Learning Tasks with JavaBars
Leslie P. Steffe & John Olive
We suggest a collection of tasks that begin with students’ first encounter with
number and proceed on through more advanced work with fractions. The tasks are
suggested to help teachers get started using JavaBars in their teaching. Our hope is
that teachers and students will generate their own tasks and use the microworld in
ways that serve their purposes. No age or grade is suggested to be most appropriate
for any task. Many similar tasks could be posed through activities with physical
materials, such as Cuisenaire Rods, Unifix Cubes, Multibase Blocks, or Fraction Bars.
A few examples of such related activities have been included, but no attempt has
been made to explicitly relate each task to physical activities. We leave that to you,
the teacher!
Mathematical tasks alone do not determine a curriculum. From the students’
side, a task never stands uninterpreted. It is the students’ ways of solving tasks with
which teachers must be prepared to deal. We sometimes attempt to describe how
students might solve a particular task based on our experiences working with
students. But no claims are made that these brief descriptions are exhaustive, nor
even typical for students at a particular age.
Introducing JavaBars
JavaBars can be used by the teacher on one computer with a whole class to
introduce a particular topic or activity. It can also be used in this “electronic
chalkboard” mode to illustrate and promote discussion of an exploration in which
students have been engaged. Many of the tasks in this manual can be adapted to
work in this electronic chalkboard mode; they have been written, however,
assuming students have access to one or more computers. An example of using
JavaBars with a whole class of upper elementary students and one computer, to
investigate area measure, has been included at the end of the manual.
When introducing JavaBars to students, you might encourage their cognitive
play in the microworld with only minimal teacher guidance. In the introductory
sessions, you can choose which buttons and menu items you want to introduce to
the students. For example, the buttons Bar, Fill, Copy, Join and ERASE from the
top row could be used. In any event, you might allow students to make designs or
whatever else they may want to make.
There are four basic reasons for encouraging students to engage in cognitive
play in JavaBars with only minimal teacher guidance. The first is to empower
students in microworld use. Students find the dynamic features of the microworld
fascinating and a continual source of motivation. The second is to encourage
students to establish their concept of rectangular region as a dynamic concept. One
of the most important aspects of using the microworld is that bars (rectangular
regions) can be made rather than simply recognized. Another is that a bar can be
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copied. The simple act of copying a bar is not available to students when physical
materials are used. Later, when students are engaged in mathematical activity, the
copying action can provide opportunities for thinking that are difficult to achieve in
less dynamic media. The third reason is to encourage students to learn to share the
mouse. The students access the possible actions of JavaBars through the use of the
mouse and they need to learn to take turns using it. The fourth reason is to
encourage students’ expectation that they can work independently when using the
microworld. This is important when we want students to engage in independent
mathematical activity.
Tasks need to be posed by teachers to transform students’ cognitive play into
mathematical activity in JavaBars. The tasks initially posed need to relate to
students’ cognitive play. These tasks can serve as starting points for further
mathematical activity with a playful orientation. The tasks presented below are
designed to engage students in mathematical activity with a playful orientation.
This orientation must come from the students; the tasks can only support it.
Constructing Numerical Meanings and Strategies.
The following activities are designed for students who know how to count.
For example, to find how many more marbles must be placed with nine marbles to
make 15 marbles, we assume that a student can count “nine; 10, 11, 12, 13, 14, 15.
Six”. These students are ready to engage in constructing arithmetical meanings and
strategies. To begin, we advocate that students be encouraged to count, because
counting is the basic way they have of establishing meaning for number words or
combinations of number words. Our intention in formulating the tasks for the
numbers through 20 is to encourage children to use counting to establish pattern
meanings and relations among these numbers. We also intend for the children to
construct strategic reasoning.
Numbers Through 20.
Making Number Bars. A number bar is a row of unit bars joined together. A
6-bar, for example, is six unit bars joined together into a row. The goal is for
students to make number bars for the numbers through ten and to establish linear
spatial patterns for these numbers. The only buttons necessary to make number bars
are Bar, Copy, and Join (although use of the Repeat button could be used for
efficiency). The phrase “unit bar” should be used to refer to the bar from which the
number bars are made. This bar can be designated as the unit using the Set Unit Bar
button.
After some agreement on what number bars mean and how to make them,
encourage students to make their own number bars. Figure 1 contains a screen
showing the first ten number bars. The pictures alone should not be considered to
be number bars unless the students have established how many unit bars there are in
each number bar.
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When making the number bars with more than five unit bars, encourage the
students to look for combinations of the bars through five. For example, to make a 7bar, a student might start with a 4-bar and think: “four: 5 is 1; 6 is 2; 7 is 3. A 3-bar!”
and then copy the 4-bar and the 3-bar and join them together.
Figure 1. Number Bars Through Ten.
After students have made their own number bars, ask them if they could use certain
bars to make others; for example, could they use the 2-bar to make the 6-bar? Also
ask them how certain bars can be related to other bars; for example, how the 3-bar
can be related to the 9-bar.
Physical Materials for Making Number Bars. Unifix cubes can be used to make
number bars and used in ways similar to the above tasks. The teacher with only one
computer could introduce the tasks to small groups or the whole class (if a large
display device is available) using JavaBars, and then ask students to construct their
own number bars using Unifix cubes. Number bars could be made also from square
grid paper pasted on thin card. Students could be given a card grid and asked to
make their own set of number bars from one to ten by cutting strips of squares from
the card. Cuisenaire rods can be used also as number bars although the unit bar is
not discernible in each of the longer bars -- they are like number bars with the parts
wiped clean. Students working with Cuisenaire rods would have the added
challenge of assigning a numerical value to a rod based on length comparisons with
the designated unit rod.
Making Number Tiles. A number tile has at least two rows and at least two
columns of unit bars. To increase the students’ awareness of number patterns, they
might be asked to try making number tiles using each number bar in Figure 1.
Encourage them to break each number bar and try to make number tiles using the
unit bars. Figure 2 contains a screen where the number bars for 4, 6, 8, 9, and 10
were broken and joined into number tiles.
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Figure 2. Number Tiles.
Note that when one number bar is placed beneath another, they cannot be joined
unless they are equal. So, number tiles could not be made for the numbers 1, 2, 3, 5,
and 7. After making the number tiles, ask students open questions like: “Can you
find which number tiles can be made using the 3-bar? The 4-bar? The 2-bar? The 5bar?” Physical number tiles can be made from the grid cards suggested for making
number bars.
Numbers From 11 Through 20.
Making Number Bars. Using each of the number bars in Figure 1, ask students
to make their own number bars for 11. The number bars in Figure 3 were made by
copying number bars from the left column and are yet to be joined together.
Students’ could be encouraged to find--prior to copying and joining--the
number of unit bars that must be joined to each number bar in the left column to
make a number bar for 11. For example, starting with the 3-bar, a student might ask
herself which bar would be needed to make an 11-bar. The student might count: “4
is 1; 5 is 2; 6 is 3; 7 is 4; 8 is 5; 9 is 6; 10 is 7; and 11 is 8. The 8-bar”. In such cases, ask
the student to review what she has done to encourage abstracting the result “3 and 8
more is 11”. To encourage review, you might ask “3 and how many more are 11?”
after the student has made an 11-bar starting with the 3-bar.
Figure 3. Number Bars for 11.
Establishing number bars for the numbers in the second decade contributes to the
construction of numerical relations for each number. It is important for students to
review the results of their activity and abstract numerical patterns as in the next task.
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Finding Pairs of Number Bars. This task is a reverse of the task in Figure 3.
Starting with a number bar between 10 and 20, students are to find pairs of number
bars which, when joined, form the number bar. For example, using a number bar for
12, the students are to make pairs of number bars which, when joined, would form
the 12-bar. Three such pairs are displayed in Figure 4 (the pairs of number bars are
yet to be joined). Encourage students to find which number bars work, before
making them. Some students may find a systematic strategy. For example, they
might start with an 11-bar and add a unit bar, then think: “Take one from the 11-bar
and give it to the unit bar. That would make a 10-bar and a 2-bar. Then take one
from the 10-bar and give it to the 2-bar, etc.”. In these cases, encourage the students
to communicate their methods to other students.
Figure 4. Pairs of Number Bars for Twelve
Finding all pairs of numbers whose sum is 12 may not be feasible for many
students. In these cases, it is important to accept what the students do and not be
overly demanding. For some students, it may be necessary to experiment in the
microworld to produce even one pair of number bars for each number.
Number Sequence Through 100.
Our intention in the following tasks is for students to construct meaning for
the numbers through 100. In this, we do not concentrate on the construction of a
two-digit number as so many tens and so many ones. Rather, we concentrate on
students’ construction of a number sequence through 100 (1, 2, 3, ..., 98, 99, 100) and
strategic reasoning. We use adding and subtracting to achieve this goal.
Adding.
Adding and subtracting were implicit in the tasks suggested for constructing
pattern meanings and numerical relations for the numbers in the first two decades.
Adding six more to seven can be acted out in the microworld in any one of several
ways. For example, a student can make a 7-bar, a 6-bar, and then join them together.
Another way is for a student to make a 7-bar and then join, one-by-one, six unit bars
to the seven bar. By counting-on we mean that a child can find how many unit bars
are in the joined bar by counting “Seven; 8, 9, 10, 11, 12, 13. Thirteen”. The student
knows that she can count to “seven” and does not need to run through the activity.
It is symbolized by saying “seven”.
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In Figure 5, a child has joined a 6-bar to a 7-bar and is counting the unit part
of the joined bar starting from “one”. Counting the unit bars in the visible joined
bar is an act of adding, but students who can count-up-to are capable of more.
Figure 5. Counting the Unit Bars in a Joined Bar Starting with “One”.
Discussion: Children working in pairs may help each other keep track of their
counting acts as they count on from the first number. Keeping track of counting is a
way to encourage children to count their counting acts rather than simply put up
fingers or some other more rudimentary means of keeping track.
Tasks like the above can be used to encourage children to construct a number
sequence to 100.
Using Physical Materials. Similar tasks can be posed using counters or beans
and paper cups. Unifix cubes can be connected end-to-end to make a 100-bar.
Constructing a 100-bar out of Unifix cubes can be an enjoyable group activity with
lots of opportunities for estimating how long the final bar will be as it grows.
Several children could each make a 10-bar with Unifix cubes and then join these
together, keeping track of how many cubes are in the growing bar. Try other
number bars besides ten. If you have 20 students in your class each could make a 5bar and join these end to end, counting by fives as they do so.
Subtracting
There are two versions of subtraction that are considered; take-away and
difference. Take-away subtraction is usually explained by taking some items of a
collection away from the collection. The idea of a difference is usually explained by
asking how many more items are in one collection of items than in another. The idea
of a difference is the more sophisticated of the two. Students who can double count
and can make number bars for the teens as described in Figure 3 can profitably
engage in subtraction activities.
Take-Away Subtraction. Take-away subtraction can be difficult for some
students because they may not have learned to say their number word sequence
backward. For these students, start with a number bar, say a 5-bar, small enough so
that they can generate the number words just before “five”, “four”, “three”, and
“two” by dropping back to “one” and saying the number words forward. Then,
gradually increase the size of the number bars so that the students learn to say their
number word sequence backward.
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Removing unit bars from the end of a number bar can serve as an initial basis
for students’ take-away subtraction. But the goal is for this to be coordinated with
counting. For example, after a student makes a 14-bar, she might be asked to find
how many unit bars would be left if she erased six unit bars. Encourage the student
to use the Erase button after breaking the 14-bar to erase one unit bar at a time.
After erasing each unit bar, the student should find how many unit bars are left. The
student might count-down, “One and 13 left; two and 12 left; three and 11 left; four
and 10 left; five and 9 left; six and 8 left”, which is also called taking-off-from when it
refers to taking items from a collection (see Figure 6).
An alternate way of enacting the taking-off-from strategy in JavaBars is to use a
cover on the screen and to move the unit bars under the cover as they are taken off
of the broken number bar (see Figure 6). The goal is for the students to be able to
take any single-digit number from any other greater number through 99.
Figure 6: Removing unit bars from a 14-bar
The Cut action can also be used to enact taking-off-from. For example, in
taking seven unit bars from a 34-bar, a student could cut one unit bar at a time from
the 34-bar and count-down “1 is 33; 2 is 32...”. Proceeding in this way, the student
arrives at 27 left. Encourage students to curtail the cutting actions and cut seven unit
bars off from the 34-bar using only one cut. The students should then count-down
without cutting unit bars off from the number bars. This is a crucial step.
Subtraction as Difference. Subtracting as taking-off-from can easily become
laborious. Just consider the situation of taking 29 from 35. If students use countingdown, do not discourage them. The activity serves as its own discouragement. But
it is important to ask the students if they can find an easier way! You might suggest
to first make the 29-bar and then ask what must be done from that point to make the
35-bar. For those students who can count from one number up to another, it is easy
to find that six more unit bars need to be joined to the 29-bar to complete the 35-bar.
After completing the 35-bar, the students can be asked what they would need to do
to make the original 29-bar.
In Figure 7, a partially visible 29-bar with six unit bars at the end of it are
shown. Also shown is a student cutting six unit bars off from a partially visible 35bar that was formed by joining a copy of the six unit bars to a copy of the 29-bar.
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Figure 7. The Difference of 29 and 35 (the bars continue beyond the screen on the left).
In this, the student is encouraged to see the 29-bar as contained in the 35-bar and to
focus on the remainder of 29 in 35 (the unit bars cut off from the 35 bar), which is the
difference of the two numbers. To find the difference, the student might count from
35 down to 29 or from 29 up to 35.
Strategic Reasoning.
Strategic reasoning in adding and subtracting should be encouraged by using
a series of related tasks. One of the first ways to encourage strategic reasoning is to
concentrate on the additive property of numeration. The property is that a decade
number increased by a digit (a single-digit number) is named by the conjunction of
the two original number names. For example, thirty increased by eight is thirtyeight. This property can be introduced before working explicitly with tens.
Adding a Digit onto a Decade. To begin, ask students to make a 10-bar and
then ask them how many more unit bars would need to be joined to the 10-bar to
make a 17-bar. Before completing the 10-bar to make a 17-bar, the students should
find how many more unit bars are needed. The activity should be completed for
each number in the ‘teens. Figure 8 shows the work of a student completing a 17-bar
after double counting “Ten; 11 is 1; 12 is 2; 13 is 3; 14 is 4; 15 is 5; 16 is 6; 17 is 7.
Seven more.”
Figure 8. An additive property of ten.
Students might then be asked to start with a 10-bar and join a 7-bar to the 10-bar.
Both types of related tasks should be completed for other decade numbers (20, 30, ...,
90).
Students should also be asked to develop a strategy called “adding up to a
decade”. For the strategy to be effective, students should first learn what numbers
added to a digit complete 10. The strategy is to add the same number to, say, 42 to
complete 50 as was added to 2 to complete 10.
Adding Ten More. Using a number bar less than ten, say a 4-bar, ask students
to copy a 10-bar, join it onto the end of the 4-bar, and find the number of unit bars in
the joined bar. In a series of related tasks, then copy another 10-bar, join it on the
end of the 14-bar, and find the number of unit bars in the joined bar. Continue
copying, joining, and finding the number of unit bars in the joined bar until reaching
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the tenth decade. The students can check their results using the Measure button at
each step. The goal is for the students to be able to predict how many unit bars will
be in the next joined bar and to abstract the pattern 4, 14, 24, 34, ..., 94 and possibly
beyond. This task can be repeated for each number bar through ten.
Figure 9 shows a student joining a 10-bar onto a 24 bar. Note that the first 13
parts of the 24-bar are off the left side of the screen to make room for the 10-bar.
Before joining, a student might think: “25 is 1; 26 is 2; ...; 34 is 10. Thirty-four!” then
join the 10-bar onto the 24-bar, and check how many units are in the joined bar using
the Measure button.
Figure 9. Ten More than 24.
Making Two-Digit Numbers Using a 10-bar and a Unit Bar. Abstracting a unit of ten is
involved in constructing an additive property of numeration and adding ten more.
To make a unit of ten explicit, a series of tasks can be generated which ask students
to find how many numbers are in a decade. Students may be asked to find how
many numbers there are from 10 through 20. Students could make a 10-bar and then
find what number bar would be needed to make a 20-bar. They then could find
what number bar would be needed to make a 30-bar, and so on.
In Figure 10, students were asked to make 57 using a 10-bar and a unit bar.
Figure 10. Making 57 Using a 10-bar and a Unit Bar.
A student counted “10, 20, 30, 40, 50 and 7 more is 57” while copying the10-bar and
the unit bar.
Number Tiles for the Decades. Students may be asked to make number tiles for each of
the numbers 20, 30, 40, 50, 60, 70, 80, 90, 100. Number tiles are shown in Figure 11
for 20, 30, 40, 50, and 100. When making the number tiles using COPY and JOIN,
students should find how many 10-bars are needed prior to making each tile. For
example, to find how many 10-bars are needed to make a number tile for 20, a
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student may think: “Ten: 11 is 1; 12 is 2; 13 is 3; ... ; 20 is 10. One more ten, so two
tens.”
Figure 11. Number Tiles for the Decades
Objective: To find how many 10-Bars can be made from a two-digit number.
Situation: Start with 34 unit bars, preferably covered.
Student Directions: Find how many 10-bars could be made using the 34 unit
bars under the cover.
Discussion: For students who need to actually make the 10-bars, let them
proceed by making a 10-bar and copying it. However, encourage students to find
out before actually making the 10-bars how many could be made. In any event, they
should not be discouraged from actually making 10-bars if for no other reason than
verification. For example, when trying to find the number of 10-bars that could be
made from 34 unit bars, one could easily imagine the student thinking as follows:
“Twenty is two tens: 21 is 1; 22 is 2; 23 is 3; 24 is 4; 25 is 5; ...; 30 is 10. Three 10bars.” and then go on, copying another 10-bar and double counting to 40 before
realizing that one more 10-bar would be too many. Repeat the above situation with
different numbers of unit bars under the cover.
Adding Two-Digit Numbers.
Using strategies for adding two-digit numbers should precede work with
computational algorithms. Encourage students to make number bars for two-digit
numbers by joining copies of 10-bars and unit bars end to end.
Objective: Counting on by tens.
Situation: Finding the sum of 24 and 32.
Student Directions:
1. Make a 24-bar using copies of a 10-bar and a unit bar.
2. Add 32 to the 24-bar by joining copies of the 10-bar and the unit bar to the
24-bar.
3. Keep track of how many unit bars are in your new bar.
4. Check your result using the Measure button in the Menu bar.
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Discussion: The student knows that three 10-bars and two unit bars must be
joined to the 24-bar. The student might be thinking as follows when copying and
joining: “24 and 10 more is 34; 34 and 10 more is 44; 44 and 10 more is 54; 55, 56.
Fifty-six ”. The student’s strategy is called counting-on by tens and ones, and should
be a fundamental way children find sums for two 2-digit numbers. Other strategies
are possible and we encourage students to use their own strategies.
The two digit numbers could also be made using number tiles for the tens and
unit-bars for the units. In this case the three ten-bars could be joined to the bottom
of a 2x10 number tile for 20 and two unit bars could be joined to the 4-bar. Students
who are developing a sense of place value for tens and units would probably use
this strategy (see Figure 12).
Figure 12: Joining 3 ten-bars to the 20-tile
Strategies for subtracting. When taking seven away from, say, 34, encourage
students to use the inversion of the additive property of decades. In this, students
would need to decompose seven into four and three, take 4 away from 34, arriving at
30, and then take 3 away from 30.
To find the difference of 35 and 29, counting from 29 up to 35 or from 35
down to 29 is much more efficient than taking 29 off from 35. However, these
methods have their limitations. Consider the difference of 72 and 46. Although the
methods work, none of them are efficient. Using them will serve as their own
discouragement as well as a context for searching for more efficient strategies.
As before, encourage students to ask how many unit bars need to be added to
46 unit bars to make 72 unit bars. Also encourage the children to consider using a
10-bar as well as a unit bar in their work. Asking the question “How many 10-bars
and how many unit bars need to be added to the 46 unit bars to make 72 unit bars?”
can lead to counting-up-to by tens and ones. Figure 13 shows a student copying 10bars onto a 4x10 number tile and a 6-bar while counting “forty-six: one more ten is
56, two more tens is 66, (three more tens is 76 -- that’s too many); so two 10-bars.
Then copying unit bars onto the 6-bar: 67 is 1; 68 is 2; 69 is 3; 70 is 4; 71 is 5; 72 is 6.
So, two 10-bars and six unit bars. That makes ten, twenty -- twenty-six!” Note that
the 46 was made using the 10-bar to make a 4x10 number tile and the unit bar to
make a 6-bar.
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Students may use more efficient strategies that are modifications of countingup-to by tens and ones, such as curtailing double counting to find the number of unit
bars. Students might know that it takes four more unit bars to complete another 10bar to make a 70-tile and then two more unit bars to make 72.
Figure 13. Finding the Difference of 72 and 46 by Counting-up-to by Tens and Ones.
Constructing Composite Units.
For a majority of students, reasoning with units of ten is the first major shift
from reasoning with units of one to reasoning with composite units. The
construction of composite units does not occur all at once and needs to be facilitated
in the context of multiplying and dividing. In fact, this is the most basic reason for
teaching multiplication and division.
Multiplying.
One meaning of multiplying a number by another is to iterate the first the
number of times indicated by the second. For example, to find how many pieces of
pizza when cutting each of four pizza’s into five pieces, iterate five four times.
Students who can double count can learn to multiply in this way.
Multiples of Two, Three, Four, and Five. Using a number bar, say a 4-bar, ask
students what number tiles can be made using that bar. Figure 14 shows a student
experimenting, finding what number tiles can be made.
Figure 14. Making Number Tiles Using a 4-Bar.
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The student is copying the 4-bar beneath the number tile being made. The student is
thinking: “4 and 4 more is eight; 4 more is 9, 10, 11, 12; 4 more is 13, 14, 15, 16. So,
four 4-bars in a 16-tile.” In this activity, it is not enough for the students to find the
number of unit bars in the tile at each step. They should also keep track of how many
4-bars they have used as well as how many unit bars are in the tile at each step. The
activity in Figure 15 should be done using the 2-bar, the 3-bar, the 4-bar, and the 5bar, making tiles with up to ten number bars.
Finding How Many Number Bars in a Number Tile. This task is the reverse of the
task in Figure 14. Starting with a partially covered 24-tile, ask the students how
many 4-bars are needed to make the 24-tile. Figure 15 shows a student making 4bars. The student found how many unit bars are in the 24-tile, using the Measure
menu. The student is thinking as he makes the 4-bars: “Four. 2 fours is 8; 3 fours is
9, 10, 11,12; 4 fours is 13, 14, 15, 16; 5 fours is 17, 18, 19, 20; 6 fours is 21, 22, 23, 24.
Six 4-bars!”
Figure 15. Finding How Many Four Bars are Needed to Make a 24-tile.
It is important to encourage the students to review what they found at each step; 2
fours is 8, 3 fours is 12, 4 fours is 16; 5 fours is 20; 6 fours is 24. Remembering these
facts is best done in the context of meaningful mathematical activity.
Follow the task in Figure 15 with a covered 20-tile and ask the students how
many 4-bars would be needed to make the 20-tile. This is a good way for the
students to not only remember their multiplication facts, but to use them in the
solution of other tasks and to regenerate them if necessary. By using their
multiplication facts, the students will have a reason to want to remember them.
Encourage the students to form their own goals and proceed independently.
If the students have the goal to find and to use the number facts for four and have
built up an appropriate way to act, this will empower the students and will
encourage them to take responsibility for learning the number facts on their own. In
the process, they construct meanings for multiplying on which they can build.
Multiples of Six through Nine.
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Multiples of six, seven, eight, and nine are conceptually no more difficult than
multiples of numbers less than six. Rather than make number tiles, ask students to
make number stacks as shown in Figure 16. The three-dimensional picture formed
by the four 7-bars is called a number stack.
Objective: To construct multiples of six through nine.
Student Directions:
1. Make a 7-bar from a unit bar.
2. Make copies of the 7-bar, stacking them on top of one another as in Figure
16.
3. Keep track of the total number of units in your stack. Try to use as many
as ten 7-bars in your stack.
4. Repeat this exercise with 6-bars, 8-bars and 9-bars.
Example: In Figure 16 the student is repeatedly copying the 7-bar while keeping
track of how many copies made and how many unit bars in the copies. The student
is thinking: “Seven. 2 sevens is 14; 3 sevens is--15, 16, 17, 18, 19, 20, 21; 21; 4 sevens
is--22, 23, 24, 25, 26, 27, 28. Twenty-eight. So, 2 sevens is 14; 3 sevens is 21, and 4
sevens is 28”.
Figure 16. Making a Number Stack Using a 7-bar.
Discussion: It would be very helpful if the student had constructed strategies for
adding. For example, if the student knew the additive property of decades and
number facts for ten, to find 3 sevens, he might add six onto 14 and then one more
onto 20 rather than count by ones. To add seven onto 21, the student might add
seven onto 20 and then add one. Other students might reason that seven plus seven
is 14; and 14 plus 14 is 28.
Using additive strategies makes learning the multiplication facts much more
efficient and enjoyable. Making number stacks is in itself pleasurable, and coupled
with finding how many unit bars in a particular stack, the student’s motivation to do
mathematics comes from within the student. In this, it is particularly important that
students are in control of their own learning and can independently work together.
Extension: The reverse of the task in Figure 16 could be presented by asking
one student to make a number tile using so many 7-bars and then hide all but the
first row of the tile under a cover, and for the other student to find how many 7-bars
are covered. The students could find how many unit bars are in the number tile
using the Measure menu. Figure 17 shows such a task. One student has just made a
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number tile using six 7-bars and hidden most of it under the cover. The second
student has used the Measure menu to find how many unit bars are in the number
tile. The task is to find how many 7-bars were used to make the number tile without
uncovering the 7-bars.
Figure 17. Finding How Many 7-Bars are Covered.
Squares. In multiplying, the squares (2x2, 3x3, ...) can serve as anchor points
for finding products just as the doubles do in addition. A task that can be used to
introduce squares is shown in Figure 18. The task is to partition a large, square bar
into seven horizontal parts using Left/Right, PARTS, break the partitioned bar into
seven horizontal bars using BREAK, partition each sub-bar into seven parts
vertically, and then find how many of these small parts would be in the large, square
bar if the original seven sub-bars were joined back together. A student has just
partitioned three of the seven sub-bars into seven vertical parts each. Students
should learn the squares 2x2; 3x3; ...; 10x10 as a result of finding them through
activities such as shown in Figure 18. [Note: The bar doesn’t have to be square but
should have approximately the same dimensions vertically and horizontally.]
Figure 18. Making seven times seven using PARTS.
Making Products Using PARTS.
Tasks like the one shown in Figure 18 encourage using or regenerating
multiplication facts.
Objective: To construct a product by projecting sub-parts into a partitioned bar.
15
Situation: Two students work together using horizontal and vertical partitions
of a bar to produce products.
Students’ Directions:
1. One student makes a bar and partitions it into eight parts horizontally.
2. The second student pulls out one of the parts from this bar (using
PULLOUT) and partitions it into four parts vertically.
3. The first student covers the original 8-part bar.
4. How many small parts would there be in the original bar if each of the
parts in the 8-part bar were partitioned into four parts like the one pulled
out?
5. What would the covered bar look like if you were to put four parts in each
of the eight parts?
Discussion: As a result of the above task, a student found a new method for
making products. To make 9 multiplied by 8, the student used horizontal PARTS to
partition a whole bar into nine parts. She then used vertical PARTS to partition the
whole bar into eight parts (see Figure 20). Her work curtailed the whole process of
making a 9-bar and repeating it eight times to make a number tile for the product.
This curtailment indicated that she might be ready to engage in strategic reasoning
in multiplication.
Figure 20: Making the product of 9x8 using horizontal and vertical parts
Constructing Strategic Reasoning. Like in addition, the methods students use to find
products for two one-digit numbers might be used to find products involving a twodigit number. For example, to multiply 13 by 8, students could make a 13-bar and
count the unit bars as they copy eight 13-bars and join them together. However, the
reason for asking students to find the product of 13 and 8 is not simply to find the
number of unit bars in eight 13-bars joined together. If that were the only reason,
students should use the Measure menu to find the result!
Objective: To encourage students to reason about products. This is not an easy goal
to accomplish, and it involves asking different kinds of questions.
Student Directions:
1. Make a number tile for 13 multiplied by 8.
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2. Cut the number tile using vertical CUTS and give a product for the
number of parts in each sub-tile.
3. Join the two tiles together and give a product for the joined tile.
4. Repeat the exercise with different cuts.
Example: (See Figure 21.) Using a unit bar, a student made a 13-bar starting
with a unit bar using REPEAT (to the left) and then made a number tile for the
product also using REPEAT (downwards). (The student could have started with a
large bar and used PARTS to partition the bar into 13 parts vertically and 8 parts
horizontally.)
Figure 21. Partitioning the product of 8 and 13 using CUT
The student counted by fours using the 4-bars in the left number tile and said
“thirty-two”. After being asked how many fours would make that answer, the
student said “eight”. Together with the teacher, the student said that the number tile
was 4 multiplied by 8. When asked what the product was for the right number tile,
the student said 9 multiplied by 8. Note that the students were not asked to find
how many unit bars in any number tile because reasoning with products was being
stressed.
Objective: To make two number tiles that could be joined together to make the
number tile for a two-digit product.
Student Directions:
1. Make two number tiles that when joined together will represent 13
multiplied by nine.
2. Find other possible pairs of number tiles for 13 multiplied by nine.
Example: (See Figure 22.) One student chose a 7-bar and a 6-bar. He then used
REPEAT to make a number tile for 7 multiplied by 9. He is in the process of making
a number tile for 6 multiplied by 9. The student joined the two number tiles together
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and was then asked if any other pairs of number tiles could be used to solve the
problem.
Figure 22. Making a number tile for 13 multiplied by 9 using a 7-bar and a 6-bar.
Discussion: A common response to this sort of task is for students to make a bar
or tile for each number in the product and then say something like “I need to put an
X between them.” What they are trying to model is the symbol structure of
multiplication encountered in their textbooks. We need to emphasize that what is to
be modeled are partial products (two tiles that when joined together will have the
total number of unit bars in the product) rather than the two numbers in the
product. Referring students back to the prior task (illustrated in Figure 21) may help
focus their attention on partial products.
Using Strategic Reasoning to Find Products. For those students who can make a
number tile for a product by making two number tiles and joining them together, it
might be possible to use this strategy to find the number of unit bars in a particular
product in an easy way. To find how many cents 8 kids have if each one has 15
cents, students could make a number tile using a 10-bar and a 5-bar. A student’s
work is shown in Figure 23. She made a number tile with eight 10-bars. She then
proceeded to make a 5-bar. She is shown using REPEAT to make a number tile with
eight 5-bars. She knew that 10 eight times is 80 and 5 eight times is 40. After joining
the two number tiles, she then counted by tens four times starting with 80.
Figure 23. Finding 15 Multiplied by 8 Using Strategic Reasoning.
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Reasoning strategically for products involving two-digit numbers 20 or
greater is more complex, but not beyond students who can reason as in Figure 23.
For example, to find 24 multiplied by 8, students can proceed in several ways. They
might make two 10-bars and a 4-bar and then use REPEAT to make three number
tiles, two for 10 multiplied by 8 and one for 4 multiplied by 8 as shown in Figure 24.
Students could also make a 20-bar and a 4-bar and then make number tiles for 20
multiplied by 8 and 4 multiplied by 8.
Figure 24. Finding 24 multiplied by 8.
Some students might see 24 as a double and make two 12-bars and then make
two number tiles for 12 multiplied by 8. These student-initiated strategies should
not be discouraged.
Reasoning with Tens and Hundreds. If the size of the numbers (factors) of a product
are increased, strategic reasoning becomes even more important. For example, to
find 72 multiplied by 8, it is advantageous for the students to have already met 70
multiplied by 8 as well as other such products involving decade factors. To find 70
multiplied by 8, prematurely teaching students to find 7 multiplied by 8 and then to
add a zero to this product discourages strategic reasoning. Students should
experience finding the product as a genuine problem. But, most students need some
preparation.
Objective: To count by tens beyond 100.
Student Directions:
1. Create a 100-tile using a 10-bar.
2. Create tiles for 20, 30, 40 and 50 using the 10-bar.
3. Join each of these tiles to a 100-tile and find how many unit bars in each of
the joined tiles.
Example: Figure 25 shows a number tile being formed by joining the 20-tile to
the 100-tile. The student who joined them counted “100. Ten more is 101, 102, 103,
..., 110 (putting up ten fingers, one for each number word said); and ten more is 111,
112, 113, ..., 120 (again putting up ten fingers). 120!” Abstracting an additive
property for the centuries is critical for the student’s progress in reasoning with tens
and hundreds. Toward that end, ask the student to review what he has done--100
and 10 more is 110; 100 and 20 more is 120; and so on.
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Figure 25. Joining a 20-Tile to a 100-Tile.
Extensions: Transforming so many 10-bars into number tiles encourages
students to establish relations between tens and hundreds. In Figure 26, there were
17 10-bars in a stack. The student is finding how many 100-tiles could be made
using seventeen 10-bars and what number tile could be made from the remaining 10bars. The student has made a 100-tile and has found that there were seven 10-bars
remaining. The student knows that seven 10-bars make a 70-tile and is in the
process of making the 70-tile. Joining these two number tiles together, the student
made a 170-tile and knew how many unit bars were in this number tile.
Figure 26. Making 100-Tiles from 10-Bars.
Ask students to complete tasks like the one in Figure 26 for 10-bars up to 100. In
this, the students should make number stacks like in Figure 27, which was made
using fifty 10-bars.
Discussion: Transforming fifty 10-bars into five 100-tiles is a crucial mental
operation. A student might learn to reason: “Ten 10-bars make a 100-tile, so because
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five tens is fifty, fifty 10-bars make five 100-tiles, which makes 100, 200, 300, 400, 500;
500 unit bars”.
Figure 27. Five 100-Tiles in a Number Stack.
When students can reason in this way, they are ready to find products like 70
multiplied by 8. A basic meaning of 70 is to make seven 10-bars. So, by multiplying
the seven 10-bars by 8 yields fifty-six 10-bars. The students then can find how many
100-tiles can be made from these fifty-six 10-bars and how many 10-bars would
remain. In Figure 28, a student has made a stack of fifty-six ten bars and is in the
process of finding how many 100-tiles could be made using the stack of 10-bars. The
student has made one 100-tile and is copying it, counting the 10-bars “10, 20, 30, 40,
50”.
Figure 28. Multiplying 70 by 8.
Upon reaching five 100-tiles, the student then knew there would be six 10-bars left.
So, the student reasoned that five 100-tiles made 500 unit bars and six 10-bars made
60 unit bars, so 70 multiplied by 8 is 560.
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Finding Divisors.
Long division is not discussed because the topic should be met by students
later in their mathematics education. Finding divisors of a number is preferred.
Make a 4-bar and designate it as a unit bar. Then, ask the students to make
multiples of the unit bar as in Figure 29.
Figure 29. Making Multiples of a Unit 4-Bar.
When making the multiples of the unit bar, encourage the students to find
intermediate products as they are making them. In the figure, a student is thinking;
“2 fours is 8; 3 fours is 12; 4 fours is 16; 5 fours is 20”.
After completing the multiples tiles, the students can verify the totals for each
tile by re-setting the single unit-bar as the unit bar and using the Measure menu.
Reset the 4-bar as the unit bar. The divisor is the number of parts in the unit bar (4)
and the dividend is the number of unit bars in the tile -- this can be verified using the
Measure menu button. Encourage the students to designate other number bars as
the unit bar and to make multiples of these partitioned unit bars up to a unit 10-bar.
The goal of the next set of tasks is for students to find partitioned unit bars
that could be used to make specified number bars or number tiles. In Figure 30, a
student is trying to find a partitioned unit bar that could be used to make a 27-bar.
The student has pulled a 4-bar from the 27-bar and is starting to repeat it to see
whether it will work. She then tried a 3-bar and counted by three before repeating.
Afterwards, the teacher asked her to designate the 3-bar as a unit bar and wipe the
bar. The student made a 9-bar using the 3-bar and found that it was the same size as
the 27-bar. The teacher encouraged the students to try to find other unit bars that
would work.
Figure 30. Finding a Partitioned Unit-Bar to make a 27-Bar.
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Partitioning and Sharing
There are two processes for creating composite units. One is to start with an
individual unit like a unit bar and then make so many unit bars and join them
together. Another is to start with a unit that can be broken into pieces or parts and
then create a composite unit using the pieces or parts. Both of these processes are
crucial in students’ mathematical education and each should be emphasized. In fact,
many of the tasks involving composite units above could begin with breaking a
whole bar rather than making multiples of a small bar.
Sharing a Bar Equally.
Objective: To share a bar equally among several mats without using PARTS.
Situation: Sharing a bar between two mats.
Use only the Pieces buttons in JavaBars for this activity.
Student Directions:
1. Create two mats in the bottom part of the screen. Pretend that the mats are
plates or friends.
2. Create one bar in the top part of the screen. Find ways to share the one bar
equally (or fairly) between the two mats.
Discussion: Students may use visual estimation to make a vertical cut in the bar
using the CUT button and then directly compare the two parts to find if they are
equal. If they are not equal, encourage the students to join the bars back together
and make further estimates until they find approximately equal parts.
Rather than cutting the bar to find equal shares, the students can use the
Up/Down Pieces action to mark a piece with a vertical segment and then pull out
one piece to compare it with the other piece. If the two pieces are not the same size,
the student can clear the vertical mark (using the Clear button) and make a new
mark, using the pulled out piece as a guide (see Figure 31).
Figure 31: Pulling out a piece when sharing a bar between 2 mats
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Situation: Sharing a bar equally among more than two mats.
Start with three mats in the bottom part of the screen and one bar.
Student Directions:
1. Share the bar among the three mats so that each mat gets a fair share.
[When sharing a bar into three equal pieces, students usually use direct
comparison as in the prior activity. To encourage students to make a visual transfer
of a piece, ask them to cut off (or mark and pullout) a share for only one of the three
mats, where each mat is to get an equal amount.]
2. Make the share for just one of the three mats.
3. Find a way to check if your one piece is a fair share.
Discussion: Students may use various ways to check if the one piece is a fair
share. Figure 32 illustrates one possible method. The student first made a visual
estimate for one share and cut the bar. She then copied the cut piece three times
beneath the bar to find if the three pieces together made a bar equal to the original
bar, but it was too short (the blue bars).
Figure 32. Attempting to Cut One of Three Equal Parts From a Bar.
She then joined the original bar back together and made another estimate, this time
cutting the bar just beyond the piece which she had first made (purple bar). Her
second estimate was much closer and was satisfactory. Other students made only
two copies of the piece they cut off beneath the remaining part while other students
dragged the piece cut off across the remaining part to find if two of them fit exactly.
All of these methods indicate that the students could construct meaning for a
fractional part of a whole bar. It is at this point that the term “one-third” can be used
meaningfully to refer to the one share.
Marking and Filling Unit-Fractional Parts of Bars.
Objective: The goal is for students to make estimates of fractions without using
PARTS.
After developing the appropriate fraction language through sharing bars
equally among mats (or friends), ask students to use the Up/Down Pieces action to
mark a fractional part and then fill this part with a different color in order to color
fractional parts of a bar. Use horizontal (Left/Right) marks as well as vertical marks
24
(together with Fill) to color different fractional parts. Encourage students to check
their accuracy by pulling out the filled piece and then copying it along the bar to find
if three filled pieces create a bar exactly the same as the original bar (Figure 33).
Figure 33. Filling an estimate for 1/3 of a Bar and Checking Using Pullout and Copy.
Reasoning with Fraction Bars
The process of breaking bars and then joining the parts or pieces back
together into composite units is the process underlying fractional reasoning. If the
original bar was broken into equal-sized parts we call the result of the process a
fraction bar. Fraction bars are distinguished from number bars by the process used
to make them. Rather than start with a unit bar and then make copies of the unit bar
to join together, students start with a unit bar and then find a part of the unit bar so
that multiple copies of this part joined together equal the unit bar.
Fractional Meanings.
The goal in these activities is to help students develop meaning for fraction
words.
Objective: To generate a set of Fraction Bars.
Only use the Pieces actions in JavaBars initially.
Student Directions:
1. Create a long thin unit bar across the top of the screen.
2. Make a set of fraction bars: that is, a set of bars equal to the unit bar that
will show “halves,” “thirds,” “fourths,” etc. [Encourage students to use their
strategies for making equal shares to create each of the fraction bars.]
Example: A student making the fraction bars in Figure 34 has made a halves-bar,
a thirds-bar, and a fourths-bar and is trying to make a fifths-bar. He used Up/Down
Pieces to make estimates and then filled his estimate with a different color. He then
used Pullout and Copy to place copies of his estimate end-to-end under the unit bar
to test his estimate. The estimate for a third was too long, so the student made
another estimate using his first estimate as a guide. To test his estimate for a fifth of
the unit bar, the student used Repeat rather than Copy. His estimate was a little
short.
25
Figure 34. Making Fraction Bars.
Objective: To generate fraction language.
Fraction bars can be used to help students generate fraction language.
Student Directions:
1. Use PULLOUT to make copies of different parts of the fifths-bar. Name
the fraction each part is of the unit bar.
2. Make a bar twice as long as one part of the fifths-bar. What fractional part
of the unit bar is this?
Example: A student used PULLOUT to make one-fifth of the fifths-bar. After
being asked to make a bar twice as long as one-fifth, she said that it was two onefifths of the unit bar without previously hearing the term “two-fifths”. Using this
method, she then generated the terms “three one-fifths”, “four one-fifths”, and “five
one-fifths” on her own. These terms reflect an iterative meaning of fractions. “Threefifths” means one-fifth three times as well as three out of five equal parts.
Students can be asked relational questions like “How many eighths does it
take to make a 1/4-bar?” “How many ninths are a bit longer than 7/8?” and “How
many sevenths are a bit longer than 5/6?”
Common Fractions.
Students who can make fraction bars and generate fraction language are
ready to use PARTS as an aid in making equal parts of bars. Introducing PARTS
too quickly can inhibit the construction of unit transfer and the iterative meaning of
fractions.
Objective: To develop facility with making fractional parts of a bar.
For these activities, use the Parts actions in JavaBars.
Student Directions:
26
1. Make a unit bar and copy it.
2. Break the copy into seven equal parts and join five of those parts together.
3. How many sevenths have been joined together?
[The language “five-sevenths” can be introduced at this point if the students
do not generate it themselves.]
4. Join all seven-sevenths of the bar together.
5. Pull 3/7 of the bar from the bar using PULLOUT.
[Figure 35 illustrates the result of pulling three parts from an interior part of
the bar. This should be encouraged because always pulling the first three parts may
lead to a restricted concept of fraction.]
6. Pull out 1/7, 2/7, and so on up to 7/7.
7. Make other common fractions like 4/5 and 3/4 from clean copies of the
unit bar.
Figure 35. Pulling 3/7 from a bar.
Finding the Complement of a Fraction
Objective: To find the complement of a part in a whole.
Student Directions:
1. Make 13/27 of a bar.
2. Cover the remaining parts so they cannot be seen.
3. What fraction of the bar is covered?
Discussion: A student may count from 13 up to 27 and say that 14 parts are left,
but may not be able to say what fractional part is left. In this case uncover the
remaining parts and ask the student to join them together (or pull them out if they
are still joined).
Ask students to make up problems like this for each other.
Joining Fractional Parts of Fraction Bars
Objective: To find the fraction formed when joining two parts of a fraction bar.
Student Directions:
1. Make two fractional parts of the sixths-bar (a unit bar partitioned into six
equal parts).
2. Join the two parts together, and find out what fraction of the unit bar is
formed by the joined bar.
27
Discussion: Extending to fractions greater than one.
These tasks offer the possibility for students to naturally encounter fractions
greater than one. For example, one student made 5/6 and 4/6, and joined them
together. The student was stymied, because the resulting bar was not a part of the
unit bar. Up to this point, the student had formed a strong concept of a fraction as a
part out of a whole, which is one of two meanings of a common fraction. The other
meaning is the number of times a unit fraction is repeated to make the fraction (in
5/6, 1/6 is repeated five times to make 5/6).
The student was asked to take a unit fraction from the six-part unit bar using
PULLOUT, and repeat it until making a bar the same length as the joined bar. The
student’s work is shown in Figure 36. After making the repetitions, she knew that
the bar made was nine one-sixths and so was the joined bar. But, it still wasn’t a
fractional part of the unit bar. So, she was asked how many unit bars was in nine
one-sixths and how many sixths (parts) were left over. She said that one whole bar
was in the nine one-sixths bar and that three-sixths was left over. So, nine one-sixths
now meant one whole bar and three more sixths.
Figure 36. Repeating 1/6 to make a bar the same length as 5/6 joined to 4/6.
From similar situations arising from the students’ activity with fraction bars,
students should be encouraged to develop the concept of, say, nine one-sixths as
nine-sixths (9/6), and it should mean one whole fraction bar and a part of the
fraction bar as well as nine one-sixths.
Finding Missing Parts of Whole Bars.
Objective: To find what fractional part of a whole bar must be joined with another
fractional part to make the whole bar.
Student Directions:
1. Create a small bar and partition it into two parts.
2. This is 2/7 of a unit bar. You need to make the missing part of the bar.
3. Join the missing part to the 2/7 to make the whole bar and make it the Unit
Bar using Set Unit Bar.
4. Pull out your original 2/7-bar, and Measure it to check if you have the
correct Unit Bar.
5. Can you find a different way to make the Unit Bar from the 2/7-bar?
28
Ask students to pose their own “missing parts” problems.
Extension: If students are successful with the above tasks when presented with a
partitioned part of a bar, present them with an unpartitioned bar as representing
(say) 2/5 of a bar. This is considerably more difficult!
Sharing More than One Bar
Objective: To develop strategies for sharing multiple bars among different numbers
of mats.
Student Directions:
1. Make a bar and then copy it.
2. Make three mats across the bottom of the screen.
3. Share the two bars equally among the three mats.
4. What fractional part of one bar is on the middle mat.
5. Remake the two bars and then fill the share that was on the middle mat
using only one bar (as shown in Figure 38).
Examples: Figure 37 shows one possible solution. Each bar has been partitioned
into three pieces using vertical PARTS and then broken using BREAK. The figure
shows the second bar about to be broken. One part of the second bar is then
dragged to each mat.
Figure 37. Sharing Two Bars Equally Among Three Mats.
Another possible solution is to break each bar into two equal parts, give one
half to each mat, and then break the fourth half into three equal parts and share
these among the mats.
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Figure 38. Filling the Share for the Middle Mat.
Discussion: Similar partitioning activities can be presented to students involving
two bars and four mats, three bars and five mats, or any other suitable selection of a
number of bars and a number of mats. If more bars than mats are selected, it should
be for students who find the sharing tasks with fewer bars than mats easy. Some
students may join the bars together prior to sharing. In these cases, it may prove to
be difficult for students to find what part of one bar is on each mat. One way to
alleviate this difficulty is to keep a reference copy of the original unit bar in another
part of the screen.
Marking Fractional Parts of Bars.
Objective: To Mark estimates of fractions and verify them using PARTS.
Student Directions:
1. Create a unit bar and use copies of this for the following activities.
2. Mark 1/2 of a bar using Pieces. Use PARTS to check your estimate.
3. Mark 1/3, and 2/3; use PARTS to check.
4. Mark 1/4, 2/4, and 3/4; use PARTS to check.
5. Mark 1/5, 2/5, 3/5, and 4/5; use PARTS to check.
Figure 39 shows one student’s attempt at marking 4/5 of a bar, filling his estimate
and checking with the Parts action. His estimate is just a little short of 4/5.
Figure 39: Estimating 4/5 and Checking using Parts
Comparing Fractional Parts of Bars.
When students become reasonably proficient at making estimates of fractional
parts of bars, ask them to compare different fractional parts.
Objective: To develop an intuitive sense of relative size of fractional parts of a bar.
Student Directions:
1. Fill 1/2 of a bar and 1/3 of a copy of the bar and then compare 1/2 and
1/3. Which fraction takes up more of the bar?
2. Choose pairs of unit fractions like 1/3 and 1/4, 1/4 and 1/5, 1/5 and 1/6,
1/6 and 1/7, and so on and find out which of each pair is the greater
fraction (more of the bar).
3. Which is greater, 1/13 or 1/14 and why? [1/13 is greater because each of
14 parts is smaller than each of 13 parts.]
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Extensions: Select pairs of non-unit common fractions that encourage student
reasoning for comparison. Select pairs like 2/3 and 2/4, 3/4 and 3/5, 2/7 and 3/7,
3/4 and 4/5, 6/7 and 7/8, and 3/4 and 5/6. In cases like 3/4 and 4/5, students can
compare the complements of the fractions (1/4 and 1/5) and on that basis, compare
the fractions.
Making Fractional Wholes from Fractional Parts
Objective: To use a given part of a bar in iteration to reconstitute the unit bar.
Student Directions:
1. Think of a bar and then make a bar that is 1/3 of your imagined bar.
2. Now make your whole bar from this 1/3 of a bar.
Repeat the task for other unit fractions.
Extensions: Presenting the task for non-unit common fractions should also be
tried for those students who find the task for unit-fractions easy. For example, ask
the students to start with a bar that is 3/5 of their imagined bar and then make the
whole bar. In Figure 40 the student partitioned her 3/5-bar into 3 equal parts and
then pulled one of those parts out of the 3/5-bar, repeated it to make a 5/5-bar as the
unit bar.
Figure 40: Making a unit bar from a 3/5-bar.
Establishing the Inverse Relation for Unit Fractions.
In the above fraction tasks, the unit fraction was established as a part of a
whole. The inverse relation of the unit fraction one-third to the whole bar is to find a
bar, three of which would be as much as the whole bar. Another way to express this
relation is to say that the whole bar will be as much as three of the bars that you are
trying to find. This inverse reasoning is quite difficult for students to establish, but
is necessary for students to construct a strong unit fraction concept.
Objective: To establish a unit bar as a multiple of its unit fractions.
Situation 1: Making Multiples of a Unit Bar.
Student Directions:
1. Make a bar and designate it as a unit bar using Set Unit Bar.
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2. Make a bar that is as much as five times the unit bar.
3. What fraction is the unit bar of this new bar? (see Figure 41)
4. Make multiples of the unit bar through the teens.
(Note: these multiples of a unit bar are the same as number bars – see earlier work
with whole numbers.)
Figure 41. Making a Bar as much as Five Times a Unit Bar.
Situation 2: Making Partitions of a Unit Bar.
Student Directions:
Starting with a unit bar, make a bar so that the unit bar would be as much as
five times that bar. (This is the inverse of situation 1.)
Discussion: Figure 42 illustrates one student’s way of solving this problem. She
made an estimate on the unit bar using Left/Right Pieces, pulled this out of the bar
(using PULLOUT) and then repeated it five times to find if five times the estimate
was the same size as the unit bar. Another student used Parts rather than Pieces to
find the part to pull out of the unit bar. Both students understood that the bar they
were trying to make was one-fifth of the unit bar and that the unit bar was as much
as five times the bar they were trying to make. These students had constructed a
concept of unit fraction. These unit fraction tasks could be related back to the
sharing tasks, where the goal was to find the share of the bar that one out of five
people would get.
Figure 42. Making a Bar so that a Unit Bar is as much as Five Times that Bar.
32
Fractions of Composite Units
Making Fractional Parts of Composite Bars.
Objective: To find fractions of quantities greater than one.
Student Directions:
1. Make a unit bar and then a bar twelve times as large as the unit bar.
2. PULLOUT 1/2, 1/3, 1/4, and 1/6 of the 12-bar.
[Figure 43 illustrates pulling four parts to make 1/3 of the 12-bar.]
3. Look at the bar that you pulled out that is 1/3 of the 12-bar. What fraction
of this bar is the unit bar?
4. What fraction of the 12-bar is the unit bar?
5. How does this relate to finding 1/4 of 1/3 of the 12-bar?
Figure 43. Finding Fractional Parts of a 12-bar.
Extension: Extend the task in Figure 43 to other fractions of the 12-bar. Ask
students to make their own composite bars using the unit bar and to pose their own
questions.
Fractions Involving Composite Units.
Objectives: To combine students’ iterative and part whole meaning of fractions and
to begin their explorations of multiplying a fraction times a whole number.
Student Directions:
1. Make a set of number bars from a unit bar through a 10-bar.
2. Make a bar six times larger than the 4-bar. [see Figure 44]
3. What fraction is the 4-bar of the 24-bar. [one-sixth]
4. Find 1/6 of 24 and name another fraction equal to 1/6.
5. What other number bar could be repeated to make the 24-bar?
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Figure 44. Making a 24-Bar Using a 4-Bar.
Example: A student found that the 3-bar could be repeated eight times to make a
24-bar by counting by threes up to 24. She was then asked what fraction the 3-bar
was of the 24-bar, and she said 1/8 and 3/24. She also knew that 1/8 of 24 is 3. She
went on, exploring other possibilities.
Extensions: The students could also make number tiles using the number bars
similar to the activities for multiplication and factoring. Thus a 24-tile could be
made by repeating the 4-bar vertically six times, producing a 6x4 number tile. The
same questions as above could be asked of such number tiles.
Using the number bars, ask students to find 3/4 of 20; 2/3 of 18; and so on.
Multiplicative Operations on Fractions
Sharing Shares of Bars
Objective: The tasks in this section are designed to bring the students’
multiplicative knowledge of whole numbers into the context of fractions.
1. Present tasks like the following: A bar is shared equally among seven girls. Each
girl shares her share equally with a boy. What fractional part of the bar does each
boy have?
2. Ask the students to enact the task. One possible result is shown in Figure 45.
Figure 45. Sharing Shares of a Bar.
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Finding a Fraction of a Fraction
Objective: To introduce multiplication of fractions as a composition of fractioning
acts.
After successfully completing several tasks like the one described in Figure
45, pose the following task as shown in Figure 46:
Figure 46. Making 1/2 of 1/3 of a Bar.
Student Directions:
1. Make and fill 1/2 of 1/3 of a bar.
2. What fractional part of the bar is filled?
3. What fractional part of the bar is unfilled?
Extensions: Similar tasks could be presented for other unit fractions, such as
making 1/3 of 1/5 of a bar or 1/6 of 1/7 of a bar. If students find these tasks easy,
ask them to make 1/4 of 2/5 of a bar. For example, in Figure 47, 1/4 has just been
cut from 2/5 of a bar. What fraction of the whole bar has been cut off?
Figure 47. Finding 1/4 of 2/5.
Paper Folding and Recursive Fractions
Objective: To extend the results of paper folding to recursive breaking actions in
JavaBars.
Student Directions for Paper Folding:
1. Fold a regular sheet of paper in half and then in half again.
2. What fraction of the whole sheet is the folded paper?
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3. Fold the folded paper once more in half. What fraction results from this
fold?
4. Figure out the result of the next fold before attempting it.
Discussion: Students may need to unfold the paper and count the number of
parts formed by their folds to verify their fraction result. Have them fill in one of the
parts in the unfolded paper and label the filled part with a fraction name. For
students who have no problem with folding in halves, suggest folding into three
equal parts. Ask them to fold the result again into three parts and name the fraction
formed. (This may challenge even your best students!)
JavaBars: Do not use the Pieces actions for these tasks.
Student Directions:
1. Create a bar that fills almost half of the screen vertically (it should look like
a regular sheet of paper). Set this as the UNITBAR and then make a copy of it in the
other half of the screen. The unit bar may be moved partially off the screen, as it is
only needed for measurement reference. All operations will be done on the copy.
2. Break the bar in half and fill one half a different color.
3. Continue the process on the filled half only, each time naming the fraction
of the whole unit bar that is to be filled in a new color.
[Students should take turns in halving the next filled part and try to anticipate
the fraction that will be formed by their action. They should end up with a figure
similar to Figure 49.]
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Figure 49: Recursive Halving
4. [Referring to Figure 49] How many light blue parts would it take to fill a
red part? How many 16ths would it take to make a half of the bar?
5. [For advanced students] What fraction of the half bar is the 1/16 bar?
Ask similar questions using different intermediate fractions.
Discussion: Students will want to carry the process as far as possible until they
cannot see the parts they are making. They could use the computer calculator to
calculate the next fraction when they get into numbers that are hard to double in
their heads. The MEASURE button can be used to verify the fraction measure of
even very tiny parts. Ask students how this activity relates to their paper folding.
The more advanced students could be challenged to figure out how many folds it
would take to get to their smallest fraction.
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The above activity can be repeated using recursive acts of “thirding,”
“fourthing” or “fifthing” rather than halving.
Combining Halving and Thirding. This activity is best done in pairs.
1. Designate one student as the “halver” and one as the “thirder.” The “halver”
will always break the newly filled part in half, the “thirder” will always break
the newly filled part into thirds (see Figure 50).
2. Start with the “halver” and each takes a turn, working on the result of the
other person’s turn. After each turn both students are to name the result as a
fraction of the unit bar.
3. After two turns each, have the students anticipate the fraction name of the part
resulting from their next turn.
4. Ask questions comparing intermediate results as in the halving task.
Discussion: Use other pairs of fractions such as thirds and fourths. Students
may wish to record their results at each stage in order to keep track of what is going
on. Provide paper and pencil for this and encourage the use of the calculator. This
is not a test of mental arithmetic! Students may create their own recording and
representation systems during this task. These should be encouraged and valued.
Ask students to explain any unusual representations or symbol systems. The
following symbol system was created by a fifth grade student to record the process
of “thirding” and “fourthing” shown in Figure 50:
1 3 1/3 4 1/12 3 1/36 4 1/144 . . .
She was able to use her recording system to anticipate the results of several more
turns.
More advanced students could be challenged to figure out if the same or
different fractions would be formed if the other person started first, or if the result of
taking four turns each would change if they took their four turns all at once!
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Figure 50: Recursive Thirding and Fourthing
Making Different Fractional Parts of the Same Unit Bar
Objective: To be able to generate a fraction of a unit whole from some portion of the
whole.
Problem: Share a bar among three people. One person is to get 1/2 of the
bar, another 1/3 of the bar, and the last person is to get what is left. How much does
the last person get?
Discussion: Unlike the previous situations where the task is to take a fraction of
a fraction (1/3 of 1/2 for instance), this problem asks to share a bar using different
fractions of the whole bar. Many students, however, will attempt to solve the task in
a similar way to the previous situations; they will give half of the bar to the first
person and then give one third of the remaining half to the second person, leaving 2/6
(2/3 of 1/2) for the third person (see Figure 51).
Figure 51. Making 1/2 and 1/3 of a Bar.
It is very difficult for some students to understand that they are to make 1/3
of the whole bar rather than 1/3 of one-half of the bar. Sometimes the difficulty is a
practical one--there is no whole bar left to take 1/3 of. In these cases, it may help to
suggest to the student to make a copy of the bar. It may also help to suggest to
partition the bar in such a way that they can take out both one half and one third of
the bar. Figure 52 illustrates one possible solution using 6 parts in the bar.
39
Figure 54. Making 1/2 and 1/3 of a Bar Using PARTS and PULLOUT.
40
Fraction Families (Equivalent Fractions)
Making Fraction Families
The members of a fraction family are commonly referred to as “equivalent
fractions”. We say that one fraction is equivalent to another if each can be
transformed into the other. The following examples illustrate possible transforming
actions.
Objective: To make fraction families by repartitioning the “root” fraction.
Situation 1: Fraction family for 1/3.
Student Directions:
1. Make a bar, partition it into thirds, and pull out one of the thirds.
2. Partition the 1/3-bar in different ways and determine another fraction
name for this partitioned 1/3-bar.
Example: Students may choose to partition the 1/3-bar into two parts. They
would then need to find into how many parts the whole bar would be partitioned if
each third was partitioned in the same way. Some students may partition the 1/3bar in the same direction as the unit bar (say vertically) others may choose to
partition the 1/3-bar in the opposite direction. The latter method may eventually
lead to more powerful strategies but it may be easier for some students to see the
1/3 as 2/6 if the partitions are in the same direction, as shown in Figure 53.
Figure 53. Transforming 1/3 into 2/6.
Discussion: The students may want to make several copies of the 1/3-bar in
order to keep a visual record of each fraction they generate. For example, two
students working together made four copies of the 1/3-bar and systematically
partitioned them into 2, 3, 4, and 5 parts. Proceeding in this way, the students
generated the fraction family {1/3, 2/6, 3/9, 4/12, 5/15, and so on}. The students
41
saw that the number of parts of the partition of the 1/3-bar was the numerator of the
fraction and three times that number was the denominator (the number of parts
needed in the unit bar to generate the fraction).
Situation 2: Ask students to make fraction families for the unit fractions through
1/10, using up to 10 partitions in the unit fraction.
Situation 3: Ask students to make fraction families for common fractions like
2/5 and to learn to write the families. Students who find making fraction families
easy, can be asked to use numbers in the teens and the twenties to partition the
fraction bar. You might also ask them to make fraction families for fractions greater
than one.
Objective: To make fraction families by repartitioning the unit bar.
Another approach to generating fraction families is to focus on the partitions
of the unit bar rather than on the partitions of the unit fraction pulled out of the unit
bar.
Situation 1: Generate the fraction family for thirds.
Student Directions:
1. Create a unit bar and pull out one third from the bar. [They will most
likely partition the unit bar into three parts and pull out one of the three
parts.]
2. Partition a copy of the unit bar using a different number of parts but you
must still be able to pull out one third of the unit bar.
3. Find as many different partitions as you can of the unit bar from which
you can still pull out one third of the unit bar. [They should be
encouraged to organize their solutions as shown in Figure 54.]
Figure 54. Fraction Family for One Third
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4. Continue this pattern verbally or on paper, naming each fraction
equivalent to the one third of the unit bar.
5. How many fractions do you think could be in the one third family?
Situation 2: The Four-Piece Pizza Parlor.
Student Directions:
1. Create a unit bar across the top of the screen and partition it into four parts.
This represents a four-piece pizza.
2. Copy the partitioned unit bar and break the copy into its four parts. These
are the four slices in each pizza. This Pizza Parlor only serves pizza cut
into four slices like the one on the screen (see Figure 55).
3. The first task is to decide how many people can easily share one whole
pizza fairly. Obviously, four people can easily share the pizza and have
one slice each, two people could have two slices each. But what if more
than four people want to share one pizza?
4. Determine the number of people that can easily share one pizza and then
make the share of one person from only one of the four slices.
5. How much of the whole pizza does each person get?
6. How many people can share one slice of pizza (remember, there are four
slices in a whole pizza).
7. What is another fraction name for one slice of pizza (other than 1/4)?
8. How many people can share two (or three) slices of pizza?
9. What are other fraction names for 2/4 or 3/4?
Figure 55: The Four-Piece Pizza Parlor
Extension: A follow-up question to the activities for making fraction families is
to ask, for example, into how many parts students would need to partition a onethird bar so that each part was, say, 1/120 of the unit bar. Or in the Four-Piece Pizza
Parlor, students could be asked how many parts would one slice be cut into if 120
people were to share the whole pizza!
Finding Families for Fractions
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Objective: Every fraction belongs to a fraction family. The goal of these tasks is,
given a fraction not in simplest form, find the family to which it belongs. That is, the
goal is to find a larger unit fraction that can be used to transform the given fraction.
Finding a fraction family for a given fraction is difficult for most students, but
it is essential for progress. Start with a fraction like 3/9 or 2/10. These fractions
make finding unit fraction bars easier, but they do not make constructing the
method any easier.
Situation 1: Start with fractions that simplify to unit fractions.
Student Directions:
1. Make a 9-part bar and pull out three of the ninths.
2. What fraction of the bar has been pulled out? [Students will probably
answer 3/9. Ask for a simpler name for that amount of the bar.]
3. Clear the 3/9-bar (using Clear in the Parts section) and use this cleared bar
to make a unit bar. [They can repeat it three times.]
4. Pull out six of the ninths from the unit bar. What fraction is this of the unit
bar?
5. Pose similar tasks for 2/10, 4/10 and 5/10 of a bar.
Situation 2: For those students who have difficulties with the tasks in Situation
1, start with a partitioned unit fraction bar.
Student Directions:
1. Make a small bar and partition it into three parts. Call this 3/12 of a bar.
2. Make the whole bar and then make 9/12 of the whole bar as in Figure 56.
Figure 56. Making 9/12 using a Partitioned Fraction Bar.
3. What would 9/12 look like if the 3/12-bar was cleared? Make a bar the
same size as the 9/12 by actually clearing the 3/12-bar and making the
whole bar again with the unpartitioned fraction bar.
4. Pull out the bar that is the same as the 9/12-bar.
Situation 3: Using horizontal (Left/Right) and vertical (Up/Down) PARTS and
COMBINE to generate Fraction Families.
Students who are successful in finding fraction families for fractions, should
practice their newly found methods. Practicing could include using horizontal and
vertical PARTS to make fraction families.
Student Directions:
44
1.
2.
3.
4.
5.
Make a bar partitioned into five parts vertically.
Fill two of the parts and name the fraction of the bar that is filled.
Partition the bar horizontally into two parts as in Figure 57.
How many parts are filled? Give a new fraction name for the filled parts.
Use the Combine action to erase the horizontal segments (assuming the
last action on the bar was to make a horizontal partition). The two filled
vertical parts, representing the 2/5, will have to be refilled after the
vertical parts have been combined.
6. Partition the whole bar into three horizontal parts to generate 6/15 as the
filled part, use Combine to erase this horizontal partition, and proceed on
in this way to generate the fraction family {2/5, 4/10, 6/15, 8/20, and so
on}.
Figure 57. Using Combine in Generating a Fraction Family.
Discussion: It is not easy to use horizontal and vertical partitions to make
fractions like 18/27. Students usually make such fractions by partitioning a bar into
27 parts either horizontally or vertically and then filling or pulling 18 parts. So,
when practicing finding the families for given fractions, if a student partitions a bar
into nine horizontal parts and three vertical parts to make the fraction 1/27,
encourage the student to communicate their method with the other students. This
way sometimes makes finding a larger unit fraction easier.
Figure 58. Finding a Fraction Family for 18/27 using Horizontal and Vertical Parts.
In Figure 58, a student has partitioned a bar into nine horizontal and three
vertical parts and has filled 18 parts of the bar. The student then pulled out the filled
parts, and by inspection saw that one column could be used as a unit fraction bar. In
this way, the student established 2/3 as the simplest fraction of the family and went
45
on to generate the fraction family using Combine or Clear and horizontal parts on a
2/3 bar as in Figure 59. After a certain amount of practice, if the students haven’t
noticed that they are dividing the numerator and the denominator by the same
number, ask them how what they are doing relates to divisors.
Figure 59. Using 2/3 and Horizontal Parts to Generate a Fraction Family.
The basic fraction families for which the students should learn patterns are as
follows:
1/2, 2/4, 3/6, 4/8, 5/10, 6/12, 7/14, 8/16, 9/18, and 10/20.
1/3, 2/6, 3/9, 4/12, 5/15, 6/18, 7/21, 8/24, 9/27, and 10/30.
2/3, 4/6, 6/9, 8/12, 10/15, 12/18, 14/21, 16/24, 18/27, and 20/30.
1/4, 2/8, 3/12, 4/16, 5/20, 6/24, 7/28, 8/32, 9/36, and 10/40.
3/4, 6/8, 9/12, 12/16, 15/20, and 30/40.
1/5, 2/10, 3/15, 4/20, 5/25, 6/30, and 10/50
2/5, 4/10, 6/15, 8/20, 10/25, 12/30, and 20/50
3/5, 6/10, 9/15, 12/20, and 30/50
4/5, 8/10, 12/15, 16/20, and 40/50
Comparing Fractions
Students should construct an operational method for comparing two fractions
not in simplest form. One method is to find the simplest fractions of each of the
fraction families to which the two fractions belong and compare these fractions.
Students may use visual comparisons before constructing such a method. To
compare fractional parts of a whole, students should begin with the same fractional
whole; this is critical when using visual comparisons as a starting point.
Objective: To use knowledge of fraction families together with visual comparisons
to quantitatively compare fractions.
Student Directions:
1. Make a unit bar and make two copies of the bar.
2. Make 10/12 of one copy and 12/18 of the other copy of the unit bar.
3. Which is greater, 10/12 or 12/18?
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4. By how much (of the unit bar) is the larger fraction greater than the smaller
fraction?
Discussion: In this case, students can compare the parts visually (as in Figure 60)
and conclude that 10/12 is greater than 12/18. Using JavaBars, this method of
comparisons is convenient and allows all students to make reasonable comparisons.
Figure 60. Comparing 10/12 and 12/18 using copies of the same unit bar.
To answer the question, “By how much greater is 10/12 than 12/18?”
students may simply drag the 12 parts over the 10 parts and use CUT to make the
difference. If students do this, they could be asked to find the fractional part of the
unit bar formed by the difference. Finding such a fraction is not easy. The following
steps could be suggested to students:
1. Find the simplest form of the two fractions.
2. Find a way of partitioning the unit bar so that both fractions can be pulled
from the one partitioned copy of the unit bar. Use your knowledge of fraction
families to help find the number of parts.
Example: 10/12 and 12/18 simplify to 5/6 and 2/3. So the problem reduces
to finding the difference between 5/6 and 2/3. The task is to make one partition of a
unit bar so that both thirds and sixths can be pulled from the unit bar. If students
have made a fraction family for 1/3 and know that 1/3 and 2/6 are in the same
family, they can partition the bar into six parts and pull two parts for 1/3 and one
part for 1/6. They can repeat the 1/3 to make 2/3 (or 4/6), and repeat the 1/6 five
times to make 5/6. The students may then realize that 5/6 is 1/6 larger than 2/3
and that 2/3 + 1/6 = 5/6.
Students need to practice comparing fractions over a rather extended period
of time using various examples in order for strategies of comparison to be
adequately learned.
Adding Fractions.
Adding fractions is based on the method of comparing fractions.
Objective: To add fractions with unlike denominators using knowledge of fraction
families.
Problem: Find the sum of 1/3 and 2/5.
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Student Directions:
1. Make a unit bar and then two copies of the bar.
2. Make 1/3 of one copy and 2/5 of the other copy and join them together.
3. What fractional part of the unit bar is formed by the joined bar?
Example solution: In Figure 61 a student has joined the two fraction bars
together. She tried to repeat the joined bar to make a bar the same size as the unit
bar, but was unsuccessful.
Figure 61. Joining 1/3 and 2/5 of a Bar.
Having worked with 1/3 and 2/5 when comparing fractions, she tried to find
a common partition of the unit bar that she could use to make 1/3 and 2/5. Using
horizontal parts, she partitioned the thirds fraction bar into 5 parts and the fifths
fraction bar into 3 parts to make 15 parts in each. She then broke the joined bar,
remade the 2/5 and the 1/3, and partitioned each fraction as shown in Figure 62.
Figure 62. Partitioning into Fifteenths.
She then reasoned that she was adding 5/15 and 6/15 because each part was a
fifteenth even though they did not look alike. She then knew the sum was 11/15.
Being able to argue that the fifteenths in each bar were the same fraction of the unit
bar, even though they had different shapes, illustrates powerful mathematical
reasoning! And that is surely what we want all our students to achieve.
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Investigating Area Measure: A Whole-Class Activity
This whole-class activity is presented as an example of using JavaBars to
extend an investigation that begins with students’ physical explorations. JavaBars is
used as an electronic chalkboard by the teacher in order to help students make
abstractions from their physical explorations that could lead to some understanding
of area measure. The teacher is able to carry out actions with JavaBars that would be
impossible or very difficult and time consuming with physical materials. The
activity also provides an excellent opportunity to apply the fraction knowledge that
students have been constructing through the tasks in this manual.
Covering Surfaces
Students could work in groups to find the area measure of surfaces in the
classroom using covering actions. The teacher might divide the class into small
groups of two or three students and assign each group to a surface (e.g. a work table,
a desk top, a floor rug). Each group should choose a set of congruent objects with
which to cover the surface (e.g. text books, sheets of paper, square tiles). Initially,
students should have enough unit objects to cover their surface. This task can
become purely a counting task for students. To add more challenge and to
encourage multiplicative strategies, the teacher could arrange the situation so that
not enough copies of an object are available to completely cover the surface. The
students need to work with this constraint and find ways to calculate the number of
unit objects that would be needed to completely cover their surface. Finding the
area of the whole classroom floor in terms of a unit floor tile (if it is a tiled floor), or
in terms of a textbook, or a cardboard one-foot square presents its own constraints
that can lead to interesting strategies.
Points to discuss:
1. The covering object (unit) needs to tile the surface. That is, copies should
fit together without overlapping and without leaving gaps, like tiles in a bathroom
floor. Thus, a circular object would not be an appropriate covering unit.
2. What to do with units that go beyond the edge of the surface. Encourage
students to estimate the fractional part of the covering unit that should be counted in
calculating the area measure. It is in this part of the task that students will have an
opportunity to apply their fraction knowledge and strategies for operating with
fractions.
In order to promote discussion among the students, the groups should record
their results and report them to the whole class. In the case of groups that have
worked with the same sized surface (e.g. tables or desk tops), encourage discussion
of differing results. Were the covering units the same? If not, what conclusions can
the class draw concerning relative sizes of covering units and the measures of the
surfaces?
Using JavaBars to Cover Mats.
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In order to use JavaBars as an electronic chalkboard and conduct this followup activity with the whole class, a large display device with one computer is
required. If a large display device (such as a large monitor or overhead projector
panel or Smart Board) is not available, we suggest conducting these follow-up
activities with groups of five or six students. We have found that a small computer
screen is not adequate to conduct these activities with the whole class.
The following steps are offered as a possible way to engage the students in
thinking about and making abstractions of their covering actions in the prior
explorations. The objective is to encourage strategies for refining estimates of the area
of a mat in terms of a unit bar. The possible actions in JavaBars provide ways for
doing this that are not easily available with physical materials.
1. Create a fairly large mat on the computer screen using the MAT button in
JavaBars.
2. Make a small bar approximately one centimeter square and designate this the
unit bar using the Set Unit Bar button.
3. Ask the students to estimate how many of these unit bars it will take to
completely cover the mat. They should write their estimates down on paper.
Ask for several students to share their estimates.
4. If students are familiar with the actions of JavaBars ask them to suggest ways
in which you could cover the mat with the unit bar. Some may suggest
making several copies of the unit bar and lining them up on the mat. Others
may suggest using COPY and REPEAT to create a row of unit bars. Use an
appropriate suggestion to make one row that fits across the top of the mat.
Ask students to make a second estimate for the area of the mat based on the
number of units that fit across the top of the mat. Hopefully, the unit bars
will not fit exactly across the top of the mat so that some fractional part of a
unit bar will be hanging off the mat. Alert students to this problem (see
Figure 63).
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Figure 63. A row of unit bars on a mat.
5. Ask students how they might use this first row of unit bars to completely
cover the mat. One suggestion might be to COPY or REPEAT this row down
the mat. Follow the students’ suggestions and completely cover the mat. Ask
them to write down a third estimate, reminding them not to count the
fractional parts of unit bars that are not on the mat.
6. Ask students to share and justify their estimates with the class. Different
estimates should be anticipated.
7. Ask for actions that you could carry out on the covering to refine students’
estimates. One way would be to mark the edges of the mat on the bar that
now covers the mat using vertical and horizontal marks using the Pieces
Actions (see Figure 64), fill the excess pieces of the covering bar, and pull out
the filled pieces using PULLOUT - FILL, from the Pieces actions (see Figure
65).
8. Ask students to suggest ways to find out how much unit bars had been pulled
out, and then use this information to refine their estimate of the area of the
mat in terms of unit bars. One way would be to mark the pieces using
IMAGES of the unit bar, break the pieces apart on these marks and
reassemble them to estimate the amount of unit bars in each piece.
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Figure 64. Marking the edges of the mat on the covering bar.
Figure 65. Pulling the filled excess out of the covering bar.
Ask students to find a way that they can estimate the amount of these extra
pieces in terms of a unit bar (the Measure function will not work on pieces of a bar).
The goal is not to get THE exact answer, but to encourage strategies for getting a
reasonable result. For instance, the width of the vertical extra piece is approximately
1/3 of a unit bar and its length is 7 units long, so it is approximately the same as 2
and 1/3 unit bars. The thin horizontal extra piece is approximately 1/5 of a unit bar
in height and 8 and 2/3 unit bars long. Thus, it is approximately the same as 1 and
3/5 unit bars plus 2/3 of 1/5 of a unit bar (or 2/15 of a unit bar). The covering bar
consists of 7 rows of 9 unit bars, and so the problem is reduced to finding the
amount of unit bars remaining when 3 unit bars, 1/3 of a bar, 3/5 of a bar and 2/15
of a bar are removed from 63 unit bars.
52