MathNotations

Here are the last two math challenges I just tweeted for middle schoolers and beyond. You may want to use this as a fifteen minute activity to improve reading, review basic terms and concepts, develop reasoning and writing in math. There was an error on the 2nd question as it originally appeared on Twitter. I then corrected it.

List the nine 2-digit primes which produce prime numbers when their digits are reversed.




List the SIX 3-digit primes which produce primes when their digits are written in ALL possible orders. 137 fails b/c 371 is not prime.

For both questions students should work in teams of 2-4.

For the first question, students should not be allowed to use a calculator!

For the second one, have them experiment with a calculator for a few minutes. If a student thinks they found one, their teammates must verify it! After 3 minutes ask: "Have you noticed that the numbers you're looking for cannot contain certain digits like 2. What digits and why? Discuss it and one member of the team must record the team's findings and provide a written explanation!

After 3-4 more minutes, have them refer to a table of primes online (or print it and hand out a copy to each team). If they don't find it within the 15 min time limit, have them finish it for extra credit for the next day.

Here is one of the numbers: 113. Good luck!


"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught." --from South Pacific

Posted by Dave Marain at 10:22 AM 2 comments

Labels: explorations, investigations, math challenge, middle school, primes

The full version in one of its many many variations:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!

Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.


The Problem in the Video Below:


If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.

[埋込みオブジェクト:http://www.youtube.com/v/P_VCYl0zdts&hl=en_US&fs=1&rel=0]


More Advanced and Efficient Algorithms


(1) We assume from the "constant rate" assumption in the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.

Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)

Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.

(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:

(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!

(3) I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!

I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.



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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

Posted by Dave Marain at 2:37 PM 3 comments

Labels: average rates, math videos, mathnotationsvids, middle school, proportions, ratios, work problems

Since pi day fell on a Sunday this year, we should still be celebrating it today. Besides, March should be declared pi-Month!


It is always fascinating to see how readership (or should I say one-time viewership) always picks up around March 14th every year! I feel obligated to add another pi Day activity or exploration in addition to those I've posted the past three years. By the way, the pi Day Scavenger Hunt is the most popular post by far and I'm not even the one who thought of that idea!


Despite the title of this post, I did not upload a video for this activity. However, there is another video on the MathNotationsVids Channel on YouTube.

Here is an investigation/exploration/activity for middle and secondary:

Part (A)
(i) List all ordered pairs of positive integers (m,n) such that
(1) 1 ≤ m ≤ 10 and 1 ≤ n ≤ 10
(2) m and n are divisible by the same prime p

For example, (m,n) could be (6,9) since 6 and 9 are each divisible by the prime 3.

(ii) Should (9,6) also be counted?

(iii) Another way of expressing Condition (2) is:
The _______________ of m and n is ________ one.
Answer: gcf; not equal to or greater than

(iv) If you listed and counted correctly, you should have found there are 37 ordered pairs which satisfy both conditions. If not, have a partner check your list. Each of you should be checking each other's lists routinely.

Part (B)
(i) Explain, using the multiplication principle, why there are 100 ordered pairs which satisfy Condition (1) above.

(ii) ) What % of all the possible ordered pairs from Condition (1) are relatively prime. If you have immediate access to the internet, research this term before asking your teacher what it means!

(iii) In probability terms, you could say:

If one of the 100 ordered pairs (m,n) from Part (A) is selected at random, the probability that
m and n are relatively prime is ____%.

Part (C) (more advanced)

If you have access to a graphing calculator, such as the TI-84 or TI-Inspire, enter the following program into memory (call it RELPRIME):

:ClrHome
:Prompt N
:0 → K
:For (X,1,N)
:For (Y,1,N)
:If gcd(X,Y) ≠ 1
:K+1 → K
:End
:End
:Disp K
:Stop

Using this program, complete the following table:

N..........Total # ord. prs..........# of not rel prime prs........% rel prime prs

10.........100.............................37....................................63%

20.........400............................ 145.................................

30

40

50

100

Notes:
K represents the count of ordered pairs which are not relatively prime
N represents the greatest value for the integers
gcd is found by going to MATH, then NUM, then 9:gcd(
The program slows down considerably as N increases. For N = 10, it checks 100 ordered pairs which may take only 2-3 seconds. For N = 100, it checks 100^2 pairs, which could take up to 4-5 minutes. Be patient!!

Conclusion: So what does all of this have to do with π ?
Well, as N increases without bound in the program, the probability that a randomly chosen ordered pair of positive integers (with values up to an including N) will be relatively prime approaches 60.7% rounded.

From out of the blue, compute 6/π²...
Want to know why? Well, that requires some advanced machinery involving infinite products, infinite series, and the Riemann Zeta Function! Perhaps, I'll do an informal development in a video. I love this stuff...


----------------------------------------------------------------------------------
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Posted by Dave Marain at 9:07 AM 6 comments

Labels: activities, explorations, investigations, middle school, pi, pi day, probability, relatively prime

UPDATE: SEE THE NEW VIDEO BELOW EXPLAINING THE PROBLEMS IN THIS POST. PLS SUBSCRIBE TO THE NEW MathNotationsVids Channel and share your comments and ratings!




The following video is available on my new MathNotations Videos Channe l.


This particular video is a 10 minute discussion of developing the Multiplication Principle of Counting. It is designed more for the instructor than the student although it may be helpful in clarifying this important concept. The focus is on using multiple representations to reach the widest variety of learning styles. It is appropriate for any teacher of mathematics but particularly for the middle school teacher or those who work with students who struggle with math concepts.


After watching the video (or skip it if you wish) scroll down to the two problems below. These are more sophisticated than the one in the video and they require application of other concepts as well. I believe they are appropriate for 8th graders through high school. A full investigation with questions is provided for each problem. Feel free to edit them to your own tastes or as needed for your students.

[埋込みオブジェクト:http://www.youtube.com/v/IAhJHV-qmlM&hl=en_US&fs=1&rel=0]

Comment: I haven't mentioned how the Pigeonhole Principle can be applied to these two problems. I'll leave it to my astute readers to comment on that!




Ok, here's another video explaining the two problems above. I hope you will subscribe to my new channel on YouTube, MathNotationsVids .


[埋込みオブジェクト:http://www.youtube.com/v/9_P_wyk7N4M&hl=en_US&fs=1&rel=0]
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Note: I've been asked why I'm using these signatures on my posts, particularly the 2nd one. Well...
"It's my party and I'll try what I want to!"
(Apologies to Lesley!)

I'm sure some of my devoted readers can figure out why I included Schopenhauer's quote and the 2nd one is really all about education, isn't it?


"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Posted by Dave Marain at 7:38 PM 2 comments

Labels: counting problems, Instructional Strategy Series, middle school, multiplication principle

I've been working on a new website which I will share with you when ready but I haven't forgotten my faithful readers who may have forgotten me!

There are so many issues in mathematics education that it would take forever to update you on all of them, however, I know that you are already aware of most of these.


Some Significant Current Issues in Math Ed

• Moving Inexorably Towards Common Standards in Math
• Teachers Need a Clear Curriculum Map/Content Guide rather than Standards!
• Rapid Push Toward Including Several Open-Ended Questions on State or Common Assessments is Slowing Down. Can you think of the major reasons for this?
• Joel Klein's Education Equality Project whose goal is to close the Achievement Gap

Of course, most of you have already skipped down to the Challenge Problems!

The first can be tackled by middle schoolers, although many high schoolers may find it interesting and fall into a trap if not careful. The wording is challenging but your students may benefit from working in small groups.

Challenge Problem #1
a, b, c, d and e are positive integers with a ≤ b ≤ c ≤ d < e.
If a + b + c + d + e = 143, what is the least possible value of e?

Comments:
Is this merely a guess-test-revise question or is there a strategy/method your students can come up with? How would you extend this problem? change the "143" to a larger value? Change the set of integers to 4 values (a,b,c,d)? 6? k? This is an important issue. Otherwise students may see each problem as an isolated quickly solved puzzle!


The goal of the next question is to review geometry and algebra skills and concepts and to encourage a variety of approaches. I will give the answer -- the challenge for your students is to find AT LEAST THREE METHODS! The teacher may want to submit the best team's efforts to me for acknowledgment on this site.



Challenge Problem #2

P(5,1), Q(8,2) and R(a,b) determine an isosceles right triangle with point R above line PQ and ∠ PRQ the right angle. Determine the coordinates a and b. In your group, you must devise at least THREE methods!

Answer (6,7)
Methods???






"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

"You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught."
--from South Pacific
Note: These lyrics provoked considerable criticism back in 1949-50 but Rodgers and Hammerstein would not take them out. Do they still have relevance today?

Posted by Dave Marain at 8:10 AM 10 comments

Labels: geom, math challenge, middle school, update

BTW, the italicized symbol in red is my submission for the name we should give to the past 10 years. What do you think of it? Let me know if you came up with one of your own. According to Time Magazine, no one has yet created a name which has caught on (and dozens were listed!). Also, I will avoid debating those who strongly believe that the first decade of the 21st century ends a year from now!





As MathNotations begins its 4th year, it has become an annual tradition for math ed blogs to challenge their readers to discover interesting facts about the number symbol representing the new year, in this case, 2010, or Twenty-Ten, for those who are as committed to multiple representations as I am!

Those who know me can anticipate that I would recommend making this an exercise for our middle schoolers. Here are a couple of ideas:


"In your group, list as many observations as you can about the number, 2010. Your team's score will be based on both quality and quantity. For example, an observation like "2010 is even" would only earn 1 pt, whereas "2010 must be divisible by 3 because the sum of its digits is divisible by 3" would earn 2 or 3 points since it contains both a fact and an explanation."

Another idea might be to have students write interesting word/number problems involving 2010 for the class to solve. Of course, to obtain credit the student posing the questions must provide correct answers and solutions!


Your turn...



A final note ---
Some of you may have noticed that I've enabled Comment Moderation due to the number of spam comments which have gotten through. I held out for as long as I could. I do check throughout the day, so, hopefully, this should not prove problematic for my readers.

Posted by Dave Marain at 6:44 AM 25 comments

Labels: investigations, middle school, new year

Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school


List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.

Comments:
(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?

(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?

Posted by Dave Marain at 6:55 AM 6 comments

Labels: divisibility, middle school, reasoning, SAT strategies, warmup

My 500 or so subscribers may not have seen the following anagrams which have been in the right sidebar of my home page for the past month or so. No one has yet taken the time to solve them. They're not that hard! Pls email me at dmarain at gmail dot com with your answers.


VORTEC SCAPE


(1) Hidden Steps OR


(2) General Arrows



The following problems are similar to ones I posted recently...



Mental Math and No Calculator!


1) The following sum has a trillion terms:


0.01 + 0.01 + 0.01 + ... + 0.01 = 1000...0
How many zeros will there be in the sum?




2) The following product has a trillion factors:


(0.01) (0.01) (0.01) ... (0.01) = 0.000...1
How many zeros after the decimal point will there be in the product?



A Few Comments...
(a) You may want to adjust the "trillion" for your own groups but I'm intentionally using this number for a few reasons, not the least of which is to review large powers of 10 (Will most think: "A million has 6 zeros, a billion has 9 zeros, so a trillion has..."?).


(b) The second one is more challenging and intended for Prealgebra students and above but, using the "Make it simpler" and "Look for a pattern" strategies, make it possible for younger students.


(c) How many of you are reacting something like: "Is Dave out of his mind? My students don't know their basic facts up to 10 and he wants mental math with a trillion!" I have found that large numbers engage students since they know there is a way of doing these without a lot of work if you know the "secrets"! Besides, we either push our students or we don't. You decide...


(d) These questions review several important concepts and skills. You may want to use these to introduce or review the importance of exponents and their properties.

Posted by Dave Marain at 6:54 AM 0 comments

Labels: exponents, middle school, patterns, problem of the day, reasoning, warmup

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students.

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.). If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.


Update: Answers/comments are at the bottom...

1. NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!

2. If 10^-1000 - 10^-997 is written as a decimal, answer the following:


(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.



--------------------------------------------------------------------------------------------


ANSWERS


1. 9√3


2. (a) 1000 (b) 3 (c) 997


An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10^-1 = 1/10 = .1 Note that there is one decimal digit.

10^-2 = 1/10² = 1/100 = .01 Note that there are two decimal places, etc.


10^-1000 - 10^-997 = 1/10¹⁰⁰⁰ - 1/10⁹⁹⁷
Using 10¹⁰⁰⁰ as the common denominator, we obtain
1/10¹⁰⁰⁰ - 10³/10¹⁰⁰⁰ =
-999/10¹⁰⁰⁰ from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10^-1000 but I chose rational form for the younger student.

Posted by Dave Marain at 9:53 AM 2 comments

Labels: 30-60-90, geometry, math challenge, middle school, SAT strategies, SAT-type problems, warmup

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

BASIC RULES
* Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
* Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
* The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
* Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
* Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
* Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
* Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.


Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

Comments
The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)




My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_

...Read more

Posted by Dave Marain at 6:29 AM 3 comments

Labels: combinatorial math, math contest problems, MathNotations Contest, middle school, more

Example I
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students left, how many were in the class to start?

Solution without explanation or discussion:

0.4x = 240 ⇒ x = 600


Example II
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students were left, how many were in the class to start?

Solution without explanation or discussion:

0.6x = 240 ⇒ x = 400


Thinking that the issues in the problems above are more language-dependent than based on learning key mathematics principles or effective methods? I would expect that many would say that using the word "left" in both problems was unnecessarily devious and that clearer language should be used to demonstrate the mathematics here. Perhaps, but when I taught these types of problems I would frequently juxtapose these types of questions and intentionally use such ambiguous language to generate discussion - creating disequilibrium so to speak. If nothing else, the students may become more critical readers! Further, the idea of using similar but contrasting questions is an important heuristic IMO.

Even though I've been a strong advocate for a standardized math curriculum across the grades, I fully understand that the methods used to present this curriculum are even more crucial. Instructional methods and strategies are often unpopular topics because they seem to infringe on individual teacher's style and creativity. BUT we also know that some methods are simply more effective than others in reaching the maximum number of students (who are actually listening and participating!). I firmly believe there are some basic pedagogical principles of teaching math, most of which are already known to and being used by experienced teachers.

Percent word problems are easy for a few and confusing to many because of the wide variety of different types.

Here are brief descriptions of some methods I've developed and used in nearly four decades in the classroom.

I. (See diagram at top of page)
The Pie Chart builds a strong visual model to represent the relationships between the parts and the whole and the "whole equals 100%" concept. How many of you use this or a similar model ? Please share! There's more to teaching this than drawing a picture but some students have told me that the image stays longer in their brain. I learn differently myself but I came to learn the importance of Multiple Representations to reach the maximum number of students.

II. "IS OVER OF" vs. "OF MEANS TIMES"
The latter is generally more powerful once the student is in Prealgebra but, of course, the word "OF" does not appear in every percent so many different variations must be given to students and practiced practiced practiced practiced over time. The first method can be modified as a shortcut in my opinion to find a missing percent and that may be its greatest value. However many middle schoolers use proportions for solving ALL percent problems. I personally do NOT recommend this!

Well, I could expound on each of these methods ad nauseam and bore most of you, but I think I will stop here and open the dialg for anyone who has strong emotions about teaching/learning per cents...

Posted by Dave Marain at 6:52 AM 9 comments

Labels: heuristics, instructional strategies, middle school, pedagogy, percent, percent word problem, SAT strategies, SAT-type problems

The following examples also provide practice for open-ended questions and a view of the Explain or Show type questions on our next Online Math Contest to be held in 5 weeks (see info below). Since formal proof is not the goal here, students are encouraged to write a logical chain of reasoning in which they can use/assume basic knowledge about odd and even integers. Further, these questions strongly suggest the strategies consider a simpler case first and patterning.

Another benefit of these types of questions is to review important terminology and to help students improve reading comprehension, a major obstacle for many youngsters in math class (and everywhere else!). Some middle schoolers and high schoolers will have difficulty making sense out of what the question is asking because of both the wording and the information load in the problem. We need to help them group key phrases together and, yes, I guess that means we are also reading teachers!

Example 1
Is the sum of the squares of the first 2009 positive integer multiples of three odd or even? Explain your reasoning.

Example 2
Is the sum of the squares of the first 2010 positive integer multiples of three odd or even? Explain your reasoning.

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REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Posted by Dave Marain at 7:23 AM 6 comments

Labels: deductive reasoning, explain, logic, middle school, number theory, odd/even problems, open-ended

Have you ever inserted batteries in a device only to find that it didn't work? You reverse the batteries and try again, but no luck. You can't find the polarity diagram to guide you and you're dealing with 3 or 4 batteries and all the possible combinations! Well, that just happened to me as I was inserting 3 'C' batteries into a new emergency lantern I just purchased. There was no guide that I could see. I knew there were 8 possibilities but it was late and my patience quickly ran out. I tried it again the following morning, shone my small LED light on it and saw the barely visible diagram.

After seeing the lantern finally operate, I realized I should have used a methodical approach -- practice what I preach!! Then I thought that this might be a natural application of the Multiplication Principle one could use in the classroom. Of course, it would work nicely if you happened to have the identical lantern but you might have some of these in the building or at home which take 2 or more batteries. IMO, there's something very real and exciting about solving a math problem and seeing the solution confirmed by having "the light go on!" I'll avoid commenting on the obvious symbolism of that quoted phrase...

Instructional/Pedagogical Considerations

(1) I would start with a small flashlight requiring only one battery to set up the problem. For this simplest case, students should be encouraged to describe the correct placement in their own words and on paper.

(2) Would you have several flashlights/lanterns available, one for each group of 2-4 students or would you demonstrate the problem with one device and call on students to suggest a placement of the batteries? Needless to say, if you allow students to work with their own flashlights, they will look for the polarity diagram so you will need to cover those somehow. That is problematic!

(3) Do you believe most middle school students (if the polarity diagram is not visible) will randomly dump in the batteries to get the light to go on and be the first to do so? Is it a good idea to let them do it their way before developing a methodical approach? Again, if a student or group solves the problem, it is important to have them write their solution before describing it to the class. If there is more than one battery compartment, students should realize realize the need to label the compartments such as A, B, C , ... Once they reach 3 or more batteries, they should recognize that a more structured methodical approach is needed so that one doesn't repeat the same battery placement or miss one. One would hope!

(4) Is it a drawback that the experiment will probably end (i.e., the light goes on) before exhausting all possible combinations? How would we motivate students to make an organized list or devise a methodical approach if the light goes on after the first or second placement of the batteries?

(5) I usually model these kinds of problems using the so-called "slot" method. Label the compartments A, B, ... for example and make a "slot" for each. For two compartments we have

A B
_ _

Under each slot, I list the possibilities, e.g., (+) end UP or DOWN (depending on the device, other words may be more appropriate). Here I would only concern myself with labeling the (+) end, the one with the small round protruding nub. For this problem I would write the number (2) on each slot since there are only TWO ways for each battery to be placed. Note the use of (..). In general, above each slot I would write the number of possibilities. For two compartments (or two batteries), the students would therefore write (2) (2). They know the answer is 4 but some will think we are adding rather than multiplying. Ask the class which operation they believe will always work. How would you express your questions or explanation to move students toward the multiplication model? The precise language we use is of critical importance and we usually only learn this by experimentation. If one way of expressing it doesn't seem to click with some students, we try another until we refine it or see the need for several ways of phrasing it. This is the true challenge of teaching IMO. We can plan all of this carefully ahead of time, but we don't know what the effect is until we go "live" (or have experienced it many times!).

Perhaps you've already used a similar application in the classroom - please share with us how you implemented it. Circuit diagrams in electronics also lend themselves nicely to this approach. Typically, I've used 2, 3 or more different coins to demonstrate the principle but the batteries seem to be a more natural example, although I see advantages and disadvantages to both. At least with the batteries, students should not question the issue of whether "order counts!"

I could say much more about developing the Multiplication Principle in the classroom, but I would rather hear from my readers.
If you've used other models to demo this key principle, let us know...


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Posted by Dave Marain at 9:53 AM 8 comments

Labels: combinatorial math, instructional strategies, middle school, multiplication principle, pedagogy

I have an equal number of pennies, nickels and dimes. I also have some quarters which have the same value as the pennies, nickels and dimes combined. If I have no other coins, what is the fewest possible total number of coins I could have? What is the value of all the coins?

Comments
(1) An opening day problem?
(2) Would you have students working alone or in small groups?
(3) Would you allow the calculator?
(4) Appropriate for prealgebra students? Students below grade 6?
(5) Is zero a possible answer?
(6) Wording too confusing for most students? Is it ambiguous or clear?
(7) Do you feel there are important underlying concepts and ideas embedded here or is it just a fun puzzle to engage students?
(8) Do students have difficulty in separating number of coins from their value?


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates:
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Posted by Dave Marain at 8:01 AM 6 comments

Labels: coin problem, middle school, puzzle, starting the year, warmup

After 4 tests, Barry's average score was 5 points higher than Michelle's. After the 5th test, Michelle's overall average was 5 points higher than Barry's. Michelle's score on the 5th test was how many points higher than Barry's?

Can you find at least three methods for solving this?
Algebraic, "plug-in", conceptual, etc...

As teachers we need to have a deep understanding of these kinds of problems and familiarity with several approaches. Of course, our students will show us a variety of methods, both right and wrong, when we open up the dialog!


Comments
Students from middle school on see many problems relating to means. However, they need to see a variety of problems of increasing difficulty. This question is certainly not a highly challenging math contest problem but I believe it demonstrates some important principles of averages and can be used to review different problem-solving strategies. Middle schoolers would struggle with the algebraic approach (a system of two equations), however they should be thoroughly comfortable with the underlying ideas.

Since the focus is on concept and method, I will give the answer: 45

Posted by Dave Marain at 6:23 AM 9 comments

Labels: averages, middle school, SAT strategies, SAT-type problems, standardized tests

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...


Comments
1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!

...Read more

Posted by Dave Marain at 6:12 AM 12 comments

Labels: math contest problems, middle school, more, SAT-type problems, warmup

Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...

Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.

BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...

What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...


1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).

3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?


These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...

To see a more challenging version of the examples above, click Read more...



You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!

The lcm of 100 and N is 500. What is the least positive integer value of N?

...Read more

Posted by Dave Marain at 7:30 AM 16 comments

Labels: higher-order questions, lcm, middle school, more, number theory, teaching for understanding

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!



The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.


The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!

Posted by Dave Marain at 6:41 AM 4 comments

Labels: compound probability, investigations, math challenge, middle school, probability

As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:

• How to introduce this
• Asking questions to provoke deeper thought
• Drawing conclusions and further generalizations
• Connecting this problem to other circle or geometry problems
  
• Maximizing student involvement
  

I told TC I would need some time to rework the original problem for the younger students so here goes...





Diagram for Parts I and II







Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.

If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).

Show that Option 2 is shorter than Option 1.

Part II (middle and secondary students)
Same diagram but now the radii are R and r with R> r.
Show algebraically that Option 2 is shorter.


Part III (secondary students)












To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R> r.

As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).

Show that Option 2 will be shorter provided π ≥ θ> 2.

Click Read More for further discussion...


Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?

Thanks TC for providing us with another stimulating challenge!

...Read more

Posted by Dave Marain at 8:15 AM 7 comments

Labels: circles, geometry, middle school, more, real-world applications, tc'sTotal Challenges

Number puzzles always intrigued me and, perhaps, they are one way we can invite our students into the wonderful and exciting world of mathematics. Oh, alright, maybe that's a bit of a stretch, but, I suspect that if you give the following famous puzzle to your students in Grades 5 and up, they will try it even if you don't offer food or a 10 point bonus! Yes, calculators are allowed but after a few minutes of frustration they will be begging for a hint.

(Oh, and if you give them this problem at the beginning of class, you may as well forget the lesson!)

Find two 5-digit numbers whose product is 123456789.


If you solve it, don't post your answer immediately. I will probably publish a hint or the answer in a day or so. You can always email me with your solution at "dmarain at gmail dot com."

Click Read more for a hint and comments...


HINT: Rather than pressing random numbers into the calculator as some would do, encourage them to find the prime factors of 123456789. It's easy to show that this number is divisible by 3 and 9, but find finding the other factors will be challenging. I'll post another hint if you request it...

COMMENT: This beautiful puzzle was invented by Y. Yamamoto and has intrigued many puzzle enthusiasts for awhile now. Is there some profound meaning behind the solution or is it just a curiosity? Perhaps we'll have to wait for Dan Brown's next book to unlock the mystery! I will probably post the answer if I don't get a response within 24 hours. Probably...

If any of your students solve it, email me at "dmarain at gmail dot com" and let me know if I can post their names.

...Read more

Posted by Dave Marain at 6:31 AM 3 comments

Labels: math challenge, middle school, more, puzzle

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