Another Algebra Problem for the "Ages!"

Forgive the lame attempt at "punnery" in the title but I hope you will see the many layers in the following discussion about a 'typical' word problem involving people's ages. Perhaps, not so typical but I hope it will be topical!

Please bear in mind that this blog is geared essentially toward the K-12 math teacher. Most of the problems I write are not intended to stump people. Rather they are vehicles to engage a discussion of instructional practice as we move toward a leaner, deeper and more coherent curriculum in this country. 

To reiterate a point made many times on this blog:


Most common teacher reaction to these problems and investigations:
"Dave, These are nice but you know we don't have time for this, particularly at the secondary level. What exactly do you want me to delete from the curriculum!"


My Response: "That is the point of why we need to reduce the number of topics covered and alter our paradigms of math education.  The underlying ideas in these problems and explorations include some of the essential ideas of mathematics, algebra in particular. Who am I to make such a pretentious claim? Well, it isn't pretentious at all! Everyone knows that most students go through the motions of learning, concentrating solely on what they believe will be tested. They have little interest in anything else (yes, there are exceptions!). If assessments focus on procedures and superficial ideas, then there is no reason for students to make the effort to actually use their 'reasoning brain'!"

As teachers of mathematics we need to know and understand what the "big ideas" of math are before we can hope to convey this to our students. In the problem below, the essential idea is NOT how to solve a contrived word problem. 


I.  One of the big ideas here is of course to develop a problem-solving approach which can be applied to many problems. 


II. Secondly, the important notion of comparing the number of unknowns (variables) to the number of relationships (equations in this case). If the number of variables exceeds the number of relationships, then we should NOT expect a unique solution to the problem. If the number of equations exceeds the number of variables then there might not be solution at all because of inconsistent conditions. These are huge themes in using algebra to model the real world. How many variables are involved in weather forecasting!


Mr. Oldman is currently five times as old as his great-grandson. In how many years will he be four times as old as his great-grandson will be? Assume ages are positive integers.

Fairly straightforward algebra problem? Easy to guess the "answer" like Mr. Oldman is currently 75 and the young man is 15. In five years, the ages will 80 and 20. Voila! You know in upper elementary or middle or high school there will always be some student who "guesses" these values and sits there complacently with the "I'm done!" look on her face.

But there is so much more here for us to discuss with the students. A much bigger picture with deeper concepts than guessing an answer or setting up algebraic equations.

I will get the ball rolling and leave the rest to my astute readers whose insights always surpass mine...

Why did my hs algebra teacher, Mrs. Hill, always require a chart or table for these problems? Even when I didn't feel I needed it, I had to use it. Decades later, I still do out of force of habit and I've been carrying on her legacy with my own students ever since! Rather than import some html table codes, I will resort to brute force for displaying the table. Pls forgive any problems with the display on your screen.

............................NOW .............. FUTURE (y yrs from now)

MR. OLDMAN.........  5x ................  5x + y


YOUNG MAN.........  x ...................  x + y

Equation:  5x + y =   4(x + y) → x = 3y

Of course, I could have used different variables or more variables and produced more equations but there is an essential truth underlying all of this. As math people and from our experience we could sense from the beginning that there would be no unique solution to this problem. But how do we develop this conceptual understanding in our students?   Some students I believe would intuit this, but many might not.

There's no earth-shattering revelation here! But IMO every student of algebra should experience this and engage in discussion about it.

Ok, teachers of algebra in middle or secondary grades -- how would you develop this into a meaningful structured lesson and, at the end, assess the learning which hopefully took place?

• Simply show them how to make the chart, set up the equations, solve and note that there is more than one solution? 
• OR would you allow the students to discover this for themselves? Would you begin by asking them to intuitively guess if there is exactly one solution in positive integers, more than one or none? 
• What questions can you compose ahead of time? What questions can you anticipate coming from the students? 
• Can we make the problem into more a puzzle-solving experience. Do you believe the word "puzzle" evokes a different visceral reaction from "word problem"?

Anxiously awaiting your thoughts which is really the point of having a blog and not simply a web site. Don't leave me twisting in the wind...

"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

Does Learning/Playing Chess Develop The 'Mathematical Mind'?

Nothing revolutionary about this question but it's been on my mind for some time and, now, that I have a six-year old grandson who seems to be analytical like me (we speak the same language!), I've decided to introduce him to chess via chess puzzles.

I'm not writing this because I believe one should develop a math curriculum around chess! I'm simply suggesting that any activity which requires problem-solving, analysis and creativity develops the mind. Chess is only one of many such vehicles which drive the student toward deeper thought processes. I could make a similar argument for writing and other artistic activities, but I chose chess because it's a game I love even though I'm still a novice.

Perhaps the most important aspect of all of this for children is that learning chess can be fun provided it is introduced properly. I believe many young children would quickly become frustrated and bored if they begin by learning proper openings and the concepts of positional play in the beginning stages.

There are well-known analogies between playing chess and mathematical problem-solving so I will only mention a few that occurred to me a priori, some of which have been validated by the writings of the brilliant mathematician, Paul Halmos:

• Chess has an elaborate technical language and is completely deterministic.
• Chess - like mathematics - requires problem solving, evaluation, critical thinking, intuition and planning

Halmos also believed, as I do, that mathematics is a creative art

"because mathematicians create beautiful new concepts; it is a creative art because mathematicians live, act, and think like artists; and it is a creative art because mathematicians regard it so.”

A New Jersey senator introduced and helped to pass bills in the Senate regarding chess instruction in schools, declaring:

"Chess increase strategic thinking skills, stimulates intellectual activity and improves problem-solving ability."

For learning how individual pieces moves and for experiencing more immediate feedback and satisfaction, I recommend Chess Puzzles, which involve contrived board positions requiring the solver to find a mate in one or more moves. Starting with Mate in One Puzzles seems developmentally appropriate in the Piagetian sense, although I would introduce Mate in Two early on as an extra challenge. That is, I wouldn't spend two weeks on Mate in One to the exclusion of other puzzles. For me, the best online site for doing these puzzles on a daily basis is the Mate in Two Chess Puzzles website developed by John Bain, MS Education, a special education teacher, certified USCF tournament director, and a specialist in textbook and materials modification. John is a teacher in the truest sense of the word!

 I'm not suggesting that children should not also be allowed to try a full game with a parent, instructor or with peers. I would allow the child free exploratory play when starting a game, correcting only invalid moves. This is not a purely constructivist approach however -- I do not expect students to invent the the theory of positional play and the deeper aspects of the game! Guided instruction must be introduced strategically just as it is with mathematics instruction, and, as with math, there is never a substitute for repetition and practice!

I'm opening the "board" for you to jump in. Do you agree with some of my thoughts about chess and math? Do you see other analogies I've missed? Have you personally used chess as a way to develop thinking in elementary, middle or secondary schools? Please share!!




"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

Odds and Ends -- Week of 2-15-10

Catchup time...

• So where are the answers and followup discussion to the previous two problems I posted? They're coming. I plan to make a video of the investigation in the "143" problem. 
• I think it would be so useful to post videos of actual lessons that demonstrate how some teachers implement explorations in the classroom at all levels from K-12. I'm sure there are some of these already online but they seem to be few and far between and I'm really referring to "best practices" here to serve as models for others. We have NCTM Illuminations for example but I'm really looking for something else here.
• More important than investigations is to see models of daily lessons which incorporate the best of what we know about effective practice. Lessons which show HOW to blend procedural and conceptual understanding, help students develop skills mastery while engaging in rich problem-solving. Easy to do? Of course not, but if we want the US to be academically competitive, we had better move quickly in this direction and use international models to guide us.
• How are you, the mathematics teacher, dealing with all of the confusing and overwhelming issues in math education today? OR have you learned to ignore all the "noise", close your door and simply go about the business of teaching? If you're able to, that is! Unfortunately some of the decisions which are being made independently of your input will have significant impact on how well you will be able to do your job today and in the future...  Issues like
  
• "Algebra for All?" So where has that experiment gone?
• States joining consortia to develop a common standards and assessments in math
• How many consortia should there be? Will most eventually merge into one or two?
• Will common math standards ultimately lead to more consistency in content  -- i.e., that which is actually taught in the classroom?
• Are math ed departments in colleges and universities adapting rapidly enough to prepare preservice teachers for the paradigm shifts which are occurring? 
• Will "methods" courses in ed schools increase focus on actual content, e.g.,
  
  "You will all prepare a lesson on the effect of changing the parameter "b" in a quadratic function. Your lesson should utilize multiple representations and include a carefully planned series of Socratic questions which develop meaning and conceptual understanding for the algorithms. Specify what actions you took to balance procedural learning with conceptual understanding. Also, be prepared to answer the essential question: "WHAT ACTIONS DID YOU TAKE AND HOW DID YOU ASSESS THAT LEARNING TOOK PLACE?"
  
• How can we all use emerging technologies to enhance our teaching repertoire. Regardless of whether you "tweet", "Buzz" or Facebook, or all of the above, how can networking and sharing make us more effective teachers? This is sort of obvious, but how many of us are using these on a daily basis in our planning and in our classrooms? 25% 50% More?
• What are the most effective online learning tools for mathematics, particularly middle and secondary math? Many excellent bloggers have researched and compiled excellent lists of resources, but specifically, what are the best online video sites (interactive or not)? This is crucial for me and it's something to which I want to dedicate myself.
• I choose not to get into NCLB or Charter Schools debates at this time...
• So where is my new website I've alluded to? It's taking forever to set up, but I want to do it right. More to follow...

Ok, so we have to have a little problem for your students to think about. This is part of a well-known genre of "puzzles" which frequently travel across the web and are always intriguing for students and adults alike. You've probably seen it...
For this blog, the essential question is, how we can make this a teachable moment in our math classes?

MIND GAME

2% or 98%

This is strange...can you figure it out?

Are you the 2% or 98% of the population?

Follow the instructions! NO PEEKING AHEAD!

* Do the following exercise, guaranteed to raise an eyebrow.

* There's no trick or surprise.

* Just follow these instructions, and answer the questions one at a time

and as quickly as you can!

* Again, as quickly as you can but don't advance until you've done each

of them ..... really.

* Now, scroll down (but not too fast, you might miss something).

Think of a number from 1 to 10

Multiply that number by 9.

If the number is a 2-digit number, add the digits together.

Now subtract 5.

Determine which letter in the alphabet corresponds to the number you

ended up with

(example: 1=a, 2=B,  3=c,etc.)

Think of a country that starts with that letter.

Remember the last letter of the name of that country

Think of the name of an animal that starts with that letter.

Remember the last letter in the name of that  animal.

Think of the name of a fruit that starts with that letter.

Are you thinking of a Kangaroo in   Denmark  eating an Orange  ?

I told you this was FREAKY!! If not, you're among the 2% of the

population whose minds are different enough to think of something else.

98% of people will answer with kangaroos in  Denmark  when given this

exercise.

COMMENTARY

Is there some basic number theory here? 

What would you want your middle schoolers to do with this after they play it a few times?

"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

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):