:
Recognizing Student Achievement
Math Log or
Exit Slip
Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess
students’ ability to predict the outcomes of an experiment and test the predictions
using manipulatives. Have students write about how the predicted outcomes for
the cube-drop experiment compare with the actual results. Students are making
adequate progress if their responses include the following:
䉯 Some predictions are closer than others to the actual results.
䉯 Predictions are expected results, not exactly what will happen.
Some students may be able to use the results to predict future events.
For example, what might happen if the cube were dropped 500 times?
[Data and Chance Goal 4]
Student Page
Date
Time
LESSON
Place Value in Whole Numbers
7 12
䉬
2 Ongoing Learning & Practice
䉴 Reviewing Place Value in
1. Write these numbers in order from least
to greatest.
964
96,400
400,960
5
7
0
9
8
94,600
964
9,460
94,600
96,400
400,960
INDEPENDENT
ACTIVITY
Whole Numbers
9,460
5
0
7
hundreds place,
ten-thousands place,
ones place,
thousands place, and
tens place.
9
9
5
,
8
0
4. What is the value of the digit 8 in the
numerals below?
800,000
80
8,000
80,000
a. 807,941
2
975,320
Students solve problems that involve ordering numbers,
identifying place value, determining the values of digits, and
reading and writing whole numbers.
the
the
the
the
the
7
with the following digits:
3
in
in
in
in
in
Write the number.
3. Write the greatest number you can make
(Math Journal 2, p. 216)
4
2. A number has
b. 583
c. 8,714
d. 86,490
5. Write each number using digits.
6. I am a 5-digit number.
a. four hundred eighty-seven thousand,
sixty-three
䉬 The digit in the thousands place is the
result of dividing 64 by 8.
487,063
䉬 The digit in the ones place is the result
of dividing 63 by 9.
b. fifteen thousand, two hundred
䉬 The digit in the ten-thousands place is
the result of dividing 54 by 6.
ninety-seven
15,297
䉬 The digit in the tens place is the result
of dividing 40 by 5.
䉬 The digit in the hundreds place is the
result of dividing 33 by 11.
What number am I?
9
8
,
3
8
7
216
Math Journal 2, p. 216
Lesson 7 12
䉬
635
Student Page
Date
䉴 Math Boxes 7 12
Time
LESSON
䉬
Math Boxes
7 12
䉬
a decimal.
sentence true.
a. fraction:
3
a. 8
63
100
78
56
1
1
15
4
500
8
1,000
16
6
19
7
20
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 7-10. The skill in Problem 6
previews Unit 8 content.
5
b. 12
b. decimal:
c.
0.63
d.
27 61
3. Write 6 fractions equivalent to
3
18
5
30
100
600
2
12
4
24
6
36
(Math Journal 2, p. 217)
2. Write , , or to make each number
1. Name the shaded area as a fraction and
e.
1
.
6
53 54
4. Divide. Use a paper-and-pencil algorithm.
51 R4, or
769
15
4
5115
Sample answers
Writing/Reasoning Have students write a response to
the following: Explain the strategy you used to decide
which fraction was greater in Problem 2e. Sample answer:
19
6
1
1
19
1
1
is away from 1. is away from 1. is less than , so 20
7
7
7
20
20
20
6
is closer to 1 than 7 is.
22 23
179
49–51
5. Multiply. Use a paper-and-pencil algorithm.
9,476
INDEPENDENT
ACTIVITY
6. Compare.
46 ⴱ 206
a. 1 day is
b. 6 years is
4 times as long as 6 hours.
36 times as long as
12
䉬
INDEPENDENT
ACTIVITY
(Math Masters, p. 239)
2 months.
c. 3 gallons is
䉴 Study Link 7 12
times as much as
4 cups.
d. 8 cm is
16 times as long as 5 mm.
10 times
e. 1 meter is
18 19
Home Connection Students predict the outcome of a cointoss experiment, check their predictions by performing the
experiment, and express the results as fractions.
315
as long as 10 cm.
217
Math Journal 2, p. 217
3 Differentiation Options
READINESS
䉴 Renaming Fractions as Percents
INDEPENDENT
ACTIVITY
5–15 Min
(Math Masters, p. 240)
To explore renaming fractions as percents, have students
represent fractions with base-10 blocks and rename them as
percents. Ask students to discuss how they solved Problems 5
and 6.
Study Link Master
Name
Date
STUDY LINK
7 12
1.
2.
Time
What Are the Chances?
䉬
You are going to toss 2 pennies 20 times. How many times do you expect
the 2 pennies will come up as
a.
2 heads?
times
c.
1 head and 1 tail?
b.
times
2 tails?
81
times
Answers vary.
A Penny Toss
Now toss 2 pennies together 20 times.
Record the results in the table.
Results
Answers vary.
Number of Times
ENRICHMENT
䉴 Comparing Actual and Expected
SMALL-GROUP
ACTIVITY
15–30 Min
Results of 1,000 Cube Drops
2 heads
(Math Masters, pp. 241, 242, and p. 388 or 389)
2 tails
1 head and 1 tail
3.
What fraction of the tosses came up as
a.
4.
5.
2 heads?
b.
Answers vary.
2 tails?
c.
1 head and 1 tail?
Sample answers:
Suppose you were to flip the coins 1,000 times.
What fraction do you expect would come up as
250
1
250
, or ,
4
a. 2 heads? 1,000
b. 2 tails? 1,000
500
1
, or 2
c. 1 head and 1 tail? 1,000
or
1
4
To further explore the effect of sample size on actual
results, have students combine the class results of the
cube-drop experiment to generate actual data on how
many times a cube landed on each color for 1,000 cube drops.
Explain how you got your answers for Problem 4. Sample answer:
I think it will be the same fraction for 1,000
times as it is for 20 times.
Practice
6.
7 ⴱ 48 8.
3,870
336
45 ⴱ 86
7.
9.
874 ⴱ 9 7,866
4,828
34 ⴱ 142
Math Masters, p. 239
636
Unit 7 Fractions and Their Uses; Chance and Probability
Teaching Master
Randomly select 20 students to report the results of their
cube-dropping experiment on a “Results” slip, cut from Math
Masters, page 241. Students combine the data into a “Class
Results” table on Math Masters, page 242.
Name
Date
LESSON
Time
Fractions and Percents on Grids
7 12
䉬
47
100
Fractions and percents can be
modeled with base-10 blocks.
47 out of 100 47%
62
Build each fraction with base-10 blocks. Shade the grid, and fill in the missing numbers.
1.
2.
NOTE Select the results for 20 students, because data for 1,000 cube drops
(20 ⴱ 50) can be easily converted into percents. If your class has fewer than
20 students, select an even number of students; that way, the total number of
cube drops (even number ⴱ 50) will be a multiple of 100.
30
100
In a Math Log or on an Exit Slip, have students compare the
results of 1,000 cube drops with the predictions they made on
journal page 214. The actual results should be very close to the
predicted outcomes. Ask students to explain why they think the
actual results are closer to the predicted outcomes when the cubes
are dropped 1,000 times. The larger the sample size is, the closer
the predicted results will be to the actual results.
䉴 5-Minute Math
30
out of 100 76
out of 100 76
%
99
99
99
out of 100 99
%
76
100
%
Create your
own.
Sample
answer:
4
100
4
out of 100 4
%
100
These grids are the whole. Find the percent of each grid that is shaded.
5.
6.
50
This activity is an exposure to the concept of sample size. Explaining how sample
size affects results is a Grade 6 Goal.
30
4.
10 20
Links to the Future
EXTRA PRACTICE
3.
100
20
out of 100 35
20 %
50
70
100
70
out of 100 70 %
Math Masters, p. 240
SMALL-GROUP
ACTIVITY
5–15 Min
To offer students more experience with probability, see 5-Minute
Math, pages 42–47.
5–15 Min
Teaching Master
(Differentiation Handbook)
Name
Date
LESSON
100%
Percent
50 50 50 50 50 50
50 50 50 50 50 50 50
50 50 50 50 50 50
1,000
of drops
S7
S5 S6
S4
S3
green
red
yellow
Total
50
S2
S1
Color
Remind students to complete Study Link 7-11, Math Masters,
pages 235 and 236, in time for Lesson 8-1.
Students
Planning Ahead
Most of the percents you calculate will not be whole-number percents. You can record them as percents in tenths or
round them to the nearest whole percent. For example, 96 out of 1,000 is equivalent to 9.6 out of 100. This could be
recorded either as 9.6% or 10%. If the answers are rounded, the total might not add up to 100%.
7 12
S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 Number
To provide language support for probability, have students use the
Word Bank Template found in the Differentiation Handbook. Ask
students to write the terms predicted and actual, draw pictures
relating to each term, and write other related words. See the
Differentiation Handbook for more information.
Time
Class Results for 1,000 Cube Drops
䉬
white
䉴 Building a Math Word Bank
SMALL-GROUP
ACTIVITY
blue
ELL SUPPORT
Math Masters, p. 242
Lesson 7 12
䉬
637