Ash’s Puzzle
This problem gives you the chance to:
• find numbers that obey given rules
• find rules for sets of numbers
Ash has a book of number puzzles. This is one of the puzzles.
PUZZLE !
Find a two-digit number such that its
digits are reversed
when 9 is added.
1. Solve this puzzle for Ash.
_________________
Show that your answer works.
Ash wonders if there are other answers to this puzzle.
2. Are there other correct answers to the puzzle?
__________________
If there are more correct answers list them all. If not explain how you know that there is only one
correct answer.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
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Ash’s Puzzle Test 9
Ash decides to try to find a three-digit number such that its digits are reversed when 99 is added.
He finds that there are a lot of numbers that work.
3. Write four three-digit numbers that Ash could have found.
______________
______________
______________
______________
Show your work.
Ash thinks that there must be rules that would make it possible to find all of the three-digit
numbers that are reversed when 99 is added to them.
4. Find these rules for Ash.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
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Ash’s Puzzle Test 9
Task 3: Ash’s Puzzle
Rubric
The core elements of performance required by this task are:
• find numbers which obey given rules
• find rules for sets of numbers
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives a correct answer: 12, 23, 34, 45, 56, 67, 78 or 89
and
Gives correct calculation for their answer:
such as 12 + 9 = 21
1
1
2.
Gives correct answer: yes
and
lists the other seven possible answers (ignore their answer to question 1
repeated) 12, 23, 34, 45, 56, 67, 78 and 89
Partial credit:
(1)
An extra 4, 5 or 6 correct answers with no incorrect ones.
3.
2
2
Gives 4 correct answers:
any 3 digit numbers with the last digit 1 greater than the first e.g. 152, 798
etc.
Shows some correct work for their answers such as:
1
1
152 + 99 = 251
4.
2
Gives correct rules such as:
The last digit is one more than the first.
1
The middle digit can be any number.
1
Total Points
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2
7
Ash’s Puzzle Test 9
Ash’s Puzzle
Work the task. Look at the rubric. What are the big mathematical ideas being assessed
by this task? What types of generalizations do we want students at this level to be able to
make?
Look at student work for part 2 of the task, finding additional answers. How many of
your students:
• Listed all the possibilities?_____________
•
Omitted one possibility?___________
•
Gave 2 or a few possibilities?______________ Gave only one
possibility?__________
•
Said yes with no work to back it up?________________
•
Said no, there aren’t other solutions?________________
• Showed evidence of not understanding the constraints of the task?_____________
What opportunities have your students had this year with problem-solving involving
making an organized list?
What are some of the habits of mind that you want for students in an algebra class? Do
they include questions on self-talk like(Taken from Fostering Algebraic Thinking by
Mark Driscoll):
• Am I able to abstract from computation?
• How are things changing?
• Do I have all the solutions? Could there be others? How are the solutions
related? What are the important properties of the solutions?
• Is there information that helps predict what is going to happen?
• Can I justify why a rule works? Will it always work? What are the cases for
when it is true and when it isn’t true? Have I tested enough different cases to be
convinced that I have all the possibilities?
• What process reverses the one I am using?
• When I do the same thing with different numbers, what still holds true? What
changes?
• Am I willing to struggle through or persevere when faced with an unfamiliar or
nonroutine problem? What do I do when I get stuck?
How do these habits of mind relate to the quality of work you saw on students’ papers?
How might some of these habits of mind helped students improve their performance?
How do you help students to develop these habits of mind in a consistent way throughout
the year? How do you help students develop these habits of mind? How can you promote
students to use self-talk?
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Now look at student work in part 3. How many of your students:
• Gave 4 numbers that fit the constraints of the task?
• Gave numbers that were incorrect and didn’t check their answers to see that they
didn’t fit the constraints?
• Showed work that did not match the constraints of the task (e.g. answers didn’t
reverse the numbers, trying to get answers totaling 99, used operation besides
adding)?
• Weren’t willing to attempt this part of the task?
One of the powerful aspects of algebra is the ability to use it to make and to prove
generalizations. In part 4, students were asked to formulate rules for finding solutions to
meet the constraints of reversing the digits in a three-digit number by adding 99. Look at
student work:
How many of your students could give rules relating the hundreds digit and the units
digit?
How many of those rules gave imprecise language about the digits (first digit, last digit so
that the reader had to make assumptions about reading from the left or right of the
number)?
How many quantified the fact the hundreds digit had to be lower than 9 (quantified the
domain)?
How many of your students thought the solutions were all in the 100’s and ended in 2?
How many students could give a rule about the middle number being equal to any
number?
How many thought the middle number had to be a specific number, such as 0 (i.e. they
didn’t test enough cases to discover all the situations for which the numbers would
reverse?)
How many of your students thought the middle number had to be a repeat of either the
hundreds digit or the units digit?
How many tried to give a pattern based a previous correct solution, such as add 20 or add
202 to the last answer?
How many had a partially correct idea, but couldn’t articulate the attributes of numbers or
think through all the relevant features of the pattern (e.g. split a part any answer from 1)?
How many were unwilling to attempt writing a rule?
What are other misconceptions that you saw in reading through students’ attempts to find
a rule or pattern?
What activities or problems do you give students to help them develop the ability to make
and articulate generalizations?
How often do students in your class conduct investigations with number and number
properties?
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Looking at Student Work on Ashe’s Puzzle
While some students were able to meet the full demands of the task, their language
around place value is imprecise. Notice that students A and B talk about first and third
digits rather than hundreds and units. This is typical of most student work. Student A
shows thinking about the constraints, what does it mean to reverse digits, before giving a
solution. The answers for part two are in an organized list. In part 3 the student only tests
cases where the middle number is zero, but Student A is still able to generalize about the
middle or tens digit. Also notice that the student establishes the domain for the hundreds
digit.
Student A
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Again, Student B only tests a limited variety of cases in part 3, where the hundreds digit
is always 1 and the units digit is always 2. This set allows the student to see and to verify
that the middle number can change. The student is also able to make the leap to seeing
how the pattern would work for different numbers in the hundreds or units places.
Student B
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Student C only tests cases where the tens digit is 0. 8% of all student agreed that the
middle number had to be 0. 13% gave rules stating a specific single digit for the tens
digit. Student C is starting to see that there is a possibility of other choices, but can’t or
doesn’t do enough investigation to really see if he has found all the solutions. He tries to
jump to a generalization too quickly. What other questions would you want the student to
ask himself to help push his thinking? How do we help students develop skills in
investigating and testing a variety of cases or conditions?
Student C
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Student D finds a pattern using a limited variety of cases. The student understands the
relationship between the hundreds and units digits, but doesn’t see that the tens digit does
not effect the solution. 13% of the students thought the tens digit should match either the
hundreds or the units digit.
Student D
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Student E struggles with language for discussing the mathematics of the task. The
student seems to actually connect the hundreds and units digit with the solutions to part 2,
which is a powerful idea not noticed by other students in the sample. Like Student D,
this student limits the tens digit to a single possibility. The student doesn’t investigate to
see what other solutions might work.
Student E
Student F, like 23% of the students, did not think there could be multiple solutions to a
problem. How do we help students build a productive work ethic, a willingness to
persevere when solutions are not immediately obvious? How do you work on this in your
classroom? Student F, like 6% of all students, thought the hundreds digit had to be one
and the units digit had to be two.
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Student F
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Student G looks at a variety of cases that work, putting solutions into an organized list.
This allows the student to then make a generalization from the emerging pattern, that the
units digit is the hundreds digit plus one. The student does not address the possibilities
for the middle digit, although the student has evidence to help support a correct
statement. How do we help students to focus on all the relevant parts of a pattern? In
Fostering Algebraic Thinking, Mark Driscoll talks about the importance of abstracting in
algebra. “. . . a good case can be made that thinking algebraically involves being able to
think about computations freed from the particular numbers to which they are tied in
arithmetic.” He also talks about the internal dialog that students can develop to be more
productive thinkers, such as “When I do the same thing with different numbers, what still
holds true? What changes”?
Student G
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Although Student H received no points, the student is starting to think about the number
system and why the pattern works in her first bullet. However Student H does not finish
the thought by making a rule. The student can’t distinguish between the relevant parts of
the information from the irrelevant parts. When do the numbers repeat? When do the
numbers not repeat? What does this mean for writing a rule for any number? The
student also notices a recursive rule, which only helps when you already know one value
in the set and actually doesn’t hold true when moving from a number with 9 in the tens
digit to the next solution in the sequence. Again how do we help students develop a set of
helpful questions for doing investigations and making generalizations? Does this pattern
hold true for all cases? Have I tested enough possibilities?
Student H
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Ash’s Puzzle
Algebra
Student Task
Core Idea 3
Algebraic
Properties and
Representations
Core Idea 2
Mathematical
Reasoning
Task 3
Ash’s Puzzle
Find numbers that obey given rules or constraints. Find rules for sets
of numbers. Use understanding of place value to solve problems in
context.
Represent and analyze mathematical situations and structures
using algebraic symbols.
• Use symbolic expressions to represent relationships
arising from various contexts.
• Compare and contrast the properties of numbers and
number systems including real numbers
Employ forms of mathematical reasoning and proof appropriate
to the solution of the problem, including deductive and inductive
reasoning, making and testing conjectures and using
counterexamples and indirect proof.
• Show mathematical reasoning in a variety of ways, including
words, numbers, symbols, pictures, charts, graphs tables,
diagram and models.
• Explain the logic inherent in a solution process.
• Use induction to make conjectures and use deductive
reasoning to prove conclusions.
• Draw reasonable conclusions about a situation being modeled.
Mathematics in this task:
• Investigating a relationship in number calculations
• Identifying relevant information using place value and number theory to discover
a pattern in the solution
• Generalizing from arithmetic to a pattern for all solutions
Based on teacher observations, this is what algebra students knew and were able to do:
• Find most solutions which will reverse a two-digit number by adding nine
• Give examples of three-digit numbers that will reverse the digits when 99 is
added and show supporting evidence
Areas of difficulty for algebra students:
• Making an organized list or check for “all” solutions that meet a set of constraints
• Testing different cases of numbers, investigating enough options or choices
before making a generalization
• Recognizing all the relevant information needed to make a convincing set of rules
for all numbers
• Vocabulary for place value
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The maximum score available on this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Many students could find one solution of adding 9 to a two-digit number that would reverse the
digits. More than half the students, 63%, could Find one solution for reversing a two digit
number and four examples for reversing a three digit number and give supporting evidence to
verify the solutions. Almost half the students could also find at least 5 solutions for reversing the
two-digit number. About 16% could find all the solutions for reversing the 2 digit number and
could verbalize a rule about the units digit being one larger than the hundreds unit. About 2% of
the students could meet all the demands of the task including noticing that the middle digit could
be any number from 0 to 9. Almost 20% of the students scored no points on this task. 76% of the
students with this score attempted the task.
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Ash’s Puzzle
Points
Understandings
76% of the students with this
0
score attempted the task.
1
Students could find 1 solution of
a two-digit number that would
reverse when 9 was added.
3
Students could find at least one
solution for a two-digit number
and four solutions with work for
reversing the digits in a 3-digit
number.
4
Students could find at least five
solutions for reversing a twodigit number and four examples
for reversing a 3-digit number
with supporting work.
6
Students could find all the
solutions for reversing a 2-digit
number and give 4 examples for
reversing a 3-digit number. They
also noticed that the units digit
was one larger than the hundreds
digit.
7
Students could find solutions for
reversing 2- and 3-digit
numbers, investigate an
arithmetic situation and
generalize to a rule for finding
all solutions. Students generally
had a system of making
organized lists and testing a
variety of cases.
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Misunderstandings
Some students did not understand the
constraints of the task. They tried to find
numbers that added to 9, listed multiples of
9, or gave answers that did not reverse the
digits, such as 20 + 9=29.
23% thought there were no other solutions.
5.5% only gave one further solution, but did
not attempt to find all possible solutions.
Another 5.5% gave two or three of the
seven additional solutions. 18% of the
students didn’t attempt part 3 of the task.
Students didn’t have strategies like making
an organized list to help them check for all
possible solutions in part 2. Students didn’t
check enough variety of examples in part 3
to help them make a good generalization in
part 4.
25% of the students did not attempt to write
a rule for part 4 of the task. About 5.5% of
the students thought the hundreds number
needed to be one and the units digit needed
to be 2. About 5% of the students thought
all 3 of the numbers should be consecutive.
7% of the students tried to give a recursive
rule, such as add 10 or add 101.
Many students did not address the issue of
the middle number. They did not connect
that feature to describing how to write the
number. 8% of the students thought the
middle number had to be a 0. 13% gave a
specific single value to the middle digit.
13% thought that the middle digit should
match one of the other two digits.
All students in the sample struggled with
using place value language, usually
referring to first and last digits rather than
hundreds and units digits.
65
Ash’s Puzzles
Students need more rich tasks that allow them to investigate numbers, number patterns.
They need to learn how to organize information from their investigation in a systematic
way that will help them to see the relevant information in the pattern and make
generalizations. Part of investigating patterns is to think about questions, such as “When
does this work? Will this always work? If I change one of the numbers what will change
and what will stay the same? Through experience and discussion students develop habits
of mind that help them to form generalizations and understand what makes a convincing
rule or generalization. There is a whole set of logic skills that develop through productive
math talk and discussion. As students get feedback about their ideas or critique the
solutions of others, they learn to focus in on the relevant properties and become more
specific in the rules they generate. Math talk or discussion also gives them practice in
using mathematical vocabulary, thus developing mathematical fluency with academic
language that doesn’t happen from just reading and memorizing definitions. Students
should be encouraged to try and justify why the patterns work using arguments about
place value and algebraic expressions.
Ideas for Action Research
Using Problems of the Month
Try giving students some longer investigations, like problems of the month. Problem
solving is the cornerstone of doing mathematics. George Polya, a famous mathematician
from Stanford, once said, “a problem is not a problem if you can solve it in 24 hours.”
His point was that a problem that you can solve in less than a day, is usually a problem
that is similar to one that you have solved before or at least recognized that a certain
approach will lead to the solution. Students need more exposure to problems with
extended reasoning chains, that require perseverance, and use a variety of strategies.
Give some students some problems, like Courtney’s Collection, which look at number
theory and algebra.
(taken from the Noyce Website:www.noycefdn.org/math under resources)
Encourage your students to restate the problem in their own words. Show their
calculations and what they tried. Then write about their process:
• How did you get started? What approaches did you try?
• Where did you get stuck?
• What drawings, charts, graphs, or models did you use?
• What was your solution? What did you discover during the investigation?
Emphasis should be on the process of solving the problem.
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Courtney’s Collection
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Understanding Struggling Students
What are the needs of students who are performing at a low level? How are their needs
for support different than those of the students who achieved some success but still need
help some specific elements of the task?
Sit down with a group of colleagues and look at the student work below. Do these
students have any skills to build on that would help them succeed on this task? If not
what is the one major target where their thinking breaks down or hole in arithmetic skills
that they would need substantial instruction on? Where would you go with this student?
As students progress through the grades, they need to develop an understanding of
operations with whole numbers and computations with whole numbers. Then, they
should start to be able to understand operations and computations with rational numbers.
Finally in eighth grade students should start to understand number systems. Do you see
evidence of understanding whole numbers? Whole number operations? Understanding of
place value and number systems? Is there evidence that these students are ready to start
making generalizations about number and number systems? Where does their thinking
breaking down?
What are the implications of this work for your school across grade levels? What are the
implications for you as a teacher of these students? Do you have examples from you own
class that would be useful to add to the discuss?
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Agatha
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Bruce
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Clara
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Dennis
Evelyn
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Evelyn, part 2
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