Download 1:00 Activity: Grade Level Exploration of the Standards

June 10, 2008
page 1
Facilitator Notes
Professional Development: K-8 Mathematics Standards
Grades K-2: Algebraic Thinking
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 K-2 teachers understand
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 students learn more. This quote may be useful to you
in establishing the need for teachers to have deeper content knowledge: “For teachers to be able to teach (in these ways) …, they need to walk the
edge between the structure of mathematics and child development, between the community and the individual. They need to be willing to live on the
edge. They need to be willing to challenge themselves mathematically and to be willing to journey with their children.” (Fosnot & Dolk, 2001,
Young Mathematicians at Work, Constructing Number Sense, Addition, and Subtraction, p. 18)
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 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 discussions related to those materials. Be
careful, however, not to lose the emphasis on deepening participants’ knowledge of mathematics.
June 10, 2008
page 2
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 halfdays. 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.
Alternately, you may choose to omit some Problem Sets completely.
8. Video clips are from the CD that accompanies Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic
and algebra in elementary school. Portsmouth, NH: Heinemann. It is sometimes helpful to view a video clip twice, but be sure that the extra
investment in time will pay off for the participants.
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 K-2. (There are different sets of problems for teachers from grades
3-5 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. Discussion among participants is 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, as might some manipulatives, such as counters or tiles.
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.
You may want to use this slide at the beginning of each content session.
Bookkeeping activities (e.g., sign in sheets) can be done now.
Participants may wonder why the materials focus on only ONE content area.
These materials will model the depth at which learners are expected to understand
mathematics. If too much content is covered superficially, it is less likely that the
learning will be sustained over time.
Remind participants that the K-8 Mathematics Standards lay out a set of
FOCUSED expectations for student learning. We want participants also to
develop FOCUSED understanding of the content, so these materials delve
deeply into algebraic reasoning. Additional professional development sessions
may be needed to address teachers’ understanding of other mathematics ideas.
Be sure to change the slide so that the names of the facilitators are correct.
Introduce the facilitators. The next slide provides an opportunity for participants
to introduce themselves.
June 10, 2008
Flow of Activities
page 4
Slides
Notes
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 at their table or “close by.”
Survey the group if you wish to know the approximate distribution of teachers by
grade or by school or by school district.
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.
It is very important for participants to feel “safe” about making mistakes as
they work on problems. Keep the “tone” as light as possible so that
participants have fun as they are learning.
Problem Set 1
about 30 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal of this problem set is to understand both the notion of equality and the
equal sign as a symbol. Additional background information can be found in
Carpenter et al., 2003, Chapter 2.
Ask participants to work the problems in Problem Set 1. Encourage work with a
partner or a small group. Encourage conversation among participants.
June 10, 2008
Flow of Activities
Debriefing
page 5
Slides
Notes
Equality means “the same as.” The language that we as teachers use about
equality is critical in influencing what students learn. When we talk about
numbers, we say “6 plus 2 is 8” or “6 plus 2 equals 8”, but the underlying
mathematics idea is really that 6 plus 2 and 8 have the same value. “6 + 2” and
“8” are different symbols, so at the symbolic level they are not the same. But their
values are the same. We don’t want to be too pedantic about these issues with
students, but as teachers we have to be alert for possible confusions in students’
minds about what equality really signifies. Students need experience with both
kinds of language.
It is correct to write 2 + 6 = 8, but is not correct to use the equal sign to connect the
numeral 6 with a set of six objects. Six cookies is NOT “the same as” the numeral
6. (See Carpenter, et al., 2003, p. 20.)
Sometimes students will write “run-on equations;” for example in calculating
5 + 3 + 2, some students will write 5 + 3 = 8 + 2 = 10. This string of symbols
creates a statement that is mathematically incorrect, even though the student’s
thinking is probably correct; that is, first add 5 and 3, and then add 2 to the result.
Many students interpret the equal sign as a direction to write down an answer. For
example, many young children will say that 3 + 4 = 7 is correct, but 7 = 3 + 4 is
“backwards.” It typically takes a lot of discussion over several weeks (or even
months!) for students to learn that equations such as 7 = 3 + 4 and 5 = 5 are
mathematically correct. Internalizing a correct understanding of the equal sign is
very important background for students to have before they encounter algebra
ideas in middle grades and high school.
There are two slides for this problem. The chart showing the data is on the next
slide.
This is the focus question for next slide: What do you notice in the data? What
conclusions can you draw?
June 10, 2008
Flow of Activities
page 6
Slides
Notes
Focus question: What do you notice in the data? What conclusions can you
draw?
The data in Problem 3 suggest that students do not easily move beyond
understanding the equal sign as a signal to write down the answer to a
computation. If the equation had been written with a variable rather than a box
(e.g., 8 + 4 = N + 5), the pattern of responses might have been different. The use
of a box seems to elicit very different responses than the use of a letter for the
variable. Yet, we want students to understand that the symbol used for a variable
is not the important feature of the variable. The important feature is what the
variable might represent. In this case the variable represents a number.
You may want to ask, “What misconceptions might students have that would lead
them to say that 12 is the number that should go in the box? Why might students
say that BOTH 12 and 17 go in the box?”
After participants have written learning targets for Problem 4, you may want them
to discuss strategies for teaching those targets. Don’t spend a lot of time on this
issue, however, since instructional practice is not the primary focus of this
professional development.
Related Performance Expectations: K.1.C, K.1.F, 1.2.B, 2.1.F, 2.2.G, 3.5.A
Note to Facilitators: There is no Reflection for this problem set. The Reflections
begin after Problem Set 2.
Problem Set 2
about 45 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal of this problem set is to explore how variations of number sentences
affect mathematical reasoning. Additional background information can be found
in Carpenter et al., 2003, Chapter 3.
June 10, 2008
Flow of Activities
page 7
Slides
Notes
Whenever appropriate, help participants focus on reasoning that leads to decisions
about true/false rather than simply on computation. Since adults know that 8 + 7 is
15, there will be a temptation for them to make decisions based on the
computations. If we think relationally, however, we can argue that 8 + 7 = 7 + 8 is
true because of the commutative property of addition. The same reasoning would
apply to this equation:
57,899 + 98,432 = 98,432 + 57,899
Participants may refer back to the corresponding examples in Problem 1.
Encourage them to create explanations based on reasoning. Also, encourage the
development of multiple solutions; for example, in 2a, the box might be
replaced by 15 or 8 + 7 or 7 + 8 etc.
Using larger numbers tends to push students (and adults) to use number
relationships rather than computation to solve problems. This is one argument for
using “bigger” numbers than students might be comfortable computing with.
Participants should recognize that the equations here are mathematically the
same as the equations in Problem 2; only the symbols are different. However,
children may not interpret a box and a variable in the same ways, so many children
would see these problems as quite different. Ask participants if they can describe
different ways that children think about a box and a letter used as a variable. Some
children will want to “fill” the box but may treat a letter in a more “mathematical
way” as a variable.
Kindergarten teachers may want to delay the use of letters as a variable for
students who are not yet differentiating numerals and letters.
June 10, 2008
Flow of Activities
page 8
Slides
Notes
Participants may not be sure what is meant by “sequence.” The task asks for a
series of carefully selected number sentences that will gradually expand children’s
understanding of the equal sign. For example,
8 + 7 = BOX + 6 might precede 8 + 7 = 6 + BOX, since positioning the BOX at
the end the sentence may require more sophisticated thinking from students.
See Carpenter, 2003, Chapter 2, pp. 9-24, for examples of sequences of true/false
and/or open equations used to engage students in thinking about the equal sign.
The intent in the K-8 Mathematics Standards is that “equation” is one kind of
“number sentence”. The word, “equation,” was chosen for the sake of
consistency.
It is good to be aware that some mathematics programs use the term “number
sentence”. Number sentences include equalities (e.g., 2 + 3 = 5) and inequalities
e.g., 3 < 5).
Related Performance Expectations: K.1.C, K.1.F, 1.2.B, 2.1.F, 2.2.G, 3.5.A
Problem Set 3
about 90 minutes for
working on problems and
debriefing, with extra time
for the Reflection
Problem Set 3
The focus of Problem Set 3 is making number
sentences true.
You may work alone or with colleagues to
solve the problems in set 3.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 18
The goal of this problem set is to solve number sentences and understand
children’s thinking about number sentences. Additional background information
can be found in Carpenter et al., 2003, Chapter 5.
Viewing videos is part of this set of problems, so it may be easier to do these
problems one set at a time.
June 10, 2008
Flow of Activities
page 9
Slides
Notes
This problem is separated across two slides.
Participants may find a value of the variable by computing rather than reasoning.
Of course that is acceptable, but try to highlight participants’ solutions that rely on
reasoning. Facilitators may prefer to word the problem differently: “Can each
equation be made true? If so, how? If not, why not?”
Answers: (a) d = -3, (b) g = -1, (c) a = 0, (d) no solution, since if you “subtract
out” the d’s from both sides, you would have 13 = 15, which is FALSE.
Note that parts d and e have no solution. Ask participants if they ever intentionally
ask students to solve problems that have no solutions. You may want to ask if a
teacher should give students a warning that there might not be a solution.
Answers: (e) no solution, (f) d = 2, (g) d = -2, (h) d = 2
You may want to ask participants how parts f, g, and h are related. “Can you use
the solution for (f) to help you find the solutions for (g) and (h) without reworking
each problem?”
Reminder: After several problem sets, some participants may be feeling anxious
about the mathematics and begin to “push back” or “check out”. Gently remind
them of the purpose and focus of the session:
The goal of these professional development sessions is to help participants deepen
their personal understanding of some of the 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
students learn more.
June 10, 2008
Flow of Activities
page 10
Slides
Notes
Answer: d = 12. One solution is to “add 20 to both sides” of the equation and then
divide the sum by 3, since there are 3 d’s on the left side.
The videos for Problems 3.2 (Allison16.mov) and 3.3 (Cody.mov) are on the CD
accompanying Carpenter et al. (2003). Help participants analyze the reasoning
used by the children; participants may be surprised at the high level of
sophistication of that reasoning.
Allison first adds the 20 and 16 to get 36, and then she divides 36 into three equal
groups. But she uses a combination of number facts and direct modeling to do
this. Her explanation of why 20 and 16 should be added illustrates her use of
problem solving skills; that is, “backtracking” or working backwards.
Answer: k = 7. One solution is to “ignore” one k on each side of the equation and
solve the equivalent equation, k + 13 = 20. The answer to this is “obviously” 7.
Cody solves the problem quickly, without writing anything down. He appears to
understand that k + 13 has to be 20; that is, he recognizes that he can ignore the
“extra” k on both sides of the equation.
Be sure that participants recognize the excellent questions that the interviewer used
to reveal Cody’s thinking. You may want to have a discussion of other questions
that the interviewer might have asked.
Allison writes her work on paper, but Cody does his work in his head. When
students do not show their work, it may be more difficult for a teacher to
understand what thinking a student is using. On the other hand, requiring students
to always write down work may interfere with their thinking.
Allison uses computation in a variety of ways, while Cody reason about the
structure of the problem.
June 10, 2008
Flow of Activities
page 11
Slides
Notes
This reflection will help participants share their perceptions about how problems
like these might be used with students.
Facilitators may want to refer to terms “compose” and “decompose” which are in
the Performance Expectations and Explanatory Comments and Examples in K.1.C
and 1.1.F. Those ideas also appear in 2.1.E.
Problem Set 4
about 60 minutes for
working on problems and
debriefing, with extra time
for the Reflection
The goal of this problem set is to use relationships among numbers to justify true
or false number sentences. Additional background information can be found in
Carpenter et al., 2003, Chapter 5.
NOTE: Before participants begin working on problems, show the next slide
to help them understand what “relational thinking” means.
Be sure that participants understand that computing is not the only way to justify a
relationship.
For example, in the first equation, we could take 1 away from 25 and add it to 17
to get 24 + 18. In the second equation, taking 1 away from 25 and adding 1 to 18
decreases the difference by 2, so this equation is false.
After making their arguments, have participants talk in their table groups
and then share out: What is relational thinking?
June 10, 2008
Flow of Activities
page 12
Slides
Notes
Anxiety may elevate again when moving from 1- and 2-digit numbers to 3and 4-digit numbers. Another gentle reminder about the purpose (teachers
learning mathematics) may be helpful!
This problem focuses attention on place value relationships along with
understanding of addition and subtraction. Encourage the use of reasoning that
does not depend solely on computing.
a. true: 3,565 is 200 less than 3,765 and 3,187 is 200 more than 2,987, so the sum
is constant
b. true: add 30 to each number on the left to get the numbers on the right, so the
difference is constant
c. false: add 10 and subtract 10 to the numbers on the left, so the difference on the
right is greater than the difference on the left. This helps illustrate one way that
addition and subtraction are not the same.
d. true: add 3,000 and subtract 3,000 to the numbers on the left, so the sum is
constant
e. true: subtract 6,000 from each of the numbers on the left, so the difference is
constant
You may want to simplify this task by asking participants to identify the
easiest and the most difficult problems (or the two easiest and the two most
difficult) for students. Rank ordering all of them may take too long.
Answers will vary, but there is like to be considerable agreement on the 2 or 3
easiest problems (e.g., often involving addition) and the 2 or 3 most difficult
problems (often involving subtraction). Ask participants which operation is easier
and which placement of the variable will cause students the least (or greatest)
difficulty.
Probe participants’ thinking about their categorization. “How were your choices
influenced by the operation used? By the location of the unknown/variable? By
the particular numbers?”
June 10, 2008
Flow of Activities
page 13
Slides
Notes
Discussion of students’ thinking will help participants understand their own
thinking about these equations. Problem (f) is related to the question, “How many
tens are there in 246?” Problems (i) and (j) differ in only one symbol; (i) is false,
but (j) is true. Ask participants whether presenting these two problems might help
students become more careful in reading the symbols in an equation or might tend
to confuse students. Students need to learn to read mathematics symbols carefully,
but it is sometimes difficult to get students to slow down and attend to what is
written (rather than what they think is written).
Participants will have different responses to this task. Their responses may be
influenced by the curriculum materials they are using in instruction, so be careful
that the debriefing does not turn into a debate about which curriculum is best for
students. Help participants pay careful attention to the particular numbers that
they include in their problems for students. Should the numbers be in the range of
computation skills of students or should the numbers intentionally be much larger
so that students are discouraged from computing? How does the choice of
numbers affect the kind of information teachers get about students’ understanding
of relational thinking?
Relational thinking includes the use of number relationships to compare and order
quantities. Relational thinking builds on computation skill, but computation skill
alone is usually not enough to support relational thinking.
Related Performance Expectations: K.1.F, 1.1.E, 2.1.F.
June 10, 2008
Flow of Activities
Problem Set 5
about 60 minutes for
working on problems and
debriefing, with extra time
for the Reflection
page 14
Slides
Problem Set 5
The focus of Problem Set 5 is properties of
operations.
You may work alone or with colleagues to
solve the problems in set 5.1.
Notes
The goal of this problem set is to identify explicitly the properties of the different
operations with whole numbers. Students need to understand these properties as a
basis for learning algebra later. Additional background information can be found
in Carpenter et al., 2003, pp. 38-40, 53-57, and 123-130.
Viewing videos is part of this set of problems, so it may be easier to do these
problems one set at a time.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 32
Be sure that the properties listed include attention to the special role of 0 and 1 for
each operation. You may want participants to write the properties on chart paper
or an overhead transparency so that everyone can see the properties as they are
discussed, or you may want to use a document camera to display participants’
work. Be sure that any incorrect properties (e.g., a ÷ 0 = 0) are challenged; if other
participants do not challenge them, then you should do so yourself. Properties that
participants might list include: a + 0 = 0 + a = a, a – 0 = a, a + b – b = a, a x 1 = 1 x
a = a, etc.
If participants are not forthcoming with properties, you may begin by posing one:
“I heard a group suggest a – 0 = a. Do you agree?”
The commutative property for addition is typically written as a + b = b + a, and the
associative property is typically written as (a + b) + c = a + (b + c). Similar
equations can be written for multiplication. There are no commutative or
associative properties for subtraction or division; for example, 5 – 3 ≠ 3 – 5.
If the commutative and associative properties were discussed during the
debriefing of Problem 5.1, you should skip over this Problem.
June 10, 2008
Flow of Activities
page 15
Slides
Notes
Asking participants to read the problem aloud will help you assess how they think
about properties of addition and subtraction. One way to read this equation
(without simply reading the symbols!!) is to say that if you add and subtract the
same number, the answer does not change.
Push participants to justify this through the use of models or drawings; just
showing examples is not mathematically sufficient, though we might let students
show examples as an explanation. For example, if an addition sentence is
represented with blocks, the equation indicates that we put one more block with
the pile representing “a” and take away one block from the pile representing “b.”
Adding on one block and taking away one block leaves the total number of blocks
unchanged.
The video 3.3 (Kenzie.mov) is on the CD accompanying Carpenter et al. (2003).
Help participants analyze the reasoning used. Kenzie uses her knowledge of
number facts and number relationships to determine which equations are true. It is
particularly interesting how she connects 4 x 6 and 4 x 7. Discuss how the
interviewer deals with Kenzie’s computational error (when she says that 4 x 6 is
32). Some participants may say that the interviewer should have corrected the
error, so discuss what may have motivated the interviewer not to do that
immediately. Choosing a problem to verify a hypothesis about what students
know (or do not know) is a critical aspect of instructional planning.
Problem 5.4
Look at video 3.3 (Carpenter, et al., 2003).
Focus your attention on the strategies the student
uses.
What, if anything, do you think the student learned
during this interview?
What problem would you pose to check your
hypothesis?
2008 May 29
Algebra: Grades K-2: slide 36
K-2 students can begin to develop an understanding of the properties of addition
and subtraction. For example, zero added to any number gives that number, or
adding and subtracting the same number leaves the total unchanged.
You may want to ask participants to find where properties of operations appear in
the K-2 Standards: Performance Expectations 1.2.D and 1.2.E.
June 10, 2008
Flow of Activities
Problem Set 6
about 60 minutes for
working on problems and
debriefing, with extra time
for the Reflection
page 16
Slides
Problem Set 6
The focus of Problem Set 6 is justification and
proof.
You may work alone or with colleagues to
solve the problems in set 6.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 39
Notes
The goal of this problem set is to practice making convincing arguments to justify
(or verify) true statements. Additional background information can be found in
Carpenter et al., 2003, Chapter 6.
Viewing videos is part of this set of problems, so it may be easier to do these
problems one at a time.
After working with the first problem have participants talk in their groups about
the question, “What is justification?" (See the note on “Levels of Justification” in
next frame.)
This statement is true. Some participants likely will first check the statement by
computing examples. Encourage them to go beyond whole numbers by using
fractions and decimals (and even negative numbers). A justification, however,
needs to go beyond simply checking examples. Models or drawings may be
useful for some participants. For example, for the left side of the equation, start
with a pile of blocks representing “a” and then remove a pile of blocks
representing “b” and a pile of blocks representing “c”. For the right side of the
equation, start with a pile of blocks representing “a”, then create two piles
representing “b” and “c”, and finally remove those two piles at the same time. In
both cases, the numbers of blocks remaining would be the same.
After working with the first problem have participants talk in their groups about,
“What is justification?"
Levels of Justification (See Carpenter, 2003, pp. 87-92, for details and examples.)
• Appealing to authority helps students avoid justification: “Last year my
teacher told us when you multiply by 10 you just put a zero at the end.”
• Justification by Example: Students use a variety of examples to justify a
conjecture. While they may be unconvinced it works in all cases that is all
they know to do.
• Generalizable Arguments may include restating a conjecture, concrete
examples that are more than examples, building on already justified
conjectures, and use of counterexamples.
June 10, 2008
Flow of Activities
page 17
Slides
Notes
If everyone gets half a soda, then 10 people can drink soda.
N ÷ 1/2 = 2xN is true. If everyone gets 1/2 of an object, the number of shares is
twice the number of objects that you started with. Alternately, the number of
times you can repeatedly subtract 1/2 from N is 2 times N.
Some participants may use proportional reasoning: “Every object can be broken
into 2 halves, so that are twice as many halves as there are objects.”
N ÷ 1/3 = 3xN is true. If thirds of an object are shared, the number of shares is
three times the number of objects that you started with.
You may want to remind participants that N ÷ 3 and N ÷ (1/3) are very
different, even though students sometimes confuse these two ideas.
N÷3≠3xN
The video7.2 (Suzie.mov) is on the CD accompanying Carpenter et al. (2003).
(Transcript is on p. 97-101.) The thinking of the student on this video is
exceptionally sophisticated, especially for a child so young. Of course, this level
of sophistication is unusual, but it does provide an “existence proof” that some
young students can reason mathematically. It is really important for teachers to
provide opportunities for children to demonstrate what they know.
Otherwise, we will never know what they know.
Reflection
What do you look for in a child’s argument
about whether something is true or false?
How sophisticated can you expect children’s
arguments to be?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 43
Children often base their reasoning on examples. When this happens, a teacher
can ask, “Will this be true for every possible example?” Children may not be able
to answer this question precisely, but it is important for them to begin thinking
about this issue.
The level of sophistication that a teacher accepts will depend on the teacher’s
knowledge of the ways that the child is thinking.
Related Performance Expectations: K.5.D, K.5.G, 1.6.D, 1.6.H, 2.2.B, 3.6.J
June 10, 2008
Flow of Activities
Problem Set 7
about 90 minutes for
working on problems and
debriefing, with extra time
for the Reflection
page 18
Slides
Notes
The goal of this problem set is to explore and justify particular number patterns.
In particular, participants will explore the divisibility rules for 5 and 3. (See
Carpenter et al., 2003, pp. 127-128.)
The divisibility rule for 5 is typically stated, “A number is divisible by 5 if the
ones digit is divisible by 5.” or “A number is divisible by 5 if the ones digit is 0 or
5.” A more general rule is “The remainder when a number is divided by 5 is the
same as the remainder when the ones digit is divided by 5.” This result, at least as
stated this way, is likely to be generally unknown by participants. The result can
be justified by noting that by “splitting off” the ones digit of a number results in a
number that is a multiple of 10 (which is, of course, divisible by 5 with a
remainder of 0). So the remainder of the number divided by 5 is the same as the
remainder of the ones digit divided by 5.
For example, 247 = 240 + 7, and 240 is divisible by 5.
So the remainder of 247 ÷ 5 is the same as the remainder of 7 ÷ 5.
247 ÷ 5 has a remainder of 2 and 7 ÷ 5 also has a remainder of 2.
June 10, 2008
Flow of Activities
page 19
Slides
Notes
The divisibility rule for 3 is typically stated, “A number is divisible by 3 if the sum
of the digits is divisible by 3.” A more general rule is “The remainder when a
number is divided by 3 is the same as the remainder when the sum of the digits is
divided by 3.” The result can be justified for a three-digit number by noting that if
abc represents a 3-digit number,
abc = 100a + 10b + c = (99a + a) + (9b + b) + c = (99a + 9b) + (a + b + c).
Of course 99a + 9b is divisible by 3 (that is, the remainder is 0), since 99 and 9 are
both divisible by 3. So the remainder of abc ÷ 3 is the same as the remainder of
the sum of the digits, a + b + c, divided by 3.
For example, 247 = 200 + 40 + 7 = 2x100 + 40x10 +7
= (2x99 + 2) + (4x9 +4) + 7 = (2x99 + 4x9) + (2 + 4 + 7)
Now, 2x99 + 4x9 is divisible by 3 with a remainder of 0, so
the remainder of 247 ÷ 3 is the same as the remainder of (2 + 4 + 7) ÷ 3
247 ÷ 3 has a remainder of 1 and (2 + 4 + 7) ÷ 3 also has a remainder of 1.
Problems 7.3 and 7.4 will take a considerable amount of time. If you are
short on time, you may want to omit these two problems.
This can be understood by considering the role of place value in the three-digit
numbers. For example, if a≥b≥c, then abc - cba = (a - c)*100 + (c - a) =
(a - c)*100 - (a - c) = (a - c)*99. Listing the multiples of 99 times one-digit
numbers provides an explanation for what is happening: 099, 198, 297, 396, etc.
This is a bit more complicated symbolically, but an explanation depends on place
value and properties of multiples of 99. Suppose abc and cba represent two, 3digit numbers, so that
abc ≥ cba. This means that a≥c.
abc - cba = (a - c)*99, just as in Problem 3.
The multiples of 99 are 99 (which = 099), 198, 297, 396, 495, etc.
In each case, the middle digit is 9, so the difference is x9y, with x + y = 9.
x9y + y9x = (x + y)100 + (90 + 90) + (y + x) = (x + y)*101 + 180
Since x + y = 9, (x + y)*101 = 9* 101 = 909.
909 + 180 = 1089.
June 10, 2008
Flow of Activities
page 20
Slides
Reflection
Why is knowledge of divisibility important for
students to know?
Notes
Developing flexible number sense thinking is enhanced when students understand
divisibility rules.
Related Performance Expectations: 1.1.G, 2.1.D, 2.1.E
In K-2, the ideas of odd and even are important
for children to learn. How are those ideas
related to divisibility rules?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 49
Problem Set 8
about 45 minutes for
working on problems and
debriefing, with extra time
for the Reflection
Problem Set 8
The focus of Problem Set 8 is representations.
You may work alone or with colleagues to
solve the problems in set 8.1.
The goal of this problem set is to explore representations of multiples. Additional
background is provided in Carpenter et al., 2003, p. 61 and pp. 79-84.
Viewing videos is part of this set of problems, so it may be easier to do these
problems one set at a time.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 50
The videos 7.1 (Allison.mov) and 7.2 (Mike.mov) are on the CD accompanying
Carpenter et al. (2003). The two students’ representations are mathematically
equivalent, even though they may seem different to some participants. Be sure
that participants understand that one possible representation for an even number is
a sum of 2’s; for example, 8 = 2 + 2 + 2 + 2; an odd number is a sum of 2’s with 1
“extra.” Another representation of an even number is as the sum of some number
with itself; for example, 34 = 17 + 17; an odd number is the sum of two numbers
that are the same plus 1 more; for example, 35 = 17 + 17 + 1. You may want to
ask participants to represent some even numbers in both ways, just to be sure that
they understand the two representations.
Ask participants why they think the interviewer asked Mike about whether “3 and
a half” was odd or even. Again, be sure that participants recognize the excellent
questioning of the interviewer. You may want to discuss why examples alone are
not sufficient for a proof of something.
June 10, 2008
Flow of Activities
page 21
Slides
Notes
An even number can be represented as 2xn or n + n. This is the way that Mike
represented even numbers.
A multiple of 5 can be represented as 5xn or n + n + n + n + n.
It is important to move students to generalizations – “Is that always true? Is it true
for all numbers?” When participants share their representations be sure to connect
the different kinds of representations: models, diagrams, numbers, and variables.
The sum is divisible by 3 with a remainder of 0. That is, N = 3x + 1 and
P = 3y + 2, so N + P = 3(x + y) + 1 + 2 = 3(x + y + 1); so N + P is divisible by 3
(with remainder 0).
If N≥P, then N - P = (3x + 1) -(3y + 2) = 3(x - y) + (1 - 2) = 3(x - y) + (-1) = 3(x –
y - 1) + (3 - 1) = 3(x – y - 1) + 2; the remainder for (N - P) ÷ 3 is 2.
If N<P, then N - P is negative, so we have to think slightly differently about the
“remainder” when the negative difference is divided by 3. The quotient can still,
however, be represented as 2 more than a negative multiple of 3.
For example, if N = 4 and P = 14, then N – P = 4 – 14 = -10 = -12 + 2. Since -12
is divisible by 3 (with a remainder of 0), then -10 ÷ 3 = (-12 + 2) ÷ 3 has a
remainder of 2.
Understanding the reasoning for this last part of the problem may be a
challenge for some participants.
Reflection
How can representations help students reason?
How can we help students learn to use
representations to clarify their thinking?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 54
Students need to learn to use representations themselves and learn to understand
representations used by other students. Sometimes teachers need to suggest
representations for students to use, but most of the time, students will choose
representations that make sense to them.
Related Performance Expectations: 1.1.G, 1.1.I, 2.1.D, 2.1.E, 2.2.C
June 10, 2008
Flow of Activities
Problem Set 9
page 22
Slides
Notes
The goal of this problem set is to explore number patterns. Additional background
is provided in Carpenter et al., 2003, pp. 47-63, and especially p. 62.
about 30 minutes for
working on problems and
debriefing, with extra time
for the Reflection
You may need to emphasize the use of number patterns as well as geometric
(or shape) patterns in teaching children.
Multiplication is the setting for the participants, even though K-2 students would
not solve multiplication problems as sophisticated as the problems here.
The number pattern here is based on the algebraic relationship, (N - a) x (N + a) =
N2 - a2. So 499 x 501 = 250,000 – 1 = 249,999 and 295 x 305 = 90,000 – 25 =
89,975. Some participants may be able to figure this out using simple algebraic
manipulations, but many participants will likely argue only from examples. You
can also represent this with a diagram.
When participants share their representations be sure to connect the different kinds
of representations: models, diagrams, numbers, and variables.
Problem 9.2
Guess these products. Then check your
guesses.
300 x 300 and 298 x 302
300 x 300 and 295 x 305
N x N and (N - a) x (N + a)
2008 May 29
Algebra: Grades K-2:slide
54
As participants discuss their work, help them identify when they made
conjectures about what was happening. Some participants will probably have
unproven conjectures as part of their explanations. Help identify those
conjectures, even though there is not enough time to prove them all.
You may want to read aloud the last two paragraphs on page 102 (Carpenter,
2003) – which captures much of what participants have done with conjecture and
justification as adult learners and relates it to the primary grades.
June 10, 2008
Flow of Activities
page 23
Slides
Notes
Reflection
What kinds of patterns might you ask K-2
students to explore?
Participants may suggest number patterns based on addition and subtraction.
Allow time for participants to create some number patterns that they might use
with students.
Related Performance Expectations: K.2.A, 1.2.1, 2.2.F, 3.6.J
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Problem Set 10
about 60 minutes for
working on problems and
debriefing, with extra time
for the Reflection
Algebra: Grades K-2: slide 58
The goal of this problem set is for participants to reflect on their own thinking
about the problems.
June 10, 2008
Flow of Activities
page 24
Slides
Notes
Hopefully, participants will comment that the opportunity to work with colleagues
and to share their thinking helped them clarify what they know.
A participant’s grade level may be an important influence on her/his identification
of particular problems that might be posed to students.
By changing some of the numbers, many of these problems could be adapted for
use with students.
The mathematics underlying these problems is critical foundation for the
Mathematics Standards. Instruction based on the Standards will allow students to
learn mathematics at a deep level. Hopefully, this deeper learning will result in
less “forgetting” of what has been learned.
June 10, 2008
Flow of Activities
page 25
Slides
Notes
Be sure to thank participants for their engagement in the activities.
You may need to distribute clock hours forms or complete other paperwork.