Back to Your Roots: Investigating Nested Radicals

PROBLEMS WITH PRIMES!

Here is a set of middle school problems on primes that require careful reading, knowledge of prime digits, organized listing and other skills. They can also be used to prepare high school students for SATs and other standardized tests which frequently test knowledge of prime numbers. Working without the calculator is strongly recommended. The last couple of questions require a partial list of primes which could be an internet activity. None of these questions is highly challenging but one of the goals is to make children aware of the mysteries of primes, something few appreciate! I guess you could say that learning how to read critically is also a PRIME objective!

1) List the FIVE 2-digit primes whose units' digit is 1.

2) List the FIVE 2-digit primes each of whose digits is NOT prime.

3) List the TWELVE 2-digit nonprimes (composites) each of whose digits is prime.

4) Mentally, determine the largest 3-digit nonprime (composite) each of whose digits is prime.

5) A palindrome is a number like 101 or 222 which reads the same when its digits are reversed.
(a) Explain why there are ninety 3-digit palindromes without listing all of them.
(b) (Internet activity). Search for a list of primes up to at least 1000. Use this list to answer the following: If a 3-digit palindrome were chosen at random, what is the probability that it would be prime?

6) Using your list of primes, determine the largest 3-digit prime having 3 different prime digits. Anything surprise you?

7) (Additional work outside of class) Using the list of primes, devise three problems of your own about 3-digit primes to challenge your classmates. A special prize for the best questions!

SAT-Type Challenge: Exponents and Combinatorial Thinking

These SAT-type questions provide review for the multiplication rule of exponents as well as recognizing the need for using the counting principle vs. careful enumeration in an organized list. Both questions need to be given for the effect. Target: PreAlgebra and beyond...

(a) Consider the list 1,2,3,4,5
If a, b and c are assigned different values from the list above, how many different values will result from the expression a^bc ?

(b) Consider the list 1,2,4,8,16
If a, b and c are assigned different values from the list above, how many different values will result from the expression a^bc ?



Notes:
(i) To encourage use of exponent rules, do not allow calculator. What variations would make this even more powerful?
(ii) Possible extension: For (a), ask students to make a conjecture regarding the largest possible power, i.e., is it obvious which is the greatest among 5¹², 4¹⁵, and 3²⁰ ?

Geometry SAT Problems - Do These Questions Help Students Develop Spatial Sense and Combinatorial Thinking?

Answers and discussion of problems below are now available in Comments.

Some of these questions are reprinted from copyrighted materials from the College Board. In some cases, I've modified the questions for instructional purposes. These questions are linked to more advanced topics involving polyhedra and college geometry but they are appropriate for middle and secondary students as well. When is it valuable for students to actually enumerate the objects asked for? These questions also stress the importance of the phrase 'determined by.'

1. What is the total number of right angles formed by the edges of a cube?

Modified version for classroom use: Show that there are 24 right angles formed by the edges of a cube. You and your partner must find at least TWO different methods. [Note: By giving students the 'answer', the focus is then on process.]

2. How many distinct pairs of parallel edges are there in a cube (or rectangular solid)?

3. How many different planes are determined by the vertices of a cube (or rectangular solid)?

4. How many equilateral triangles are determined by the vertices of a cube?

Another Response to a DI Proponent

The following is another reply to a highly knowledgeable proponent of the Direct Instruction program in Joanne Jacobs' ongoing discussion 'Teachers Wonder About Direct Instruction'. Mr. DeRosa has systematically countered my arguments for a more balanced view of pedagogy, allowing teachers to adapt and modify approaches for their students. I have made it clear that I see the value of the DI program for many children, but not all. I will continue to advocate for some flexibility in methodology, however, I will not bend on the issue of standardized math content. I'm sure some readers are tiring of this and were hoping to see some more Problems of the Day. Don't worry - they're coming!

More ‘bang for your buck’ is an expression that would definitely ring true for central school administrators and board members who often relate to the ‘business model’ of education and for whom the ‘bottom line’ is maximizing district scores at minimum cost to the taxpayers. From that perspective, adding a column of 15’s may be a ‘waste of time.’ From the perspective of math educators and mathematicians like Liping Ma who has called for a more ‘profound understanding of fundamental mathematics’ the following scenarios may not be:
One child who didn’t recognize 15 x 10 = 150, explained that she added up the 5’s to get 50, then moved her finger to the next column and counted on by 10’s: 60-70-80-90-…-150. Ken, you might argue that’s the result of not learning her skills well enough and you may be right. A different view is that she demonstrated an understanding of place value that many 4th graders do not have. Actually, I will never convince you of that! Another child for some reason, said that he counted by 15’s up to 75 and then doubled it to get 150. What an inefficient method and waste of time, right? Another student demonstrated his understanding of place value and emerging sense of the distributive property by explaining that ten 10’s = 100, ten 5’s = 50, then added the sums. Several students just visualized the standard algorithm in their minds, getting 50 in the one’s column, pictured writing the zero and carrying the five to the next column, counting 5-6-7-8…-15.
About 7 of the 20 students multiplied 15 x 10. Guess that class wasn’t trained properly, right? I do not believe in having these kinds of dialogues every day for 15 minutes! I do believe that when students see other ways of approaching a question, their understanding deepens. Certainly, some children will be confused by being shown alternate methods and these students need to be shown one straightforward approach in the clearest possible manner. But this is not true for all students. Let me share a dialogue I had with one of the SAT students I taught last night. Won has been in this country for about a year and he is clearly an outstanding problem-solver with very strong skills (and he expresses himself in English surprisingly well). After reviewing a challenging problem asking for the least value of y satisfying some inequality involving 2 variables, Won raised his hand and showed us a much simpler method: “Just make x and y equal”, etc… I asked Won if his teachers in Seoul allow students to discuss different ways of solving a problem and he just smiled at me and said, “No, they tell us how to do it.” Guess that supports a more direct model, right? I asked Won how he felt about following the teacher’s method and he replied, “Oh, I usually did it my way anyway and he only checked my final answer, so it was ok.” Is Won merely the exception to the rule and he’ll learn math despite the methodology. Let me guess what you might say!

One of my daughters who is a regular and special education elementary teacher told me that several years ago she taught from a scripted program for reading and spelling for a special population of children. She expressed that these children absolutely learned with this program provided the teacher strictly adhered to it. She felt confined by it but recognized its value. However, she noticed htat some children soon became bored and were able to move at a faster pace but that she couldn’t simply give them more advanced materials to work on their own nor could she move this subgroup to another class. Yes, Ken, this is an implementation problem, but probably not uncommon. Here’s how she put it to me: Dad, for children who come to us one or more years behind, this type of program helps them to catch up and that’s really good. She also said that many children may only be able to learn in this kind of structured environment, but she did not feel that it was appropriate for other children and, in fact, she thought that one of her goals was to prepare her students to move into the ‘mainstream.’ Just one point of view.
No matter how I argue my points, I know you will have a counter-argument because of your intimate knowledge of DI and because you have seen it work. I can’t argue those results. But statistics often don’t tell the whole story and everyone knows how we can prove almost anything we want by by how we present the data and by the assessments we use. I looked at the 5th grade posttest and, while I was impressed with some of the required skills, I saw questions that used some models and terminology that are non-standard and inherent to the program. I also did not see too many questions that got at assessing conceptual understanding or applying the skills in other contexts. I may not have seen enough to form a real judgment however.

In the end, there is probably room for more than one approach. HOWEVER, AS I’VE STATED REPEATEDLY, THERE ISN’T ROOM FOR DIFFERENT ESSENTIAL CONTENT IN MATHEMATICS. THE CORE MUST BE STANDARD.

Developing Conceptual Understanding of PerCents using an SAT-type Problem

The following was inspired by a question from a recently released SAT. How do you think many students would approach this? Seems like an innocent quantitative or algebraic problem, but I also see some subtleties and important ideas to discuss.

No Calculator Allowed.
(a) In a school election, 480 students voted for one of 2 candidates. Candidate A received 24 more votes than candidate B. What percent of the number of votes cast did Candidate A receive?

Notes: This problem can be done mentally if one has strong number sense and a good feel for percents. You may want to encourage this but allow pencil and paper for those who want it. When reviewing it, certainly share several methods including an algebraic approach. Is this question appropriate for middle school? I think so!

(b) Now generalize: N votes cast for 2 candidates and Candidate A wins by k votes. Derive an algebraic expression for the percent of the votes that Candidate A received.

A Comment on Joanne Jacobs' Post Re DI

The following was my comment on Joanne's stimulating discussion on 'Teachers wonder about direct instruction.'

Although my primary focus is currently on WHAT we teach rather than HOW, I must strongly endorse Mr. Strauss’ reasoned and thoughtful comments. Good teachers have always blended successful methods of the past with the best of what is currently known about the different ways that children learn. No single style can possibly meet the needs of our more and more diverse learners we encounter every day. There seems to be considerable confusion about the technical meaning of DI as developed by Mr. Engelmann. One would need to thoroughly study his rationale and approach to make an informed judgment and I suspect many are responding to the ‘label’ rather than its substance just as many react to ‘discovery learning’ as if it is a method to be used all the time. Effective math lessons I’ve observed for the past 10 years included the essential components of instructional/learning theory:
1. Motivated the lesson (a ‘hook’)
2. Articulation of the objectives of the lesson (what students will know and/or be able to do at the end of the lesson) - this must be carefully thought out during planning and conveyed clearly.
2. Connected current learning to prior learning
3. Reviewed the necessary prerequisite skills for success
4. Provided clear explanations both orally and in writing (on board, on handout or in an electronic presentation)
5. Maximized student involvement via questioning, promoting of dialogue or an activity
6. Assessed what was actually learned (e.g.,responses to questions or requiring students to complete a specific task).
When you remove all the labels, Joanne, it comes down to this: How do we know that the objectives of the lesson were achieved? When I am transmitting parcels of information directly to students, I am still engaging their minds by asking many many questions of different taxonomies to check for their understanding as well as checking if they are still conscious! When I propose a challenging problem and give them a few minutes to work on it in small groups, I am still monitoring their progress carefully and asking guiding questions.
If DI includes all of these components and allows children to explore at times and tackle unstructured open-ended questions for which there is no clear blueprint for solution, then I applaud DI and I guess I’ve been using it all along. If ‘Discovery Learning’ includes all of these components, then I guess I’ve been using it all along and I applaud that too.
Again, as Larry so ably expressed it, good teachers FIND A WAY that works for most of their students most of the time. There will always be some in the class who are not able to grasp the material for a myriad of reasons, often having nothing to do with the child’s ability. Rather than continue this general debate, perhaps we should be looking at REAL examples of effective teaching and then we can applaud these efforts and use them as models for the rest of us, rather than debate the category into which the lesson falls. Oh well, this will never happen, because real examples and pictures would obviate all of the rhetoric and we’d have nothing to blog about!

Another National Math Curriculum Statement

I posted the following comment on edspresso on 2-12-07 in response to The Poisonous Politics of Implementation by Neal McCluskey. Same old, same old??

I have been advocating for national math standards for the past three decades. Its time has come. Despite all the negativity, skepticism and distrust the bottom line is that what we currently have in ALL disciplines is chaos with the result that our children fall further behind other industrialized nations moment my moment. Fifty states with fifty different sets of standards (yes, there is overlap but they are not congruent) with fifty different sets of assessments with fifty different levels of proficiency adds up to what -- let me get my calculator to do the math...
Ultimately, most teachers, including myself, will teach essentially what is in the textbooks and will stress the content and types of questions that are on the high-stakes tests for which they are held accountable. As an AP teacher for 35 years, I am proud to say I 'taught to the test' -- one of the better quality tests I've seen constructed. However, I never felt constricted by a straight-jacket curriculum. My methods and creativity were never affected. I felt crunched by time to cover all the topics in the BC course in 8 months but somehow we got through it and I knew in the end that my students were at least as well prepared for the next level as any other student in any other BC Calc class. I knew that because of their performance in my class which was then validated by the results of the standardized AP test. I saw a reasonable correlation between a '5' on the test and an A or B in class and so on. Could I have taught the same quality course without this 'nationalized' Calculus curriculum. Quality, yes, but I don't think the content and emphasis would have been consistent with the thousands of other calculus classes around the country. Just look at how similar or dissimilar Precalculus classes are from classroom to classroom, never mind state to state. Yes, the road to a standard curriculum is a mine field but the road we're on now leads only to an abyss. I'll take my chances...

Another Quadratic Function Problem 2-16-07 through 2-20-07

You may want to read the comments for this post. Answers and possible solutions are discussed. There is also considerable discussion about teaching techniques for f(x-h).

The parabola problem from 2-15-07 generated some interesting discussion. I haven't had a chance to see it implemented with our Algebra 2 classes yet but I'll let you know if and when...
Today's problem is along the same lines. I'm trying to provide some problems that are exclusively high school math content for this time of year. There are dozens of outstanding problem-solving sites for MathCounts and similar middle-school competitions but there appears to be a dearth of secondary math problem-solving sites (or I haven't found them yet!). Again, how might one use the problem below? As a bonus or an extended in-class activity or a performance assessment or ??? How many would regard this question as suitable only for honors or accelerated students? My take is that if students are exposed to higher levels of thinking and know they are expected to learn how to do these and held accountable on an assessment, they will adjust. Not all will experience equal success but that's ok too! Many should be able to do part (a) or are my expectations way too high?

(a) Consider the quadratic function f(x) = 4(x+4)².
The graph intersects the line y = k, k>0, in 2 distinct points B and C.
The rectangle whose base is on the x-axis and 2 of whose vertices are B and C
has area 64. Determine the value of k. Show method clearly.

(b) Now let's generalize the result of (a).
Consider the quadratic function f(x) = a(x-h)², a>0.
The line y = k, k>0, intersects the graph of f in two distinct points B and C. The rectangle whose base is on the x-axis and two of whose vertices are B and C has area R.
(i) Explain graphically (not algebraically) why the area, R, of this rectangle is independent of h.
(ii) Express k in terms of a and R. Check that your formula for k gives the value you obtained from part (a).

Algebra 2 Challenge for 2-15-07

The following is an open-ended problem for Algebra 2 students...
Enjoy it but I'd really like to hear how you might implement this in the classroom. Part of homework? A bonus? An open-ended activity in class? Students working independently or in pairs? Part of an assessment? At what point would you use this? At the end of the chapter on quadratic functions?

(a) Consider the function f(x) = 6x - x².
If P and Q are the points of intersection of the graph of f with the x-axis and R is a point on the portion of the graph above the x-axis, what is the maximum area of triangle PQR?


(b) Consider the quadratic function whose x-intercepts are the nonzero numbers p and q, p > q, and whose y-intercept is -pq.

(i) Explain carefully why the graph of this function has a maximum point no matter what the signs of p and q are.
(ii) Write an expression for the y-coordinate of the vertex of the graph of this function in terms of p and q (simplified).
Note: This appears to be a standard problem using the formula -b/2a, but there are other approaches and the result may surprise you.

A Number Theory Problem for 2-12-07: Mission Impossible?

PLS READ THE COMMENTS TO SEE HOW THE LESSON FARED OVER TWO DAYS. DO YOU THINK I HAD TO MODIFY IT A LITTLE OR A LOT TO MAINTAIN THEIR INTEREST?

Thanks to jd2718 for motivating me to include the following famous number-theory conundrum. You can find many references to it on the web but I remember it from the classic text by Hardy and Wright. Give this as a challenge bonus problem or as an activity for your middle or high schoolers. Revise and modify it to make it appropriate for your students. I've tried to turn this into something students can at least attempt, but you could probably make it much more user-friendly...
Remember: My goal is to provide meaningful activities for ALL of your students, not only for honors or accelerated classes. The challenge is to modify them for students who struggle in math. The real test comes when I implement today's problem in my 'skills' class to which I frequently refer. That is my plan -- I'll let you know if it worked or was a disaster! Pls note that these activities also provide practice for those open-ended types of questions that now frequently appear on standardized and state assessments.
Finally, I hope you enjoy these challenges, but my primary target audience is students. I am really interested in reading students' reactions to these.

MISSION IMPOSSIBLE?
Now, boys and girls, we know that, despite all attempts by the most famous mathematicians, no one has yet devised a formula for primes. At one time, it was thought that 2^p-1 would always be prime if p is prime:
2^2-1 = 3; 2^3-1 = 7; 2^5-1 = 31; 2^7-1 = 127 -- all primes. But alas, 2^11-1 - 2047 = (23)(89). Imagine the disappointment!
But I think I found a method that will produce primes EVERY time! If you can do all parts below and prove I'm wrong, you get 5 bonus points. You'll get at least one point for doing part (a).

Ok, I decided to start with my favorite prime 41, since that's the age I once was!
41 + 2 = 43
43 + 4 = 47
47 + 6 = 53
53 + 8 = 61
61 + 10 = 71
71 + 12 = 83
...
Here's your mission:
(a) I listed the first 6 and they're all prime. You can see the pattern, right? Keep the sequence going until you reach 40 numbers. Check that they are all prime by researching a list of primes on the web. So, am I right? Am I famous now?
(b) My method is easy to see but harder to describe algebraically. Here's one way:
To get each number in the list after the first, you can see that I added the next even number to the preceding term. SHOW that the following recursive description produces the first 6 terms:
a(1) = 41; a(n+1) = a(n) + 2n, n = 1,2,3,... where the notation a(n) refers to the nth term. Then explain why this will generate all the terms.
(c) This one is harder but I'll give you a hint: Devise a polynomial that will generate this sequence of numbers when n = 1,2,3,....
Hint: Think of a quadratic like n^2 +.... Just remember, when n is replaced by 1 it has to yield a value of 41!
(d) Show that my amazing sequence 'blows up' when you reach the 41st term. Why? There goes my million dollars from the Clay Institute!
Extension: Is there any other sequence like this? What's special about 41? Happy web-questing...

NML's Response and My Reply on 2-11-07

2-11-07
The following is the response from the National Math Panel to the email I sent on 1-31-07. My reply is underneath. The Panel granted permission for me to print all their replies to my emails sent over the past year. I would very much be interested in reader comments.

Dear Mr. Marain:

This letter is in response to your email sent January 31. I want to thank you for your on-going interest in the important work of the National Mathematics Advisory Panel (Panel).

As we have stated previously, the National Math Panel and its staff have encouraged the public to communicate with us about the Panel’s activities. Consistent with that, we do not object to your reprinting of the e-mails we have sent you on your blog, MathNotations.

As you know, the Math Panel is committed to full and open communication with the public. Our website displays all public comments presented at each Panel meeting and invites the general public to email comments to the Panel at Panel’s Website address, NationalMathPanel@ed.gov. Your messages (and all others we receive from the public) are shared with each Panel member and each Panel member receives also receives a summary of key issues presented by the public through email messages. Further, the Panel Website invites the public to sign up for the Math Panel listserv to receive periodic announcements sent out by the Panel broadcasting activities of the Panel.

Moreover, of particular importance to the Panel is hearing from the public in regards to mathematics education. To facilitate public comment, each Panel meeting has provided the opportunity for Panelists to hear testimony from individuals, organizations, and the general public that are interested in the work of the Panel. The public testimony proceedings of all meetings held to date are available on the Panel’s website.

These various methods of communication are most helpful in keeping the Panel informed by and connected with interested stakeholders. Finally, key announcements by the Panel are published in the Federal Register in keeping with the Panel’s commitment as a federal advisory board as required by the Federal Advisory Committee Act. As you can see, the Panel and the Department of Education are committed to serving the public by ensuring an open process for communications.

In regards to your concern that Panel consist of at least one secondary mathematics teacher, we agree that it is important that a range of experience related to pre-K through 12th grade in mathematics instruction be considered by the Panel. To this end, the Panel consists of several members with extensive classroom experience that total over 75 years combined experience in the classroom as well as experience serving as principals and supervisors of instruction. Moreover, several panelists are deans of schools of education directing pre-service training for secondary teachers. Further, the panel has representation from state commissioners of education and national representation including the National Council of Teachers of Mathematics. We believe that this broad range of expertise meets the requirements of pre-K to 12th grade teaching experience.

I have had an opportunity to look at your Math Notations Blogspot, and I want to compliment you on your efforts to support the learning of mathematics through your “problems of the day” feature supported by key discussions. Thanks again for your interest in the work of the National Math Panel -- I appreciate your most recent inquiry and hope that I have allayed your concerns. I value your efforts to further mathematics education.


Best regards,
Jennifer

Jennifer Graban
National Math Panel Staff
U.S. Department of Education
400 Maryland Avenue, SW
Washington, DC 20202-1200
202-260-1491 (Telephone)
202-401-9027 (Fax)
http://www.ed.gov/about/bdscomm/list/mathpanel/index.html
--------------------------------------------------------------------------------------------
My response:

Dear Jennifer,
Thank you for your reply. Please share the following with Mr. Flawn, Ida and Panel members.
My concerns are NOT allayed for the following reasons:

1. My request clearly implied a CURRENT 9-12 mathematics teacher on this panel, not one who might have once been in the classroom. It is exceptionally important to have someone who can bring both their past and current experiences to the table, since the issues of curriculum, instruction, assessment and most of all the problems associated with teaching TODAY’s children require someone who is teaching TODAY’s children. Further, it would make even more sense to include 2-3 such teachers who represent classrooms of varying demographics. I’m sure you would agree that the problems facing urban teachers is a bit different from those who teach in the suburbs.
2. A combined 75 years of K-12 classroom experience on this panel suggests an average of 75 divided by 30 = 2.5 yrs of experience per panel member. Although the mean is obviously not the best measure of central tendency here, the 75 years seems less than significant to me.
3. Panel members’ titles, degrees, awards, number of publications, etc., do not change the fact that there is only one current K-12 math teacher on this panel. I personally feel that this omission is an expression of disrespect to myself and all other K-12 mathematics teachers.
4. While NCTM may represent a large body of mathematics teachers across the country, they do not replace a current Algebra 2 or Geometry teacher. Nor does their collective thinking necessarily reflect my points of view.
5. Who on this Panel daily confronts the challenges of motivating up to 6 classes of children who often present a myriad of problems that affect their ability to learn? Who on this Panel daily confronts children who come to us with different backgrounds in math because of differences in prior instruction and/or differences in parental support and/or differences in learning style and/or differences in attitude and motivation? Who on this Panel can bring their current experiences in helping children who have a wide range of learning disabilities and who are mainstreamed in our classes? Yes, kids are kids, however the interests and motivations of today’s children reflect today’s post 9-11 society. Teaching conditions have changed and to some degree so have our students.
6. When this Panel makes their recommendations, who will be the ones who will be told to implement the changes? Who will it most directly affect? Who on this Panel will say, STOP: “What you’re suggesting seems to make sense mathematically or pedagogically, but it simply won’t work in real classrooms and here’s why.”
7. There is an accepted truism that the further one is removed from the classroom, the less in touch one is with the realities of teaching children. Yes, we need visionaries and experts who bring a research base to the table. However, nothing replaces the research base of the teacher who has lived through all of the recommendations of all of the experts from the past 30-40 years. The new math of the 60’s following this country’s reaction to Sputnik, the new new math of the 90’s that emphasizes conceptual understanding, problem-solving and communication — yes, many of us have lived through this and much much more. Children continue to learn because of dedicated teachers who have strong convictions of WHAT children need to know despite the latest state or federal initiative or assessment program. Teachers who know how to BALANCE conceptual understanding with computational and procedural proficiency. Teachers who have brought more technology into the classroom but know when to tell children to turn off their calculators or who know how to balance the kinesthetic advantages of using the compass, ruler and straightedge with ‘virtual’ constructions in geometry.
8. I appreciate your nice comments about the Problems of the Day on my blog. However, you did not mention your feelings about all of my other postings regarding critical issues in mathematics education. I would really like your thoughts on those as well. The Problems of the Day are somewhat ephemeral. The real Problems of the Day are the problems this panel is being asked to confront and ‘solve.’

And I’m just scratching the surface here, Jennifer. My single voice on one of hundreds of math teacher blogs is not going to have a seismic effect on the National Math Panel. Of course I knew that when I wrote you. You have already received thousands of emails and statements from concerned educators. Some will agree with my positions and some will not. That is not the point. Even if EVERY email is read and taken seriously by every Panel member, it will not change the constituency of this panel and their perspectives. When a jury makes a decision about the fate of a defendant, the jury is supposed to be of one’s peers. The recommendations of this panel will impact on the ‘fate’ of our classroom mathematics teachers for many years. Where are their peers?

I will have much more to say on these issues. No, Jennifer, my concerns are not allayed...
Thank you for your thoughtful comments and the time you took to reply personally.

Sincerely,
Dave Marain

A Math Exploration 2-9-07

Important Update: Day 10 - received a reply today from Jennifer Graban of the National Math Panel - will be posting her entire statement and my reply by Monday.

Update to Activity Below: This lesson was implemented today with my group of 9th graders. Math is generally a struggle for them. Do you think they completed it in 40 minutes? If not, then how much? Do you think this activity engaged them or they lost interest after a few minutes? Do you think anyone identified what kinds of numbers are 'unsummable'? Before I tell you, I'd be interested in your best guesses!

The following investigation enables the student to explore concepts in factoring, primes, composites, odd vs. even, consecutive integers, averages, median, pattern recognition, arithmetic series, generalization and proof just to name a few ideas!

It may appear to be written for middle schoolers but it can modified for grades 9-12 using algebraic methods (particularly sequences and series) and more sophisticated reasoning. Students may discover new ideas I never imagined when I wrote it. The basic idea of this question is very well known. What might make it different is the journey you and your students will be taking. Ok, if you're not an educator, enjoy the ride (even if it may be simplistic!).

Students should work in pairs. Calculator for checking sums is optional. Allow one period for this, however, additional time may be allocated for further investigation outside of class. This problem is about much more than making an organized list! How would you modify it to make it better? Richer? More suitable for younger children? Older children? What questions might you ask to guide them through it when they appear to be 'stuck'? Is it better not to say anything and let them struggle with it? Is there a place for this kind of discovery? Is it worth all the effort and time 'lost'?

You and your partner are trying to unlock the secret of the 'UNSUMMABLES'!

The number 5 can be expressed as a sum of two consecutive positive integers: 5 = 2+3
Similarly, 6 can be written as a sum of three consecutive positive integers: 6 = 1+2+3
22 can be written as a sum of four consecutive positive integers: 22 = 4+5+6+7
9 = 4+5 but it can also be written as 2+3+4
Ah, but no matter how hard you try, the number 8 cannot be written as the sum of 2 or more consecutive positive integers (try it!!). The number 8 is one of the mysterious unsummable numbers!

(a) In a table format, express each of the integers from 5 through 35 as a sum of 2 or more consecutive positive integers if possible. If it is not possible for some integer, call it unsummable! If you are able to find more than one way to sum a number, that's even better.

(b) Write at least 5 observations and conjectures, i.e., what did you notice and what do you think will always be true. We'll start you off:
We noticed that every odd number can be ________________________.
Note: Think about primes, composite numbers, factors, ...

(c) How many unsummable numbers did you find? What did you notice about these numbers? Can you unlock their secret? A special prize if you can explain WHY they are unsummable!

2-7-07 Challenge Student Solutions!

(Images may be very small - click to magnify)

Student Solution 1



Student Solution 2

Update on National Math Panel and the Real Issues

Day 9 - still awaiting reponse from the National Math Panel...
I absolutely believe individuals from the Panel's administration have assessed my credibility and are judging whether it is worth their effort to reply to me! Again, your emails to Mr. Flawn will influence this. At this point, they see no need to respond...

Folks, again I ask you to send an email to Mr. Flawn the Executive Director, if you believe, as I do, that the panel must include at least one mathematics teacher from Grades 9-12. Even though the Executive Order called for no more than 30 members, an individual could still be included in an advisory capacity providing direct input to the members. Also I urge you to download and read some of the compelling e-mail statements coming from concerned citizens. These were not available a few months ago, but they are now. How much influence they will have on the panel is another question of course.

The most important product that could be produced by this panel will not be a determination of whether Direct Instruction is superior to Constructivism. It will be the identification of the most important math concepts and skills to which ALL children should be exposed at each grade level or course. This would be an extraordinary achievement when you consider that, currently, there are discrepancies in content between classrooms in the same district, between districts in the same county, between counties in the same state and so on. This is of course an oversimplification of the changes needed for our children to develop proficiency in the Fundamentals of Mathematics that will enable them to compete in this century. The whole issue of what defines mastery is a critical piece, tied to assessment and accountability. BUT FIRST WE MUST AGREE ON WHAT MATHEMATICS CHILDREN SHOULD BE MASTERING!

Once again, this is truly the PROBLEM OF THE DAY. Are you prepared to be part of the solution?

Challenge Problem for 2-7-07

Day 8 - still awaiting a response from the National Math Panel...

Pls read the comments re the problems from 2-6-07. There are hints for #1 and a good discussion about #3.

Something different today. This is another one of those weekly online challenges I gave last year just before the AMC Contest. Students found it difficult but those who persisted got it. Perhaps that's the best reason to give these challenges -- to teach persistence, a quality that distinguishes some of the best researchers and problem solvers from the rest. Remember to click on the image to magnify it if it's too small.

Problems 2-6-07

I'm sure most visited today hoping I would forget about my 'cause' and go back to writing problems. Well, I haven't given up the fight for what I believe, but I also will try to keep up my usual routine, although perhaps not as often.

Question #1 below is a challenge for Algebra 2 students who want to test their factoring skills with fractional exponents. This type of manipulation is rare these days in most textbooks, probably because it is seen as 'old-school'. Do our top math-science students still need exposure to these or have these questions been rendered obsolete by 'Symbolic Algebra Manipulators'?
Question #2 is a hard version of a standard SAT problem, comparing two series. Many students think only in terms of summing each series separately.
Question #3 is fairly easy using trig, but I'm encouraging strategies using only the Pythagorean Theorem and one other key idea (thinking about areas when you see a perpendicular!).

Latest Communication to the National Math Panel and a Special Request

As I promised a few weeks ago, I plan to publish my correspondences with the National Math Panel.
I will begin with an email I sent on 1-31-07. As of today, 2-3-07, I have not received a reply. In keeping with my policy of not posting email replies without the permission of the sender, I will not include any of the replies I've received from the panel' s administration.

Currently , the panel consists of only one public school educator (Vern Williams, Math Teacher, Longfellow Middle School, Fairfax, Va.). There is no K-5 or secondary school mathematics educator on this panel. Therefore, I am urging those of my readers, who believe that this commission must include more representation from our most qualified classroom educators, to email the panel:
Tyrrell.Flawn@ed.gov or
Ida.Kelley@ed.gov or
Jennifer.Graban@ed.gov.
formally requesting immediate appointment of at least one additional panel member, a secondary mathematics educator from our public schools.

The panel has ONE more year to conclude their work and issue a final report which may have a significant impact on math education for the next two decades. However, without the direct participation of frontline teachers, this report in my opinion will lack credibility and validity in the eyes of those who are most directly affected, our teachers, and, in turn, our children.

I am asking you to email this post to your colleagues in education or whomever you feel would be interested in supporting this request. Feel free to add a link to this post and/or to reprint this post. This is an opportunity for all concerned citizens, educators in particular, to participate in this process and be heard. My lone voice cannot make a difference. Together, our voices cannot be ignored.

Your comments to this posting will give me some idea of the support I can anticipate. I realize this is quite different from the Problem of the Day you've been expecting from me, but on 2-3-07, this is, for me. the only Problem of the Day that I cannot solve without your help.

My email follows:


Dear Ida and Jennifer,

First, I am formally asking on 1-31-07 for permission to reprint replies to my emails from the Panel in my blog MathNotations. Ida, you have been gracious and supportive of my requests, choosing to respond in a humane way rather than in a form letter or an impersonal invitation (as I received from others) to open meetings which most could not attend. Please reply with your decision as soon as possible. Regardless of this decision, I plan to reprint all of my emails sent to the Panel.

I have maintained from the outset that it is imperative that the Panel establish lines of communication enabling the broadest possible spectrum of views, particularly those of other K-8 educators, but, most importantly, from a secondary math perspective.

Repeated requests for an electronic forum, live chat or similar means of communication have been denied.

I’ve read the Executive order to which you referred and I will excerpt it below:

(a) the critical skills and skill progressions for students to acquire competence in algebra and readiness for higher levels of mathematics;
(b) the role and appropriate design of standards and assessment in promoting mathematical competence;
(c) the processes by which students of various abilities and backgrounds learn mathematics;
(d) instructional practices, programs, and materials that are effective for improving mathematics learning;
(e) the training, selection, placement, and professional development of teachers of mathematics in order to enhance students' learning of mathematics;
(f) the role and appropriate design of systems for delivering instruction in mathematics that combine the different elements of learning processes, curricula, instruction, teacher training and support, and standards, assessments, and accountability;
(g) needs for research in support of mathematics education;
(h) ideas for strengthening capabilities to teach children and youth basic mathematics, geometry, algebra, and calculus and other mathematical disciplines;

Nothing in this order precludes having a secondary math educator on this Panel. In fact, it is a glaring omission. The AP ‘vertical team’ concept suggests we work backwards from calculus to determine what skills and concepts students need coming out of Precalculus. This follows logically to Algebra 2 and Algebra 1 as they are often termed. A secondary educator was not deemed appropriate for this?

Therefore, my second request:
I am formally requesting that there be an immediate reconstituting of the Panel to include at least one secondary math educator. Without such representation, any report would be considered invalid by myself and thousands of other educators in my opinion. It is never too late to get the job done right, is it?

Pls confirm receipt of this email.

Sincerely,
Dave Marain

Problems 2-2-07