Can Your Students Find At Least Three Methods? Odds and Evens Week of 2-1-10

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?

If We're 'Packing", Are We Going Somewhere?

Fascinating article from today's New York Times. In 2-dimensions we talk about tessellating objects to fill the plane. Circles of course will always leave gaps. In 3-dimensions, equal spheres will also leave gaps when packed as closely as possible, but the question then becomes, "How do we arrange the spheres which would result in the densest packing. Turns out that the grocer's method of stacking oranges solves that problem! Equal cubes can be packed together without any gaps, so we can say that the densest packing for cubes is 100%, that is, identical cubes can be packed so that they use 100% of the available space.

But packing regular tetrahedrons (a pyramid whose 4 faces are congruent equilateral triangles) as tightly as possible has defied the best logical mathematical minds, including Aristotle's, for nearly 2000 years. Recently, significant strides have been made, not only by the best mathematical and scientific minds, but also by graduate students, like Ann Chen from U. of Mich.,, who have taken dozens of tetrahedral dice from Dungeons and Dragons games and are using a hands-on approach to build various configurations and then computing the density of the packing. For the past several months teams from different schools have published their latest and best attempts, but, as of this moment, Ann has found the densest packing at 85.63%. That's right, she's outdone the best theoretical mathematicians and scientists in this quest for the new Holy Grail of packing problems. Aristotle mistakenly asserted that there exists a perfect 100% packing for tetrahedrons, but this was shown to be false. Now it appears that the percent is far more than we thought. Professor Nash, we need you!

I strongly believe that there is a place for this kind of discussion in our math classes from the earliest years on. Let students know that solving mathematical problems often involves hands-on experimentation as in science! Besides, who's to say that there isn't some middle schooler out there right now who might sit in her room playing with these dice who will arrive at 86%!!



 

"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)

HAPPY 2 x 3 x 5 x 67! Let The "Problems" Begin!

May this new year and decade bring happiness and prosperity to each of you now that the 2KO's have come to an end! 

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.

A Quadratic Trinomial/Factoring Investigation for Algebra I/II

In my Christmas post, I raised the issue of how much time should be spent on factoring quadratic trinomials over the integers in light of the new ADP Standards for Algebra I and II. Hopefully, some of you will provide us with the benefit of your knowledge and experience. I may even make this into a poll or survey to be voted on but, in this post, I will appear to contradict myself and propose an investigation of this topic which requires some effort and time on the part of the student. The target audience would be the regular or accelerated Algebra I/II student.  

We all need to become more creative in the strategic use of time in our classrooms (I still think of myself as being in the classroom!). What are some alternatives to using class time for this? I'll suggest one approach and I'm hoping others will offer their suggestions:


Assign the following as an extra credit or "long-term" project to be due in a week or two. I would not even take classroom time to discuss it. Just hand it out or post it on your website or the department's website if it is to be given to all the Algebra classes. Students can easily download it or print directly if they wish. After they are collected, graded and returned, you may choose to discuss it briefly for about 10 minutes using an overhead transparency, opaque projector or via your computer and a projector. You can also post some  student solutions on the website.


THE INVESTIGATION/PROJECT

[OPTIONAL HINT OR CUE]
The following may require an application of the ac-method learned in class.

(1) Factor the following over the integers and show all steps used in your method of factoring:

(a) 12x² + 27x + 15

(b) 12x² + 28x + 15

(c) 12x² + 29x + 15

(2)
(i) List all positive integers values of b, including the ones from part (a), for which 12x² + bx + 15 is factorable over the integers.
(ii) For each value of b, factor the resulting trinomial.
(ii) How many of these trinomials produce a gcf ≠ 1 for 12, b and 15?

(3) If we knew in advance that 180 has 18 positive integer factors, explain how it follows that there are 9 values for b in part (2).

(4)
(i) If the "12" and "15" were interchanged, explain why this would not change the possible values for b in part (2)?
(ii)For each resulting trinomial such as 15x² + 28x +12, determine its factors and explain how they are related to the factors of the original trinomial (i.e., before interchanging the 12 and 15).


QUESTION FOR OUR DISCUSSION (No, these are not rhetorical! Some are quite knotty)

(1)  What do you see as the benefits of this investigation, if any?

(2) Do the new standards and assessments discourage us from investing time into this type of in-depth problem-solving?

(3) Do you believe this type of assignment should be reserved for the accelerated/honors Algebra I student in 7th or 8th grade or even for the stronger Algebra II student?

(4) With the new ADP Algebra standards, do you believe this type of investigation is reasonable, particularly since it is unlikely that any variation of this would appear on an End of Course Test?

(5) If you were to give this problem, how would you edit the investigation? Parts you would delete or change? Parts you would add?

(6) My goal for this blog has always been to provide you with useful and engaging examples of in-depth problems for your students that require going beyond the mechanical aspects of the course. These problems are developed for this blog -- they do not come from my notes from 30 years ago! Would you be interested in a supplementary resource of such problems for each course you teach? Do you already have one from the publisher or from another source which you really enjoy? Share it!

How Much Factoring In 1st Year Algebra?

SEASON'S GREETINGS

Math Notations 3rd Birthday- Thank You!

The American Diploma Project is and will be impacting on what is being taught in both Algebra I and II in the 15 states who have joined the ADP Consortium. The classic flow from Standards to Assessments to Course Content is leading to the type of content standardization in our schools which I envisioned decades ago. A natural part of this process is deciding what topics in our traditional courses need to be deemphasized or eliminated to allow more time for the study of linear and non-linear function models, one of the central themes of the new Algebra standards.This leads to curriculum questions like...

How much time should be spent on factoring quadratic trinomials in Algebra I?

My assumption is that factoring ax²+bx+c where a ≠ 1 is still taught in Algebra I. Please challenge that assumption if wrong! If we also assume there is sufficient justification for teaching this, then we move on to the issue of how much time should be devoted to instruction. Two days? More? Time for assessment?

Here are some arguments pro and con...


PRO

(1) It is required by the ADP Standards (see below).

(2) Learning only simple trinomial factoring of the form x²+bx+c is not sufficient for solving more complex application problems.

(3) The various algorithms, such as the "ac-method", which have been developed for factoring quadratic trinomials, are of value in their own right; further, the "ac-method" introduces or reinforces the important idea of factoring by grouping.

(4) Students gain technical proficiency by tackling more complicated trinomials.

(5) Students should be given the option of more than one method, not just the quadratic formula.



CON 

(1) The AP Calculus exam generally avoids messy quadratics in their problems. If such occur, students normally go directly to the Quadratic Formula.

(2) The SATs generally avoid asking students to factor such quadratics directly, particularly since it is easy to "beat the question" by working backwards from the choices. Instead, they ask the student to demonstrate an understanding of the process.

Here's a typical question they might ask:

If 6x² + bx + 6 = (3x + m)(nx + 3) for all values of x, what is the value of b?


(3)The ADP standards for Algebra I do include this topic but it does not appear to be stressed. The following are taken from the ADP Algebra I standards and practice test:

(3) Do other nations teach our traditional methods of factoring or are students told to go directly to the quadratic formula?

(4) Current Alg I texts seem to have deemphasized factoring in general and some have moved this topic to later in the book.


So I am opening the floor for your input here!


(a) How much time is spent on factoring quadratic trinomials in Algebra I in your school?
(b) Do you teach the "ac-method"? If yes, do you motivate it or teach it mechanically?
(c) Do you believe factoring quadratic trinomials is essential or should it be deemphasized?


By the way, here is an example of the ac-method:

Factor completely over the integers:   6x² + 13x + 6

Step 1: Find a pair of factors of ac = (6)(6) = 36 which sum to b = 13.
Hopefully, students think of 9 and 4 without a calculator!

Step 2:  Rewrite the middle term 13x as 9x + 4x (works in either order)
Then 6x² + 13x + 6  =   6x² + 9x + 4x + 6

Step 3: Group in pairs and factor out greatest common monomial factor from each pair:
3x(2x + 3) + 2(2x + 3)

Step 4: Factor out the common binomial factor 2x + 3:
(2x + 3) (3x + 2)

Step 5: Check carefully by distributing.


Here is a "proof" of this method (some details omitted like the meaning of h and k):

Divisibility, Counting, Strategies, Reasoning -- Just Another Warmup

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? 

Demo For Building An Investigation In Geometry For All Levels

Note: Diagram has been modified from original.

 For Figures 1 and 2, the following is given:

AD + AC = BD + BC

Perimeter of triangle = 36

AC = 15

Show that the length of AD = 3.

In other words, demonstrate that the length of AD is independent of sides AB and BC.

Instead of imposing or suggesting my way of using this question to build an investigation, how would you do it?

If you're new to this blog, I have published dozens of examples of investigations which are intended to develop process, conceptual understanding, generalization  and a different view of what mathematics is for our students. An investigation allows students to explore particular cases before attempting to generalize and abstract. Some might call this scaffolding. I see it as creating an experimental environment in the classroom, encouraging our students to become mathematical researchers! I know every argument against this approach but, remember, I'm suggesting that this type of activity only be used perhaps once a month...

The question above can be given as is to some groups of students but may not be appropriate in its present form for many others. The question can be reworded or changed completely.

What would you do?

Using WarmUps in Middle School/HS to Develop Thinking and Review/Apply Skills

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.