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I probably should start a new word game called find all the typos! 'Simpel' and rightt' are pretty impressive variations. I guess if you can't spell write rite, then you're not alright! Well, I'm trying to correct these but once they go to a reader, the mistakes are out there for all to see! Haste makes --------------.
Just another one of those square and right triangle problems. Yes, I do have a passion for these types of geometry questions (I've published similar ones previously) and you probably will recognize this one from your own experience. The difference of course is that an investigation takes students from particular cases to a general formula which is related to the harmonic mean of the legs of the right triangle.
Find the length of the side of the square in Fig. I.
This one is eminently guessable and doesn't require the use of similar triangles. Encourage students to justify their conjecture that D and F are midpoints.
Do the same for Fig. II.
Comments: Ok, the answer is 12/7. You may find students trying a Pythagorean approach, guessing midpoints or assuming special angles. Most students do not look for similar triangle solutions unless they have considerable experience with this or the problem is assigned in that unit!
Of course, we will now ask students to generalize the result:
If the legs of the right triangle have lengths a and b, show that the side of the indicated square (inscribed square, largest inscribed square with sides parallel to the legs, however you want to describe it!) has length ab/(a+b).
Friday, October 31, 2008
Thursday, October 30, 2008
A simple warmup for Grades 3-12?? Can one problem really be appropriate at many levels?
Would 3rd, 4th or 5th graders guess the obvious answers 0,0 or 2,2 provided they understand the meaning of the terms sum and product? Do youngsters immediately assume the two numbers are different? Children at that age are thinking of whole numbers, however, what if you allowed them to try 3 and 1.5 (with or without the calculator)?
For middle schoolers: After they 'guess' the obvious integer answers, what if you were to ask them: "If one of the numbers is 3, what would the other number be?" If one of the numbers were 4? 5? -1? -2? Is algebra necessary for them to "guess" the other number? Would a calculator be appropriate for this investigation? Would they begin to realize there are infinitely many solutions? What if you asked them to explain why neither number could be 1...
For Algebra students: If one of the numbers is 3, they should be able to solve the equation:
3+x = 3x; they can repeat this for other values including negatives as well. They should be able to explain algebraically why neither number could be 1. Let them run with this as far as their curiosity takes them!
For Algebra 2, Precalculus and beyond: See previous ideas. Should they be expected to solve the equation x+y = xy for y obtaining the rational function y = x/(x-1)? Analysis of this function and investigation of its graph may open new vistas for this 'innocent' problem about sums and products. Does this function really make it clear why 0,0 and 2,2 are the only integer solutions?
The original question is well-known. At any level, I would recommend that they be allowed to explore and make conjectures before more formal analysis. High schoolers enjoy coming up with 0,0 and 2,2 as much as 8 year olds! Modifying it and asking probing questions as students mature mathematically is the challenge for all of us. Have fun with this 3rd grade question!
Monday, October 27, 2008
Not the most challenging contest problem but something to give to your students to develop logical careful thinking and some "basic skills." There's a slight 'twist' but nothing that will faze our math experts out there.
(x2 - 6x + 9)(x2 - 4x + 3) = 1
Poincare deserved the extra attention but it's time to move on. Our new icon holds a special place in my heart as much for his teaching and writing ability as for his original research in a field that occupied my younger days in research...
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Sunday, October 26, 2008
[PLS NOTE: Several edits have been made. This version should be accurate!]
Two new documents are now available on Achieve's Algebra 2 Test Overview web site:
Released Items Oct. 2008
Released Items Oct. 2008 Commentaries
I strongly encourage our readers to download these pdf documents. The 2nd document is particularly useful since it contains excellent discussions of each question, scoring rubrics, sample student solutions and detailed explanations and alternate methods.
There will be endless arguments about over-testing, open and fair testing, who's making the profit from these tests, quality and authenticity of assessments, the politics of how results will be used (programmatic vs. comparing teachers, schools, states). In the end, we need to get past the rhetoric. For me the bottom line is that these released questions are high quality and require youngsters to demonstrate both mechanical skill and conceptual understanding. Further, they include several open-ended (extended response each counting 4 pts) and short constructed response items (each counting 2 pts) that give students the opportunity to display what they know, not how skillful they are at eliminating answer choices.
Released items often contain more difficult questions. I found many of these questions required some sophisticated thinking and analysis.
I'm not permitted to reproduce any of the items however I will attempt to categorize each problem and enumerate topics (you may disagree with some of these classifications so please read the documents).
- Imaginary solutions of quadratic equation
- Graph of inverse of a linear function
- Meaning of rational exponents
- Solving absolute value equation which includes a linear expression outside the absolute values (leading to an extraneous solution)
- Relationship between a polynomial function and its graph
- (Short answer) Solving a quadratic equation resulting from a Pythagorean application
- (Short answer) Constructing the graph of a simple rational function, e.g., f(x) = k/x2
- Recognizing an exponential function from its characteristics (domain, range, intercepts, asymptotes, etc.)
- Determine the slope of a linear-type function involving absolute values
- Graph of a system of linear inequalities
- Associating a function involving the greatest integer function with a problem situation
- Roots of a quadratic equation with a negative discriminant
- Matching the graph of a quadratic function with characteristics involving its coefficients
- (Short answer) Determining an expression for the volume of a cube whose original dimension is increased by a variable amount (also, expand the expression).
- Determining the zeros of an exponential function
- Solving for a variable in a literal equation involving a radical
- Solving an applied problem (physics-type) involving a given quadratic function
- Domain of a composite function
- Simplify a 'complex' fraction (Mechanical skill)
- Analyzing functions of the form cxd (including end behavior)
- (Short answer) Construct a piecewise function to model a given problem situation
- (Extended Response) Applied problem involving interpretation of a given quadratic model
- Simplifying rational expression (Mechanical skill)
- Recognizing graph of a linear programming application (simple)
- Matching a given exponential function (involving a parameter) to a function table (conceptual)
- Application of concept that the product of a complex number and its conjugate is real
- Exponential growth application
- Analyzing the effect on the zeros of a quadratic subjected to different transformations (conceptual)
- (Extended response) Applied problem involving percent increase and an exponential model
Sunday, October 19, 2008
It's that time of year folks! Here's a problem for you or your students to think about as a warm-up for this week or for the SATs or for just developing logical thinking. It might also deepen student thinking about the distribution of data. You might find this question trivial but don't be too quick to judge this until your students try it! I would allow a calculator to be used. Observe how students approach this: Guess-Test, consideration of the 'mean', etc...
In a certain high school election, there were 12 candidates for President of the Student Council. If 1600 votes were cast and Denise received more votes than any of the others (i.e., she received a plurality), what is the least number of votes she could have received?
Saturday, October 18, 2008
Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.
How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?
I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.
First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter taking measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.
Are the number of steps roughly the same?
Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole number part of the larger mixed number, convert the 1 into common denominator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.
Improper Fraction Algorithm:
(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.
Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of the steps
may not be equivalent.
I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!
I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.
Further, I suspect there are some youngsters who convert mixed numerals to improper fractions procedurally without full conceptual understanding that a mixed numeral is an addition problem!
Your experiences and thoughts...
Sunday, October 12, 2008
Update: My daughter is making a miraculous recovery and should be home soon. Your thoughts and prayers brought us all through this. Thank you...
This post is dedicated to the extraordinary health care professionals at Hackensack University Medical Center (Children's Wing) who have and are continuing to help our daughter recover from critical illness. The doctors, nurses, respiratory therapists and social workers in the Pediatric Intensive Care Unit have been remarkable. Their sensitivity and understanding of our daughter's needs (as well as her family's) have been limitless.
In particular, my wife and I would like to personally thank Mike H whose optimism and uncanny ability to make our daughter smile (her parents too!) place us forever in his debt. Mike's wisdom, humility, and selflessness are qualities that make him truly unique among a unique group of dedicated caring professionals.
Thursday, October 9, 2008
With the PSAT rapidly approaching, here are a couple of problems which require the student to review their knowledge of circles. This will be set up as an open-ended investigation with several parts, but the content is often assessed on standardized tests. As usual there are many approaches, although efficient use of ratios and proportions is the goal here. It is critical that students thoroughly read the detailed given info in the text box.
In Figure I, determine the length of minor arc PQ.
In Figure II, determine the area of sector OPQ.
A central theme here is the relationships among the ratio of the radii of the two circles, the ratio of their intercepted arc lengths and the ratio of the areas of their corresponding sectors. Students need to have a clear understanding that one is a linear relationship and the other is a direct square variation. there are many ways to set up the solutions of these problems. We will discuss this further in the reader comments.
Also, a good discussion point is to have students explain why the last piece of given information, regarding the central angles not being congruent, was not necessary. It might be interesting to have students compute the degree or radian measures of the central angles.
Wednesday, October 1, 2008
No, there's no mistake in the constant term in that equation. Imagine giving this to your Precalculus/Math Analysis/Adv Math students! Actually, 'solving' it with the TI-84 requires some effort using Solver since one needs to make an approximate guess or adjust the lower bound so that the positive root is obtained. Using the graph is no 'walk in the park' either! The TI-89 or Mathematica would have much less difficulty in displaying the exact radical form or a suitable decimal approximation but they may not be within reach. Perhaps an important issue here is that sometimes technology gives us unexpected or even inaccurate results. That's when students need some understanding of theory to recognize the limitations of the technology and adjust accordingly.
Here's the point of all this. The given quadratic is not factorable over the integers, however we can replace it with a 'nicer' quadratic that is. The roots of the desired quadratic can be shown to be approximately the same as the 'nice' quadratic and we can show that the absolute error is less than two ten-thousanths (and a much much smaller relative or % error)! Does this 'numerical analysis' have any practical value? Why approximate roots when powerful technology can produce exact answers? Do professionals who need to apply mathematics to the solution of 'real' problems ever use such approximation techniques? Could it be that theory actually provides practical application!
(1) Show that the roots of the x2-10000x-10001 = 0 are 10001 and -1 by factoring.
(2) Show that the roots of x2-10000x-10000=0 can be approximated by 10001 and -1 with an error of less than 0.0001.
(a) By direct calculation: Using the quadratic formula and, yes, you may use the calculator!
(b) (Challenging) By comparing, in general,
(*) the roots of x2-bx-(b+1)=0 and
(**) the roots of x2-bx-b=0.Here we are assuming that b > 0.
(i) First show by factoring that the roots of equation (*) are b+1 and -1.
(ii) Then use the quadratic formula to express the roots of (**) in terms of b.
(iii) Compare the positive roots of these equations by subtracting them and (after algebraic manipulation and simplication), show that the absolute value of the difference is less than 1/(b+1).
Note: For b=10000, this error is therefore less than 0.0001.
(c) Explain intuitively why the roots of the original equation and the 'approximating' equation are virtually the 'same' for 'large' values of b. One possibility here is to consider how the graphs of the associated quadratic functions are related. What do they have in common? How are they different?
Note: Subtle point here for students. Even though the difference of the function values (i.e., y-values) is always 1, this is not true of the difference between their zeros! This may be the essence of the numerical analysis in this investigation.
Ok, now "solve" x2 - (googol)x - googol = 0 without a calculator.
Without your calculator show that √(10001) - √(10000) is less than 0.005.
Does this provide us with an effective method of approximating the square root of some large numbers or is it limited and impractical?
For Calculus students: How does this compare to using linearization to approximate the square root?
For more advanced calculus students: Newton's Method? The Binomial Formula (using fractional exponents)? A Taylor Polynomial approximation? All equivalent?