Wednesday, January 31, 2007

Problems 2-1-07


Just 3! Have fun!!

Note for #1: Assume k is a positive integer.

Tuesday, January 30, 2007

Problems 1-31-07

Tomorrow's problems focus on sequences and are of varying levels of difficulty. Although #4 may be more appropriate for Algebra 1/2 students, middle schoolers should be able to handle the others. Again, read the comments later in the evening for the answers, comments and solutions. There were some profound ideas expressed about today's questions particularly that innocent-looking quadrilateral problem with the 60 degree angles!

Monday, January 29, 2007

Problems for 1-30-07

At the suggestions of readers, I'm reducing the number of daily problems to 3 or 4. I may also post problems on alternate days. Again, my main purposes in posting these questions is to supplement content, provide instructors with an additional resource of standardized test questions and encourage greater depth of understanding of important math concepts and strategies for problem-solving. Please view these questions in that light. I am very appreciative of your support, comments and suggestions for improving this process.

Sunday, January 28, 2007

A Student Solution of 1-29-07 Challenge

The following is the winning student solution from the original challenge. I know I was impressed!

Friday, January 26, 2007

Special Challenge Week of 1-29-07

The problem below (designed for Algebra 2 and beyond) was part of a weekly online challenge I established a little over a year ago. It ran for 2 1/2 months and the response from students was overwhelming. I posted the problem on our dept web page at precisely 6 PM on Mondays. Students had until 6 PM on Wed to email me their detailed solutions. One student would always wait until 5:59 PM! I would post the answers, comments and the best 2-3 student solutions. I rarely had to give my solutions - theirs were usually better! I may do this on weekends for a Monday challenge here and give you time to play with it for a day. Of course, I know some of you will have it solved in a heartbeat, but I'm hoping you will see why I created this for the students -- an opportunity to delve more deeply and learn how to deal with more sophisticated problems having several layers.

This week's problem is algebraic and not that bad. If you think you have all the answers, you can post them and your comments but hold off on detailed solutions to give others a chance! I will respond to the answers submitted at first. Sorry, it's still in hard to read format...

Thursday, January 25, 2007

Problems for 1-26-07

Uh oh!! The original post appears to have been deleted. I also thought I lost all your comments but now they're back! I may have inadvertently done this or Blogger is doing weird stuff...
I am working on a better way to display the problems and diagrams since I know these images are too small or fuzzy and difficult to copy/paste for use with your students. Pls be patient...

I am planning something new for Monday - a more extended challenge that you can give to your interested students. We used it as part of an online contest - stay tuned

Wednesday, January 24, 2007

Problems for 1-25-07

Important note: The problems are stored as an image and it is currently appearing fuzzy and too small. Guess that makes this fuzzy math! TRY CLICKING ON THE IMAGE TO MAGNIFY IT. I will try to make the image larger but no promises. Blogger was down for awhile and they're having problems with images right now.


Monday, January 22, 2007

Problems for 1-23-07

Above are 4 more problems. Questions 1 and 4 can be attempted by Grades 7 and up. The technique for the probability question is worth demonstrating in middle school (imo).

Sunday, January 21, 2007

Another Definitive Report 'Proving' That Discovery Learning Fails in Science/Math!

If you have an hour or two, you may want to skim through the technical jargon in a recent piece in the Educational Psychologist (2006) with the 21-word title:
Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential and Inquiry-Based Teaching.
If your browser can open pdf documents it will view directly or you may have to save it to your desktop.
This is of course one of the hottest topics in education (see Joanne Jacobs, edspresso and John Dewey) and I've also been addressing this issue indirectly through my own classroom examples. Here's the problem as I see it. Those who want all aspects of discovery learning (and its 17000 other names) to disappear will always cite uses of this approach in black-and-white extremist terms. That is, describing a classroom setting in which students are given a problem to work on with little or no prior instruction or explanation and the students are left entirely to their own devices for much of the classroom period to reinvent knowledge from the ground up. The research piece in question does something like this. It talks in general terms about how the learning measured in such classrooms fall short of the learning in 'direct-instruction' control groups as if there are numerous extreme examples of both kinds of environments.
I would argue that any of us on the frontlines could write our own research entitled, "Why Minimal Interaction With Your Students Does Not Work!"
Here's what I have observed from over 35 years in education from my own instruction and being in hundred of other classrooms watching highly effective and somewhat less effective lessons:
Any teacher of middle school or high school students who expects our current generation of students to sit passively for more than a few minutes while information is being hurled at them will have some issues with classroom management! Of course, the number of minutes is directly proportional to the skills, abilities, motivation, maturity and proximity of the student to the end of the marking period (and the college admissions process)! However, even my AP group will zone out if I stay in lecture mode for more than 15 -20 minutes. Yes, I know that the college professors reading this are silently screaming that these students better 'get their act together' for those college lecture halls and they had better learn how to take notes and stop whining. However, having taught at the college level also, I don't recall a majority of these 'mature' young men and women staying focused throughout the 1 1/2-2 hour presentation and that was a long time ago!
Here's the point if you haven't already exited from this polemic:
Anyone who wants to disprove some educational theory or any other theory for that matter can do so by taking extreme examples that are out of context and out of touch with the real world. For example: Let's disprove the Theory of Parenting that states "Listening to your child is part of effective parenting" by examining models in which parents are NEVER firm, NEVER make decisions and allow their child to make all decisions and learn from them ALL THE TIME! Yup, that's real. Actually, some of you reading this are probably saying, "Yeah, that's what's wrong with society today -- all those darn parents who overindulge and and give their kid everything they want!" But you're missing the point. Effective parents and effective teachers DO LISTEN TO THEIR CHILDREN AND THEN MAKE THE FINAL DECISIONS! In other words they do both.
The most effective lessons I observe often (not ALWAYS!) begin by hooking the students with a provocative question or a demonstration at the beginning to catch their eye. These lessons also connect new learnings to prior learnings, e.g., they review and/or provide a context for the Laws of Exponents by beginning with concrete numerical examples with which students can quickly identify. In another post, I described how an Algebra I teacher distributed a worksheet containing 20 or more numerical examples of exponential expressions which students had to evaluate on their calculator. They were then asked to group several examples and describe what they had in common and to formulate a generalization. This was not an honors class. Having set the stage for the general rules, this outstanding educator then DIRECTLY provided the rules both orally and on the chalkboard in the clearest of terms, then provided several guided exercises (worked-out examples) and then had students do a few Try These. She asked numerous questions and walked among the rows observing and guiding. How would you rate that lesson?
I continue to be offended and deeply disturbed by researchers who attempt to draw conclusions about any method of instruction by commenting on and observing only extreme examples! Any comments? More to follow...

Friday, January 19, 2007

Problems for 1-22-07

I am trying to post these problems before 3 PM each day, Mon-Fri. Again, I depend on your feedback to gauge whether these problems are useful for your students (grade-level dependent of course) or engaging even if you're not a frontline math teacher. I'm trying to choose problems I have written that address current math standards in most states or topics I feel are underrepresented. Moreover, I try to select questions that are nonroutine requiring more time and analysis than students usually give to their homework. Do they fit naturally anywhere in your curriculum? Should our textbooks have some questions like these throughout or just in some ancillary workbook for enrichment? Can you guess my preference!

For Monday, #1 and #3 are Algebra I/II and #2 is for Gr 7-12:

1. If n2+n = x, then, in terms of x, (2n+1)2 = ?
(A) x+1 (B) 4x (C) 4x+1 (D) 4x2 (E) 4x2+1

2. If a number is chosen randomly from the set of 2-digit positive odd integers, what is the probability that the product of its digits is even?

3. For the system x+y > -3 and x-y <5, which of the following must be true?
(A) y >-4 (B) y<-4 (C) x>1 (D) x<1 (E) none of the preceding answers is correct

Introducing Exponents (Middle School)

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

Here's an activity I recently used with my 9th grade class of students for whom math is a struggle. it is equally appropriate for middle school, modified as needed.
I gave them a copy of the famous "As I was going to St. Ives I met a man with 7 wives" nursery rhyme/riddle. This is well-known and many of you have probably used it. We modeled each stage of the poem using diagrams (tree model for example), multiplication in expanded form, exponent form and using a table to record the number of people or objects at each stage and the cumulative total. Pretty standard stuff...

I gave them two assessments.
1. Working with your partner, invent your own St. Ives poem. The first 2 lines had to rhyme and there had to be at least 4 levels as in the nursery rhyme. I received some interesting responses. Here are two:

K.G. and S.T. came up with:
Each girl had 5 designs.
Each design had 5 lines.
Each line had 5 colors.
Designs, lines, colors and girls.
How many all together?

K.C. and E.D. came up with:
Fuzzy Wuzzy was a bear.
Fuzzy Wuzzy had no hair.
Fuzzy did not care he had no hair.
So he went to town and found a crown.
Each crown had 5 points.
Each point had 5 diamonds.
Each diamond had 5 faces.
Faces, diamonds, points and crowns...
How many were going to town?

2. I gave a written individual assessment (pls excuse my lame attempt at poetry!)
As I was shopping in Value-Mart,
I found 9 special shopping carts.
In each cart I found 9 boxes.
In each box, I found 9 hats.
In each hat, I found 9 bats.
Bats, hats, boxes and carts,
How many did I find at Value-Mart?

They were expected to express the number of objects at each stage using both expanded and exponent notation. The calculator was allowed.
This is always a daily learning experience for me. I feel the need to be creative with this group to keep them involved and to change activities often. I know that any approach that engages their learning is worthwhile but I also know that activities do not easily transfer to traditional skills unless we enable that transfer. I also feel that the assessments for this activity were a critical piece to evaluate their learning and my instruction.

Any thoughts? See this as a waste of time? 'Fuzzy Wuzzy' math? Is this 'constructivist' according to someone's definition? Why not just define exponents and provide direct instruction? There are several other ways I get at the meaning of exponents and the ideas behind the 'laws' and 'rules' as well as zero and negative exponents. But that's another posting.

Thursday, January 18, 2007

Problems for 1-19-07

I deeply appreciate the positive comments and support I'm receiving for these problems. At some point my time, energy, creativity and resources will start to decay exponentially! I've been focusing more on high school topics but today's are more mixed. Read the comment for the 3rd question for 1-18-07. One of our astute readers clarified my sloppy wording -- thanks!!

1. The lengths of the sides of two triangles are 7,16,x and 9,20,x respectively. If x denotes the same integer value in both triangles, how many values are possible for x?

2. Borat averaged 150 mi/hr for the first 3 hours of his sojourn through Kazakhstan. What must be his average rate, in mi/hr, for the next 5 hours to average 200 mi/hr overall? (Will he get a speeding ticket in view of the fact that he is so famous?!?)

3. The square root of an integer added to the square root of one less than that integer is greater than 100. What is the least possible value of the integer?
[Comments: What % of your students would reach for the calculator for this one rather than estimate using number sense?? How many would try an algebraic solution?]

Wednesday, January 17, 2007

Problems for 1-18-07

Some SAT-types for today (more reasonable)...

1. A rectangle is inscribed in a semicircle so that the base of the rectangle lies on the diameter. If the diameter is 4 and the base of the rectangle is twice its height, what is the area of the rectangle? (Sorry, no diagrams today!)

2. If 1 + 2 + 3 + ... + 2006 = x, then, in terms of x, 2 + 3 + 4 + ... + 2007 = ?
(A) x - 1 (B) x + 1 (C) 2x (D) x + 2006 (E) x + 2007

3. On a certain 20-question math contest, 12 points are awarded for each correct answer, 4 points are deducted for each incorrect answer and a blank receives zero points. What is the fewest number of right answers needed to score at least 100 points if no questions are skipped?

Note: Although many students will use guess-test here (which is fine), we're looking for an algebraic approach as well.

Tuesday, January 16, 2007

Problems for 1-17-07

I may have comments about yesterday's problems later. mrc did a great job - read his comments, answers and solutions.

Today's problems (shown above) involve more geometry. They will appear as images (including the text).
These problems review 30-60 and 45-45-90 triangles, radicals, slope and/or similar triangles.

Regents-like Assessments for NJ (and other states) in Sci and Math?

As I mentioned the other day I will continue to post about the American Diploma Project developed by Achieve which, IMO, will have a significant impact on curriculum and assessment in many states (26 out of 50 are currently members of this consortium). Whether we like it or not, a more standardized curriculum for math and science is coming and you all know my opinion on this. The following is excerpted from a statement by Lucille Davy, Commissioner of Education for NJ:

...However, we believe that New Jersey’s assessments must serve a purpose greater than simply meeting federal compliance requirements. As with all of our statewide tests, New Jersey’s science assessments must help ensure that students are prepared to compete for post-secondary educational opportunities and careers in a highly demanding global economy. Our science assessments in particular must provide an authentic link to that world by embodying and measuring the science skills that students must master to succeed in college courses and in their careers, as they compete for those opportunities against highly trained and highly motivated students from throughout the world.

...More so than language arts and other content areas, mathematics and the sciences demand discipline-specific instruction and assessment. Increasingly, states such as those involved in the American Diploma Project (ADP) consortium, of which New Jersey is a member, are deciding to implement end of course measures in science aligned to specific proficiencies in biology, physics, chemistry, and environmental science. ADP is also recommending that states consider such end of course assessments in the mathematics disciplines. Several states, such as Maryland and Indiana, already have such assessments in place.

In consultation with our partners in the 25-state ADP consortium and following more recent discussions in December 2006 with an advisory panel of New Jersey science educators and stakeholders, the New Jersey Department of Education will be moving to end of course science assessments, starting with biology in 2008. This direction will entail not merely a redesign of science testing specifications, but recommendations to the State Board for establishing specific course requirements in high school science, including a requirement that all students take biology...

This is just the tip of the iceberg folks. BTW, what do you think Commissioner Davy's reply would be to educators who argue that an Algebra 2 assessment for all (currently on the table) is unrealistic given the large population of children in the state who are functionally innumerate? Do we push for higher standards now or wait until 2014 when no child is left...
Food for thought... What do you think?

Monday, January 15, 2007

Problems for 1-16-07

1. 9 x 10499 + 9 x 10501 = k x 10499; k = ??

2. On a certain trip, Paris covered 80% of the distance by car and the rest by bike. Her average car speed was 10 times her average bike speed. Her time spent biking was what % more than her driving time?


AB = AC and triangle DEF is equilateral. Points D, E and F lie on the sides of triangle ABC. The degree measure of angle A is x and the measure of angle FEC is x/2. What is the degree measure of angle DFB?

Note: Figure not drawn to scale and the answer is numerical! Show method and explain your reasoning.
[No answers/solutions for now!]

Sunday, January 14, 2007

In Observance of Dr. King...

In observance of Dr. Martin Luther King's Birthday, I will not be posting a Problem of the Day for Mon Jan. 15th.
Instead, I ask you to consider what Dr. King would say if he were by some miracle to be with us for just one day. Would he be amazed by how far human rights have come in the past 40 years or would perhaps he feel that he went to sleep for just one day...

I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin but by the content of their character.

I have a dream today!

I have a dream that ... one day... little black boys and black girls will be able to join hands with little white boys and white girls as sisters and brothers.

I have a dream today!

Friday, January 12, 2007

National Standards Dialogue with Professor Schmidt

William H. Schmidt is a University Distinguished Professor at Michigan State University and is currently co-director of the Education Policy Center, co-director of the US China Center for Research and co-director of the NSF PROM/SE project and holds faculty appointments in the Department of Educational Psychology and the Department of Statistics. Previously he served as National Research Coordinator and Executive Director of the US National Center which oversaw participation of the United States in the IEA sponsored Third International Mathematics and Science Study (TIMSS). His current writing and research concerns issues of academic content in K-12 schooling, assessment theory and the effects of curriculum on academic achievement. He is also concerned with educational policy related to mathematics, science and testing in general. He was awarded the Honorary Doctorate degree at Concordia University in 1997 and received the 1998 Willard Jacobson Lectureship from The New York Academy of Sciences and was recently elected to the National Academy of Education.

I have had the pleasure of communicating with Prof. Schmidt via email a few times this past year. He has given me permission to quote him from these emails. I discussed my campaign to promote a national math curriculum and my appeal to the National Math Panel to listen to the teachers on the frontlines. He was very gracious and gave me cause for optimism for the future. I strongly urge you to read the complete text of the PBS interview he gave to Frontline in 2001. His words seem as current today as they were then. How far have we come in developing a 'coherent vision' since his characterization over a decade ago of our math curriculum as 'an inch deep and a mile wide?' In the interview he refers to Achieve, a non-profit group of business leaders and governors. I will have more to say about this over the next week or so. This organization is already having a significant impact on math and science curriculum in my state and 25 others. I advise all our readers to take their American Diploma Project report (can be downloaded from the site) very seriously. Over the next few weeks, I will be excerpting and commenting on the high school mathematics benchmarks in this document.

Here was his reply on 2-10-06 to an email I sent him. I do not have a copy of my original email but you can read between the lines...

No, I don't disagree with your comments at all. I have been very an
outspoken advocate of national standards. I believe it is rather
unbelievable that we do not have what most other countries in the world do
have. The fact we do not have national standards leads to much of the
chaos, not only in the curriculum, but in the testing, professional
development, and teacher training, as well. I think there is a greater
tendency right now to move in this direction and I've actually been asked
to prepare some comments for a national symposium that is being together on
the pros and cons of national standards.

So, do not despair. There is hope.


* * * * * * * * * * * * * * * * * * * * * * * * * * * *
William Schmidt
University Distinguished Professor
Co Director University Policy Center
Co Director US China Center
Co Director Center for the Study of Curriculum
238 Erickson Hall
College of Education
Michigan State University

A Challenging Problem...

I encourage you to visit jd2718 and the stimulating discussion over the 'innocent-looking' geometry problem he posed. Read the comments below the problem. I struggled with this problem myself, shared it with my department and ended up constructing the counterexample using traditional tools, not Geometer's Sketchpad! The author of this excellent blog made a compelling point about our need to struggle with difficult problems ourselves as math educators. It's not only humbling, but reminds us of the need for our students to be similarly challenged. Most mathematical geniuses had inspiration but they had the one quality that separated them from the rest -- they didn't give up. Just ask Professor Wiles from Princeton!

By the way, there's another challenge geometry problem there today involving 3 circles. Guess I'm going to lose more sleep...

Thursday, January 11, 2007

Problems of the Day 1-12-07

Hope you enjoyed the 'inserting the digit 9' problem. Algebraically, 9 is indeed special:
(100t + 90 + u) - (10t + u) = 90t + 90 = 90(t + 1), where t = tens' digit and u = units' digit of the original number.

Some probability today...

For Grade 7 and above:
This just happened. A bio teacher asked the math teachers for a mathematical explanation of a standard Punnett square problem. The question asks for the probability of a Brown-eyed offspring given a Brown-eyed (dominant) male parent and a Blue-eyed (recessive) female parent. Everyone knew the answer was 75% but a clear explanation was desired.

Here are the Punnett diagrams for the two equally likely possibilities.
B = Brown; b = blue; B is dominant so any BB or Bb combination produces a Brown-eyed child. To refresh your student's memory, recall that the paternal genotype is in the left column and the maternal genotype is across the top.

Case I (BB male parent, bb female parent)
Note: Spacing may be off - the b's at the top of each table should be indented to appear as headings in the 2nd and 3rd columns. I didn't have time to enter the html code for Tables.

B Bb Bb

B Bb Bb

Case II (Bb male, bb female)

B Bb Bb

b bb bb

It's easy to see that in 6 of the 8 cases (75%), the offspring will have brown eyes. (fairly standard stuff you can find in a bio text, but it's always fun to play with this). The science teacher recognized that the probability of a Brown-eyed offspring was 100% in Case I and 50% in Case II, but didn't want to simply tell them to average these. He wanted a more precise method that would work in other problems.
See if your students can explain why it's 75%!

Comments: Here's the explanation given by one of our outstanding math teachers, who thoroughly understands the probability concepts underlying the Punnett square. She said it is like an average but the key is that the 2 cases are equally likely each with a 50% chance of occurring. Thus, (0.5)(1) + (0.5)(0.5) = 0.75 or 75%. Voila! Her explanation allows students to handle more complicated genetic problems with 3 or more cases. Hey, this stuff isn't calculus but math-science articulation is always healthy and doesn't happen as often as it should, right?
Now for the challenge problem for your hs students who want to be or believe they are probability experts. Remember, most of us feel insecure about these problems just as our students often feel about math in general.

You have 8 playing cards: 2 jacks, 2 queens, 2 kings and 2 aces. If 4 of these are dealt to you at random, what is the probability you will have at least one matching pair?

Answer (no solution yet): Approx. 77.1% or 27/35 [corrected from earlier posting - thanks to Eric!]
Send me your comments.

Wednesday, January 10, 2007

Problem of the Day 1-11-07

No comments the last 2 days but many visitors are looking at these - I think! Feedback is critical for for me so I'd appreciate any suggestions. I'm thinking about having 2 a day, one for Gr 5-8 and one for Gr 9-12, but I don't know if I can get this going or maintain it!

Grade 5-8 Activity
Goals/Standards: Those relating to prealgebra, patterning, place value, factors, multiples, making conjectures, verifying, etc.
Class Time Required for Activity: 20 minutes

Your mission, should you accept it, is to discover the effects of inserting a 9 in the middle of a 2-digit number. You and your partner have 10 minutes to make your conjecture. You will start by collecting data from several 'sample' values and recording your results in a table. Mission Control (that's me!) will start the table for you: (scroll down to see it!)

Original #

New #









  • Continue the table on your own, choosing 7 more random values in the 30's, 40's,... through the 90's.
  • Look at the differences and formulate a rule for finding the differences using only the original numbers. Your rule may use any of the following: words, numbers, algebraic expressions.
  • Make sure you verify your conjecture (that is, thoroughly TEST it).
  • Be prepared to explain your 'theory' and demonstrate it. A student or your Mission Control Commander may throw a random 2-digit number at you. You will have to correctly compute the difference mentally in 3 seconds or less to receive one extra bonus point!

Now for the extension: Ask students to think about the following outside of class and submit their thoughts to you or the class in an electronic class forum, wiki, or however:
Is there a similar rule for inserting other digits into a 2-digit number or is NINE special? Show work or explain your method.

Tuesday, January 9, 2007

Problem of the Day 1-10-07

Not much response to Tue problem, so I guess it wasn't too exciting. Those who are familiar with Euler's totient function, phi, could have approached it like this:
phi(144) = phi(2^4) x phi(3^2) = (16-8)(9-3) = 48.

Wed problem (posting early) is all about Algebra 2 (rational expressions) and an interesting connection to Myrtle's post the other day. I typed it in word using eqn editor, saved it as a png and inserted it but the size and quality are difficult to tweak. I'm still learning! Despite fuzziness you should be able to read it. Try clicking on the image - it might appear in magnified form. On the surface it's just a mechanical computation but it has an interesting application to Egyptian fractions. Try it for other odd denominators!

thanks for the links...

Thanks to Understanding at for commenting on how the hypotenuse problem was used at a department meeting. Here's a copy of my reply:
I really appreciate your comments--
I'm hoping these 'innocent' little exercises will stimulate dialogue about HOW to develop problem-solving skill and reasoning. Most of us view problems such as these as math contest problems applicable only to the few. If I ask my 'basic' class this question and do not characterize it as an 'honors' problem, would they be willing to try it? I've done it! I do know they could guess 6 and 6 for the sides, we could compute the hypotenuse, then move on to 5 and 7, 4 and 8, etc. I would model the use of a table to record the data and then suggest letting one leg be 5.9, asking them to determine the other, then applying the Pythagorean Theorem. I've already shown this group how to use the STO key on the ti-83 to enter different pairs of values for A and B, then enter a formula like SQR(A^2+B^2) into the home screen. Press ENTER to see the result. Use STO again to enter new values, then 2nd ENTER a few times to retrieve the formula, etc. I had 2 of these students teach the STO method to my BC Calculus class! Betcha think I'm fabricating this story!
BTW, it's great that you used this question for staff development. Thanks again, encourage your teachers to look for my daily Problem of the Day! I welcome constructive comments.
Dave Marain

Monday, January 8, 2007

Warmup #4 for Tue 1-9-07

If n is a positive integer from 1 to 144 inclusive, how many fractions of the form n/144 are in lowest terms? For example, 1/144 is in lowest terms and 2/144 is not.

Grades: 5 and up
Answer: 48

Comments and Possible Solution: [Change the denominator to 12, 24 or 36 to make it a quicker warm up.] This is a fairly well-known type of problem that is accessible to students at many levels. It often appears as a math contest problem, but it can be the basis for a class activity. A 5th grader can, with a partner, list every fraction and cross out the reducible ones. Most students quickly recognize that they can eliminate all even numerators. From the remaining 72 numerators, the odd multiples of 3 (3,9,15,...) can be eliminated. There are 144/3 = 48 multiples of 3 in all, half of which would be odd. Eliminating these 24 numbers leaves 48 values.

Additional comments: There are many approaches here. The idea of counting the reducible fractions and subtracting is a powerful strategy that does not occur to everyone at first. In the upper middle grades and high school, a problem like this can be used to develop number theory concepts, such as the concept of relatively prime and Euler's phi function. Number- theoretic ideas need more attention in middle school IMHO. See if you can find this in the new NCTM Curriculum Focal Points document.

Sunday, January 7, 2007

Warmup #3 for Mon 1-8-07 A Quickie??

The last one probably made some viewers nauseous, recalling that pit in your stomach when walking into math class, so I'll keep this one shorter and less technical...

This is a well-known problem but I've added some variations:

Version 1 (Grades 3-6):
Using the digits 1,2,3 and 4 exactly once and in any order, what is the largest possible product you can make? Also, list the factors used. For example, 312 x 4 = 1248 is one possibility, but that may not be the largest. You can of course use a 2-digit number times a 2-digit number.

Answer: 1312 (I'll leave it to the reader to determine the factors!)
Comments: There aren't that many combinations here so 'guess-check with the calculator' would be the method preferred by most students. Consider our role as educators when presenting such a problem, however. What can we do to maximize the benefit from problem-solving? Would having students work in teams be more effective or not? Is it a no-brainer here to allow the use of the calculator, allowing the student to focus on the ideas behind choosing factors and making discoveries? Should we recommend that students record each attempt to develop a more systematic approach? We know how most 8-11 year olds jump around and 'push buttons' hyperactively! What is our role in their development? Most educators give these 'fun' puzzle problems, but do we take the time to plan the best way to present them? Quick warmups don't allow that, now do they??

Version 2 (Grades 5-8):
Same as above, but use the digits 1,2,3,4 and 5 this time! Record all attempts.

Answer: 22,412 (if you find a larger one let me know).
Comments: Using 5 numbers dramatically increases the number of combinations, an interesting combinatorial problem in its own right! How would you extend this problem to make it richer? If we use any 5 digits such as 2,5,3,9, and 0 would there be a pattern students could recognize? Would it be worth assigning that for extra credit? Does this problem develop number sense, pattern-based thinking and estimation skills? Is the calculator a much more necessary tool for this version than for Version 1? Will I ever stop asking 'obvious' rhetorical questions?!?


Now, You Don't Have to be Registered to Comment!

Be kind, I am still catching on! Pls feel free to post your comments at any time. Well, at least I think I know why there haven't been any comments thus far!!

DY/DAN - I give it an A+

Dan Meyer's blog is bold, refreshing and energizing. Dan is a math teacher (like myself, don't hold that against him!) who proves that creative juices and eloquent writing can flow through those left-brained minds! I read his posts every day to remind myself why I am still in this profession and to be re-energized. He has a perspective on assessment and accountability that belies his age and has developed clever ratios like the DisruptiveStudent'sSatisfaction to measure his effectiveness in dealing with students whose sole purpose is teacher burnout. You may not agree with all of his views but you will appreciate his pizazz!

Saturday, January 6, 2007

Legs of a right triangle revisited...

I haven't read any comments yet about this problem, but I promised an algebraic solution...

Represent the legs as 6 + x and 6 - x. It takes experience for students to think of something like this, but once they see it, they'll use it again.
Then the square of the hypotenuse will be (6+x)^2 + (6-x)^2 = 72 + 2x^2, which is minimized when x = 0, i.e., when the legs are both 6 and 6. Yes, the algebra is challenging here for most, but isn't it wonderful to see how this innocent problem connects to quadratic functions! BTW, most students will buy into the idea that minimizing the square of the hypotenuse leads to the desired result.

Using the same representation, the area = (1/2)(6+x)(6-x) = (1/2)(36-x^2). This is maximized when x = 0. Ah, the power of algebra. Of course, this derivation is not appropriate for most middle schoolers, but many will internalize this when substituting values for the legs. This can also be facilitated by choosing values like 6.1 and 5.9 and using the calculator to show that the hypotenuse will be slightly longer than 6radical2 and the area slightly less than 18.

I'm not suggesting that problems such as these 'a curriculum make', nor can one do these frequently, nor are they appropriate for all! I am suggesting that our students need to be exposed to some of these with time provided for discovery and dialogue at the high school level as well as middle school. Do you find many examples like these in our textbooks? If you do, pls send me the ISBN #. If not, I will try to provide a few more of my own!!

Stay tuned for Monday's Warm Up Problem. It's much more of a 5-minute opener and is designed for grades 5-8.

I'm risking losing a good part of my audience here by not ranting about inequities in education and the need for a national math curriculum. Never fear, I will return to that as well. The 'inner math child' in me is dominant right now however!

Friday, January 5, 2007

'Warm Up' #2 (not a 5-minute one, perhaps)

[After writing several thousand SAT-type problems over the years and authoring many math contests here in my state, I should be able to come up with a few of these. Now if I could only locate those 250 file folders with the materials...]

Some geometry today...This question is easily guessed by those with a passing knowledge of the Pythagorean Theorem, but its usefulness may go far beyond this. Sorry, but if you were looking for a quickie, this isn't it. You could still use part of it and limit the discussion for now to making conjectures and doing some verification (not a full-blown proof).

Determine the least possible length of the hypotenuse of a right triangle whose legs have a sum of 12. Also, what would be true about the area of this triangle compared to other right triangles with the same condition on its legs?

Comments: Target audience -- Grades 7-12. Again, I apologize. This is far more than a 5-minute class opener. I usually use this kind of problem to encourage discovery and an inductive approach to problem-solving before getting into the issue of proof. Middle schoolers who have learned the Pythag rule can be shown how to begin the investigation by setting up a table of values with 3 or 4 columns (the last one could be area), computing various hypotenuses, areas and making conjectures. I'll leave the issue of calculator use up to you! Lot of meat here but most will accurately guess that the isosceles right triangle solves it. No calculus needed to PROVE that the isosceles case minimizes the hypotenuse and maximizes the area! There's a wonderful algebraic approach that proves both results with one broad stroke! What approaches will your students devise??? Pls share if you can or email me and I'll post a few.

Odds and Evens...

The WaPo recognition (one of their top 10 edublogs) has certainly put this blog on the map and I see I'm being cross-referenced by many outstanding blogs. I will add these to my roll once I get more in tune with the procedures of blogging. Remember, it's hard to teach an 'old dog ...'

Several have noted that my posts seem to be going off in many different directions and that I haven't yet chosen an identity. Absolutely and by intent! Math ed is my main passion but as I am wiping the nose of my 11-month old foster baby at this moment, and getting ready to wake up the 10, 13, and 14-year olds (my 18 yr old is like me, she gets up by herself), I realize that my interests in education will always transcend the best way to introduce quadratic functions!

The best part of what Jay Mathews did for me was to open up my eyes to the hidden talents and unique perspectives of so many in the edublogosphere (or whatever the jargon is!). To those who have recommended this blog as provocative, honest and worth a quick read, I am indebted and will return the favor. Just give me a little time! BTW, I'm not going out of my way to foment anything, I'm simply committed to calling the shots as I see them after 35 yrs in public education -- let the chips fall where they may. [My district has been very supportive of my efforts!]. I've seen every possible experiment in education and while I've survived, I wonder about the generations of children who were the subjects of these experiments. I guess they've survived too and some are now blogging like mad to tell the truth as they see it too!

Thursday, January 4, 2007

Math Warm Up #1

I'd like to post occasional 'Problems of the Day' that I've written and shared with colleagues. Pls pls share your favorites too (but leave answers and solutions to be kind!).

Here's one for today, appropriate for grades 7-12:

Let N$ denote the largest odd positive integer factor of the positive integer N.
For example, 6$ = 3, 7$ = 7 and 8$ = 1.
If N < 100, what is the largest value of N for which N$ = 3.

No calculator allowed!

and the answer is...
I'll post it around noon today (yup, that's hypocritical of me) but I know you'll have it solved way before then! BTW, the math skills and concepts needed are middle school but most students struggle with the notation (hence the name of my blog!) and the meanings of key terms like factor. There's also a lot of verbiage embedded in just a few words. Have fun!

Ok, here it is: 96
Comments: Many students who attempt this question (some will simply give up and say, "I don't understand it"), will come up with 99 since 99 is divisible by 3. However, the largest odd factor of 99 is 99, not 3! This suggests we need an even number divisible by 3. The next candidate going down by 3's is 96 = 32 x 3. All factors of 96 will be even except for 1 and 3. Seems like such an easy problem, right? Way too easy for high schoolers? Try it and comment on the results and the grade level tomorrow!

A Bad Math Pun...

Thanks to Joanne Jacobs for connecting me to math jokes via NYC Educator. Heres my lame contribution.
One-billionth of a defective iPod: A nano-second (ugh, groan,...)

Wednesday, January 3, 2007

Birth of a Blog... Chapter Zero...

I thought I would briefly sketch how this blog came to be. Educators tend to live in cloistered environments where no one but their students know what they really do in the classroom on a daily basis. I can recall many times that school districts would talk about peer observations and sharing/co-planning of lessons (aka 'lesson study' in Japanese schools), but this never seemed to get off the ground. I began to reach out to others by attending conferences and doing presentations to math teachers in the late 80's and 90's. With the advent of the WorldWide Web, when electronic discussion groups, forums, message boards and the like took off in the 90's, my sphere of contacts increased dramatically. Now my views were being critiqued mercifully and mercilessly by hundreds! I began by replying to questions posed by other math educators and then progressed to sharing instructional strategies I'd used in the classroom and my philosophy about education in general. I learned to be more selective in my posts and to become a better listener (still learning!). I lurked and entered dialogues only when I felt I had something meaningful to contribute. I also posed my own questions, seeking new ways of introducing a math topic, then sharing these ideas with my colleagues. I was an active contributor to a few math listserves, most notably the AP-Calc listserv, but also math-teach, math-learn, and more recently Math Talk. A couple of years ago I decided to create a new Yahoo! group, called MathShare, which is still in existence, although it is not nearly as active as it once was. My intent was to invite middle school and high school math teachers to share ideas and have a place to find new resources for their classrooms. A place to discuss the real issues we face every day and not get bogged down in rhetoric or philosophical conflicts.

However, I began to realize that my perspective was changing. I was becoming more concerned with national math issues such as standards, curriculum and assessment. I began to have serious concerns about what students were expected to 'know and be able to do', the buzz phrase of the 90's. I became involved in the development of my state's math content standards, curriculum frameworks and assessments at that time. I started challenging some of the directions being taken by math leaders in my state and on the national stage. I saw confusion among teachers about maintaining a balance between conceptual understanding and developing and maintaining skills. I argued in every venue that success in mathematical problem-solving is not only a result of discovery and communication, but also a result of having a strong foundation of skill and knowledge of content. At the mere mention of skills, others recoiled and I began to feel I was a stranger in my own land, but I relentlessly voiced my message that we were headed on a dangerous course because the balance between knowledge and discovery was tilting too far in one direction. This past summer I decided to make a stand regarding a movement toward a national math curriculum and posted an appeal for support to math educators everywhere on MathShare. Some leading educators and moderators of similar discussion groups expressed grave reservations about my ideas, but I had to do what I believed was right. I sent many impassioned emails to the newly appointed National Math Panel and received some kind replies and some boilerplate responses as well. I suggested that if voices like mine could not be heard directly, then someone else would listen, perhaps education journalists.
To be continued...

A Thank You to Jay Mathews and Joanne Jacobs

Considering that this blog has been in existence less than 30 days, I am deeply appreciative of the vote of support given it by Mr. Mathews and Walt Gardner in yesterday's Washington Post listing of edublogs worth looking at. Further, many of you who have discovered this blog, have come from Joanne Jacob's referral -- thank you, Joanne. I recommend that you visit the other edublogs listed in the article. I've learned much about blogging from reading these on a daily basis. I plan on posting some background of why and how I started blogging and, in particular, why I began connecting with Jay several months ago, but for now, I'd like to share a sentiment that was emailed to me by a close friend. You may have already seen this, but it does send a message about one's values that bears repeating (thank you Elisa for sharing this):

One day a father of a very wealthy family took his son on a trip to a farm with the firm purpose of showing his son how poor people live. They spent a couple of days and nights on the farm of what would be considered a very poor family.

On their return from their trip, the father asked his son, "How was the
trip?" "It was great, Dad." "Did you see how poor people live?" the father asked. "Oh yeah," said the son. "So, tell me, what did you learn from the trip?" asked the father.

The son answered: "I saw that we have one dog and they had four.

We have a pool that reaches to the middle of our garden and they have a creek that has no end.

We have imported lanterns in our garden and they have the stars at night.

Our patio reaches to the front yard and they have the whole horizon.

We have a small piece of land to live on and they have fields that go beyond our sight.

We have servants who serve us, but they serve others.

We buy our food, but they grow theirs.

We have walls around our property to protect us, they have friends to
protect them.

The boy's father was speechless. Then his son added, "Thanks, Dad, for showing me how poor we are."

Isn't perspective a wonderful thing? Makes you wonder what would happen if we all gave thanks for everything we have, instead of worrying about what we don't have.

Appreciate every single thing you have, especially your friends!

Life is too short and friends are too few.