MathNotations

So on Twitter.com/dmarain this morning I posted the following (I modified it slightly):

"Java Coffee Co." gives you a free cup after you purchase 12. The % discount is 7.7% rounded.
EXPLAIN!

Wonderful concise conceptual explanation from @MrLeiss on Twitter this morning. This is what sharing of ideas is all about!
See it on my account at

twitter.com/dmarain

So where's the formula for % discount? % change in general? Why is it NOT in that video? Uh, that's rhetorical...


As I explained in my video --
THIS IS NOT INTENDED TO BE PRESCRIPTIVE OR SCRIPTED!
It is just one old retired math teacher's way of keeping the dialog going and we all know that sharing of ideas is the only way our students will understand math better. We do believe that, yes??

SO SHARE YOUR THOUGHTS BUT PLS REMEMBER THE RAISON D'ETRE OF THIS BLOG. THE PROBLEM AND THE "ANSWER" IS FAR LESS IMPORTANT THAN THE METHODS WE USE TO DEVELOP CONCEPTUAL UNDERSTANDING.



My Core Beliefs

1. THERE IS NO ONE BEST WAY TO TEACH THIS!
2. BELIEF #1, NOTWITHSTANDING, SOME WAYS ARE BETTER THAN OTHERS.
3. IT IS NEVER TOO EARLY IN A CHILD'S DEVELOPMENT TO INTRODUCE MATH CONCEPTS. IT'S JUST A MATTER OF HOW WE SAY IT AND HOW WE SHOW IT AND HOW WE ASK THE QUESTIONS AND WHAT ACTIONS WE ASK THEM TO TAKE!
4. CONCRETE (PHYSICAL OBJECTS) TO SYMBOLIC TO ABSTRACT ALWAYS MAKES SENSE REGARDLESS OF STUDENT DEVELOPMENT.




If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is 9ドル.95. Secured pdf will be emailed when purchase is verified.

Posted by Dave Marain at 10:11 AM 0 comments

Labels: coomon core, instructional strategies, math strategies, percent discount, percent word problems

Students who have been out of geometry for a year or so and are preparing for standardized test like Math I Subject Test or SATs/ACTS need occasional review. The following is similar to several other cone problems I've posed before but even our strongest Algebra 2 through Calculus students lose their "edge" when it comes to "solid" geometry questions (yes, believe it or not, my terminal course in high school was called Sold Geometry and we covered topics like spherical trigonometry!).


A right circular cone of height 16 is inscribed in a sphere of diameter 20. What is the diameter of the base of the cone?


Reflections....

1) Are these kinds of problems somewhat hard merely because students forget? I can think of several more reasons:

• The problem itself is somewhat challenging, however it's far from over their heads!
• The student never experienced a question like this in Geometry; perhaps questions like these were in the B or C or D exercises in the text and were never assigned or only for the "honors" students? Do you recall seeing a problem similar to this in the textbook from which you taught?
• The student did not take a formal course in geometry
• The topic was covered in a cursory manner or perhaps not at all because of time crunch. That's the whole point of a standardized curriculum, isn't it? To know what is needed to be covered and plan accordingly. Of course, I'm a realist enough to know the myriad of reasons why the best laid plans oft go .........
• Students don't remember how to start because key geometry strategies were not explicitly stated and reiterated ad nauseam. Were your students asked daily to begin by reciting the key strategies such as those for circle and sphere problems? Were they placed on index cards or blocked out in a particular section of their notebook?:
• DRAW THE BEST DIAGRAM YOU CAN (and believe me, I'm no artist!)
• Always locate the CENTER of circles, spheres and label the point
• Label the measurements of all segments (angles) - I know, everyone does that!
• Successful problem-solving in mathematics is based on finding relationships! Were guiding/leading questions asked
• What do the cone and sphere have in common?
• TRUE FALSE The height of the cone is the same as the diameter of the sphere. EXPLAIN!
• Was the student exposed to the strategy of comparing the 2-dimensional analogue of the 3-D problem? Would it be a right triangle in a circle? Equilateral triangle inscribed in a circl or???
• Oh and yes...
• Draw the radius of the sphere (or circle) so that it is the hypotenuse of some right triangle!

"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 8:37 AM 1 comments

Labels: 3-dimensional geometry, cone, cone in sphere, instructional strategies, math challenge

Ok, so I don't have another anecdote from my grandson today so I'll have to get back to mathematics -- problem-solving, teaching and learning.

The video below deals with an algebraic equation in 2 variables which should be straightforward for your stronger Algebra II or Precalculus student. But will it be? I invite you to predict how many in your classes will answer it correctly, then try it out. After all, it is multiple choice, so statistically some should get it right by some means or other!

Hopefully, the purpose of the problem and the video will become clear to all of you. If we want our students to demonstrate better reasoning and an understanding of important ideas in math, we need to feel comfortable in teaching for meaning and understanding. This doesn't mean we stop teaching algorithms and procedures, however. Exactly what all this means and how to do it is the reason for this blog. I certainly never claimed to know the answers or any other mystical secrets. I only know that I never gave up trying. Sometimes my efforts failed miserably, but I hopefully learned from these attempts.

It would mean a lot to me if you share your thoughts here or on my You Tube channel, MathNotationsVids, where you will find my other videos.

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


Note: Another subtle point I should have made in the video---
y(x-4) = 0 → y = 0 OR x = 4
It is important for us to stress this point and distinguish it from "AND" logic. If the equation were in the form: y² + (x-4)² = 0, we would have (y = 0) AND (x = 4), whose graph would be the single point (4,0). Another instance where an exercise on the board can lead to a rich, fruiful and profound discussion. If all of this is seen as taking too much time away from content, remember this is precisely the kind of change in curriculum and instruction that Prof. Schmidt has been trying to tell us about for over 15 years! Well, I'm preaching to the converted, aren't I...
-------------------------------------------------------------------

"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 5:17 PM 3 comments

Labels: algebra, instructional strategies, math videos, SAT-type problems, video lesson

Posted by Dave Marain at 7:02 AM 8 comments

Labels: geometry, instructional strategies, investigations

Alright, you're teaching about the rule for slopes of perpendicular lines in Algebra or Geometry.

Here are some of the instructional strategies or approaches you may have used...

(1) State the theorem without explanation followed by 3-4 demo examples of how it's used
(2) Motivate the theorem using the lines y = (3/4)x and y = (-4/3)x, choosing the points (4,3) and (-3,4) to demonstrate why these lines are perpendicular
(3) A more abstract approach using the following diagram

Posted by Dave Marain at 6:46 AM 10 comments

Labels: instructional strategies, perpendicular lines, slopes

Have you forgotten to register for MathNotation's Third FREE Online Math Contest coming in mid-October? We already have several schools (from around the world!) registered. For details, link here or check the first item in the right sidebar!!

Before tackling a more challenging problem in the classroom, I would typically begin with one or more simpler examples. My objective was to review essential concepts and skills and demonstrate key ideas in the harder problem. This incremental approach (sometimes referred to as scaffolding) enabled some students to solve the problem or at least get started. Usually within each group of 3-4 students, there was at least one who could help the others. Some groups or classes might still not be ready after one example, so more would be needed. I never felt that this expense of time was too costly since my goal was to develop both skill and understanding.


SIMPLER EXAMPLE
Consider the following two statements about positive numbers A and B:

(1) A is 80% of B.
(2) A is 20% less than B .

Are these equivalent, that is, if values of A and B satisfy (1), will they also hold true for (2) and conversely?

How would you get this idea across to your students?

Again, depending on the students, I would often allow them to discuss it first in small groups for two minutes, then open up the discussion.

Note: If the group lacks the skills, confidence or background (note that I left ability out, intentionally!), I might first start with concrete values before giving them the 2 statements above: E.g., What is 80% of 100?

How would I summarize the methods of solution to this question. Here's what I attempted to do in each lesson. I didn't reach everyone but I found from further questioning and subsequent assessment that this multi-pronged approach was more successful than previous methods I had used. Most of these methods came from the students themselves!

INSTRUCTIONAL STRATEGIES
I. Choose a particular value for one of the numbers, say B = 100. Ask WHY it makes sense to start with B first and why does it make sense to use 100. Calculate the value of A and discuss.

II. Draw a pie chart (circle graph) showing the relationship between A and B. Stress that B would represent the whole or 100%.

III. Write out the sentence:
80% of B is the same as 100% of B - 20% of B
In other words:
80% of B is the same as 20% less than B.

IV. Express algebraically (as appropriate):
0.8B = 1B - 0.2B

Numerical (concrete values)
Visual (Pie chart)
Verbal (using natural language)
Symbolic (algebra)

Yes, it's Multiple Representations! The Rule of Four!

To me, it's all about accessing different modes of how students process. Call it learning styles, brain-based learning, etc., it still comes down to:
RARELY DOES ONE METHOD OF EXPLANATION, NO MATTER HOW CLEAR OR STRUCTURED, REACH A MAJORITY OF STUDENTS. YOUR FAVORITE EXPLANATION WILL MAKE THE MOST SENSE TO THE STUDENTS WHO THINK LIKE YOU!!


Now for today's challenge.
(Assume all variables represent positive numbers)

M is x% less than P and N is x% less than Q. If MN is 36% less than PQ, what is the value of x?

Can you think of several methods?
I will suggest one of the favorite of many successful students on standardized assessments:

Choose P = 10, Q = 10. Then...

Click on More (subscribers do not need to do this) to see the answer without details.




Answer: x = 20

...Read more

Posted by Dave Marain at 6:02 AM 1 comments

Labels: conceptual understanding, instructional strategies, more, percent, percent word problem, SAT strategies, SAT-type problems

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

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

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a^-3/2 - a^-1/2 - a^1/2 = 0

Preliminary Comments/Questions/Issues

• Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  
• Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  
• It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  
• Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
• What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  
• Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...
  

Posted by Dave Marain at 6:39 AM 7 comments

Labels: advanced algebra, complex numbers, exponents, instructional strategies, math challenge, math contest problems

WHAT WILL YOUR STUDENTS BE DOING AFTER THE AP'S?
TAKING MATHNOTATIONS 2ND ONLINE (FREE) MATH CONTEST!

UPDATE -- Registration deadline extended until Fri May 15th!

FOR MORE INFO, LINK HERE. We already have several schools registered but there's room for more!

The following is intended for all students in 2nd year algebra. Your stronger students should not find these overly challenging but there is more here than meets the eye. The purpose here is to demonstrate how we can review procedures AND develop deeper understanding of important mathematical ideas in the same lesson. The graphing calculator can be used to enhance the lesson by employing multiple representations (Rule of Four) to reinforce the essential ideas.

Note: Finding the solutions is only the tip of the iceberg. Understanding WHY one equation must have finitely many solutions and the other must have infinitely many solutions is the bigger idea here...

SOLVE EACH OF THE FOLLOWING

Equation 1:
(x-1)(x-2) = (1-x)(2-x)

Equation 2:
(x-1)(x-2)(x-3) = (1-x)(2-x)(3-x)


Click on Read More... for solutions and further discussion.




ANSWERS
Equation 1: All real numbers
Equation 2: {1,2,3}

DISCUSSION
Would most of your students eliminate parentheses in the first equation and solve by traditional methods? Even though the left and right sides of the equation appear similar, it is reasonable to expect they will distribute and solve since that is what they're used to doing. This is fine and the standard procedure should be reviewed.

Assuming students will not make "careless" mechanical errors in distributing, they should obtain:
x² - 3x + 2 = 2 - 3x + x².
This generates a nice discussion of an "identical equation" or identity since the left and right sides are mathematically equivalent (if they recognize that!). The instructor may or may not want to continue the mechanical approach of moving all terms to one side producing 0 = 0 to reinforce that the equation is satisfied by all real numbers. Your stronger student will not have much difficulty with this.

Before moving on to the 2nd equation, we can develop a deeper conceptual understanding by asking students to approach the problem another way. We know that some students will wonder about the form of the original equation. Could we have predicted that the two sides would be identical without removing parentheses? Could we also have determined by inspection that both x = 1 and x = 2 are solutions? Asking them to revisit the original equation to see this is critical. Now what about trying some other real number, say x = 5. This should strongly suggest that all real numbers will satisfy the equation. Using the graphing calculator will also drive this point home visually. Store the left side of the equation Y1 and the right side of the equation in Y2. Change the appearance of Y2 (make it bold for example) and have them observe on the viewscreen that the graphs are identical.

So, how come the 2nd equation only has 3 solutions! I'll leave that to my readers to elaborate on...
How can we generalize this?

These kinds of lessons seem to involve way too much overhead, stealing so much valuable time away from other content. BUT these are precisely the kinds of problems students are expected to grapple with in Japan and other countries. Do you really believe "Less is More?"

...Read more

Posted by Dave Marain at 6:02 AM 4 comments

Labels: algebra 2, conceptual understanding, instructional strategies, more

As I await the blizzard of '09 here in the Northeast, some updates...


RSS FEED ISSUES FOR MATHNOTATIONS
Any problems getting my RSS feed via Google Reader or other aggregators? Blogger, which recently purchased Feedburner, required all users of Feedburner to transfer their feed. This may have caused a temporary disruption of the feed for MathNotations. I updated the feed address for redirecting my readers as of 2-28-09 and I inserted a new 'gadget' into the sidebar allowing for re-subscribing if needed. I noticed that the number of subscribers was cut in half when I transferred over so I'm hoping it will correct itself in a couple of days. Please email me at dmarain at gmail dot com to let me know if you're having any problems getting the feed. Either way, let me know. Apparently other Bloggers are having similar problems with their feed.


SAT Day is March 14th. How appropriate. May all of your students score at least 250π on the math section!

An SAT Ratio Problem
Here's a common SAT type of problem (above-average difficulty) which all students should know how to approach. I've chosen this because it demonstrates different mathematical approaches and test-taking strategies:

In Mr. Jonas' AP Stat class, the number of left-handed students is three times the number of right-handed students. If one-fourth of the lefties and one-third of the righties in the class play an instrument, what fractional part of the class plays an instrument?

(A) 7/12 (B) 5/16 (C) 11/36 (D) 5/18 (E) 13/48

Notes, Comments, Solutions, Strategies,...

(a) To make the wording more convoluted and difficult for students, I could have used the noxious "three times as many as" phrase. I chose to avoid this as it would weaken the reliability of this question IMO. Your thoughts?

(b) How would you categorize this problem? Prealgebra as it can be handled by ratio considerations? Algebra since it can be solved algebraically? Are questions like these typically included in middle school texts here in the US? Singapore texts in their primary materials?

(c) In your opinion, would most juniors in HS approach this algebraically or would they use the most common SAT strategy, 'Plug in'?? Is this question so obvious that it's rhetorical! More importantly, do most students really know how to use this method effectively? How many students know how to organize the information in the problem using a tree model? A Punnett Square model?

(d) How would teachers of Singapore materials explain this question? What model would their students be encouraged to use?

(e) What % of your students would have the level of ratio sense to do this:
(1/4)⋅(3/4) + (1/3)⋅(1/4) = 13/48.
As an aside, how many would attempt the arithmetic without a calculator!

(f) What is your opinion of the distractors? Could the intuitive student eliminate some answer choices quickly, narrowing the options to one or two and making an educated guess? Would this question make a better grid-in (student-constructed response)?

(g) If you felt that the proportion of left-handed students in this problem was highly abnormal, I can only reply that since I'm left-handed, I must be in my right mind!


I would like to share some other methods I've seen successful students use as well as delve further into the conceptual and skill foundations for this problem but I'll stop here for now. If you would like me to go further with this, let me know by commenting or emailing me directly.

Posted by Dave Marain at 4:41 PM 7 comments

Labels: instructional strategies, ratios, SAT-type problems

Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?

Note: It may be highly instructive to ask the students what natural assumptions (stated below) are being made here before you tell them!
(1) All beagles work at the same rate. (If you understand beagle behavior intimately, you might question this). Seriously, it's the underlying assumption of constant "rate of work" that is so fundamental here.
(2) All holes are the same size.

Instructional Commentary
Well, at least, I didn't ask the classic: "How many eggs can 1.5 hens lay in 1.5 days (my all-time favorite word problem)!

The focus of this post will be on the first two stages of concept development using a concrete numerical example. You may take strong exception to the approach below of combining both direct and inverse variation in the same lesson, but, remember, the goal here is concept development, not proficiency with an algorithm! The algebraic stage will be deferred or left to the reader. The algebraic relationships are extremely important and worthy of extended discussion but that needs to be a separate discussion.

Stage I: Building on Intuition
Before developing a strict mathematical procedure involving direct, inverse or joint variation I feel it is critical for students to trust their "math sense." Encourage this with comments like:
"Forget calculations here, boys and girls, just think about this problem, use commonsense, and you might be able to arrive at the answer in less than 10 seconds!"
Don't think they can? No harm in trying...

I believe that when we tell them to trust their intuition, some will arrive at the correct answer of 5 days. Encourage those who "see" it to share their reasoning: WHY will the number days not change! This will vary according to the ability level and confidence of individuals in the group but, even more importantly, according to the environment you create in the classroom (accepting non-judgmental climate leads to greater risk-taking).

When review of homework, content coverage and time for guided practice (before the assignment is given) are the highest priorities of our lessons, then it is natural to question the wisdom of the above strategy.
This is obvious from typical comments like:

"Very nice, Dave, but who has the time for that, it's not going to be tested on the State Test and, moreover, I'm not teaching gifted kids like you must have had."

I won't react to my own Devil's Advocate arguments. Those you who know my philosophy of education know what my response would be!

Stage II: Beyond Intuition - Developing Proportionality Concepts via a Systematic Approach
"Well, boys and girls, now that we believe the answer is still FIVE days, let's try to approach this more mathematically, that is, more logically and systematically, in case the answer cannot be 'guessed' so easily."

I have found over the years that the following TABLE or matrix approach is a powerful model for devleoping proportionality concepts before the student sees a single algebraic relationship:

EVERY DOG HAS HIS DAY!
Beagles	Holes	Days
3	4	5
3	8	??
6	8	???

Note how this approach avoid changing both the number of holes and the number of dogs in the same step! By keeping one quantity fixed, the student may better be able to focus on the relationship between the other two. Thus, in the second row I kept the number of dogs constant, changing only the number of holes:
"Boys and girls, if the number of dogs stays the same and we double the number of holes, then what will happen to the number of days ?"(they will double).
[Note that I asked for the effect on the the number of days before I asked for the actual number of days, namely 10 days.]

This approach develops the idea of direct variation before we express the relationship algebraically: As one quantity increases, so does a second quantity proportionately.

Now that we have filled in the second row (replace the ?? with 10 days), we can move on to another relationship:
"Boys and girls, look at the 3rd row. What quantity (variable) did we not change (keep constant)? What quantity did change? If we double the number of dogs, what should happen to the number of days needed to dig the same number of holes?"
(Yes, some will think 'double', since direct variation is often the initial reaction of many students).

Thus we are literally constructing direct and inverse variation via numerical computation before we develop any general relationships. Yes, this is time-consuming, but hopefully you will see the payoff in comprehension.

Stage III: Expressing Relationships Algebraically
Not in this post!
Important Note:
Normally, we would be very reluctant to mix both types of variation in one lesson, choosing to develop mastery of just direct variation first, then inverse much later on. Yes? Therefore you might feel that combining these will lead to confusion on the part of most students in most classes. Remember, though, the intent here was to develop a strong intuitive base for different types of variations before attempting to formalize any of this! You may not agree, but I'm proposing it anyway. I have done this with good results. Once the concept foundation is laid, students can take off with all the formulas!

Posted by Dave Marain at 8:03 AM 7 comments

Labels: direct variation, instructional strategies, Instructional Strategy Series, inverse variation, variation

The following is the first in a series of strategies for teaching concepts that often prove difficult for many students from middle school on. These are not based on carefully controlled research studies following clinical methodology for a dissertation. They are based on 30+ years of learning how to do it better!! I suspect that's why we refer to the practiceof teaching. Our readers are encouraged to share their own favorite methods that have been helpful to their students or to themselves. These ideas are intended only as suggestions. Each teacher will, of course, bring her/his own ideas and style to bear on the lesson.

Most of you know the classic algebra word problem type that has appeared frequently on standardized tests and math contests:

THE BIG QUESTION
Jack averaged 40 mi/hr going to school and 60 mi/hr returning from school over the same route. What was his average speed in mi/hr for the round trip?

Since there has been a decrease over the past 25 years in the number of word problems to which our students are exposed, some youngsters may not get to see one of these until reviewing for SATs or in their physics class.

From watching how students approach this type of question, I'm getting a sense that we need to introduce the basic concepts earlier on in middle school, which I am sure already occurs in some programs. In planning to teach methods of solving these kinds of problems, I usually tried to return to basic principles of math pedagogy - keep it simple and start with concrete numerical exercises that built on prior knowledge. What does all this jargon mean?


Start with a review of averages, then move on to combined averages before attempting to explain the round-trip rate problem!

[Concerned that such development will take too much time? There won't be enough time to review homework and provide enough practice for the homework assignment? My supervisors never threatened to fire me if a lesson lasted for more than one day and if, heaven forbid, I did not assign homework that first evening! Some ideas just cannot be rushed.]

Suggested Question #1:
Jack had a 70 avg on some tests and a 90 average on some other tests. Can his overall average be determined?
More specifically: When do you think 80 will be the correct answer? When will it not?

Comment:
Question 1 is intended to provoke thought and encourage an intuitive response, not a calculated answer!

Suggested Question #2:
Jack had a 70 average on his first 4 tests and a 90 average on his next 6 tests. What was his overall average for the 10 tests?

Comments
Note that I am suggesting beginning with problems to which middle school students may better be able to relate than a rate-time-distance question. The first question above is fundamental in developing the concept of the original rate problem.

These questions should help many students focus on the essential idea that we need to know how many are in each sub-group!

Since most students connect average to dividing a TOTAL by some quantity, they should feel comfortable in solving the average grade question as follows:

(TOTAL PTS)/(TOTAL NUMBER OF TESTS) to arrive at an average of 82.

BUT DON'T STOP THERE! Stress the UNITS of this result to build the rate concept:

Since students generally do not attach units to the 82, stress that the combined average is 82 PTS PER TEST or 82 PTS/TEST! BTW, not a bad time to mention that PER MEANS DIVIDE!!

Suggested Question #3:
Jack averaged 40 mi/hr for 2 hours, then 60 mi/hr for the next 2 hours. What was his average speed (rate), in mi/hr, for the 4 hours?

[Note the incremental development (commonly termed scaffolding in today's world!). Rather than jump to the abstraction of the original problem, we move on to the next logical step - giving them both the rates and the times for each part of the trip. In this case, we use equal times to provoke their thinking about why the result is also the simple arithmetic mean of the two rates. Each of us needs to make decisions about how many of these examples are needed before moving on to the main question.

Depending on the background and ability level of the group, you may be able to skip one or more of these suggested questions.Further, you may already be thinking of placing these questions on a worksheet for students to try alone or in pairs, stopping and reviewing as needed.

Suggested Question #4:
Jack averaged 40 mi/hr for 4 hours, then 60 mi/hr for 2 hours. What was his average rate, in mi/hr, for the 6 hours?

Suggested Question #5:
Jack averaged 40 mi/hr for the first 120 miles of a trip, then 60 mi/hr for the remaining 120 miles. What was his average rate, in mi/hr, for the entire trip?
Key question: Why does it turn out that the answer is NOT 50 mi/hr?

Comments
Do you think your students would now be ready for the BIG QUESTION near the top of this post? OR do you think they would need at least one more interim problem? Again, could these questions have just as effectively been placed on a worksheet and given to students, working in pairs?

I'll leave the rest to our readers. Pls feel free to share your ideas, comments, thoughts and questions. There's no question in my mind that some of you would develop these ideas differently! Remember you can always email me personally at dmarain at geemail dot com (the last 4 words misspelled intentionally of course!). Unfortunately, I typically get little response from posts about instruction since most readers prefer to solve a challenging problem!

Final Comment: Note that I didn't once suggest that students use a short-cut for the original round-trip problem. Ok, so it is the harmonic mean of the two rates, and can be calculated
from the formula: 2R₁R₂/(R₁+R₂).
But who would want to use that (uh, SATs, GREs, GMATs,...)???

Posted by Dave Marain at 7:28 AM 9 comments

Labels: average rates, averages, instructional strategies, Instructional Strategy Series, pedagogy

One of the reasons I began this blog was to share the collective wisdom of experienced math teachers as a benefit to the novice. Well, here I am 18 months into MathNotations and I don't believe this has yet been specifically addressed. I expect the comments or follow-up posts to be even more beneficial than what I'm writing below.

Here's what I'm asking my readers --
In this post, I will begin enumerating one or two instructional components which I believe should be an integral part of most (math) lessons. Since I have strong antipathy towards jargon, I will try to avoid technical phrases like 'set', 'hook', although closure is ok.

Note that I put math in (..) to emphasize the point that I regard many of these suggestions as integral to effective lessons in general!

Note: These lesson components should be independent of teacher style, makeup of the class, content, etc.


Background
I do know that newbies often feel overwhelmed by all of the differing expectations coming from their immediate supervisor, colleagues, principal, other administrators, courses of study/syllabi, district technology initiatives, state standards, state standards, NCTM Standards/Curriculum Focal Points, standardized test specs -- just to name a few! I haven't even mentioned what they learned from their methods classes, the influence of their math teachers in their formative years, advice from just about everybody. When all is said and done, it seems that the number one concern on the part of most evaluators in the beginning is classroom management, effective delivery of content being number two. Of course, evidence of content knowledge becomes of greater importance if there is an immediate supervisor who has math certification.

How does one navigate through this morass without losing one's mind? Prioritize! Less really is more! Rather than attempt to build the perfect lesson to please the observer, be guided by what you know will lead to demonstrable evidence of learning. Yes, planning is critical. I will comment on that further.

Here then is just the beginning of what I expect to be an extended discussion and one which I am considering publishing as a pamphlet. Please adhere to the Creative Commons License in the sidebar if reproducing any of this.

DISCLAIMER
I am stating unequivocally that these are my own personal ideas of what makes an effective math lesson. I do not want anyone to say that I am telling anyone how to teach!

Each of you out there will have your own list, although I'd be surprised if there wasn't considerable overlap. The order of course will vary. These are the principles by which I was guided both as a classroom teacher and as a supervisor. At the beginning of the year, I would meet with teachers to discuss what I was looking for in the lesson. For clinical observations, I would also have a preconference to discuss specifics. This was particularly of critical importance before observing the non-tenured teacher.

THE BEGINNING
1) Class Opener - Critical first 5 minutes - Establishment of Routines

a) Allow students to socialize/decompress for a couple of minutes as they enter, but let them know what is expected of them; close door at late bell. Establish iron-clad routines for students to follow if they arrive after that - stick to it!

b) Math Warmup/Problem of the Day already on the board or projected on a screen using the overhead or PowerPoint (or Word) from the computer; the warmup can be used to review prerequisite skills for the upcoming lesson, SAT review, an opportunity for students to practice their communication (e.g., writing) skills in math, etc.

c) Answers to some or all of the homework exercises can be written on the board or projected on a screen from overhead or computer. Virtually every publisher of current texts provides ready-made transparencies both for WarmUps and answers to homework, not to mention PowerPoint presentations for every lesson! Some educators object to displaying answers like this as it invites students to quickly copy these on their paper. You may want to have selected answers displayed rather than all. There is no foolproof method here, so use your own judgment. The important thing is to busily engage students from the outset. While students are working on their warmup problem, the teacher is circulating, checking homework and engaging students. This personal interaction with students means so much (e.g., Lily, I saw you in the play on Thu night -awesome!).

Ok, folks, this is just a beginning...
Please contribute your suggestions!

Posted by Dave Marain at 2:10 PM 5 comments

Labels: instructional strategies, pedagogy

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