OVERVIEW
Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.
THE PROBLEM
Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.
(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.
Answer: (b) 6+2√2
REFLECTIONS
•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...
I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...
Consider the following problem:
If -5 ≤ x ≤ 4, and f(x) = 2x2 - 3, how many integer values are possible for f(x)?
One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).
Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...
WITH YOUR LEARNING PARTNER(S):
1. Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.
WRITE your conjecture for the answer to the problem: ____
2. Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table. Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1? YES NO
3. The following statement is plausible but FALSE.
The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.
Explain why this is wrong. There is more than one error!
4. The correct answer is 51. Depending on the class, a few, if not several, students should be able to come up with the correct answer and provide a thorough explanation.
5. Group Discussion:
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. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!
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I decided to post a video solution of the Twitter problem I posted on 8-25-10:
4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red?
Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly. Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!
There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:
A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?
I'm sure my astute readers can improve on this wording but we'll leave it at this.
A few questions naturally pop up:
(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.
(2) For whom is the video intended? Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.
(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).
SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS
(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems: DOES ORDER COUNT!
(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!
Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...
(3) The basic definition of probability should always be in the forefront of your mind:
P(an event) = TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.
As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes. In the Twitter problem it is 15 if order is disregarded. Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from "How many ways are there to get 2 red and 2 blue cards?"
(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.
An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!
Look for my other videos on my YouTube channel MathNotationsVids. Look for all of my Twitter SAT Problems on twitter.com/dmarain.
As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.
Please note correction to 2nd problem in the video. The correct answer is 4096 "real" values. The original answer, 13, applies to rational solutions only. Thanks to Nick Hobson for pointing out my careless error. Haste makes waste!!
Please vote in the poll at the right. Be candid in your opinion of these videos. It will guide me in the future to improve. Don't hesitate to share your opinions on MathNotationsVids and rate each video there as well. If you subscribe to my feed, please vote directly on the site. Only a few days left...
The title says it all so here is the video as promised:
Note: See above correction to 2nd problem! The video has not been corrected so beware!
Comments on 2nd problem:
If x is greater than or equal to 0 and less than or equal to 3, for how many values of x will 16^x be an integer?
As mentioned above, Nick pointed out my error. I should have restricted x to be of the form a/b, where a and b are integers, b ≠ 0. Normally, SAT questions avoid use of the term rational so they would spell it out. This problem however is very questionable for SATs. If real solutions were sought, this question would be more appropriate for a math contest. Here's one way of explaining why the answer is 4096 for real solutions:
16^x = k, k an integer → 2^(4x) = k
3 ≥ x ≥ 0 → 12 ≥ 4x ≥ 0 → 4096 ≥ 2^(4x) ≥ 1 since the exponential function 2^(4x) is increasing. This argument is reversible, so there are 4096 solutions for x, one of each integer value of k from 1 to 4096 inclusive. This solution could be written more concisely using log base 16 or log base 2 as Nick did, but I wanted to show a method without the log symbol.
Again, the video solution is WRONG as it shows only rational solutions! Well, at least i was thinking "rationally!"
I fully realize that the school year is over for some and about to end for others but these SAT Problems will be around for you or your students in perpetuity! Let me know if you like the questions. They are now appearing in the right sidebar of my blog so you will need to visit the page to see them.
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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) 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
Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school
List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.
Comments:
(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"
(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?
(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?
(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!
(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?
Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students.
Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.). If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.
Update: Answers/comments are at the bottom...
1. NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!
And the seasons they go round and round
And the painted ponies go up and down
We're captive on the carousel of time
We can't return we can only look behind
From where we came
And go round and round and round
In the circle game...
Oh, how I love Joni Mitchell's lyrics made famous by the inimitable Buffy Sainte-marie. Oh, how The Circle Game lyrics above describe my feelings about the state of U.S. math education. I feel I've been on this carousel forever. But I do believe that all is not hopeless. I do see promise out there despite all the forces resisting the changes needed to improve our system of education.
Our math teachers already get it! They get that more emphasis should be placed on making math meaningful via applications to the real-world, stressing understanding of concepts and the logic behind procedures, reaching diverse learning styles using multiple representations and technology, preparing their students for the next high-stakes assessment, trying to ensure that no child is ... They've been hearing this in one form or another forever. BUT WHAT THEY NEED IS A CRYSTAL CLEAR DELINEATION OF ACTUAL CONTENT THAT MUST BE COVERED IN THAT GRADE OR THAT COURSE.
The vague, jargon-filled, overly general standards which have been foisted on our professional staff for the past 20 years is frustrating our teachers to the point of demoralization. THIS IS NOT ABOUT THE MATH WARS. THIS IS NOT AN IDEOLOGICAL DEBATE. JUST TELL OUR MATH TEACHERS WHAT MUST BE COVERED AND LET THEM DO THEIR JOB!
BY "WHAT MUST BE COVERED" I AM INCLUDING THE SKILLS, PROCEDURES AND ESSENTIAL CONCEPTS OF MATHEMATICS. NONE OF THIS CONSTRAINS TEACHER STYLE OR CREATIVITY. BUT WITHOUT THIS STRUCTURE THERE IS ONLY THE CHAOS THAT CURRENTLY EXISTS. AND IF YOU DON'T THINK THERE IS CHAOS OUT THERE, TALK TO THE PROFESSIONALS WHO HAVE TO DO THIS JOB EVERY DAY.
UPDATES...
Results of MathNotation's Third Online Math Contest
The Common Core State Standards Initiative
NCTM's latest response to the Core Standards Movement - the forthcoming Focus in High School Mathematics
Validation Committee selected for draft of Core Standards
The results of the latest round of ADP's Algebra 2 and Algebra 1 end of course exams
It will take several posts to cover all of this...
RESOURCES FOR YOU
MODEL PROBLEMS TO DEVELOP HIGHER-ORDER THINKING AND CONCEPTUAL UNDERSTANDING
Consider using the following as Warm-Ups to sharpen minds before the lesson and to provide frequent exposure to standardized test questions (SAT, ACT, State Assessments, etc.). I hope these problems serve as models for you to develop your own. I strongly urge you to include similar questions on tests/quizzes so that students will take these 5-minute classroom openers seriously.
I've provided answers and solutions/strategies for some of the questions below. The rest should emerge from the comments.
MODEL QUESTION #1:
For how many even integers, N, is N2 less than than 100?
Answer: 9
Solution/Strategies:
Always circle keywords or phrases. Here the keywords/phrases include
"even integers"
N2
"less than".
This question is certainly tied to the topic of solving the quadratic inequality, N2 "<" 100 either by taking square roots with absolute values or by factoring. Of course, we know from experience, when confronted with this type of question on a standardized test, even our top students will test values like N = 2, 4, 6, ... However, the test maker is determining if the student remembers that integers can be negative as well and, of course, ZERO is both even and an integer! Thus, the values of N are -8,-6,-4,-2,0,2,4,6, and 8.
MODEL QUESTION #2
If 99 is the mean of 100 consecutive even integers, what is the greatest of these 100 numbers?
ANSWER: 198
Solution/Strategies:
There are several key ideas and reasoning needed here:
(1) A sequence of consecutive even integers (or odd for that matter) is a special case of an arithmetic sequence.
(2) BIG IDEA: For an arithmetic sequence, the mean equals the median! Thus, the terms of the sequence will include 98 and 100. (Demonstrate this reasoning with a simpler list like 2,4,6,8 whose median is 5).
(3) The list of 100 even consecutive integers can be broken into two sequences each containing 50 terms. The larger of these starts with 100. Thus we are looking for the 50th consecutive even integer in a sequence whose first term is 100.
(4) The student who has learned the formula (and remembers it!) for the nth term of an arithmetic sequence may choose to use it: a(n) = a(1) + (n-1)d. Here, n = 50 (we're looking for the 50th term!), a(1) = 100, d = 2 and a(100) is the term we are looking for.
Thus, a(50) = 100 + (50-1)(2) = 198.
However, stronger students intuitively find the greatest term, in effect inventing the formula above for themselves via their number sense. Thus, if 100 is the first term, then there are 49 more terms, so add 49x2 to 100.
MODEL QUESTION #3: A SAMPLE OPEN-ENDED QUESTION FOR ALGEBRA II
If n is a positive integer, let A denote the difference between the square of the nth positive even integer and the square of the (n-1)st positive even integer. Similarly, let B denote the difference between the square of the nth positive odd integer and the square of the (n-1)st positive odd integer. Show that A-B is independent of n, i.e., show that A-B is a constant.
MODEL QUESTION #4: GEOMETRY
If two of the sides of a triangle have lengths 2 and 1000, how many integer values are possible for the length of the third side?
MODEL QUESTION #5: GEOMETRY
There are eight distinct points on a circle. Let M denote the number of distinct chords which can be drawn using these points as endpoints. Let N denote the number of distinct hexagons which can be drawn using these points as vertices. What is the ratio of M to N?
Answer: 1
Solution/Strategies: The student with a knowledge of combinations doesn't need to be creative here but a useful conceptual method is the following:
Each hexagon is determined by choosing 6 of the 8 points (and connecting them in a clockwise fashion for example). For each such selection of 6 points, there is a uniquely determined chord formed by the 2 remaining points. Similarly, for each chord formed by choosing 2 points, there is a uniquely determined hexagon. Thus the number of hexagons is in 1:1 ratio with the number of chords.
MODEL QUESTION #6: GEOMETRY AND THE ARITHMETIC OF PERCENTS
If we do not change the angle measures but increase the length of each side of a parallelogram by 60%, by what per cent is the area increased?
(A) 36% (B) 60% (C) 120% (D) 156% (E) 256%
UPDATE #2: Answers to the quiz are now provided at the bottom. If you disagree with any answers or would like clarification, don't hesitate to post a comment or send an email to dmarain "at gmail dot com".
UPDATE: No comments from my faithful readers yet -- I suspect they are giving students a chance to try these! I will post answers on Friday 9-25. However, students or any readers who would like to check their answers against mine need only email me at dmarain "at" gmail "dot" com and I will let them know how they did!
With the SAT/PSAT coming in a few weeks, I thought it would be helpful to your students to try a challenging "quiz". Most of these questions represent the high end level of difficulty and some are intentionally above the level of these tests. Then again, difficulty is very subjective. A student taking Honors Precalculus would have a very different perspective from the student starting Algebra 2!
Also, these questions can also be used to prepare for some math contests such as the THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST! Yes, another shameless plug, but time is running out for your registration...
A Few Reminders For Students
(1) Do not worry about the time these take although I would suggest about 30 minutes. The idea is to try these, then correct mistakes and/or learn methods/strategies. It's what you do after this quiz that will be of most benefit!
(2) I added strategies and comments after the quiz. I suggest trying as many as you can without looking at these. Then go back, read the comments and re-try some. I will not provide answers yet!
(3) Don't forget these problems are copyrighted and cannot be reproduced for commercial use. See the Creative Commons License in the sidebar. Thank you...
PRACTICE PSAT/SAT QUIZ
1. If n is an even positive integer, how many digits of 1002n - 1002n-2 will be equal to 9 when the expression is expanded?
(A) 2 (B) 4 (C) 8 (E) 2n (E) 2n - 4
2. The sides of a triangle have lengths a, b and c. Let S represent (a+b+c)/2. Which of the following could be true?
I. S is less than c
II. S > c
III. S = c
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III
3. The mean, median and mode of 3 numbers are x, x+1 and x+1 respectively. Which of the following represents the least of the 3 numbers?
(A) x (B) x - 1 (C) x - 2 (D) x-3 (E) 2x - 2
4. (10/√5)500 (1/(2√5))500 = _________
5. A point P(x,y) lies on the graph of the equation x2y2 = 64. If x and y are both integers, how many such points are there?
(A) 4 (B) 8 (C) 16 (D) 32 (E 64
6. Each side of a parallelogram is increased by 50% while the shape is preserved. By what percent is the area of the parallelogram increased? __________
7.
AB is parallel to CD , AB = 3, CD = 5, AD = BC = 4. If segments AD and BC are extended to form a triangle ABE (not shown), what would be the length of AE?
Ans_________
Figure not drawn to scale
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STRATEGIES/COMMENTS
1. Most students learn to substitute numbers for n here although it can be done algebraically by factoring. However, the real issue here is figuring out what the question is asking. Reading interpretation - ugh!!
2. When you are not given any information about what type of triangle it is, just choose a few special cases and draw a conclusion. O course, if one recalls a key inequality theorem from geometry, this problem can be done in short order.
3. If you don't feel comfortable setting this up algebraically (preferred method), PLUG IN A VALUE FOR x...
4. Your calculator may not be able to handle the exponent so skills are needed. The large exponent also suggests a Make it Simpler strategy. This is a "Grid-In" question so if one is guessing remember that most answers are simple whole numbers! Finally, if one knows their basic exponent rules and basic radical simplification, none of the above strategies are needed!
5. Possibilities should be listed carefully. It is possible to count these efficiently by recognizing the effect of reversals and signs. Easy to get this one wrong if not careful.
6. For those who do not remember or want to apply a key geometry concept about ratios in similar figures, there are a couple of essential test-taking strategies which all students should be aware of of:
(a) Consider a special case of a parallelogram
(b) choose particular values for the sides.
In the end, even strong students often make a different error, however. That darn ol' percent increase idea!
7. Should you skip this if you have no idea how to start? Absolutely not! Draw a complete diagram and even if you don't recognize the similar triangles, make an educated guess! It's a grid-in and there's no penalty for guessing. Further, answers tend to be positivc integers!!
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ANSWERS
1. B
2. B
3. C
4. 1
5. C
6. 125
7. 6
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
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...
After 4 tests, Barry's average score was 5 points higher than Michelle's. After the 5th test, Michelle's overall average was 5 points higher than Barry's. Michelle's score on the 5th test was how many points higher than Barry's?
Can you find at least three methods for solving this?
Algebraic, "plug-in", conceptual, etc...
As teachers we need to have a deep understanding of these kinds of problems and familiarity with several approaches. Of course, our students will show us a variety of methods, both right and wrong, when we open up the dialog!
Comments
Students from middle school on see many problems relating to means. However, they need to see a variety of problems of increasing difficulty. This question is certainly not a highly challenging math contest problem but I believe it demonstrates some important principles of averages and can be used to review different problem-solving strategies. Middle schoolers would struggle with the algebraic approach (a system of two equations), however they should be thoroughly comfortable with the underlying ideas.
Since the focus is on concept and method, I will give the answer: 45
Well, the June SATs have arrived today so these problems come too late for that, but these kinds of questions can be used to review basic ideas while strengthening thinking skills. Both questions below are appropriate for both middle and secondary students, although the second requires knowledge of a fundamental geometry principle regarding the sides of triangles.
There are other important principles embedded in these problems as well. In the end, I believe that students need to be exposed to many of these "contest-type" challenges to improve reading skill, learn how to pay attention to detail and think clearly. As a separate issue, performing well under testing conditions requires extensive training. You may not feel this is an important objective for math teaching in the classroom, but testing is a reality for the student...
These questions may appear fairly straightforward at first but be careful! I believe the second is more challenging than the first. These are not so different from the "gotcha" problem on our latest online contest.
1) The dimensions of a rectangle are odd integers and its perimeter is 100. How many different values are possible for its area?
2) The perimeter of an isosceles triangle is 96 and the lengths of its sides are even integers. How many noncongruent triangles satisfy these conditions?
For my "unofficial" answers, click on Read more...
Unofficial Answers (no solutions):
1) 13
2) 11
Feel free to challenge these answers or express agreement!
Comments
Which of the following do you believe would cause the most difficulty for students?
In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?
Comments
SAT "Grid-In" Type
Level of Difficulty: 5 (High)
Content: Algebra 2, precalculus
The function F satisfies the condition
F(N + 6) = F(N) + 8, for all integers N.
If F(7) = -2, what is the value of F(25)?
Click on Read more to see the answer, solution, discussion.
Answer: 22
Suggested Solution:
Replace N by 7 since F(7) is known:
Therefore, F(7 + 6) = F(7) + 8 or F(13) = -2 + 8 = 6
Next, replace N by 13 since F(13) is known:
F(19) = F(13) + 8 = 6 + 8 = 14
Finally, F(25) = F(19) + 8 = 14 + 8 = 22.
Comments
(1) Too difficult for the SATs? Not really! A similar problem recently appeared. There aren't that many "hard" questions (Level 5) on the SAT but, if a student wants to score over 700 they will need exposure to these types in practice.
(2) Consider writing some variations of these function-type problems for additional practice. At first, change the constants, then consider changing the operations (from addition to multiplication for example). One could raise the bar even higher by asking the question in reverse:
If F(25) = 22, what is the value of F(7)?
(3) There is considerable advanced theory in functional equations and recurrence relations underlying these problems. However, the student needs only to feel comfortable with the function symbolism (or should I call it "Math Notations!"). Starting by "plugging in' N = 7 seems simple in retrospect but most students are too intimidated to consider it. Even the precalculus student may be able to get started, but, without experience, they will often get lost. This is all about exposure, but isn't it always?
(4) One could rewrite this problem using sequence notation:
aN+6 = aN + 8. By expressing the problem in the context of the Nth term of a sequence, students may grasp it a bit better, but, in the end, it's all about interpreting function notation.
1) The latest "biweekly" edition of our new Carnival, Math Teachers at Play, is currently being hosted by Misty over at Homeschool Bytes. I enjoyed the "step by step" approach, progressing from primary math activities like Candy Math through middle school posts like Division of Fractions Conceptually to secondary articles like Ten 16th Century word problems. I contributed a post on Function Questions for the SATs. I'm not including any direct links to these articles. Go to Misty's site to enjoy this carnival!
2) I'm putting a lot of effort into the SAT Math Tips feature in the sidebar (readers of my feed won't get to see this of course unless they visit the site). I realize a good part of the country takes the ACT but the suggestions may be applicable to any standardized test and the math content is valid for anyone who wants to use it.
3) The Read More feature I recently instituted has some bugs as I'm sure you noticed. It doesn't work in RSS or Atom feeds of course and it sometimes doesn't work properly even on this site. I'm a coder at heart but I'm not sure it's worth all the effort. Clearly the Blogger developers are not interested in making things easy for us. You may see Read more... even if there is nothing else to see or it may not work all! Too late for me to migrate to Wordpress at this point... Please be patient with me here as I work through this.
4) The Math Problems of the Day in the sidebar are of good quality and are challenging but I'm seeing more repetition of questions. I will evaluate this and decide if I want to keep this feature.
5) No, I haven't forgotten about the next MathNotations contest I promised for April or May. Stay tuned...
6) Ever hear of Krypto or 24 (the game not the TV show!) or the more well-known "Four Fours Game"? I played many years ago and have always enjoyed the challenge of this tantalizing arithmetic game. I've used it effectively in upper elementary and middle school classrooms when I was a math staff developer to reinforce order of operations and basic fact recall (no calculator allowed!). Ok, so I'm now driving you crazy with another KenKen-like game!!
Click on Read more to learn more. (if this works!).
Here is a sample play of Krypto:
Suppose you're dealt the following five number cards:
5, 9, 4, 11, 1.
A 6th or objective card is turned up, say 2.
Using some or all of the four basic arithmetic operations and the 5 numbers exactly once, produce the objective number. This one is straightforward using only addition and subtraction:
11 + 5 - 9 - 4 - 1 = 2. Krypto! You've Won!
It is possible to be dealt an "impossible" hand for which there is no solution or there could be many solutions for the same hand! The original game did not allow the use of parentheses but you could choose any variation you wish, including reducing the number of cards to 4. An ordinary deck could be used modifying the face cards to be 11,12, etc.
If you want to play the online version from mphgames, go here for instructions and play. Your browser must be java-enabled but most are. For more background on the game and a discussion of the underlying combinatorial mathematics and a discussion of the computer program which generates it, look here.
ENJOY KRYPTO WITH OR WITHOUT YOUR STUDENTS!!
Version I (Level of difficulty 3 - medium)
With a special promotion, Al received a 60% discount on a new stereo system and paid $x. Sylvia bought the same system (same original price) but only received a 20% discount. In terms of x, how much (in dollars), did Sylvia pay? Assume x > 0 and disregard sales tax.
(A) 4x (B) 3x (C) 2x (D) 4x/3 (E) x/3
Version II (Level of Difficulty 4 - medium/hard)
Grid-In Type
Maury purchased a new electronic game system with a 25% off coupon. His friend bought the same system (same original price) with a 40% off coupon. If his friend paid $45 less for the system, how much did Maury pay (disregard sales tax)?
For the answers, suggested solutions, strategies and discussion, click Read more...
Level I problem
Answer: (C) 2x
Possible Solutions:
Method I ("Plug-in" SAT Strategy - Student-preferred?)
Let original price = $100.
Then Al's discount was $60, so he paid $40. Thus x = 40.
Sylvia's discount was $20, so she paid $80, which means she paid twice Al's price or 2x.
Method II (conceptual)
Al paid 40% of the original price, Sylvia paid 80%, therefore Sylvia paid twice as much as Al.
Method III (traditional - Algebraic)
Reasoning as in Method I, Al paid 40% and Sylvia paid 80% of the original price.
Let y = original price (before discounts).
Then Al paid 0.4y = x. Solving, y = 2.5x.
Sylvia paid (0.8)(2.5x) = 2x.
Level II Problem
Answer: $225
Methods???
Note: There is a mental math method which will be discussed later.
FOOD FOR THOUGHT
Do you remember the problem I posted a couple of days ago at the bottom of one of my updates:
What is the greatest possible number of Saturdays in a string of 100 consecutive days?
Well, here's a new feature that I hope will work. Click "Read more" and, hopefully, the answer and solution(s) will appear! If it doesn't work, then you will see the entire post.
Let me know if this works by posting a comment or emailing me (dmarain at geemail dot com)!
Answer: 15
Suggested Solutions
To maximize the number of Saturdays it is logical to start with 1 as the first Saturday, then the next Saturday will be day #8, then day #15, and so on. Each term of this sequence can be described by the expression 7a+1, that is, the positive integers which leave a remainder of 1 when divided by 7. The largest multiple of 7 less than 100 is 14x7 = 98, thus our sequence of Saturdays proceeds: 1,8,15,22,...99. Note that the first term 1 is actually 7x0+1 and the last term 99 = 7x14+1, for a total of 15 Saturdays.
Students should also recognize that if a sequence can be described by a linear function of the form s(n) = kn+b, then the sequence is arithmetic and we can apply the well-known formulas for arithmetic sequences. Thus 99 = 1 + (n-1)7 leading to our result of n = 15. Here n represents the number of terms of our sequence starting with a value of 1.
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