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Objective
To investigate the relationship between sample size
and the reliability of derived results.
1
materials
Teaching the Lesson
Key Activities
Students take different size samples from a population and create circle graphs for these
samples. They determine the more trustworthy of the samples and make predictions.
Key Concepts and Skills
• Convert fractions to percents.
Math Journal 1, p. 180
Study Link 6 4
Teaching Aid Master
(Math Masters, p. 430; optional)
Geometry Template
Class Data Pad
small pieces of colored candy
[Number and Numeration Goal 5]
• Construct circle graphs of class data.
crayons or markers (optional)
calculator
[Data and Chance Goal 1]
• Make predictions based on sampling.
[Data and Chance Goal 2]
slates
Key Vocabulary
See Advance Preparation
sample • population
Ongoing Assessment: Recognizing Student Achievement Use the Math Message.
[Number and Numeration Goal 2]
2
Ongoing Learning & Practice
Students continue to collect data by playing Finish First.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
Students practice identifying the whole for
use as the denominator when converting
fractions to percents.
ENRICHMENT
Students predict the outcome of an
experiment, test the predictions, and
summarize the results.
Additional Information
Advance Preparation For Part 1, you will need candy of different colors, at least 5 pieces per
student. Place the candy in a bowl near the Math Message. Draw a large circle on a sheet of
paper and mark the center.
400
Math Journal 1, pp. 170, 171, and
181
Study Link Master
(Math Masters, p. 167)
number cards 4–8 (4 of each;
from the Everything Math Deck,
if available)
materials
Differentiation Options
READINESS
materials
Unit 6 Using Data; Addition and Subtraction of Fractions
Teaching Masters
(Math Masters, pp. 168 and 169)
calculator
Per partnership: coin or six-sided die
Technology
Assessment Management System
Math Message
See the iTLG.
Getting Started
Mental Math and Reflexes
Math Message
Fraction-of problems:
The bowl contains pieces of candy of several
colors. On a half-sheet of paper, explain how
you would find the percent of each color in the
bowl.
1
If 5 counters are 10 of the set, what is the whole? 50 counters
1
If 8 counters are 2 of the set, what is the whole? 16 counters
3
A set has 40 counters. How many counters are in 8 of the
set? What is the whole? 15 counters; 40 counters
Study Link 6 4 Follow-Up
2
If 20 counters are 10 of the set, what is the whole?
Partners compare answers and resolve any
differences. Ask a volunteer to share a strategy
for matching the plots and data sets.
100 counters
4
A set has 25 counters. How many counters are in 5 of the
4
set? 5 is what percent of the set? 20 counters; 80%
2
If 12 counters are 3 of a set, what is the whole? 18 counters
Ongoing Assessment:
Recognizing Student Achievement
Math
Message
Use the Math Message to assess students’ understanding of the concept of
percent. Students are making adequate progress if their written responses
name the whole as 100% and refer to pieces of a given color as a fraction or
percent of the whole.
[Number and Numeration Goal 2]
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Students share how they would find the percent of each color in
the bowl. Most will probably suggest counting the total number of
candies in the bowl as well as the number of each color in order to
find the fraction and then the percent of each color.
If looking at a sample is not mentioned, ask:
●
How could we find the percent of each color in the bowl without
counting every piece?
●
Would it help to look at a sample of candies from the bowl and
count the number of each color in the sample?
●
How many candies should we include in the sample?
Tell students that they are going to do a candy-counting
experiment. The results will give them information they can use to
predict the percent of each color in the bowl. Explain that the total
number of candies in the bowl is called the population. To
support English language learners, write sample and population
on the board and explain their meanings in this
mathematical context.
Lesson 6 5
401
Student Page
Date
6 5
Taking a Small Sample
Time
LESSON
Sampling Candy Colors
1. You and your partner each take 5 pieces of candy from the bowl. Combine your
candies, and record your results in the table under Our Sample of 10 Candies.
Sample answer:
Candy Color
red
yellow
green
orange
brown
Our Sample
of 10 Candies
Combined
Class Sample
Count
Percent
Count
Percent
1
5
2
2
10%
50%
20%
20%
13
28
22
12
25
13%
28%
22%
12%
25%
2. Your class will work together to make a sample of 100 candies. Record the counts
and percents of the class sample under Combined Class Sample in the table.
PARTNER
ACTIVITY
of Candy Colors
(Math Journal 1, p. 180; Math Masters, p. 430)
Have each student take 5 pieces of candy from the bowl without
looking (10 candies per partnership). Ask partners to count and
record the number and percent of each candy color in their small
sample on journal page 180. Tell them to put the candy back in
the bowl when they have finished recording their results.
3. Finally, your class will count the total number of candies in the bowl and the
number of each color.
a. How well did your sample of 10 candies predict the number of each color in
Answers vary.
the bowl?
b. How well did the combined class sample predict the number of each color in
Answers vary.
the bowl?
c. Do you think that a larger sample is more trustworthy than a smaller
sample?
Yes
Sample answer: Smaller samples
vary more. A larger sample can be
trusted to give a better picture of the real
situation than a smaller one.
Explain your answer.
180
Math Journal 1, p. 180
With samples of 10, percents are easy to calculate because the
fractions easily convert to percents. For example, 4 yellows out of
4
40
10 candies is 10 100 40% of the sample. Point out that sample
sizes are often chosen to make calculations easy.
Have each partnership make a circle graph to show their sample
result for 10 candies. They can color the sections of the graph with
the candy colors. Students may use the Percent Circle on their
Geometry Templates and circles you provide from copies of Math
Masters, page 430.
NOTE Counting a whole population is rarely
possible, which is why one is concerned with
practical sampling and sampling sizes. Even
if it is possible to count a whole population, it
is often tedious and time-consuming.
Yellow
Green
Orange
Red
Brown
Each circle graph shows the result for one sample of 10 candies.
Graphing and Predicting on
WHOLE-CLASS
ACTIVITY
the Basis of a Sample
(Math Journal 1, p. 180)
Ask 10 partnerships to display their circle graphs, report their
results, and record them on the board. Discuss the variations in
these results. Individual small samples of 10 may have many of
one color, few of another color, and none of some colors.
Combine the data from the small partner samples to make a large
sample. Tally the results by color on the board. Ask students to
record the tallies on journal page 180.
By choosing 10 pairs of students, the combined sample total is
100—making percent calculations simple.
Make a circle graph of the combined sample. Use the large circle
that you drew previously. (See Advance Preparation.) Ask
402
Unit 6 Using Data; Addition and Subtraction of Fractions
Student Page
volunteers to use a Percent Circle to mark the sections showing
percents of colors in the combined sample.
Date
Time
LESSON
Math Boxes
6 5
1. Find the median and mean for each set
2. Estimate the sum.
3.1 0.72 34.7
of numbers.
a. 17, 13, 27, 33, 25
25
23
median:
Yellow
Green
Orange
Red
Brown
Choose the best answer.
mean:
1,370 and 1,380
440 and 460
b. 47, 29, 53, 46, 43, 32
44.5
41.6
苵
median:
mean:
35 and 40
13 and 15
247
119–121
Fifth Graders’ Pets
3. Draw a circle graph that is divided into
the following sectors: 32%, 4%, 22%,
18%, and 24%. Make up a situation for
the graph. Give the graph a title. Label
each section.
Circle graph of the combined samples of 100 candies
(title)
Sample answer:
Pets owned by fifth
graders
Discuss which is more trustworthy—a sample of 10 candies or the
combined sample of 100. Prompt students with questions like
the following:
●
●
How do the small-sample results compare with one another?
Sample answer: The results for our small samples jump all over.
One of them shows 50% yellow, and one shows 10% yellow.
Guide students to predict and agree on what percent of each color
is in the bowl, and record these predictions on the Class Data Pad.
Then have them count the total number of pieces of candy in the
bowl and the number of each color. Record the results on
the board.
Ask students to find the percent of each color:
1. Name each color as a fraction of the total number of candy
pieces.
18%
none
22%
gerbil
4%
bird
125 126
5. Write or to make the sentence true.
4. Solve.
5.7
7.81
a. 34.2 6 b. 39.05 / 5 203,467
699,842
521,369
a. 662
b. 7,341
231.2
c.
c. 3冄6
苶9
苶3
苶.6
苶
How do the results of the larger combined sample compare with
the small ones? Sample answer: The combined sample of 100 is
better. Every color is represented. The sample for 100 gives a
better picture than the sample for just 10.
24%
dog
32%
cat
Description:
d.
42 43
e.
626
7,347
203,764
699,428
9
531,399
181
Math Journal 1, p. 181
Adjusting the Activity
Have students calculate the
combined class sample using all partnerships
in the class. If there are 17 partnerships, the
whole is 170 pieces of candy. Compare these
results with the results for 100 pieces of
candy.
AUDITORY
2. Use a calculator to divide the number of each color (the
numerator) by the total number of candies in the bowl
(the denominator).
KINESTHETIC
TACTILE
VISUAL
Study Link Master
Name
3. Multiply the result by 100.
Date
STUDY LINK
65
Expect the percent for each color in the bowl to be reasonably close
to the percent on the circle graph for the large sample of 100.
1.
Time
Constructing a Graph from Landmarks
Make up a list of data with the following landmarks:
Sample answers:
119 122
mode: 15
minimum: 5
median: 10
maximum: 20
Use at least 10 numbers.
5, 7, 7, 8, 8, 9, 10, 13, 14, 15, 15, 15, 20
2.
Draw and label a bar graph to represent your data.
Minutes Needed to Get Ready for Bed
Playing Finish First
Number of Students
2 Ongoing Learning & Practice
PARTNER
ACTIVITY
(Math Journal 1, pp. 170 and 171)
Partners continue to collect the needed data for the specified
number of games of Finish First. They record their results on
journal page 171 and on the classroom tally sheet. For detailed
instructions, see Lesson 6-2.
Reminder: The game data will be used in the next lesson.
3.
8
7
6
5
4
3
2
1
0
(title)
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Minutes
Describe a situation in which these data might actually occur.
The number of minutes it takes to get ready for bed
Practice
4.
305 29 8,845
5.
524 81 443
6.
671 132 88,572
7.
7,356 4 1,839
Math Masters, p. 167
Lesson 6 5
403
Teaching Master
Name
Date
LESSON
Time
Math Boxes 6 5
Identify the Whole
65
(Math Journal 1, p. 181)
In the following number stories, find the whole using parts-and-total diagram.
Write the fraction for the given part, and rename the fraction as a percent.
Example: Two girls each have 5 hats. Three of their hats are purple. What percent
of the hats are purple?
3
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-7. The skill in Problem 5
previews Unit 7 content.
3
Solution: 2 º 5 10 hats; 3 out of 10 1
0 ; Rename 10 as a fraction with 100 as
30
30
10 º 3
the denominator, 10 º 10 100 ; 100 0.30, or 30%.
Reminder: To use a calculator to convert a fraction to a percent, divide the
numerator by the denominator. Use your fix key to round to the nearest
hundredth, or multiply the decimal by 100 to display the percent.
Lamont, Jose, and Kenji are recycling soda cans. Lamont collects 13 cans.
Jose collects 20 cans, and Kenji collects 17 cans. What percent of the cans
does Jose collect?
1.
Unit:
cans
Fraction:
Whole:
20
50
Study Link 6 5
13 20 17 50
Percent:
40%
Unit:
pieces
Fraction:
17
50
Whole:
Percent:
INDEPENDENT
ACTIVITY
(Math Masters, p. 167)
Jacqui and Edna decide to share their hot lunches.
They put together their fried potatoes and their
onion rings. There are 33 pieces of fried potatoes
and 17 onion rings. What percent of the lunches
are the onion rings?
2.
INDEPENDENT
ACTIVITY
Home Connection Students make up a list of data to fit
a given set of landmarks. Students then construct
a bar graph.
50
34%
The boy’s club is having a popcorn sale. Each of the
10 members of the club is given 5 boxes of popcorn,
but Edward sells only 3. What percent of the
5 boxes remain for Edward to sell?
3.
Unit:
boxes
Fraction:
Whole:
2
5
Percent:
5
40%
3 Differentiation Options
Math Masters, p. 168
READINESS
Identifying the Whole
SMALL-GROUP
ACTIVITY
15–30 Min
(Math Masters, p. 168)
To explore converting fractions to percents, have students
find the parts and totals contained in number stories,
write the fractions, and convert them to percents.
Students can use calculators but should not use the percent key at
this time.
ENRICHMENT
Teaching Master
Name
LESSON
65
1.
Date
Investigating Sample Size
Time
Choose a specific outcome or event for one of the following actions.
1
2
Example: The coin will land heads up.
Rolling a die
Example: The die will land with a 4 on the top.
1
6
Predict the results of 10 trials and 100 trials. Report your predictions as the
fraction of the total you think will result in a favorable outcome, or favorable
event. For example, the coin will land heads up about 12 of the time, or the die
will land with a 4 on the top about 16 of the time.
Event
10 trials
100 trials
1,000 trials
Sample answers:
Prediction
Result
Prediction
Result
Prediction
Coin will
land heads up
6
10
5
10
60
100
51
100
500
1,000
3.
Perform 10 trials. Record the results first with tally marks on a separate piece
of paper and then in the table as a fraction.
4.
Repeat for 100 trials. Record the results first with tally marks on a separate
piece of paper and then in the table as a fraction.
5.
How do your predictions compare with the actual results?
Sample answer: They are very close.
6.
To apply students’ understanding of the relationship
between sample size and the reliability of predictions,
have students conduct the experiment on Math Masters,
page 169. Students choose an event with random outcomes—
flipping a coin or rolling a six-sided die. They predict and compare
the results of 10 and 100 trials. Then they predict the results for
1,000 trials, explain their prediction, and design an approach to
collect the actual results.
Discuss students’ predictions and their prediction methods.
Consider implementing one of their designs for collecting the
actual results for 1,000 trials. Have partners report their findings
to the class.
Predict the results for 1,000 trials, and explain your prediction.
Sample answer: Because the results have been very close
1
1
to 2, I predict the coin will land heads up about 2 of the time.
7.
15–30 Min
(Math Masters, p. 169)
Investigating Sample Size
Flipping a coin
2.
PARTNER
ACTIVITY
On the back of this page, name two ways you and your partner could get data
on the actual results for 1,000 trials.
Math Masters, p. 169
404
Unit 6 Using Data; Addition and Subtraction of Fractions