293
11
PLACE-VALUE REPRESENTATIONS OF
FRACTIONAL PARTS: DECIMAL FRACTIONS,
DECIMALS, AND OPERATIONS ON DECIMALS
TEACHING TIPS
AIMS AND SUGGESTIONS
Unit 11•1: Decimal Fractions &
Decimals
One aim of this unit is to help readers appreciate why decimals were invented, reflect on decimal notation, and explicitly identify the types of
decimals and how they are connected to number
concepts. A discussion of Probe 11.1: Understanding Decimals (page 11-3 of the Student
Guide) can help achieve these aims. In solving A
Tale of Two Rain Totals, students quickly recognize
that adding six fractions with different denominators is a difficult and time-consuming task. While
some undertake the laborious process of finding a
common denominator and equivalent fractions,
others simply skip this task in the hope that one of
their classmates will do the job. Yet others proceed to convert the fractions to decimals. By asking such students to explain to the class why they
are performing this procedure, an instructor can
underscore that decimals were invented because
they are typically easier to operate on than are
fractions. Alternatively, an instructor can wait
until all students have solved the problem, ask
them whether it was easier to determine the rain
total for Urbana (with fractions) or Champaign
(with decimals), and then point out that the six
fractions listed for Urbana are equivalent to the six
decimals listed for Champaign. (Investigation
11.A: Another Need for Decimals on page 297 of
this guide can help students see that this symbolic
notation makes operating on irrational numbers
easier also.) Discussing the other questions in Part
I of Probe 11.1 typically leads to disagreements
and opportunities to illuminate students' understanding of decimal notation (e.g., 30 and 3 are
base-ten numbers or "decimals" and the ones
place, not the decimal point, is the center of the
decimal system).
Part II was designed to help students explicitly distinguish among terminating decimals, infinitely repeating decimals, and infinitely nonre-
peating decimals and link them to rational and irrational number concepts and common fractions
and other fractions. For example, by asking students to predict what will happen when 38 is converted to a decimal by means of long division and
checking their prediction, they should see that the
division ends at some point with no remainder
and, thus, the quotient (the equivalent decimal)
terminates. By converting a fraction such as 13 (or
1
7
) to a decimal in the same manner, students
should quickly recognize that the same remainder
(or pattern of remainders) will go on forever and
that the quotient (equivalent decimal) will never
terminate.
A second aim of this unit is to help foster an
explicit understanding of the concepts underlying
decimal notation, the value of using decimal fractions as a bridge between fractions and decimals,
and the importance of using base-ten blocks to
model decimals. Investigation 11.1: Using BaseTen Blocks to Introduce Decimals (page 11-6 of
the Student Guide) was designed to help achieve all
three of these aims. Discussing how to design a
place-value mat to keep score of a game involving
decimals (Question 2 of the Questions for Reflection), in particular, can serve to underscore that
decimals are a way of representing fractional parts
in terms of place-value notation (see also Box 11.1:
Discovering How Decimal Fractions Can Be Represented as Decimals on page 11-7 of the Student
Guide).
To simulate the investigative approach to decimal instruction, an instructor can opt to use Investigative 11.B: Zorkian Quadal (Base-Four)
Numeration (page 298 of this guide). This investigation can serve as an especially powerful
demonstration, partly because students typically
have practically no comprehension of "quadals,"
and many assume that there is no way they could
understand them. Working through this reader inquiry can highlight all the aims noted above:
highlight (a) the grouping and place-value concepts underlying place-value representations of
fractional parts, (b) how "decimal" fractions can
294
serve as a bridge, and (c) the value of manipulatives.
A third aim of Unit 11•1 is to foster number
sense for decimals. Investigation 11.2: Exploring
Decimal Patterns (pages 11-9 and 11-10 of the Student Guide) can help students see how calculatorbased activities can serve to link fractions and decimals and to reveal interesting decimal patterns.
Investigation 11.C: Converting Fractions to Decimals with a 10 x 10 Grid (on page 299 of this
guide) can provide an alternative way of linking
fractions and decimals.
Unit 11•2: Operations on Decimals
One aim of this unit is to help readers see that
the same meaningful analogies used to make sense
of operations on whole numbers and fractions can
be useful in making sense of operations on decimals, particularly multiplication and division of
decimals less than one. (If adult students have
constructed a reasonably sound understanding of
operations on fractions, operations on decimals
should be readily comprehensible. This is why the
topic, unlike operations on fractions did not rate a
separate chapter.) As in chapter 10, Unit 11•2 begins with efforts to encourage operation sense and
informal solution procedures, namely Investigation 11.3: Some Challenging Decimal Tasks, Investigation 11.4: Gauging Understanding of Decimal Multiplication and Division, Investigation
11.5: Thinking About Decimal Multiplication,
Part I of Investigation 11.6: Constructing the
Conceptual Basis for Symbolic Decimal Multiplication, Questions 1 and 2 of Investigation 11.7:
Dividing a Decimal by a Whole Number, and
Parts I and II of Investigation 11.8: Dividing by a
Nonunit Decimal (pages 11-11, 11-12, 11-15, 11-16,
11-20, and 11-21 and 11-22 of the Student Guide, respectively). Investigation 11.D: Concrete Decimal Addition and Subtraction on page 300 of this
guide illustrates the conceptual phase for adding
and subtracting decimals.
A second aim of Unit 11•2 is to illustrate how
symbolic expressions can be linked to meaningful
analogies and concrete models or solutions (see
Part II of Investigation 11.6, Questions 3 to 7 of
Investigation 11.7, and Part III of Investigation
11.8; pages 11-16 and 11-17, 11-20, and 11-22 to 1124 of the Student Guide, respectively). A third aim
of the unit is to help readers see how children can
be guided to reinvent formal algorithms (see Part
III of Investigation 11.6 and Part IV of Investiga-
tion 11.8 on pages 11-17 and 11-24 in the Student
Guide, respectively). Note that special emphasis
on the rationale for each step of the distributive
algorithm (e.g., why it involves multiplication and
subtraction) will probably be needed (see Figure
11.6 on pages 11-25 of the Student Guide).
SAMPLE LESSON PLANS
Project-Based Approach
Using the SUGGESTED ACTIVITIES on
pages 302 and 303 of this guide as a menu, an instructor can assign a required project or allow students to choose a required or an extra-credit project. Option 1 requires them to compare and contrast concrete models of decimal division. Options
2 and 3 involve developing bulletin board displays, including one of everyday applications of
decimals. Options 4, 5, and 6 entail developing,
using, and/or evaluating a lesson plan based on
the investigative approach. Options 7 and 8 require compiling and critically analyzing resources
related to decimal instruction. Option 9 involves
developing a worthwhile task, trying it out, and
evaluating it.
Single-Activity Approach
One possibility would be to use Investigation
11.B: Zorkian Quadal (Base-Four) Numeration
(page 298 of this guide) to simulate the investigative approach to decimal instruction (as previously
discussed). To extend this lesson, an instructor
could have adult students consider how operations on quadals could be introduced and have
them develop one or more lessons for doing so.
Alternatively, the instructor could introduce, for
example, multiplying quadals using activities
analogous to Investigation 11.5 and 11.6 (pages
11-15 and 11-16 of the Student Guide, respectively).
Multiple-Activities Approach
For relatively comprehensive coverage of
chapter 11 material, an instructor could choose the
following sequence of activities:
1. To underscore the reason for decimals, have
students complete Question 2 in Part I of Probe
1.1: Understanding Decimals (page 11-3 of the
Student Guide). Completing Part II of this probe
can help them explicitly distinguish among the
three types of decimals and their relationship to
rational or irrational numbers.
295
2. Touching on Investigation 11.1: Using
Base-Ten Blocks to Introduce Decimals (page 116 of the Student Guide) and Box 11:1: Discovering
How Decimal Fractions Can Be Represented as
Decimals (page 11-7 of the Student Guide) can illustrate how a teacher can use the investigative
approach to introduce decimal numeration in a
purposeful, meaningful, and inquiry-based manner.
3. Engaging students in Part I and at least one
activity in Part II of Investigation 11.2: Exploring
Decimal Patterns (page 11-9 of the Student Guide)
can illustrate how a calculator-based activity can
serve as a basis for introducing infinitely repeating
decimals and the many interesting patterns that
decimals can embody.
4. Question 5 of Investigation 11.4: Gauging
Understanding of Decimal Multiplication and
Division (page 11-12 of the Student Guide) can help
to underscore the importance of thinking about
operations on decimals in terms of a meaningful
analogy. For example, thinking about 4.55 x 6.108
in terms of a groups-of analogy (about four groups
of about six is about 24) can lead them to identify
where the decimal point in the product 277914
should go.
5. Completing and discussing Cases E and H
on page 11-15 of the Student Guide (Investigation
11.5: Thinking About Decimal Multiplication)
can further underscore the point above and highlight the value of qualitative reasoning.
6. To further model the investigative approach to introducing decimal multiplication, students can be encouraged to use base-ten blocks to
devise their own informal solutions (models) for
Problems B, C, and E in Part I of Investigation
11.6: Constructing the Conceptual Basis for
Symbolic Decimal Multiplication (page 11-16 of
the Student Guide). Instructors will probably have
to remind them not to use their formal procedures
or calculators to determine the answer first and
then work backwards. The idea is to simulate how
children, who have not learned formal solution
methods, might solve such problems. Moreover,
students need to become accustomed to thinking
in terms of meaningful analogies and letting this
guide their actions with manipulatives. Questions
6 and 7 in Part II (on page 11-17 of the Student
Guide) require students to analyze and evaluate
manipulative-based solutions in terms of a meaningful analogy. The latter question, in particular,
illustrate what can happen when children learn
and apply a meaningless manipulative procedure
(without relating it to a meaningful analogy).
Specifically, Adrienne did not think clearly in
terms of an area model for decimal multiplication.
In effect, she treated units of length and units of
area as interchangeable: treated a cube as one (as
both one linear unit and one square unit) and a
long as ten (as both 10 linear units and 10 square
units). If she had first devised a key for area (and
defined square units in terms of the surface of
blocks) and length (and defined linear units in
terms of the length of blocks), she would have
stood a better chance of responding correctly
(answering 0.48 square units). Part III can help
adult students see how children might be guided
to discover the rule for placing a decimal point in a
product.
7. Instruction on decimal division can be handled in a manner analogous to that outlined for
decimal multiplication above. Modeling the conceptual phase can begin with qualitative-reasoning
tasks that require drawing on a meaningful analogy (see, e.g., Cases E and H in Part I of Investigation 11.8: Dividing by a Nonunit Decimal (page
11-21 of the Student Guide). Next, students can be
encouraged to use base-ten blocks to invent their
own informal strategy for measure-out and area
problems (see, e.g., Problem D and G in Part II of
Investigation 11.8, page 11-22). To model the
connecting phase, ask students to explicitly relate
symbolic expressions to a meaningful analogy
(e.g., Questions 2 and 9 in Part III of Investigation
11.8 on pages 11-22 and 11-23 of the Student
Guide). Again, to underscore the need to think
clearly in terms of an analogy and to begin with a
key, students can be asked to grapple with Question 10 (on page 11-24 of the Student Guide).
SAMPLE HOMEWORK
ASSIGNMENTS
Read: Chapter 11 of the Student Guide.
Study Group:
• Questions to Check Understanding: 1, 13c, 14,
16b, and 18a (pages 303 to 306).
• Writing or Journal Assignments: 4, 6, and 10
(pages 306 and 307).
• Problem: Which Option? (page 308).
• Bonus Problem:
(page 308).
Decimals in Other Bases
296
Individual Journal: Writing or Journal Assignments
5 and 8b (page 307).
FOR FURTHER
EXPLORATION
ADDITIONAL READER INQUIRIES
Investigation 11.A (page 297)
Another Need for Decimals supplements
Question 2 in Part I of Probe 11.1: Understanding
Decimals (page 11-3 of the Student Guide) by illustrating that, without decimals, computation with
irrational numbers would be difficult.
Investigation 11.B (page 298)
Zorkian Quadal (Base-Four) Numeration can
be used to model the investigative approach for introducing decimals and to help students construct
a general understanding of the grouping and
place-value concepts underlying decimal numeration.
Investigation 11.C (page 299)
Converting Fractions to Decimals with a 10 x
10 Grid illustrates how 10 x 10 grids can be used
to translate common fractions to decimals. The activity begins with the most obvious case (decimal
fractions) and ends with least obvious case (where
the prime factors of denominators are not exclusively 2 and/or 5). Note that trying to represent
fractions such as 13 on a 10 x 10 grid can provide
one way of introducing the concept of infinitely
repeating decimals. Attempting to divide a 10 x 10
grid into thirds would result in three equal parts,
each consisting of three columns (tenths) and one
column leftover. Dividing the leftover column
into thirds would produce three equal parts, each
consisting of three 1 x 1 grids (hundredths) and
one 1 x 1 grid leftover. If the leftover hundredth is
divided into 10 one-thousandths minigrids and
these minigrids were divided into three equal
parts, each consisting of three one-thousandths,
one one-thousandth would be left. Students
should quickly recognize that this process could
go on forever.
Investigation 11.D (page 300)
Concrete Decimal Addition and Subtraction
(Conceptual Phase) illustrates that problems involve renaming can be introduced almost immediately and the following two teaching tips:
✑ Encourage children to discover for themselves how base-ten blocks can be used to find
decimal sums and differences involving renaming. Pupils who understand how to represent
decimals with base-ten blocks should have little
difficulty drawing on their understanding of
whole-number addition and subtraction to construct a strategy for using the blocks to solve
decimal addition and subtraction problems, such
as Problems 1 and 3 in Part II of the investigation.
Problems that do involve renaming, such as Problems 2 and 4, will probably pose a greater challenge but should not be overwhelmingly difficult
(an enigma) for properly-prepared pupils. If children need a hint, have them consider a similar but
familiar situation—situations with whole numbers
that require renaming.
✑ Although it may help some children to
begin with problems that involve even decimals
with tenths, meaningful instruction for most can
probably move quickly to problems involving
uneven decimals that include hundredths and
thousandths. Note that solving uneven decimal
problems (such as Problems 11.1 and 11.2 below)
concretely sets the stage for "lining up the decimals." With a concrete base-ten blocks model, it
makes sense to add cubes to cubes (units to units),
flats to flats (tenths to tenths), and so forth. Problems that involve money, such as Problem 11.2 below, can provide a meaningful context for adding
and subtracting numbers involving tenths and
hundredths. (Note also that Problems 11.1 and
11.2 below are multistep problems.)
■
Problem 11.1 Snow Day? (◆ 4-8) In the
school district of Deep South, 1.75 inches
of snow constitutes a snow day. Last
night it snowed 2.25 inches. At dawn 1.4
inches melted, but the weatherman predicted more snow by 8 am. How much
must it snow to have a snow day?
■
Problem 11.2: The Parking Fee (◆ 4-8).
The fee for parking a car at a downtown
parking garage is 85 cents for the first halfhour, 70 cents for the second half-hour,
and 45 cents for each additional half-hour
or portion of a half-hour. Mr. Newton
parked his car in the garage from 9:35
(problem and text continued on page 301)
297
Investigation 11.A: Another Need for Decimals
◆ Decimal numeration ◆ 6-8 ◆ Any number
Consider the following problem:
A
Auden's Fishing Pond. To get
from his house
(Point A in the
figure to the right)
to a fishing pond
(Point F), Auden cut
across a field 2
Part II. Another approach is to estimate the
2
x
3
4
3
4
each side. How far
in all did he have to
walk?
1
4
F
Students who have studied the Pythagorean
theorem may recognize that the diagonals x and y
may each be computed by the formula c2 = a2 + b 2 :
x2 = 22 + 22 = 4 + 4 = 8;
y2
=
12
+
12
= 1 + 1 = 2;
x2 =
2
y
=
8;x=
8
2;y=
2
Therefore, the distance from Auden's house to
the fishing pond is 8 +
2 + 34 . "Great, but
what is
8 +
2 + 34 ?" a student might well ask.
Quite likely, a student will suggest using a calculator—the square-root function—to determine
the values of 8 and 2 . Have the students consider how such a problem might have been solved
before decimals were invented.
Part I. One approach would be to estimate
the fractional value of 8 and 2 , using one of
the methods described on pages 8-17 and 8-18 in
chapter 8 of the Student Guide. * For example, 8
is between what perfect squares? A more precise
estimate could be determined by asking how far
* Some students may suggest substituting
8 +
2 . Does
8 +
value of 8 and 2 using a number line. Using a
ruler or the edge of a paper to measure the distance in the drawing to the left, the distances of 1,
2, and 2 can be marked off on a number line:
2
miles on each side,
then cut across another
1 field 1 mile on each
side, and finally
walked along the side
of a field 3 mile on
y
along is 8 from the next smaller perfect square to
the next larger perfect square? Using this method,
what is a fractional estimate of 2 ? What is the
total distance from Auden's house to the pond?
2 =
10 ?
10 for
0
1
2
2
Now use this number line to estimate the fractional
value of 2 . Some students might suggest that
2 is about a third of the way between 1 and 2
and, thus, about 1 13 . How could you check
whether 2 is one-third of the way between 1 and
2? Does your check suggest that 1 13 is a good estimate of 2 ? Is a more accurate estimate possible? How so? Try estimating the fractional value
of 8 using a number line. What is the total distance between Auden's house and the pond?
Part III. If a ruler is not available, another
way to estimate square roots is to divide the number line constructed in Part II into tenths. Using
this method, 2 appears to be about 1 104 . Locate
8 and estimate its value. What is the total distance between Auden's house and the pond?
2
0
1
2
An even more precise estimate could be derived
from dividing the number line into hundredths.
This would necessitate changing our scale.
Questions for Reflection
1.
Would it be easier to compute the distance
from Auden's house to the pond using the
decimal fractions derived from Part III than it
was to compute this sum from the common
fractions derived in Part I or II?
2.
Would it be easier to compute this distance using decimals as opposed to common fractions?
298
Investigation 11.B: Zorkian Quadal
(Base-Four) Numeration
◆ Decimal-numeration enrichment ◆ 4-8 ◆ Any number
Exploring how fractions and decimals would
be represented in other base systems can help students see the patterns underlying our decimal numeration system and deepen their understanding
of it. Consider the following problem:
An Off-Base "Decimal." What base-ten number would the Zorkian number (quadal) 1.2
equal? How would the base-ten rational num1
ber 4 or 0.25 be written in the Zorkian (a basefour) numeration system?
■
Interpreting decimals in other bases may well
seem an impossible task—an enigma—to many
students. Learning to decipher decimals in other
bases is a challenging problem, one that can give
many students' sense of mathematical power a
boost. To understand Zorkian quadals, a student
needs to understand (a) quadal fractions (the basefour counterpart of decimal fractions such as
1
,
100
and
1
)
1000
1
10
,
and the Zorkian place-value system.
Part I: Conceptual Level—Quadal (BaseFour) Fractions
1. If a (base-four long) = 1, then (a) , (b)
, (c)
, (d)
and (e)
would equal what
decimal fractions and what equivalent quadal
(base-four or Zorkian) fractions?
2. If a
(base-four flat) = 1, then (a) , (b)
, (c)
, and (d)
would equal what decimal
fractions and what equivalent quadal (base-four
or Zorkian) fractions?
3. If a
(base-four large cube) = 1, then (a)
, (b) , (c) , (d)
, (e)
, (f)
, and (g)
would equal what decimal
fractions and what equivalent quadal (base-four
or Zorkian) fractions?
4. How are the quadal fractions (base-four) coun1
1
terparts of the decimal fractions (fractions 10, 100 ,
1
and 1000) expressed (a) as a base-ten fraction and
(b) as a base-four fraction?
Part II: Connecting Level—Quadals
1. Complete the place-value chart at the bottom of
the page. (a) How should the unlabeled arrows
going to the right be labeled? (b) What is the
equivalent base-ten value of a zorkth? A
superzorkth? A superduper zorkth? (c) What
quadal fraction represents each?
2. If
= 1, then
would represent what (a)
base-ten fraction, (b) decimal, (c) base-four
fraction, and (d) quadal?
3. (a) What base-ten fraction and decimal does the
quadal 1.2 represent? Illustrate how Zork
(base-four) blocks could be used to represent
this number and determine its value. Do the
same for the quadals (b) 0.01 and (c) 2.003.
4. (a) What base-ten fraction and decimal does the
quadal 0.11 represent? How this could be
determined concretely with Zork (base-four)
blocks? How so for (b) 0.23 and (c) 1.011?
5. Would the Zorkians use a decimal like 1.5 or
1.9? Why or why not?
6. What quadal would represent the base-ten
fraction (a) 34 , (b) 165 (c) 1 647 (d) 13 , (e) 15 ?
299
Investigation 11.C: Converting Fractions to Decimals with a 10 x 10 Grid
◆ Modeling fraction-decimal equivalents ◆ 3-5 ◆ Any number
This investigation illustrates how students can concretely convert fractions to decimals with
the aid of a 10 x 10 grid. Work through the following questions to see what is involved. It may
help to work in a group.
1. Figures A and B below illustrate
1
10
and
1
,
100
respectively. Illustrate below how a
10 x 10 grid can be colored in to represent
the following decimal fractions: (a) 103
(use Figure A), (b)
153
100
3
100
(use Figure B), (c)
(use Figure C), and (d)
23
100
3. (a) Try representing each of the following
fractions on a 10 x 10 grid: 13 and 29. (b)
What did you discover by trying to answer Question 3a? (c) Why is the task of
1
2
representing 3 and 9 on a 10 x 10 grid
3
different from representing 15 and 4?
(use Figure
D).
Figure A
Figure B
Figure C
Figure D
2. Devise a strategy for using a 10 x 10 grid
to represent the following cousins to
decimal fractions: (a) 15 and (b) 34 .
4. (a) Can the fractions below be represented
on a 10 x 10 grid? (b) How are the
fractions below different from those in
Questions 1 to 3 above?
(i)
2
3
(ii)
3
2
(iii)
5
7
(iv)
10
8
300
Investigation 11.D: Concrete Decimal Addition and Subtraction (Conceptual Phase)
◆ Constructing concrete renaming procedures for adding
and subtracting decimals ◆ 3-8 ◆ Ideally in groups of four
Part I: Qualitative Reasoning and Estimation. Instruction on decimal addition and subtraction should begin with a focus on thinking rather
than determining the correct answer. This can be
prompted by problems like Problem A below.
■
Problem A: Money Down the "Drain" (◆
3-8). Ruffus accidentally knocked Mary's
piggybank onto the floor, breaking it wide
open. Some of the $5.35 worth of coins
were scattered about. A number of coins
worth $2.19 fell down the heat vent and
another $.22 was lost elsewhere in the
room. Ruffus also managed to carry away
another $1.20 in change. About how much
did Mary lose? (a) Was it more than $5.35
or less than $5.35? (b) Did she lose more or
less than half her money? (c) Did she lose
between $1 and $2, $2 and $3, $3 and $4, or
more than $4?
house. It had already rained a total of 6.85
inches. Today it rained another 1.25 inches.
Now how much rain had fallen altogether?
■
Problem 3: A Bailing Effort That Fell Short.
Flooding left water 3.75 inches high in Ruffus'
doghouse. After bailing water for an hour,
Mary reduced the water level by 1.25 inches.
What was the new water level in Ruffus' doghouse?
■
Problem 4: Further Bailing Efforts. The next
day, the water in Ruffus' doghouse was only
2.0 inches deep. How high would the water
level be if Mary bailed out another 1.25 inches
of water?
■
Problem 5: A Rod Reduced (◆ 4-8) A wood
rod was 20 cm long. Three pieces were cut
from the rod in lengths of 12.2 cm, 2.99 cm,
and 3.4 cm. Because the width of the blade
was reduced to sawdust, each cut of the rod
results in a waste of 0.12 cm. How much of the
original 20 cm rod was left?
Part II: Solving Problems Concretely. Next,
children can be encouraged to use manipulatives
such as base-ten blocks to solve decimal addition
and subtraction problems. As the problem below
illustrates, problems that involve renaming can be
introduced almost immediately. Consider how
base-ten blocks could be used to model each of the
following word problems.* For the questions below, assume that large cubes, flats, longs, and
cubes but not chips (tenths of a cube) are available.** Illustrate your concrete procedure. Be prepared to describe and defend your procedure to
the class.
■
■
Problem 1: Accumulating Rain. Ruffus was
worried; the water was rising quickly around
his dog house. Yesterday, it had rained 2.54
inches. Today it rained 3.01 inches. How
much rain had fallen the past two days?
Problem 2: More Rain. Ruffus was worried;
the water was rising quickly around his dog
waste
Questions for Reflection
1.
The written algorithm for adding decimals is
based on what two key principles? Hint: They
are the same principles for the written algorithm for adding whole numbers.
2.
How can using base-ten blocks for decimal
addition concretely model these principles?
3.
Why is it important to ask students questions
like 1 and 2 above?
* With elementary-age students, a teacher might wish to begin with relatively simple problems—ones that
involve like decimals with only units and tenths.
** Chips are now commercially available and can extend the range of problems and expressions that can
be modeled with base-ten blocks from four places to five places.
301
a.m. until 2:10 p.m. When he got back to
his car, Mr. Newton discovered he had
only $5.00 left in his wallet. Would he be
able to pay his parking fee and reclaim his
car?
10N = 3.3333
- N = 0.333 3
9N = 3.000 0
3
1
N =
=
9
3
QUESTIONS TO CONSIDER
1.
Cuisenaire rods are another manipulative that
can be used to introduce decimal fractions.
Compare using Cuisenaire rods with using
base-ten blocks for this purpose. What are the
advantages and disadvantages of each?
2.
27
329
Decimal fractions such as 103 , 100
, or 1000
can be
converted into terminating decimals, decimals
with no rounding error. Some other common
fractions such as 12 , 34 , 25 , 58 , 203 , and 251 can also
be converted into terminating decimals, decimals without rounding error. Yet other common fractions such as 13 , 56 , 27 , 19 , 112 , 121 , and 154
can be converted into infinitely repeating decimals—decimals that involve rounding error.
What do common fractions that convert to
terminating decimals have in common? How
are they different from common fractions that
convert to infinitely repeating fractions?
3.
4.
Why must common fractions that do not convert to terminating decimals convert to infinitely repeating decimals? Hint: Consider
fractions with a denominator of 3. How many
possible remainders can there be if you divide
by 3? Consider also fractions with a denominator of 7. How many possible remainders
can there be if you divide by 7? Now consider
fractions with a denominator of 9, 11, 12, or 14.
In each case, is there a specific number of possible remainders or not? What is the implication of the answer to this question for the
question originally raised?
The key to converting an infinitely repeating
decimal to a fraction is to eliminate the repeating pattern so that an algebraic equation involves only integers (and a variable). Consider the decimal 0.333 3 . If 0.333 3 equals N,
then 10N equals 3.3333 , and subtracting the
two equations eliminates the repeating decimal portion. The resulting algebraic equation
with integers can be expressed in fraction
form. This process is summarized at the top of
the next column.
Miss Brill's class was practicing the decimalto-fraction conversion method just described.
Michelle noted that for 0.2727 or .3555 , this
method did not eliminate working with decimals. How could the method described above
be modified to accommodate these decimals?
5.
Mr. Ma'kwaganda asked his class to convert
the following decimals to fractions: .09, .17
9 17
25
and .025. Sonya answered 10, 10, and 100 . Paz
answered
0
9
,
1
7
, and
0
.
25
Analyze the errors
made by each of these students. (a) Briefly describe what systematic errors each student
made. (b) Hiebert and Wearne (1986) found
that one of these errors was made by about
one-fourth of their fifth-grade sample. The
other error was the most common error
among older students. Whose error indicates
a more complete understanding of decimals—
Sonya's or Paz's? Why?
6.
Ms. Simons gave her fifth-grade class the following question:
Sandor, a new student, chose a for Item 1 and
c for Item 2. Josie chose d and a. Gary chose e
and b. (a) Presumably, what is the basis of
each student's error? (b) Which child has the
most complete understanding of decimals, and
which has the least complete understanding?
Justify why or why not.
7.
Mae Marie had a reputation for being a smart
aleck. She saw a sign in Mr. Tillson's window
that said, SALE ON RIBBON .99¢ A ROLL.
Mae Marie was delighted. She entered Mr.
Tillson's store, picked out the roll of ribbon she
wanted, dropped a penny in Mr. Tillson's
palm, and said cheerily as she left, "Keep the
change." Mr. Tillson angrily called out, "Come
back here you little shoplifter and pay for that
ribbon or I'll call the police." Did Mr. Tillson
have legal grounds for arresting Mae Marie on
a shoplifting charge? Why or Why not?
302
8.
Miss Brill asked her class to represent .341
with base-ten blocks. One group offered the
solution below. What can Miss Brill conclude
about their understanding of decimal numeration? Does the model indicate an understanding of decimal numeration or can't you
tell? Why?
11. Asked to divide 131.92 ÷ 4, 10-year old Arianne asserted, “I’m doing it my own way.”
She proceeded to make the notations below.
Was Arianne using a divvying-up or a measure-out meaning to guide her actions? Why?
9.
Lina, a student teacher, introduced a groupsof meaning of decimal multiplication with
base-ten blocks in the following manner:
SUGGESTED ACTIVITIES
1.
(a) Which is easier to model using base-ten
blocks, a measure-out or an area meaning for
decimal division? Construct a chart comparing the two models. You may wish to include
items such as:
• Any block can be used to represent 1, 0.1,
0.01, or 0.001.
• Can represent divisors involving hundredths or thousandths as well as tenths,
when a flat equals a whole.
(b) Delineate your conclusions about the relative flexibility and ease of each model. (c) Present your data and conclusions to your class.
2.
Describe your own idea for a bulletin board
display involving decimals. Indicate the learning aims of the display and illustrate how it
might look.
3.
Examine newspapers and news magazines for
everyday applications of decimals. Illustrate
how you could create a bulletin board to illustrate decimal applications.
4.
(a) Devise an integrated lesson that could be
used to introduce a decimal concept or operation. Specify the aims of the lesson for other
content areas as well as for mathematics. Indicate clearly how the lesson will create a need
for the decimal concept and operation. (b) Try
out your lesson with a group or class of elementary-level children. If possible, videotape
the lesson. Assess the children's progress in
constructing an understanding of decimals or
operations on decimals.
Evaluate the
strengths and weaknesses of your lesson. (c)
Share your lesson plan and your assessment of
it with your class.
5.
(a) Find examples of children’s literature in
which decimal concepts or decimal arithmetic
play a key or interesting role. (b) Consider
0.5 x 0.2: Let a long = 0.1; 2 longs = 0.2; one
half group of 0.2 is 1 long or 0.1
0.25 x 0.8: Let a long = 0.1; 8 longs = 0.8; onequarter group of 0.8 is 2 longs or 0.2
0.2
0.5 {
answer = 0.1
Compare Lina's approach to the approach illustrated in Figure 11.4 on page 11-18 of the
Student Guide. In terms of a number sense for
decimals, what does Lina's approach underscore that the illustrated approach does not.
Consider expressions such as 0.7 x 2.1 or
0.3 x 0.9. What limitations does Lina's approach have?
10. a.
Express the fraction equal to X below in
simplest terms.
X = 2.25 + 2.25 + 2.25 + 2.25 + 2.25 + 2.25
.72 + .72 + .72 + .72 + .72 + .72
b. If you answered 13.5/4.32 for part (a)
above, consider whether or not your answer is in simplest terms.
c.
Why does the expression in part (a) reduce
to the simplest fraction that it does?
d. Would the fraction equal to Y below reduce to simplest terms in the same way?
Why or why not?
Y = 2.25 + 2.25 + 2.25 + 2.25 + 2.25 + 2.25
.72 + .72 + .72 + .72 + .72
303
how a lesson or unit that embodies the investigative approach could be built around one or
more of the books on your list.
6.
7.
8.
9.
(a) Using one of the reader inquiries in the
Student Guide or this guide, plan a lesson or
unit. (b) Try out your lesson with a group or
class of elementary-level children. Evaluate
their progress in constructing an understanding of decimals or operations on decimals.
Assess the strengths and weaknesses of your
lesson. (c) Share your lesson plan and your
assessment of it with your class.
(a) Using publishers' catalogues, reference
books on teaching, the internet, or other
sources, make a list of instructional resources
that would be helpful in teaching decimal concepts and operations. (b) Obtain one or more
of the resources. Evaluate a resource in terms
of what approach to mathematics instruction it
seems to suggest. Indicate how one or more
recommended activities could serve, or be
adapted to serve, as a worthwhile task and the
basis for the investigative approach. (c) Share
your data, evaluation, and teaching ideas with
your class.
(a) Try out one of the examples of educational
software listed on page 11-32 of the Student
Guide. (b) Evaluate the software in terms of its
ease of use and instructional value. (c) Share
your evaluation with your class, including a
demonstration of particularly interesting or
useful aspects of the program review.
Consult Activity File 11.4 (on page 11-31 of the
Student Guide). (a) Then construct your own
scrambled message for an estimation exercise.
Keep in mind that for such an exercise, the answers for the various expressions should be
far enough apart that good estimates would
result in a correct answer. For example, using
both 0.5 × 0.326, which is equal to 0.163, and
0.175 – 0.011, which is equal to 0.164, would be
fine for an exercise in which children compute
the exact answers but would not be appropriate for this estimation exercise. To give your
students practice ordering decimals, ensure
that most of your answers are decimals rather
than whole numbers. Use a variety of operations and different types of expressions (e.g., a
larger decimal divided by a smaller decimal, a
smaller decimal divided by a larger decimal, a
decimal divided by a whole number, and so
forth). (b) Try out your scrambled message
with a group or class of elementary-level children. Evaluate their reaction and learning. (c)
Share your activity and your evaluation with
your class.
HOMEWORK OR
ASSESSMENT
QUESTIONS TO CHECK
UNDERSTANDING
1.
According to the Student Guide, which of the
following items are true? Circle the letter of
any true statement; change the underlined
portion of any false statement to make it true.
a.
All decimals, including those such as
3.15155155515555..., can be converted into
a common fraction.
b. If the fractional equivalent of an infinitely
nonrepeating decimal is divided out,
sooner or later the division will result in a
remainder of 0.
c.
Decimals include .3, 2.3, 2.25 and .03 but
not 2.0, 3, or 45.
d. The correct way to write $2,312 is Two
thousand three hundred and twelve dollars.
e.
The center of the decimal system is the
units (ones) place.
f.
The whole number 3 can be expressed as a
terminating decimal.
g. Developmentally, it makes sense to introduce decimals first, then decimal fractions,
and finally common fractions in general.
1
1
h. Decimal fractions include 1.2
3 , 0.5 7 , and
_____________
3
0.44 11 .
______
i.
Using different base-ten blocks to represent one is confusing to children and
should be avoided.
j.
Instruction on decimal operation should
begin with tasks involving qualitative reasoning and estimation.
k. In a groups-of interpretation of 0.3 x 0.8,
both 0.3 and 0.8 are fractions of a whole
(i.e., 1 unit).
304
l.
Multiplying by a decimal always results in
a product that is smaller than the size of
the group (multiplicand).
represented by a common fraction?
c.
m. In an area interpretation of 0.3 x 0.8, both
0.3 and 0.8 represent the length of a rectangle's side, and the product .24 represents the area.
Some rational numbers can be reduced to
an integer. Would such numbers be categorized as terminating decimals, infinitely
repeating decimals, or infinitely nonrepeating decimals? Why?
n. Children can be encouraged to rediscover
the rules for placing the decimal point in a
product by prompting them to use a repeated-addition interpretation of multiplication.
d. The Math Book from Hell noted that terminating decimals involve no rounding error, whereas infinitely repeating decimals
do. Unfortunately, the text did not clearly
explain why this is the case. How would
you explain these facts to a student?
o.
e.
Makato pointed out that his textbook
noted that all rational numbers can be expressed as a terminating decimal or an infinitely repeating decimal.
Using his
1
calculator, he found that 23 is equivalent to
0.0434783. "It doesn't repeat. Is it possible
that the textbook is wrong?" he asked.
How could you help Makato resolve this
conflict?
f.
Using the numbers 9.99, 9.09, 9.909, 9.099,
9.009, and 0.999, find the positive difference between the smallest and largest of
these numbers?
Dividing a decimal by a whole number
can be easily interpreted in terms of a
divvy-up or measure-out meaning; dividing a decimal by a decimal, in terms of a
divvy-up meaning.
p. In an area interpretation of 0.9 ÷ 1.2, both
0.9 and 1.2 represent the length of a rectangle's side, and the quotient represents
its area.
2.
Construct a concept map of decimals outlining
the connections discussed in chapter 11 of the
Student Guide.
3.
(a) Draw a Venn diagram to illustrate the relationship among decimals, integers, natural
numbers, and rational numbers. (b) Asked to
represent the relationships among decimals,
integers, natural numbers, and rational numbers with a Venn diagram, Doris' group drew
the figure below. Evaluate the drawing.
natural
g. Batteries normally sell for $1.00 each, and
you see a SALE sign that reads, "Batteries
for .99¢ each." (a) About what percent off
are the batteries on sale? (b) Is this sale a
significant bargain? Why or why not?
5.
decimals
Deb, a student teacher, taught a lesson on
converting quotients to fractions and fractions
to decimals. She used the following example:
÷
integers
Confused, the children asked, "How could a
remainder of 1, which is a whole number,
convert to a fraction ( 13 ) or a decimal ( .333 )
rationals
4.
a.
less than one?" Deb explained that r 1 was not
really a one, not really a whole number. (a)
Why might the children have been confused?
Consider why students commonly do not
comprehend such conversion procedures. (b)
Evaluate Deb's explanation. Is it accurate?
Explain why or why not.
When a decimal is rounded off to the tenths
place, the result is 0.3. Can you tell if the
original decimal is a terminating decimal,
an infinitely repeating decimal, or an infinitely nonrepeating decimal? Briefly
justify your answer.
b. The real number 0.732050808... is an infinitely nonrepeating decimal. Can it be
6.
As early as 2000 B.C., the Babylonians wrote
fractions in a place-value form (without a de-
305
nominator), using a sexagesimal (base 60) system but no "sexagesimal point": no symbol to
separate the whole-number portion and fractional-number portion (Bunt et al., 1976;
Mainville, 1989). (a) Taking into account that
the Babylonian numeration system is an
(incomplete) base 60 system, the symbol
would have represented what fraction as
well as 1 or 60 (depending on its position)? (b)
How would the Babylonians have represented
half of something? (c) How would they have
represented two wholes and a quarter of
something?
7.
(a) The base-five numbers 0.215 and 1.04 5
would be equal to what decimals? (b) The
decimals 0.8 and 0.25 would equal what basefive numbers?
8.
(a) Demonstrate or illustrate how base-ten
block models of 3.017, 3.91, and 3.8 could be
used to help children decide which of these
decimals is the largest. (b) Do the same using
a 10 x 10 grid.
9.
Demonstrate or illustrate how base-ten blocks
could be used to determine the difference for
each of the following: (a) 1.45 - 0.89 and
(b) 2.003 - 0.015. Illustrate each step in the
process and briefly describe how each step in
the blocks procedure models a corresponding
step in the written renaming algorithm.
(or score of 10) and subtracted however many
blocks their spins indicated. The first player to
get rid of all his or her blocks (to get to zero)
won. Rodney's previous total was 2.062. On
his next turn, he spun a 5 in the tenths spinner,
a 9 on the hundredths spinner, and a 3 on the
thousandths spinner (.593). (a) Demonstrate
or illustrate how Rodney could use base-ten
blocks to compute and to represent his new
score. (b) Explain how each step of the written algorithm corresponds to a step in the concrete model you illustrated in Part a.
12. a.
To check her understanding of symbolic
decimal multiplication, Stasha was asked
to use base-ten blocks to create a groups-of
model of 3.5 x 0.2. She responded with a
model unlike anyone else in her class.
Evaluate Stasha’s model below. Consider
whether it is a correct or an incorrect representation of 3.5 x 0.2. Consider whether
the model indicates that her number sense
for decimals is more or less advanced than
her classmates. Briefly justify your answers.
and
, where
= 1.0.
= 0.1
b. Asked to show how base-ten blocks could
be used to make a groups-of model of
1.3 x 0.2, Bobbette made the model illustrated below. Evaluate Bobbette's solution.
10. To practice the written decimal-addition algorithm meaningfully and purposefully, Miss
Brill had the class keep score of a game called
Decimal Spin. On their turn, players spun a
tenths, hundredths and thousandths spinner
and collected the appropriate size and number
of base-ten blocks. These were then added to
a player's existing collection (score). A player
recorded each turn with written symbols in
order to keep a running score. The first player
to collect 10 large cubes (obtain a score of 10)
is the winner. Rodney's existing score was
.235 to which he added another .326 points.
(a) Demonstrate or illustrate how Rodney
should use base-ten blocks to compute and to
represent his new score. (b) Explain how each
step of the written algorithm corresponds to a
step in this concrete scoring procedure.
13. Demonstrate or illustrate how base-ten blocks
could be used to model a groups-of interpretation of the following expressions: (a)
0.3 x 0.21, (b) 2.4 x 1.56, (c) 0.12 x 0.34, (d)
1.02 x 0.16, and (e) 2.01 x 0.36.
11. To practice decimal subtraction, Miss Brill had
her students play Reverse Decimal Spinner. In
this game, players began with 10 large cubes
14. Raina made the following area model for
0.2 x 0.3 and concluded the product was 0.6.
(a) Did she set the model up correctly? Briefly
1 group
of .2:
.3 group
of .2:
306
justify. (b) Is her answer correct or incorrect?
According to the Student Guide, how might she
have arrived at this answer?
a.
b.
c.
d.
0.6 ÷ 0.2
0.4 ÷ 0.8
0.3 ÷ 0.5
0.1 ÷ 0.8
e.
f.
g.
h.
1.19 ÷ 2.38
1.5 ÷ 5
1.28 ÷ 0.4
0.84 ÷ 0.4
i. 7.9325 ÷ 5
j. 1.368 ÷ 12.4
WRITING OR JOURNAL
ASSIGNMENTS
15. In an area model of decimal multiplication or
division, is it practical for the surface of a flat
to represent anything other than 1 square
unit? More specifically, can it be used to represent 10 square units, 0.1 square units, 0.01
square units, or 0.001 square units? Why or
why not?
1.
In Part I of Investigation 11.1 (page 11-6 of the
Student Guide), a unit was defined as a long, a
flat, or a large cube. Miyoshi, a prospective
teacher, argued that this might be confusing to
children and that a unit should be defined as a
large cube from the start. What advantage is
there in changing how a unit is defined?
2.
In one study (Hiebert & Wearne, 1986), intermediate-level students were asked to write a
decimal to represent the shaded part in grids
like those below. The most frequent errors for
Item A was 3.10; for Item B, 4.100; and Item C,
1.5. What is the basis of these errors?
3.
(a) Beryl wrote one-tenth as
; two-tenths as
; three-tenths, as
and so forth. Briefly
explain why he might be making this error.
(b) Tai wrote three hundred forty-two thousandths as .0342. What might account for this
error? (c) Does Tai have more, less, or the
same amount of understanding about decimal
notation as Beryl? Why?
4.
While Mrs. Katzenmoyer's class was exploring
decimals, a number of questions arose. For
each of the following cases, (a) what is the answer to each query? (b) How could Mrs.
Katzenmoyer help children determine the answers to these questions for themselves?
16. Demonstrate or illustrate how base-ten blocks
could be used to model an area interpretation
of (a) 2.4 x 1.6 = 3.84 and (b) 1.2 x 0.4 = 0.48.
Note which part of each model represents
each partial product of the written algorithm.
17. If X is a number from 0.1 to 1.0 and Y is a
number from 0.001 to 0.01, what is the maximum value possible (a) when X is divided by
Y? and (b) when Y is divided by X? Why?
18. Demonstrate or illustrate how base-ten blocks
could be used to model a measure-out interpretation of (a) 1.2 ÷ 0.3, (b) 1.44 ÷ 0.24,
(c) 2.02 ÷ 0.505, (d) 1.4 ÷ 0.4, (e) 0.11 ÷ 0.03, (f)
1.24 ÷ 0.4, (g) 0.4 ÷ 0.8, (h) 1.5 ÷ 2.5, and (i)
0.03 ÷ 0.12. Express answers in decimal form.
19. Demonstrate or illustrate how base-ten blocks
can be used to model an area interpretation of
(a) 1.5 ÷ 0.3, (b) 0.8 ÷ 0.2, (c) 0.96 ÷ 0.3, (d)
0.12 ÷ 0.4, (e) 0.42 ÷ 0.7, (f) 2.4 ÷ 1.2, (g)
0.6 ÷ 1.2, (h) 0.35 ÷ 0.5, (i) 1.26 ÷ 1.8. Express
answers in decimal form.
20. Which interpretation of division would be appropriate for using base-ten blocks determining the quotient of the expressions below?
Write M if a measure-out interpretation can be
modeled; write A if an area interpretation can
be modeled; write MA if both a measure-out
and an area can be modeled; write —
0 if neither would be useful. Circle a M or an A if
interpolation would be required. Assume that
large cubes, flats, longs, and cubes, but not
chips (tenths of a cube), are available.
307
Case I: Chelsey pointed out that decimals such
as 0.75 and 3.0 could be written with an infinite numbers of zeros after them (0.75 0 and
3. 0 ) and asked, "Should they be considered
infinitely repeating decimals too?"
Case II: Meg asked, "How can π, which is an
irrational number, be represented by a com22
mon fraction 7 ?"
Case III: After the class agreed that square
roots such as 2 and 3 were represented by
infinitely nonrepeating decimals and were
irrational numbers, Na'il asked, "Can we
conclude from our examples that this is true
for all square roots?"
5.
6.
Lorry was surprised when he used his calculator to determine the products of 0.1 x 3.2,
0.2 x 4.5 and 0.5 x 1.24. "Good grief, multiplying by decimals always makes the product
smaller." (a) What kind of reasoning was
Lorry using to draw his conclusion? (b) Evaluate the accuracy of his conclusion. (c) How
would you help Lorry understand the results
he got (e.g., why 0.2 x 4.5 = 0.9, a number less
than 4.5 and, indeed, less than 1).
Write a realistic word problem that would fit
each of the following number sentences:
(a) 135 ÷ 4 = 33 r 3
(b) 135 ÷ 4 = 33 43
(c) 135 ÷ 4 = 33.75
7.
(d) 135 ÷ 4 would be 33
(e) 135 ÷ 4 would be 34
(f) 135 ÷ 4 is about 30
Rosemary used a calculator to determine the
quotient of 0.945 ÷ 0.15. She was flabbergasted
by the quotient: 6.3. "How did you get six
and three tenths units by dividing two numbers less than one?" she wondered. How
could a teacher answer Rosemary's question in
a meaningful fashion?
8.
Write a realistic word problem for each of the
following equations: (a) 0.75 x 0.8 = 0.6 and
(b) 0.75 ÷ 0.05 = 15.
9.
Somerset divided 105.5 by 4 and got a quotient
26.37 r2. Is his answer correct or not? Briefly
explain why.
10. The new teacher on your grade-level does not
understand the rationale for the distributive
algorithm for decimals. Explain and illustrate
how you would help him to understand such
a procedure, using 3 )17.05 as an example.
PROBLEMS
■ A Nongoal-Specific Problem About the
Pioneers Game (◆ 6-8)
For the game Pioneers (described in Activity
File 11.A below, specify an era, mode of travel,
starting point and destination. Assume a 10-sided
die will be used to determine the distance traveled
per turn. For the mode of travel chosen, specify a
reasonable time frame for each turn (e.g., a turn
represents one hour, a day, 48 hours, 1 week, a
month, or a year) and reasonable unit of distance
for the mode of travel. For example, defining each
turn as a day and the unit of distance as miles
would be appropriate for horse-drawn covered
wagons over rough terrain. It would not be reasonable to define a turn as 1 hour because such
transport did not travel more than 10 miles per
hour.
Activity File 11.A: Pioneers—DecimalFraction Version
◆ Decimal-fraction equivalents ◆ 2-4
◆ Two to five players
This game is a variation of the trading games
with base-ten blocks introduced in Activity File 6.2
on page 6-8 of the Student Guide. The players each
pretend they're a pioneer group whose mission is
to be the first to reach and settle a new region. The
context (times, starting place, mode of transport,
goal) can be varied and can correspond to a current Social Studies lesson. On their turn, players
draw a card that can specify how far they traveled
7
3
(e.g., 10
miles or 1 10
miles). (Note that travel,
particularly slow travel, provides a natural context
to discuss tenths of a mile at least. To introduce
hundredths and thousandths, have the students
47
47
play Space Pioneers where 100
would represent 100
of a light year.) The players would keep track of
their mileage with base-ten blocks, trading up for
a larger block whenever possible. (What blocks
would be used and their assigned values would
depend on whether the game involved tenths and
units only or other denominations.)
To make the game more interesting, the cards
could include items such as STUCK IN THE MUD,
LOSE TURN; IMPASSABLE TERRAIN, TURN
BACK, LOSE 108 MILE; USE RIVER RAPIDS, GO
9
AHEAD 10
MILES; and DRAW AGAIN.
308
■ Which Option? (◆ 6-8)
Mr. Burris offered his class the following grading options:
Option 1:
Option 2:
■ Decimals in Other Bases (◆ 6-8)
How would a computer, which processes in
base 2, represent the decimals (a) 0.125, (b)
0.25, (c) 0.5625, and (d) 0.875? Recall how it
represents whole numbers and consider how
this representation could be extended to show
decimals.
23
8
2.
1.
Complete the following addition sentences
written in Zork (base four). Write your answer in both base four and base ten.
homework worth .25, midterm worth
.25, and a final worth .50
homework worth .33, final worth .67
Claudine was not confident in her test-taking
ability. She had been doing reasonably well on her
homework, averaging roughly 8.5 points (out of a
possible 10 points per assignment). Assume she
will continue to perform at the same level on her
homework for the rest of the semester. If a 9 or
above is an A, 8 to 8.9 is a B, 7 to 7.9 is a C, and 6 to
6.9 is a D and the exams are also based on a 0 to 10
scale, which option should Claudine take if she is
concerned with passing the course (i.e., getting a D
or above)? Justify your answer.
1.
■ Arithmetic with Zorkian Quadals (◆ 5-8)
22
4
21
2
20
1
Shoe sizes are in eighths. (a) The following
shoe size is written in a base 8 decimal: 12.38 .
What size is the shoe in base 10? (b) What
fraction is represented by the following base 8
decimal: 12.318 ?
3.
In base ten, the fractions 12 , 15 , and 101 can all be
represented by terminating decimals, whereas
1 1 1
10
3 , 6 , 12, and 12 are all represented by infinitely
repeating decimals. Which of these seven
common fractions would be represented by a
terminating decimal in base 12?
4.
What would the decimal equivalents of the
seven fractions listed in Question 3 be in base
12?
5.
In base 12 fractions, what determines whether
a fraction will be represented by a terminating
decimal or an infinitely repeating decimal?
Hint: Consider the answer to Question 2 of
Questions to Consider on page 301 of this
guide.
2.
(a) 1.2 4 + 0.1 4 =
(e) 1.203 4 + 0.110 4 =
(b) 1.3 4 + 0.1 4 =
(f) 1.312 4 + 0.022 4 =
(c) 1.2 4 + ____ = 2.0 4
(g) 1.231 4 + 0.1334 =
(d) 1.34 + ____ = 2.2 4
(h) 2.132 4 - 0.234 =
What is the product of the following Zorkian
multiplication expressions: (a) 1.14 x 2 4; (b)
0.3 4 x 34; and (c) 0.24 x 1.2 4. Write your answer
in both base four and base ten.
■ Discs (◆ 7-8)
A stack of blue and red discs is exactly 10 cm
high. The blue discs are 0.6 cm thick, and the red
discs are 0.8 cm thick. What is the greatest number of red discs that could be in the stack?
(Assume that only whole discs are used.)
■ An Open-Ended Decimal Problem About
Sevenths (◆ 6-8)
List the decimal value of the fractions
4
7
,
5
7
, and
6
7
1
7
,
2
7
,
3
7
,
. Use a calculator to obtain the digits.
These decimals repeat digits in sequential patterns.
Show each repeating cycle at least twice
1
(minimum of twelve digits). Example: 7 =
.142857142857... Write the decimals so that common place values are lined up vertically: tenths
under tenths, hundredths under hundredths, and
so forth. Carefully investigate the patterns. List as
many conclusions as you can find concerning the
patterns, sequences, or any other interesting observations.
■ A Challenging Application of the Area
Model for Fraction Division
Illustrate (or demonstrate) how base-ten
blocks could be used to create an area model of
0.8 ÷ 1.2 = ? and, with only the aid of the blocks
(i.e., without using a calculator or a formal algorithm) determine the quotient. Assume that cubes,
longs, flats, large cubes, but not chips, are available.
309
ANSWER KEY for Student
Guide
3
2
1
10 10 10 10
0
-1
10
10
-2
10
-3
Key for Probe 11.1 (page 11-3)
Part I
1.
Do all fractions, including mixed numbers such as
4 104 and fractions that reduce the integers such
This symmetry would be lost if the decimal
point were the center.
5.
How should and be used? Many people respond
to Question 5 by writing .342; many others
write 300.042. In the first case, and is used as a
connector between hundreds and tens (e.g.,
read 342 as three hundred and forty-two) or
between hundredths and thousandths (e.g.,
read .342 as three hundredths and forty-two
thousandths). Unfortunately, using and in this
way leaves room for confusion. For instance,
three hundred and forty-two thousandths
could mean either 300.042 or .342. To avoid
this confusion, and should only be used to
mark the decimal point. For instance, 342 and
.342 should be read without the and, as three
hundred forty-two and three hundred forty-two
thousandths, respectively. According to convention, three hundred and forty-two thousandths
should be written as 300.042. This explains
why the conventional procedure for writing
checks is to use and only between the dollar
and the cent amounts (e.g., $1,423.56 would be
written out
).
6.
The decimal 2.86 may read as (a) Two and
eighty-six hundredths; (b) Two units, eighttenths, and six hundredths; (c) Two point
eighty-six, and (d) Two point eight six.
7.
Three-tenths may be written as .3 or 0.3. By
international agreement, 0.3 is preferred. This
notation clearly indicates that the quantity
represented is less than a whole.
7
1
as , have decimal equivalents? Indeed, all common fractions, including all those listed in
Question 1, can be converted to a decimal by
dividing the numerator by the denominator
(see Example A below). In the case of a mixed
number such as 4 104 , the quotient of the numerator and denominator is simply added to
the whole number portion of the mixed fraction (see Example B below).
Example A:
3
16
Example B:
3.
4.
4 4
10
➞
➞
0.1875
16) 3.000
16
140
128
120
112
80
80
0.4
4 + 10 ) 4.0
40
➞
4.4
What is a decimal? For Question 3, many people circle only 3.3, 0.3, and 0.003—only numbers that are subdivided into tenths, hundredths, and so forth. Some circle these numbers and 3.0 as well—any number with a decimal point in it. In fact, a decimal is any baseten number (e.g., 30 can be expressed as the
fraction 301 or 30.0). Students develop a narrow concept of decimal when instruction
refers to decimals only when discussing numbers such as 3.3, 0.3, 0.003, and 3.0.
What is the center of the decimal system? The
decimal point serves as an important marker.
It separates the whole-number portion (on the
left) from the decimal or fractional portion (on
the right). Intuitively, many students feel that
the decimal point is the center of the decimal
system. In fact, the ones place is the center of
the system. As the following figure shows, the
result is a symmetrical system:
Part II
3.
(a) terminating (b) infinitely nonrepeating
4.
T → integers & some other common fractions;
R → the remaining common fractions; N → irrational numbers
Key for Problem 11.1 (page 11-8)
For this rate problem (r x t = d), the r is given,
the d is given, and the t is unknown. Because t =
d/r, 93,000,000 miles/186,000 miles/seconds
equals 500 seconds. Because there are 60 seconds
310
in a minute and 60 minutes in an hour, the unrounded answer is:
1 minute
1 hour
500 seconds x
x
60 second 60 minutes
500 hours
=
= 0.1388 hours.
3600
for an answer of 1.42857142857....
Key for Investigation 11.3 (page 11-11)
1.
See Figure 11.1 on the next page. Note that the
concept map shown in this figure specifies
that common fractions can also be written as
terminating decimals or as infinitely repeating
decimals—all of which represent rational
numbers. Drawing in the links between irrational numbers and other fractions and infinitely nonrepeating decimals can also help
underscore that some fractions and decimals
represent irrational numbers.
5.
(a) 4,375.6. (b) Chandra's error is a common
one (Hiebert & Weane, 1986). She may have
overgeneralized her whole-number rule for
multiplying by 10—add a zero to the right of
the multiplicand (e.g., 10 x 437 = 4370). Chandra did not recognize that 437.560 was equivalent to 437.56 and, thus, was not a plausible
answer. Disregarding the digits to the right of
the decimal, Sasi also applied whole-number
rule for multiplying by 10.
6.
(a)
Rounding to the nearest hundredth yields an answer of 0.14 hours.
Key for Investigation 11.2 (pages 11-9 and
11-10)
Part II
Ninths. The numerator of the ninth is the sole
4
repeating digit in the decimal value (e.g., 9 =
7
.4444...; 9 = .7777...).
Elevenths. The two digits that form the repeating cycle of the decimal equivalents of the
elevenths can be found by multiplying the nu5
merator of the elevenths fraction by 9 (e.g., 11 =
.45454...).
Sevenths. (1) If Andy were correct (that the
sevenths could not be represented by infinitely repeating decimals), then they would have to be represented by terminating decimals. There was no
evidence such decimals would terminate. (2) Like
many children, Andy did not take into account his
calculator's rounding error. For example, the
2
decimal equivalent of 7 indicated by his calculator
was .2857143, which is .28571428...rounded off. In
using calculators to uncover decimal patterns,
students should be aware of the effects of
rounding errors. (3) The decimal equivalents of
sevenths involve the repeating sequence 142857
1
(e.g., 7 = .142857142857142857... and 2/7 =
.2857142857142857...). This is relatively easy to
remember because 14 is twice 7 and 28 is twice 14,
5 is the fifth digit, and the last digit of the repeating pattern is 7 for sevenths. The key to the trick is
determining the first digit in the decimal equivalent—where in the six digit repeating pattern a
particular decimal begins. When Miss Brill paused
before giving a decimal, she was mentally calculating the tenths digit. An improper fraction such
as 10
can be converted to a mixed fraction 1 37 . The
7
3.0
7
tenths place is
or 4. She then stated the
remaining portion of the sequence (2857) and continued with the next complete sequence (142857)
7
10
= 0.7 (b)
7
4
= 1.75 (c)
22
7
= 3.142857
Key for Investigation 11.5 (pages 11-16 and
11-17)
Part II
1.
(a) A groups-of model for 3 x 0.2 (three
groups of two-tenths) could be modeled in the
following manner:
Let a
= 0.1 (Note that any block, including
the large cube could have been defined as a tenth
and used instead of the cubes.)
first group of 0.2:
second group of 0.2:
third group of 0.2:
(b) To create an area model of 3 x 0.2, let the
surface of a flat = 1 sq. unit, the surface of a
long = 0.1 sq. unit, and the surface of a cube =
0.01 sq. unit. The length of a flat or long, then,
would be 12 or 1 linear unit and the length
of a cube would be 0.1 linear units. One side
of the rectangular would be equal to the
length of three longs (3 linear units); the other,
311
Figure 11.1: Concept Map Showing the Relatonships Among the Types of Real Numbers,
Fractions, and Decimals
equal to the length of two cubes (0.2 linear
units). Filling the rectangle would require six
longs, which represents 6 x 0.1 sq. units or 0.6
sq. units.
2.
X X X
(a) Assuming that chips (tenths of a cube) are
not available, 0.3 x 2 (three-tenths groups of
two) cannot be modeled if the cube is defined
as one. For the solution illustrated below, a
long was chosen to represent one because it
can be subdivided into 10 parts. The answer is
six cubes or .6.
X X X
X X X
X X X
2
(a) Without chips, three-tenths groups of twotenths cannot be modeled if the long is defined
2
0.3
The answer is six cubes or 0.06.
4.
.3
3.
as one. (The two cubes representing 0.2
cannot be divided into tenths.) If a flat
represents a unit and a cube equals a tenth,
0.3 x 0.2 can be modeled as follows:
(a) To create a groups-of model 0.3 x 0.02, the
large cube could be used to represent a unit.
(If a flat represents one, then .02 would have to
be represented by two cubes, and you cannot
represent tenths of a cube.) If a large cube
equals one, then two longs can represent 0.02
and 0.3 of this is six cubes or 0.006. (b) To create an area model of 0.3 x 0.02, a new key is
312
necessary: Let the length of a long or flat = 0.1
linear units and the length of a cube = 0.01 linear units. The area of a flat, then, equals 0.1 x
0.1 linear units or 0.01 sq. units and the area of
a long equals 0.001 sq. units. One side of the
rectangle would be represented by the length
of three longs (0.3 linear units); the other side,
by the length of two cubes (0.02 linear units).
Filling in the area of the rectangle would
require six longs, which according to our key,
represents 6 x 0.001 sq. units or 0.006 sq. units.
6.
small cube. There is no way to represent hundredths of a (linear) unit and, thus, no way to
show 1.02.
Key for Investigation 11.7 (page 11-20)
2.
(a) Problems A and B are both divvy-up division problems. (b) An informal divvying-up
solution with play money is described below:
Step 0: Represent the total amount $46.17.
Put out four $10 bills, six $1 bills, one dime
(tenth part of a dollar), and seven pennies
(one-hundredth parts of a dollar).
A correct groups-of model for 0.01 x .3 is illustrated below:
Step 1A: Divvy-up the $10 bills (tens) owed.
Each boy would have to pay one $10 bill, leaving one $10 bill leftover.
Step 1B: Trade in any leftover $10 bills (tens)
for ten $1 bills (ones). The additional ten $1
bills and the original six $1 bills makes a total
of sixteen $1 bills owed.
Step 2A: Divvy up the $1 bills (ones). Each
boy would have to pay five $1 bills, leaving
one leftover.
Step 2B: Trade in any leftover $1 bills (ones)
for ten dimes (tenth-parts). The additional
ten dimes make a total of 11 dimes owed.
(a) Jasper appeared to divide .01 into .3 (found
how many tenths could fit into three-tenths).
(b) Jack made the common error of incorrectly
assuming that both .01 and .3 were fractions of
the same whole. In fact, .01 is a fraction of the
.3 of a whole. (c) Dianne arrived at the correct
answer but modeled .3 x .01 (three-tenths
groups of one hundredth).
7.
Adrienne created a correct area model of
1.2 x 0.4 but incorrectly interpreted the answer
because she did not distinguish between linear
and area units. She simply treated the long as
one unit and the cube as one-tenth of a unit.
Note that the length of a long (or flat) can be
equated to 1 linear unit and the length of a
cube can be equated to 0.1 linear unit. The top
(surface area) of a long, though, represents 0.1
square unit, and surface area of a cube, 0.01
square unit. Adrienne’s error is common and
underscores the importance of first setting up
a key.
10. Without chips, an area model is not possible.
This should be evident from inspecting the
factor 1.02. If a flat is 1 square unit, then 1
(linear) unit is the length of the flat and 0.1
(linear) is one-tenth that or the length of a
Step 3A: Divvy up the dimes (tenth-parts).
Each boy would have to pay three dimes,
leaving 2 dimes leftover.
Step 3B: Trade in any leftover dimes (tenthparts) for pennies (hundredth-parts). The
additional 20 pennies make a total of 27 pennies owed.
Step 4: Divvy up the pennies (hundredthpart). Each boy would have to pay nine pennies. Thus, the answer is one $10 bill (ten),
five $1 (ones), three dimes (tenth-parts), and 9
pennies (hundredth-parts) or $15.39 (15.39).
6.
By encouraging students to reflect on what
would happen if $46.18 was divided up
among three children, they should recognize
that the process could, in principle, be repeated indefinitely. Note that this task could
also raise the issue of rounding. (b) In 139.6 ÷
3, one tenth remains undistributed. Thus, Jodi
was correct; the quotient can be represented
46.5r.1 (c) 46.5 3 .
313
Key for Investigation 11.8 (pages 11-22 to 11-
volves five places. With 1.444 ÷ 2.5, it would
be possible to represent the dividend (1.444)
and the divisor (2.5) but not the quotient
(0.5776). (Note that it would be possible to
model both of these expressions if chips—
tenths of a cube—were used.) Miss Brill could
have recognized that dividing the 4 in the
thousandths place of 1.444 would not be
evenly divisible by the 5 in 2.5 and, thus, the
quotient would have had to extend beyond the
thousandths place.
24)
Part III
1.
Measure-out or area meaning.
2.
(a) The decimal 0.9 represents the amount: 10
of something. The decimal 0.3 represents the
3
size of each group (e.g., each share is 10 of
same something in size). The quotient 3 represents the number of groups that can be ob3
tained: If each share is 10 in size, then 3 shares
9
9.
a.
9
can be made from 10 of something. (b) A tenth
could be represented by a small block, a long,
or a flat. (c) If a long = a whole, then the
amount =
size of each share =
and the
The amount can be divided into three groups
of
:
.
3.
The model for 2.1 ÷ 0.3 requires trading. One
model is shown below:
Amount =
Size of each share =
By trading
each long for
10 cubes, the
amount can be
represented
by 21 cubes,
which can be
used to make
7 groups of
.
Because concretely modeling 2.1 ÷ 0.3 entails
trading, Miss Brill was correct in beginning
with 0.9 ÷ 0.3. With this latter problem, students could focus their attention on relating a
measure-out meaning to the concrete model,
which for many children is a difficult enough
task.
7.
This item illustrates that there are limits to using base-ten blocks for modeling decimal division. With four sizes of blocks, a measure-out
interpretation of decimal division can be modeled to four places. Thus, it is not possible to
model 25.335 ÷ 2.4, because the dividend in-
It is essential that a teacher help students
understand how each aspect of a division
expression such as 2.1 ÷ 0.3 = 7 relates to
an area model. The dividend 2.1 represents the area of a rectangle; the divisor is
the length of one side; the quotient is the
length of the unknown side.
b. Unlike the measure-out model, an area
model involves two different types of
measures (area and length) and, hence,
two types of units (square units and linear
units). Students often do not explicitly
distinguish between the two types of
units, and this can lead to confusion. It is
essential that a teacher have students explicitly state what represents a square unit
and what represents a linear unit.
c.
If a flat = 1 square unit, then a long = 0.1
square unit and a cube = 0.01 square unit.
(Actually, area is represented by the surface area of the blocks, not the blocks
themselves. The blocks third dimension is
essentially ignored.)
d. If the surface of a flat represents an area of
1 square unit, then the side (length) of the
flat must be 1 linear unit, because the
square root of 1 is 1—because a 1 by 1
(linear) unit rectangle has an area of 1
square unit. If the outside length of flat
(or a long) is 1 linear unit, then the length
of cube is 0.1 of a linear unit. (Note that 1
linear unit is not represented by the flat itself but by the length of the flat. Likewise,
0.1 linear unit is not represented by the
cube itself but by its length.)
e.
The dividend represents the area 2.1
square units. Hence, 2 flats and 1 long are
needed to represent it.
314
f.
The blocks representing the area must be
arranged into a rectangle that has a length
representing 0.3 linear units; that is, has a
length equal to the length of three small
cubes:
0.3 linear
units =
length of
3 small
cubes
This can be done by trading the 2 flats for
20 longs and arranging these and the
original long in the following manner:
Area = 1.3 = 1 flat, 3 longs = 13 longs
Known side: 0.4 = length of 4 cubes
unknown side = length of 3 longs (3 linear
units + part of another long → 3+ linear units
known side
= length of
4 cubes
(leftover)
12. b. Interpolation is necessary to determine the
exact answer of 1.44 ÷ 2.5 by using an area
model.
Dividend = 1 flat, 4 longs, and 4 small cubes
known side = 2.5 units
with trading, the
unknown side can
be shown to be 0.5
linear units
Unknown side has a length =
the length of 7 longs = 7 (linear) units
11. a.
The leftover block could be used to interpolate the answer. For 1.3 ÷ 0.4, for example, the dividend 1.3 can be thought of as a
rectangular area of one and three-tenths or
13 tenths square units (one flat and three
longs or 13 longs) and the divisor 0.4, as
the known side of a rectangle four tenths
linear units (the length of four cubes). In
building the rectangle with a known side
of 0.4 linear units, 12 longs are used up
making four rows of three longs each and
one is leftover. This demonstrates the unknown side must be between 3 and 4 linear units (see figure at the top of the next
column). Note that you have only one of
four longs needed to complete another
column. In theory, the remaining long
could be divided into four equal parts and
placed to the right side of the longs
shown. As the length of a long = 1 linear
unit, 1 linear unit ÷ 4 = 0.25 linear units.
More concretely, a student can think, "I
have one of four longs needed to complete
another column. If a completed column
represents 1.0 linear unit, then one-fourth
of a column represents 14 x 1 or 0.25 linear
units." Thus, the unknown side must be
3.0 + 0.25 or 3.25 lin. units. This process of
determining a fractional part of a whole is
called the process of interpolation.
This leaves 19 cubes. To complete the next
larger rectangle, it would take another 25
cubes (= 2 longs + 5 cubes). Interpolation
(19/25 = 0.76) indicates that the unknown
side is an additional 0.76 of a 0.1 linear
unit or 0.076 linear units. In brief, it is
possible to determine the answer with an
area model, but not without interpolation.
c. Like measure-out models, area models too
have limitations. Because the large cube
cannot be used to model an area meaning,
the dividend, divisor, and quotient are
limited to three places (if chips are not
used). Thus, Example E with a four-place
dividend 2.727 could not be modeled. If a
flat = 1 square unit, then Example F with a
thousandth place in the divisor (0.012)
could not be modeled. (A flat could not be
reassigned the value of 0.1 square unit for
the reasons discussed earlier.)
Part IV
1.
Relate an unknown problem to a known situation.
2.
Multiplying both the dividend and the divisor
by 10 yields an equivalent expression.
315
Key for Activity File 11.15 (page 11-31)
N
O
T
E
S
T
8 ÷ 0.001 = 8,000
150.8 × 25.6 = 3,860.48
120 ÷ 0.05 = 2,400
153.56 × 8.3 = 1274.548
562.48 - 75.99 = 486.49
21.63 + 33.094 = 54.724
F
R
I
D
A
Y
0.5 ÷ 0.018 = 27.7 7
0.8 × 16 = 12.8
1.085 × 3.451 - 3.744335
3.2 - 2.205 = 0.995
0.25 × 0.832 = 0.208
1.683 ÷ 30.6 = 0.055
)4
)4
Part II
1.
(a) ÷ 4 (b) zorkth =