4-5 M
Clued
in to
Decimals
by Michelle Pauls
AIMS Research Fellow
Paper towels
Clue cards
Student page
Topic
Decimals
Key Question
How can you work together using logic clues to
determine what decimal is being described?
Background Information
Logical thinking skills are important in many
aspects of life besides mathematics. This activity
provides an opportunity for students to sharpen their
logical thinking while working collaboratively. Students must learn to listen carefully to the members
of their group and work together to reach the correct
solution. No one person can solve the logic problem
alone. All four clues are needed to reach the answer.
This activity also provides an excellent opportunity for students to work with decimal fractions and
mathematical vocabulary. Clues about odd and even
or greater than and less than are applied to narrow
down possibilities. Students must be aware of the
decimal equivalents of common fractions and will
gain experience with place value for numbers less
than one.
Learning Goals
Students will:
• work collaboratively in groups, and
• use logic clues to determine what decimal fraction
is being described.
Guiding Document
NCTM Standards 2000*
• Understand the place-value structure of the
base-ten number system and be able to represent
and compare whole numbers and decimals
• Apply and adapt a variety of appropriate strategies
to solve problems
• Build new mathematical knowledge through
problem solving
Management
1. Students need to work in groups of four on this
activity. If you do not have a number of students
divisible by four, you can add a fifth student to
some groups to be the official recorder.
2. Groups must be able to use the student page to
repeatedly record the different clues. The easiest
way to do this is to copy the page onto transparency film or place it in a sheet protector and give
each group a dry erase marker. Another option is
to give each group a set of 30-45 counters with
which to cover up the numbers. These counters
can be pennies, buttons, beans, area tiles, or any
other small manipulative that will fit into the spaces
provided.
3. It is expected that students are already familiar
with decimals and know some common fractional
equivalents.
4. There are eight sets of clue cards that get progressively more difficult. Sets A, B, and C deal with
decimals to the hundredths place; the remaining
sets deal with decimals to the thousandths place.
Select the clue set that is appropriate for each
group of students. (Multiple groups may use the
same set, or each group may have its own.)
Math
Number and operations
decimals
place value
odd and even
greater than, less than
Logic
Problem solving
Integrated Processes
Observing
Comparing and contrasting
Collecting and recording data
Applying
Problem-Solving Strategies
Use logical thinking
Use manipulatives
Materials
Sheet protectors or transparency film
(see Management 2)
Dry erase markers
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©AIMS Education Foundation
5. You will need to copy and cut apart the clue sets
before beginning this activity. You may wish to
laminate the clues and/or copy them onto card
stock for repeated use. Copying each set on a
different color of paper will make them easier to
keep separate.
Extensions
1. Create your own logic clues for students, or allow
them to create clues for each other.
2. For advanced students, go to the ten-thousandths
place.
Solutions
Clue Set A: 0.05
Clue Set B: 0.13
Clue Set C: 0.97
Clue Set D: 0.875
Clue Set E: 0.337
Clue Set F: 0.048
Clue Set G: 0.209
Clue Set H: 0.461
Procedure
1. Have students get into their groups of four. Distribute one student page, one clue set, a paper towel,
and a dry erase marker to each group.
2. Instruct each student in the group to take one clue
card from the set. (If there are five students in the
group, have the fifth student take the recording
page and the pen.)
3. Inform students that the challenge is to work
together as a group to figure out what decimal
their clues describe. This is done by marking off
numbers in each column based on the clues until
only one possibility remains.
4. Explain that students will take turns reading the
clues on cards. If the clue gives information that
can be immediately used, that information can
be recorded by crossing out the numbers that are
eliminated. (For example, if the clue says “There
is no ‘7’ in this decimal,” all of the sevens on the
page can be crossed out.) Once all of the clues
have been read, they can share them as a group
and revisit clues that may become more helpful
once other information is known.
5. Give groups time to work on the clues. Encourage
them to persevere until they come up with a solution and to check that solution against every clue
to make sure that it is correct.
6. Have them justify their responses to you, or to
the entire class, by going through each clue and
showing how the answer meets the criteria.
7. Repeat this exercise throughout the year to
continue to practice logic and critical vocabulary.
*
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
Connecting Learning
1. What kinds of things did you need to know in
order to make sense of the logic clues?
2. Were some clues more useful than others at first?
Why or why not?
3. What strategies did your group use to solve the
problem?
4. How do you know that you have the correct
answer?
5. What do you know when a clue says that the
decimal is greater than one-fourth but less than
one-half? [The digit in the tenths place must be
a 2, 3, or 4.] Does this information tell you anything about the digit in the hundredths place or the
thousandths place? Why or why not? [No. Only
the tenths place digit can be known for certain.]
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©AIMS Education Foundation
Clued
we’ve found
the decimals!
can you?
WINTER 2008
in to
Decimals
tenths
0
0
0
0
0
0
0
0
0
0
20
.
.
.
.
.
.
.
.
.
.
0
1
2
3
4
5
6
7
8
9
hundredths thousandths
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set A
A
no fear!
my faithful chum
and i will locate
the
decimals!
A
no fear!
my faithful chum
and i will locate
the
decimals!
• The digit in the hundredths place
is odd.
• The digit in the tenths place is less
than five.
• There is no digit in the thousandths
place.
• The decimal is less than one-fourth.
A
no fear!
my faithful chum
and i will locate
the
decimals!
A
• There is a difference of five between
the digit in the tenths place and the
digit in the hundredths place.
WINTER 2008
no fear!
my faithful chum
and i will locate
the
decimals!
• There is no “9” in the decimal.
• The digit in the hundredths place
is greater than the digit in the
tenths place.
21
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set B
B
hmmm!
This is a tricky
one!
B
some more
sleuthing can
help solve the
problem.
hmmm!
This is a tricky
one!
some more
sleuthing can
help solve the
problem.
• There is no digit in the thousandths
place.
• There is a difference of two between
the digits.
• The digit in the tenths place is odd.
• There is no “3” in the tenths place.
B
hmmm!
This is a tricky
one!
B
some more
sleuthing can
help solve the
problem.
some more
sleuthing can
help solve the
problem.
• The digit in the hundredths place
is larger than the digit in the tenths
place.
WINTER 2008
hmmm!
This is a tricky
one!
• The sum of the digits in the decimal
is less than nine.
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©AIMS Education Foundation
Clued
in to
Decimals
Clue Set C
C
i suggest
we go looking
for clues to solve
this mystery!
C
• The tenths place digit is not six.
• The digit in the hundredths place
is less than the digit in the tenths
place.
• There is no digit in the thousandths
place.
C
• The decimal is greater than
three-fourths.
i suggest
we go looking
for clues to solve
this mystery!
C
• The sum of the digits in the decimal
is greater than 12.
•
The digit in the tenths place is odd.
WINTER 2008
i suggest
we go looking
for clues to solve
this mystery!
i suggest
we go looking
for clues to solve
this mystery!
• There is a difference of two between
the digit in the tenths place and the
digit in the hundredths place.
23
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set D
D
hmmm...
i see no decimals
in here.
the
mystery thickens!
D
• The digit in the hundredths
place is one less than the digit
in the tenths place.
• The digit in the tenths place is even.
• The digit in the tenths place is
greater than six.
D
hmmm...
i see no decimals
in here.
the
mystery thickens!
D
• The digit in the thousandths place
is greater than four.
hmmm...
i see no decimals
in here.
the
mystery thickens!
• The digit in the thousandths place
is two less than the digit in the
hundredths place.
• The value of the decimal is greater
than three-fourths.
WINTER 2008
hmmm...
i see no decimals
in here.
the
mystery thickens!
24
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set E
E
E
it’s all quite
elementary,
my feathered
friend!
The clues
are right under
our beaks!
The clues
are right under
our beaks!
• The digits in the tenths and
hundredths places are the same.
• The digit in the thousandths
place is greater than the digit in
the tenths place.
• All of the digits in the decimal
are odd.
E
E
it’s all quite
elementary,
my feathered
friend!
The clues
our right under
are beaks!
it’s all quite
elementary,
my feathered
friend!
The clues
are right under
our beaks!
• There is no “5” in the decimal.
• The digit in the tenths place is
four less than the digit in the
thousandths place.
• The decimal is less than one-half.
WINTER 2008
it’s all quite
elementary,
my feathered
friend!
25
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set F
F
aha!
I spy a digit!
is he in the
decimal?
F
let’s read
the clues
to find out!
• The sum of all the digits in the
decimal is 12.
aha!
I spy a digit!
is he in the
decimal?
• The decimal is less than one-third.
F
let’s read
the clues
to find out!
• The digit in the thousandths place is
double the digit in the hundredths
place.
aha!
I spy a digit!
is he in the
decimal?
let’s read
the clues
to find out!
• No two digits in the decimal are
the same.
• The digit in the hundredths
place is greater than three.
• The smallest digit is in the tenths
place.
WINTER 2008
let’s read
the clues
to find out!
• The digit in the thousandths place
is even.
• There is no “5” in the decimal.
F
aha!
I spy a digit!
is he in the
decimal?
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©AIMS Education Foundation
Clued
in to
Decimals
Clue Set G
G
G
no calculators
are needed here,
my friend!
we’ll be
using good,
old-fashioned
deductive
reasoning.
we’ll be
using good,
old-fashioned
deductive
reasoning.
• The digit in the tenths place is less
than five.
• There is a difference of seven
between the tenths place digit
and the thousandths place digit.
• There is no “7” in the decimal.
G
G
no calculators
are needed here,
my friend!
we’ll be
using good,
old-fashioned
deductive
reasoning.
no calculators
are needed here,
my friend!
we’ll be
using good,
old-fashioned
deductive
reasoning.
• The digit in the tenths place is
two greater than the digit in the
hundredths place.
• The digit in the tenths place is even.
• The digit in the thousandths place
is the largest.
• The decimal is less than one-fourth.
WINTER 2008
no calculators
are needed here,
my friend!
27
©AIMS Education Foundation
Clued
in to
Decimals
Clue Set H
H
let’s analyze
our clues at
headquarters!
H
imal
dec ives
ct
dete
let’s analyze
our clues at
headquarters!
imal
dec ives
ct
dete
• The smallest digit is in the thousandths
place.
• There is no digit greater than “7” in
the decimal.
• The digit in the hundredths place is
the largest.
• There is a difference of two
between the digits in the tenths
and hundredths places.
• The digit in the tenths place is even.
H
let’s analyze
our clues at
headquarters!
H
imal
dec ives
ct
dete
let’s analyze
our clues at
headquarters!
imal
dec ives
ct
dete
• The digit in the tenths place is not
two.
• The decimal is less than one-half,
but more than one-third.
• The smallest digit is five less than
the largest digit.
• The digit in the thousandths place
is odd.
WINTER 2008
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©AIMS Education Foundation