A Rant, An Update and Model Problems for You

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%

Another Sample Contest Problem - Counting...

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

BASIC RULES
* Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
* Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
* The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
* Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
* Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
* Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
* Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.


Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

Comments
The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)




My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_

...Read more

MathNotations Third Online Free Math Contest Update and Sample "Proof"

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

• Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
  
• Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
  
• The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
  
• Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
  
• Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
• Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
  
• Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.
  

The following is a sample of the open-ended "proof-type" questions on the test:

Explain why each of the following statements is true. Justify your reasoning carefully using algebra as needed.

The square of an odd integer leaves a remainder of 1 when divided by
(a) 2
(b) 4
(c) 8

I may post a sample solution to this or you can include this in your comments to this post.

Two Trains and a Tunnel! Is There Room For This In The Tunnel And In Your Curriculum?

At the same instant of time, trains A and B enter the opposite ends of a tunnel which is 1/5 mile long. Don't worry -- they are on parallel tracks and no collision occurs!

Train A is traveling at 75 mi/hr and is 1/3 mile long.
Train B is traveling at 100 mi/hr and is 1/4 mile long.

When the rear of train B just emerges from the tunnel, in exactly how many more seconds will it take the rear of train A to emerge?

Click on More to see answer (Feed subscribers should see answer immediately).

Comments
1. Appropriate for middle schoolers even before algebra? Exactly when are middle schoolers in your district introduced to the fundamental Rate_Time_Distance relationship?
2. What benefits do you think result from tackling this kind of exercise? If it's not going to be tested on your standardized tests, is it worth all the time and effort?
3. How much "trackwork" needs to be laid before students are ready for this level of problem-solving?
4. As an instructional strategy, would you have the problem acted out with models in the room or use actual students to represent the trains and the tunnel? OR just have them draw a diagram and go from there? Do a simulation on the TI-Inspire or TI-84 using graphics and parametric equations for the older students?
5. If you believe there is still a place for this type of problem-solving, should it be given only to the advanced classes and depicted as a math contest challenge?
6. I'm dating myself but I remember seeing problems like this in my old yellow Algebra 2 textbook? Uh, I believe this was B.C. -- before calculators! Can you imagine! Do you recall these kinds of problems? Do you recall the author or publisher?
7. Of course, the proverbial "two trains and tunnel" problems are frequently parodied and used as emblematic of the "old math"! They've been replaced by "real-world" applications. "Progress makes perfect!"

YOUR THOUGHTS...





Answer: 9.4 seconds (challenge this if you think I erred!)

...Read more

More Challenges/SAT Practice, Core Curriculum Standards, Reminders, Comments...

Additional SAT/Contest/Challenges

Challenge 1:

HOW MANY DIGITS OF 1000¹⁰⁰⁰ - 1 WILL BE EQUAL TO 9 WHEN THIS EXPRESSION IS EXPANDED?

Challenge 2:

HOW MANY 5-DIGIT POSITIVE INTEGERS HAVE A SUM OF DIGITS EQUAL TO 43?

Challenge 3:

Jorge can run a 6-minute mile while Alex can run a 5-minute mile. If they start at the same time, how much less distance, in miles, will Jorge run in 10 minutes?

(Yes, you can respond with answers and solutions to these in the comments!)
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Tired of hearing about THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST!? IF I RECEIVE 10 MORE REGISTRATIONS, I MAY JUST STOP!
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The Common Core State Standards Initiative
First look here for a quick overview and here for an index to the latest draft of the standards. Of course, this blog only discusses the mathematics part of the document.

Overview

The Common Core State Standards Initiative is a joint effort by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12.

These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills. The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards.

HIGHLIGHTS

• Core Concepts and Core Skills
• 11 Core Standards including the new "Mathematical Practice":
  
• Mathematical Practice
• Number
• Quantity
• Expressions
• Equations
• Functions
• Modeling
• Shape
• Coordinates
• Probability
• Statistics

• Each standard is broken into Core Concepts and Skills, provides research-based evidence and many illustrative examples to clarify the language
  
• Alignment of these standards to those of 5 representative states: California, Florida, Georgia, Massachusetts and Minnesota
• Standards reduce the number of Core Concepts and Skills in accordance with many recommendations to pare down the number of required topics to allow for greater depth

Example of a Standard (Standard 5)

Equations | see evidence
An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously.

An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers).

A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b₁ + b₂)/2) h, can be solved for h using the same deductive process.

Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities.

Connections to Functions, Coordinates, and Modeling. Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling.

Core Concepts
Students understand that:

1. An equation is a statement that two expressions are equal.
   see examples

2. The solutions of an equation are the values of the variables that make the resulting numerical statement true.
   see examples

3. The steps in solving an equation are guided by understanding and justified by logical reasoning.
   see examples

4. Equations not solvable in one number system may have solutions in a larger number system.
   see examples

Core Skills
Students can and do:

1. Understand a problem and formulate an equation to solve it.
   see examples

2. Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
   see examples

3. Rearrange formulas to isolate a quantity of interest.
   see examples

4. Solve systems of equations.
   see examples

5. Solve linear inequalities in one variable and graph the solution set on a number line.
   see examples

6. Graph the solution set of a linear inequality in two variables on the coordinate plane.
   see examples

FUNDAMENTAL ASSUMPTIONS AND CONSIDERATIONS

Very Important!
(Click on image to see a clearer view)




























INITIAL MATHNOTATIONS REACTIONS

1. Exceptionally clear and definitive document
2. Influenced by NCTM (Curriculum Focal Points), Achieve, College Board, ACT
3. Illustrative examples are of high quality
4. Will serve as a basis for states' revisions of current standards hopefully creating more consistency than currently exists
5. Leaving curriculum to local districts and states was a politically necessary decision, however, in my opinion, developing a reasonably consistent curriculum by grade level and/or course across districts and states from these standards may prove to be difficult and may again lead to considerable disparity. Hopefully, this will be self-correcting when standardized assessments are created as is currently being done with the End of Course Tests from Achieve
   

A Practice PSAT/SAT Quiz with Strategies!!

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 - 100^2n-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

...Read more

Demystifying Per Cent Problems Part II - Using Multiple Representations and an SAT Problem

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

Demystifying Harder Per Cent Word Problems for Middle Schoolers and SATs - Part I

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...