June 10, 2008
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Facilitator Notes
Professional Development: K-8 Mathematics Standards
Grades 3-5: Fractions
1. OSPI is pleased to provide materials to use in teacher professional development sessions about the K-8 Mathematics Standards that were
approved by the State Board of Education on April 28, 2008. These materials provide a structure for two full days focused on helping
Grade 3-5 teachers understand some of the critical content embedded in these Standards. We hope that these materials will be
used by local schools and school districts, education service districts, and university teacher educators to help inservice and preservice
teachers deepen their personal understanding of key mathematics ideas. Feedback about the effectiveness of the materials and ways to
improve them can be sent to OSPI so improvements can be made.
2. The goal of these professional development sessions is to help participants deepen their personal understanding of mathematics embedded
in the K-8 Mathematics Standards. With deeper understanding, teachers will be better able to (a) understand students’ mathematics
thinking, (b) ask targeted clarifying and probing questions, and (c) choose or modify mathematics tasks in order to help increase student
learning.
3. The problem sets are presented to participants on separate “activity pages.” Sometimes you will want participants to work on all the
problems in a set, and then you will lead a debriefing of each of the problems. Sometimes you will want participants to work on
problems sequentially so that you can debrief each problem in order. The choice will depend on the particular problem set, your
preferences about facilitating discussions, and the needs of the particular group of participants you are working with.
4. Some of the problems may be appropriate for students to complete, but other problems are intended ONLY as work for the participants as
adult learners. After participants have solved problems, you might want to discuss which ones would be appropriate for students. There
are also reflections after each problem set that can help participants begin to think about how knowledge of the underlying mathematics
ideas can help them plan more effective instruction for students.
5. Encourage participants to discuss their thinking with partners. This will help participants develop fluent language about the relevant
mathematics ideas. Sometimes you may choose to ask participants to work independently before talking with partners, but sometimes
you may choose to ask participants to work immediately with partners. Always allow sufficient time for participants to work on a
problem set (or on individual problems) before you begin the debriefing. Participants are more likely to contribute to the discussion if
they are confident about their answers and about their solution strategies. This will also model that it is important to give students ample
time to work on a problem before discussing answers to that problem.
6. Although there is no explicit attention to instructional practice in these content professional development sessions, discussing implications
for teaching will help deepen participants’ own understanding. You are encouraged to tailor those discussions to the needs of each
group of participants. For example, if participants are all using a common set of curriculum materials, you may want to lead some
June 10, 2008
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discussions related to those materials. Be careful, however, not to lose the emphasis on deepening participants’ knowledge of
mathematics.
7. Approximate times are given for each problem set, but you will need to create an agenda that responds to the specific parameters of how
you are working with participants. For example, you might schedule these sessions on two consecutive days or you might schedule
them across four half-days. Extra time will be needed for the Reflection at the end of the Problem Sets. There may be too many
problems for participants to complete comfortably in a two-day session, so you need to think carefully about which problems you ask
participants to solve in each Problem Set. That is, you may choose to work only some problems within particular problem sets.
Alternately, you may choose to omit some Problem Sets completely.
Logistics
These professional development materials were designed with the following assumptions about logistics for the meetings.
1. Participants will primarily be classroom teachers of mathematics from Grades 3-5. (There are different sets of problems for teachers
from Grades K-2 and 6-8.) Modifications may need to be made if there are significant numbers of ELL teachers or special education
teachers.
2. Participants should be seated at tables of 3-6 people each. Discussions among participants are strongly encouraged.
3. The problem sets need to be copied prior to the start of the sessions. You will also need a computer and projector for display of the
slides, chart paper and markers, and enough space for small groups of participants to work comfortably. A document camera might
also be useful, and graph paper (or grid paper) might be helpful to support thinking.
The materials were developed by a team of Washington educators:
Kathryn Absten, ESD 114
George Bright, OSPI
Jewel Brumley, Yakima School District
Boo Drury, OSPI
Andrea English, Arlington School District
Karrin Lewis, OSPI
Rosalyn O’Donnell, Ellensburg School District
David Thielk, Central Kitsap School District
Numerous other people from Washington and from across the nation, provided comments about various drafts of these materials. We
greatly appreciate all of their help.
Publication date: June 10, 2008
June 10, 2008
Flow of Activities
Introductions
page 3
Slides
Notes
It is important that participants feel comfortable about participating in professional
development sessions that address their own personal mathematical understanding.
Some participants may be a bit nervous about the prospect of possibly making
mathematical mistakes in front of colleagues. Assure participants that everyone
makes mistakes occasionally, and in these sessions there are no consequences of
doing so.
Bookkeeping activities (e.g., sign in sheets) can be done now.
Fractions is the focus of the problems, since that is an area that is often difficult for
students to learn and teachers to teach.
Be sure to change the slide so that the names of the facilitators are correct.
Introduce the facilitators.
Have participants introduce themselves. If the group is small enough, this can be
done with the whole group, but if the group is large, have participants introduce
themselves to the people “close by.”
Survey the group if you wish to know the approximate distribution of teachers by
grade or by school or by school district.
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Remind participants that since we are working on personal understanding of the
mathematics underlying the K-8 Mathematics Standards, some of the problems are
appropriate for adults but may not be appropriate for students. Teachers need to
know more mathematics, and at a deeper level, than students.
For any of the problems, you may want to take time to ask participants to
identify the Core Content area (or particular Performance Expectations) that
a problem might exemplify. Since participants should have already completed
two days of professional development on the K-8 Mathematics Standards, this
kind of activity would reinforce their understanding of the Standards.
Problem Set 1
about 75 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal is to show that appropriate representations of fractions depend on the
context of a problem.
As a group, participants will likely make different representations for the
problems. As you debrief the problems help participants make connections among
the various representations.
Having participants draw representations on chart paper would allow you to post
the representations around the room.
Comment that each of these problems deals with the fraction, 3/4, but the
situations are different so the representations may, or may not, be the same.
Keeping track of how to represent “1” for a situation is critical for
understanding fractions.
As you debrief the answers to these problems, you may want to correlate the
different models to specific Performance Expectations for Grades 3, 4, and 5.
In particular, in Grade 5 teachers may need more examples of ways that
models of fractions can help students understand the central ideas as well as
addition and subtraction of fractions.
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Answers to these problems are provided in Lamon (1999a) pp. 32-36
Many participants will draw a picture of a clock and show 45 minutes out of an
hour.
Related Core Content: 3.3
The most common representation is to draw 3 circles (for the cookies) and shade
1/4 of each cookie (for the parts eaten).
Related Core Content: 3.3
One representation is
BBB
BBB
GGGG
GGGG
BBB
GGGG
BBB
GGGG
Related Core Content: 6.3 (ratios)
The key here is to cut the package (NOT the individual cupcakes) into fourths. So
a picture would show a rectangle with three circles inside (the cupcakes), with the
rectangle cut into fourths (typically one cut horizontal and one cut vertical).
Related Core Content: 3.3
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The typical representation is to show 4 pieces of gum with 3 of them shaded (for
the ones chewed).
Related Core Content: 3.3
The unit is the number of men (12); 3/4 of 12 is 9, but those 9 need to be a
different representation, since the 9 people are women and the original 12 people
are men.
Alternately, the 12 men might be formed into 4 sets of 3 men each, and then 3 sets
of 3 women are added to the picture.
Related Core Content: 5.2, 6.3
The simplest representation is a picture of 3 quarters, but a representation that
relates the amount of money to a dollar would be to show a 10x10 grid with 75
units shaded (for $0.75) or a dollar bill with 3/4 of it shaded.
Related Core Content: 3.3, 4.4
A representation would be to break the number line from 0 to 1 into fourths and to
put a “heavy dot” on 3/4.
Related Core Content: 3.3
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One representation is to should 3 packages of four candies each, and to shade one
candy from each package. Three packages need to be shown, however, to fit the
context.
Related Core Content: 3.3
One representation is to draw a 10x10 grid and shade 75 of the 100 small parts. A
second representation is to draw a number line from 0 to 1, mark fourths, and label
this as 0%, 25%, 50%, 75%, and 100%.
Related Core Content: 6.3 (ratios)
One representation is to draw 12 quarters going in a machine and 9 tokens coming
out of the machine.
Another representation is:
QQQQ
QQQQ
TTT
TTT
Related Core Content: 6.3 (ratios)
QQQQ
TTT
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A typical representation is to draw 12 blue socks and 4 black socks to show the
relationship of the 12 blue socks to the total of 16 socks.
Related Core Content: 4.4
After all twelve have been discussed, you may use some of these questions for
general debriefing.
• Why are the representations different?
• Which contexts have similar representations? What is it about the contexts that
support similar representations?
• Which two representations are most different? What is it about the contexts that
seem to “require” different representations?
It takes students a long time and lots of experience to be comfortable
understanding and using different representations of fractions. This helps explain
why attention to fractions begins even in primary grades and extends through
middle grades.
Problem Set 2
about 45 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal is to reinforce the role of “unit” or “whole” in thinking about fractions.
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For part a, some participants may shade 5/6 of each of 4 rectangles (representing
the 4 acres). If so, ask how many acres are planted in corn. Other participants
may compute 5/6 x 4 = 20/6, which is 3 2/6 or 3 1/3 and shade 3 1/3 acres. If so,
ask why multiplication is the correct operation for this problem.
For part b, some participants may think about 2/3 of 3 cakes (or 2 cakes) and then
2/3 of the 4th cake. This kind of reasoning does not work in part a, since there are
fewer than 6 acres.
For part c, some participants may “cut” each cupcake in half so that they have 4
pieces (matching the 4 in the denominator of 7/4). Then they can draw 7 halves of
cupcakes (i.e., 7 pieces the size of 1/4 of the set of 2 cupcakes), which is 3 1/2
cupcakes.
Related Core Content: 3.3
Problems 2.2 and 2.3 are to be solved by participants as adults. When you debrief
answers, try to reveal several different strategies for solving the problem. Make
explicit the assumptions that participants typically make in solving the problem;
for example, most people assume that the line in the “middle” of the rectangle
divides the rectangle into two squares. This assumes (a) that the line is
perpendicular and connects the mid-points of the top and bottom and (b) that the
two resulting figures are squares. Indeed, the number of assumptions about the left
half of the figure is greater than the number of assumptions about the right half of
the figure, since the “main diagonal” in the right part automatically divides the
area in half; no assumption about the placement of that line is needed.
Related Core Content: 3.3, 4.2
The sum of the fractions should be 1, since all the pieces together make up the
whole. This is almost a “self-correcting” question, since we know that the sum of
the fractions for all the pieces that make up a whole has to be 1.
Related Core Content: 5.2
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The unit changes after each child takes (or eats) some of the cake. Changing the
order of the fractions in this problem would change the problem significantly.
Answers: Abby 1/6; Ben 1/5 of 5/6 = 1/6; Charlie 1/4 of 4/6 = 1/6; Julie 1/3 of 3/6
= 1/6; Marvin and Sam (each) 1/2 of 2/6 = 1/6. The shape of the cake is not
relevant.
For each person, the “size” of the cake changed. Each person had to think
about her/his fraction relative to a different cake; that is, relative to a
different representation of “1”.
Related Core Content: 3.3
Problem 3: Answer: more dogs, since the number of cats is only 7/8 of the number
of dogs. The “unit” is “number of dogs.” It does not matter what that number is,
however.
Related Core Content: 3.3
Many people subdivide the interior of the circle (by making what looks like a
“peace symbol”) without explicitly recognizing that this also subdivides the
circumference into three congruent arcs. You can ask participants why they
subdivided the interior when the problem as about walking around the perimeter
(circumference) of the circle, but it is often difficult for them to explain why.
Answers: There are two answers, depending on whether Ralph walks clockwise or
counterclockwise.
Related Core Content: 3.3
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Some people subdivide each side of the square into thirds and then count 8 of the
12 resulting segments. Others may subdivide the interior with a 3x3 grid, without
explicitly recognizing why this strategy works. Asking them why it works can
show them that they do not have a clear understanding of why this strategy makes
sense. This insight is exhilarating for some, but frustrating for others. You will
need to make some explicit decisions about how much you “push”
participants’ thinking about their strategies.
Answers: There are two answers, depending on whether Ralph walks clockwise or
counterclockwise.
Related Core Content: 3.3
Deciding what the unit is in a situation is not always easy, especially in a situation
where the unit is “changing.” But learning how to do this is critical foundation for
learning about percentages later.
Problem Set 3
about 30 minutes for the
introduction to unitizing,
working on problems, and
debriefing, with extra time
for the Reflection
The goal is to introduce and practice the idea of “unitizing.” Problem Set 3 and
Problem Set 4 should be fairly easy, so during the debriefing of these problems,
help participants learn to use the idea of unitizing in describing their thinking.
The term, unitizing, may not be familiar to some of the participants, so you should
introduce this idea with the 4 PowerPoint slides that follow.
These unitizing problems relate to Core Content 3.3
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This slide introduces the basic idea. Thinking about different “units” for
measuring the size of a set supports flexibility in thinking about fractions.
This slide illustrates how unitizing applies to a common situation.
This slide shows how unitizing can be used to solve problems.
This slide reinforces the usefulness of unitizing for thinking about situations.
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1/9 is represented by 2 cookies, so 4/9 is represented by 4 sets of 2 cookies, or 8
cookies.
1/12 is represented by 1 1/2 cookies, so 5/12 is represented by 5 sets of 1 1/2
cookies, or 7 1/2 cookies.
1/6 is represented by 3 cookies, so 5/6 is 15 cookies.
1/36 is 1/2 cookie, so 14/36 is 7 cookies.
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1/4 is represented by 4 1/2 cookies, so 3/4 is 13 1/2 cookies.
This reflection, along with the Reflection for Problem Set 4, can help participants
consolidate their understanding of unitizing.
Problem Set 4
The goal is to reinforce flexible thinking about different sizes of groups.
about 15 minutes for
working on problems and
debriefing, with extra time
for the Reflection
These problems focus on different-sized units. They probably will be easy for
most participants.
When you debrief these problems, reinforce participants’ understanding of
unitizing.
These unitizing problems relate to Core Content 4.2
16 eggs are 1 1/3 dozen; 26 eggs are 2 1/6 dozen.
Participants may draw dozens of eggs and shade in various ways. Encourage
participants to think in groups of 12 rather than in individual eggs.
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32 sodas are 5 1/3 six-packs or 2 2/3 twelve-packs or 1 1/3 24-packs.
Different representations for the three questions should highlight the role of the
changing unit.
Alternately, participants might write:
32 sodas = 16 (pairs of sodas) = 8 (four-packs of sodas)
A six pack is 3 pairs, so 16 (pairs of sodas) = 5 1/3 (six packs)
etc.
14 sticks are 2 1/3 six-packs or 1 4/10 ten-packs or 7/9 eighteen-packs. Point out
that thinking about 10-packs could reinforce place value knowledge for students.
Some participants may work symbolically:
14 sticks = 7 (pairs of sticks) = 3 1/2 (groups of 4 sticks)
etc.
2/3 of a pie is the “unit” of measure. There are 7 sets of 2/3 in 4 2/3.
If you have time, follow-up this problem by asking, “If there were 11 pies left,
how many people could she serve?” [There are 16 and 1/2 servings. That is, 16
people each get 2/3 of a pie, and the 1/3 of a pie left over is 1/2 of a serving.]
5 pies is 1/3 pie more than 4 2/3 pies, and 1/3 of a pie is 1/2 of a serving, so there
are 7 1/2 servings in 5 pies.
Some participants may struggle making the connection that 1/3 of a pie is 1/2 of a
serving; this struggle is related to keeping track of the unit for fractions. The
phrases, “of a pie” and “of a serving” are important labels to keep in the
foreground of the debriefing.
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In order for children to develop flexibly thinking about the unit, they need lots of
experiences with different sizes of units. Using multiple units for a single context
is more likely to develop that kind of flexible thinking that using different kinds of
units across different contexts.
Students do not “naturally” think about different units for measuring the size of a
set. Teachers have to be intentional about providing opportunities for students to
learn this kind of thinking.
Problem Set 5
about 60 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal is to reinforce the concept that the size of the unit is critical to
understanding fractions.
These problems relate to Core Content 3.3 and 4.2.
It is important that participants are explicit about keeping the unit the same
size when comparing equivalent fractions. If they draw pictures, the units must
be the same size. If they simplify each fraction to 3/4, there is an implicit
realization that the unit is kept constant, but you will want to make that explicit.
(For example, ask, “How do you know that 6/8 is equivalent to 3/4?”)
Possible strings of equivalences:
6/8 = 3/4 = 3x3/4x3 = 9/12
6/8 = 6x1.5/8x1.5 = 9/12
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This problem illustrates how thinking about division and can be used to develop or
extend thinking about fractions. For problems (a) and (b), the answers are whole
numbers, so comparisons are easy. For the other problems, answers are fractions,
and in order to compare two fractions, the units have to be the same.
(a) blue: 2 cupcakes; red: 3 cupcakes; red is greater
(b) blue: 2 cupcakes; red: 2 cupcakes; the same
(c) blue: 1 1/3 cupcakes; red: 1 1/2 cupcakes; red is greater
(d) blue: 5/6 cupcake; red: 3/4 cupcakes; blue is greater
(e) blue: 1/3 cupcake; red: 1/4 cupcakes; blue is greater
The units for these fractions are not the same, so the comparisons are
impossible to make. The picture is quite misleading and will encourage students
to generalize incorrectly.
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It is probably impossible to over-emphasize the importance of understanding the
unit when working with fractions. It takes students a long time to internalize this
idea.
Problem Set 6
about 75 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal is to reinforce the density property of fractions; that is, between every
two distinct fractions there is another fraction.
These problems relate to Core Content 3.3 and 4.2.
There are many possible correct answers. Participants may struggle a bit with
knowing whether their fractions are equally spaced. It is important to take time
to debrief these ideas.
For example,
3/5 = 12/20 and 4/5 = 16/20, so 13/20, 14/20, and 15/20 are in between and they
are equally spaced (each is 1/20 more than the previous one).
Some participants may draw a number line and use a visual strategy.
Some participants may be uncomfortable with the symbol, 3.5/5; this indicates a
comparison between 3.5 and 5, or it could be interpreted to mean 3.5 ÷ 5. You
may want to have a discussion about what this symbol means for adults and
whether participants would use this kind of symbol with students.
3.5/5 is halfway between 3/5 and 4/5, since the value of a fraction is directly
proportional to the size of the numerator. It is important to take time to debrief
these ideas.
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There are many possible correct answers. However, this problem is likely to be
more challenging than problem 1. It is important to take time to debrief these
ideas.
For example,
1/4 = 3/12 and 1/3 = 4/12, but a different denominator is needed to clearly find
fractions in between. Denominators of 24 or 36 are not quite large enough, but
3/12 = 12/48 and 4/12 = 16/48, so 13/48, 14/48, and 15/48 are in between and
equally spaced.
Some participants may “overgeneralize” the results from Problem 2 to Problem 4
and believe initially that 1/3.5 is actually halfway between 1/4 and 1/3. This is not
true, since the value of a fraction varies indirectly with the value of the
denominator.
It may be difficult for some participants to explain their reasoning, and this can in
itself be an opportunity to reinforce the fact that many aspects of fraction
understanding are not easy for children to learn. Be sure that you reserve
adequate time to debrief this problem. Showing a graph of y = x and y = 1/x
might help some participants, but it likely will not be convincing for others.
Some participants will use subtraction of fractions to determine if their answers are
equally spaced. Subtraction of fractions is part of the Core Content for Grade 5.
Other participants may reason with pictures or other representations. These ways
of reasoning are equally valid, so be sure that a variety of explanations has been
discussed.
You may want to ask participants if they would us any of these problems with
children, and if so at what grades.
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Problem Set 7
The goal is to explore different ways to think about addition of fractions.
about 45 minutes for
working on problems and
debriefing, with extra time
for the Reflection
These problems relate to Core Content 5.2.
Each of the (infinitely many) correct answers has to be equivalent to 3/4.
There are “common” answers (e.g., 6/12 + 9/12) and there are “uncommon” or
“weird” answers (e.g., 5/4 + 0/97). Encourage participants to think “outside the
box.” If you ask students to add 5/4 + 0/97, the first thing that many of them will
do is try to find a common denominator. THIS IS NOT REASONABLE,
however, since 0/97 is automatically 0, so the sum is simply 5/4. Teachers should
challenge students with some unusual situations occasionally so that students learn
to think flexibly.
For example, if we ask students what 5/4 + 0/97 equals, many students will
attempt to find a common denominator first; after all, they have been told over and
over again that the first thing you do in order to add fractions is find a common
denominator. However, in this instance, finding a common denominator is both
time consuming and totally unnecessary.
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Some participants will think that since the denominators of fractions are restricted
to 10, 11, 12, …, 98, 99, then there are only finitely many solutions. However, if
we allow “negative fractions,” there are infinitely many solutions; for example,
50/40 + 0/40, 51/40 + -1/40, 52/40 + -2/40, 53/40 + -3/40, etc. It is important for
participants (as adults) to recognize these solutions, whether or not they think it is
appropriate to encourage children to think about these solutions.
After the problems are debriefed, ask participants how these problems might
reveal important information about children’s thinking. For example, in solving
Problem 9.2, children are likely to use only fractions that they are comfortable
with. Having information about the comfort level of children with fractions might
be very important information for informing instructional decisions. Problem 9.3
includes a restriction on acceptable solutions that teachers do not often impose.
Imposing these kinds of restrictions may encourage children to rethink what they
know about equivalent fractions.
This is a time for participant to reflect on how their curriculum materials present
fractions to students. Do not spend too much time now on this issue, but you may
want to comment that thinking about, and adapting, curriculum materials is
something that participant may want to do over the next year.
Problem Set 8
The goal is to explore changes in value as numerators and denominators change.
about 75 minutes for
working on problems and
debriefing, with extra time
for the Reflection
These problems relate to Core Content 3.3, 4.2, and 5.2.
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For these problems, some participants will need to compute some specific
examples (with numbers) to convince themselves that they know what happens.
Examples are a good way to generate information about a problem, but examples
alone are not enough to prove a result.
(a) the fraction doubles in value
(b) the fraction is halved in value
(c) the fraction remains at the same value
(d) the fraction remains at the same value
(e) the fraction is 4 times as great in value
(f) the fraction is 1/4 the original value
Again, some participants will need to work with numbers to understand what
happens in each case. For problems (c) and (d), one COUNTEREXAMPLE is
enough to DISPROVE a conjecture, even though examples alone are not enough
prove a conjecture. This idea may need to be discussed explicitly.
(a) the fraction increases in value
(b) the fraction decreases in value
(c) indeterminate: it depends on how the numerator and denominator increase
for example, 3+N/4+N (e.g., 3+5/4+5 = 8/9) is greater than 3/4 but we don’t
know how much more, while 3+N/4+M might be greater than 3/4 (e.g.,
3+7/4+1 = 10/5) or it might be less than 3/4 (e.g., 3+1/4+7 = 4/11).
(d) indeterminate: it depends on how the numerator and denominator decrease
(e) the fraction increases in value
(f) the fraction decreases in value
Participants can easily generate all possible combinations on a spreadsheet, though
children would not be expected to do this. (There are only 24 = 4x3x2x1 ways of
assigning different values to a, b, c and d.)
Without examining or listing all possible combinations, it is more difficult to be
sure that the greatest or least value has been generated.
(a) a=6, b=3, c=5, d=4: greatest sum=3.25
(b) a=3, b=5, c=4, d=6 OR a=4, b=6, c=3, d=5: least sum=1.266666666…
(c) a=4, b=5, c=6, d=3: greatest difference=-1.2
(d) a=6, b=3, c=4, d=5: least difference=1.2
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Participants may not know immediately which of these problems could be used as
“mental math” activities, so allow them time to think about this issue and discuss it
with a partner.
Problem Set 9
The goal is to consolidate the discussions so far about fractions.
about 75 minutes for
working on problems and
debriefing, with extra time
for the Reflection
Some possible contexts include (a) distance/rate/time (e.g., miles, hours, and miles
per hour), (b) cost per unit (e.g., pounds of apples, dollars to buy them, dollars per
pound), and (c) pay (hours worked, total pay, dollars per hour). Be sure that
participants’ problems are really division problems. The debriefing of these
problems may take some time, but try to avoid having the same kind of problem
repeated too much.
Related Core Content: 5.2, 6.1
In order for the quotient to be greater than the dividend, the divisor will need to be
less than one. For example, Mark had 4 pizzas all the same size. If a portion is
3/8 of a pizza, how many portions are there? [4 ÷ 3/8 = 10 2/3 portions]
Related Core Content: 5.2, 6.1
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Some participants may say that a fraction is two numbers, rather than one number.
Certainly some children believe this. It is difficult to understand that a fraction is a
number if too much emphasis is placed on the fact that a fraction has a numerator
and a denominator.
Related Core Content: 3.3, 4.2
Depending on the time available, you may choose to omit Problems 9.3 and
9.4.
This problem helps participants understand the difference between “exactly the
same” and “has the same value as.” For example, 3/4 and 9/12 have the same
value, but they are different fractions. If there is time, you may want to lead a
discussion about what language is appropriate in instruction.
• How precise do we as teachers need to be?
• How precise should we expect children to be?
These discussions can help teachers be better prepared to pose problems for
students.
Be sure to thank participants for their engagement in the activities.
You may need to distribute clock hours forms or complete other paperwork.