MathNotations

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?

Posted by Dave Marain at 6:55 AM 6 comments

Labels: divisibility, middle school, reasoning, SAT strategies, warmup

My 500 or so subscribers may not have seen the following anagrams which have been in the right sidebar of my home page for the past month or so. No one has yet taken the time to solve them. They're not that hard! Pls email me at dmarain at gmail dot com with your answers.


VORTEC SCAPE


(1) Hidden Steps OR


(2) General Arrows



The following problems are similar to ones I posted recently...



Mental Math and No Calculator!


1) The following sum has a trillion terms:


0.01 + 0.01 + 0.01 + ... + 0.01 = 1000...0
How many zeros will there be in the sum?




2) The following product has a trillion factors:


(0.01) (0.01) (0.01) ... (0.01) = 0.000...1
How many zeros after the decimal point will there be in the product?



A Few Comments...
(a) You may want to adjust the "trillion" for your own groups but I'm intentionally using this number for a few reasons, not the least of which is to review large powers of 10 (Will most think: "A million has 6 zeros, a billion has 9 zeros, so a trillion has..."?).


(b) The second one is more challenging and intended for Prealgebra students and above but, using the "Make it simpler" and "Look for a pattern" strategies, make it possible for younger students.


(c) How many of you are reacting something like: "Is Dave out of his mind? My students don't know their basic facts up to 10 and he wants mental math with a trillion!" I have found that large numbers engage students since they know there is a way of doing these without a lot of work if you know the "secrets"! Besides, we either push our students or we don't. You decide...


(d) These questions review several important concepts and skills. You may want to use these to introduce or review the importance of exponents and their properties.

Posted by Dave Marain at 6:54 AM 0 comments

Labels: exponents, middle school, patterns, problem of the day, reasoning, warmup

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!

Posted by Dave Marain at 9:53 AM 2 comments

Labels: 30-60-90, geometry, math challenge, middle school, SAT strategies, SAT-type problems, warmup

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%

Posted by Dave Marain at 6:42 AM 0 comments

Labels: core curriculum standards, national math curriculum, reasoning, SAT strategies, SAT-type problems, update, warmup

I have an equal number of pennies, nickels and dimes. I also have some quarters which have the same value as the pennies, nickels and dimes combined. If I have no other coins, what is the fewest possible total number of coins I could have? What is the value of all the coins?

Comments
(1) An opening day problem?
(2) Would you have students working alone or in small groups?
(3) Would you allow the calculator?
(4) Appropriate for prealgebra students? Students below grade 6?
(5) Is zero a possible answer?
(6) Wording too confusing for most students? Is it ambiguous or clear?
(7) Do you feel there are important underlying concepts and ideas embedded here or is it just a fun puzzle to engage students?
(8) Do students have difficulty in separating number of coins from their value?


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates:
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Posted by Dave Marain at 8:01 AM 6 comments

Labels: coin problem, middle school, puzzle, starting the year, warmup

With the school year starting for some and soon for others, here are a couple of ideas to set the tone in our math classes early on. Do not assume these are intended only for your advanced youngsters!


Middle School

1) (No calculator!) What is the average of ninety-nine 1's and one 2?

2) (No calculator!) Find 5 different sets of 5 numbers each of which has a mean of 5.
Note: The wording will be problematic here since students often associate the adjective different with the numbers themselves. Basic grammar, cough, cough...


High School (or advanced middle schoolers)


(No calculator!)
Set S consists of 100 different numbers each of which is between 0 and 1.
Which of the following could be the mean of these 100 numbers?

I. 0.01
II. 0.5
III. 0.98

(A) I only (B) II only (C) I and II (D) I and III (E) I, II, and III

[Yes, there will always be some discussion of "between!"]

A few comments...
(1) These problems are intended to be a springboard for your own creativity. You can do better!!

(2) Each of you probably has your own favorite resources of problems so that you don't have to reinvent the wheel. However, finding high-quality Problems of the Day which are matched to your curriculum is not always easy despite the abundant ancillaries supplied by the publisher and resources on the web.

(3) From the previous comment you can guess that I feel strongly about giving more challenging warm-ups to our students - all of our students (adjusted for backgrounds, abilities, skills). Don't worry that discussion of these will destroy your lesson. Students can work together for 5 minutes while you're taking attendance, checking homework, etc. I usually invited students who solved some or all of these to go to the board and explain their methods. To encourage students to look these over, tell them you will include a variation of one of these questions on the next quiz or test. Start by having it as an Extra Credit problem, then worth a couple of points, gradually increasing their value.

(4) Imagine if our students were exposed to these higher-order types of questions about 180 times a year from middle school on. By the time they take their college-entrance exams or other state assessments (or tests like the ADP End of Course Exams), they will have a much higher degree of comfort and should perform better, although we know that there are so many other factors that go into performance on high-stakes tests.

(5) Yes, the above high school problem is in SAT format. Why do you think I included these kinds on my daily warm-ups? By the way, I'm not promoting ETS but middle and high school teachers may well want to invest in (or ask their supervisor to order) the College Board's book of
10 Real SATs. There is no better source for these kinds of problems and many questions are appropriate for middle schoolers.

Posted by Dave Marain at 8:07 AM 6 comments

Labels: averages, mean, Problems of the Day, SAT-type problems, warmup

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...


Comments
1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!

Posted by Dave Marain at 6:12 AM 12 comments

Labels: math contest problems, middle school, more, SAT-type problems, warmup

Posted by Dave Marain at 8:38 PM 3 comments

Labels: combinatorial math, logic, multiplication principle, SAT-type problems, warmup

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!

Posted by Dave Marain at 8:33 AM 0 comments

Labels: arithmetic sequence, investigations, pedagogy, warmup

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.

SOLVE

(x² - 6x + 9)^{(x2 - 4x + 3)} = 1

Posted by Dave Marain at 8:31 AM 14 comments

Labels: algebra 2, exponents, math contest problems, warmup

While we're waiting for the 60° integer triangle problem, here's an easier one for both middle schoolers and secondary students. The only fact from geometry that is needed is the all-important triangle inequality:
Any side (in particular, the largest side) of a triangle is less than the sum of the other two sides.
Of course this refers to the lengths of the sides and one can express this in other forms, but I'll leave it at that.
This type of question has become a favorite on the SATs and other standardized tests but, more importantly, it develops clear systematic thinking - the organized list....

How many different triangles have integer side lengths and a perimeter of 5? 10? 15? 20? 25?

COMMENTS/INSTRUCTIONAL HINTS:

• There are really five separate questions here. The instructor can give some or all of these depending on the time allotted. To help the group get started and for clarification, it may be helpful to demonstrate the first question for the group: For a perimeter of 5, there is only one possible triangle, which we can symbolize as {2,2,1}. If these are older students who are comfortable with the triangle inequality, you do not necessarily have to model this one, but that's your call. By modeling the first one, you eliminate some of the ambiguity of ordering the sides.
  
• Since a primary objective here is to make an organized list, you may want to stop after the perimeter of 10 and discuss it at the board. Depending on the ability level of the group, I usually have students work independently, then check each other's work in pairs after they do a couple of these questions. Sort of a think-pair-share approach. Also, don't be afraid to provoke their thinking with questions as they begin to develop their systematic lists (which can get boring for some): "So, do you expect more triangles for a perimeter of 10? Twice as many?"
  
• As each question is reviewed, encourage students to record their results in a table:
  Perimeter..................Number of Triangles
  ......5........................................... 1 ................
  ....10.......................................... 2 ................
  This is critical for middle schoolers in particular, since tables are a basic model for functions! At some point, you can use n or p for the perimeter and symbolize the number of triangles having perimeter n or p as T(n) or T(p).
• Naturally, some students will assume there is a pattern and guess there are 3 possible triangles with a perimeter of 15 - NOT! However, it is natural for all of us to ask: "WHAT'S THE FORMULA?" Well, there is one. It's fairly sophisticated and related to partitions of numbers, but I'll let our readers do their own research for this...
  

Posted by Dave Marain at 6:09 AM 0 comments

Labels: combinatorial math, geometry, organized list, SAT-type problems, systematic counting, triangles, warmup

There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.

STUDENT PROBLEM/READER CHALLENGE

This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).

The number 12 has 6 positive integer factors: 1,12;2,6;3,4.
(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.

Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?

Posted by Dave Marain at 7:52 AM 5 comments

Labels: factors, number theory, primes, warmup

Many math educators use warmup problems to review, challenge or set the tone as students walk in the room. Routines like this are effective in having students 'hit the ground running'. These mini-problems can be on the board, while an overhead transparency of selected answers are displayed. The instructor then has time to circulate, check homework, engage students, and get a feel for the difficulties they had with the assignment. By the way, do most of your students take these warmups seriously?

Here is a warmup that requires more active participation on the student's part. I may suggest a different one in a later post, but I'd really like readers to share some of their favorites!

Math Bee
All students stand at their seats. They are told they will be have to give the next number in sequence, according to some rule that will be explained. They will have 3 seconds to respond (can be adjusted but no more than 5). If incorrect or time runs out, they will be instructed to sit down and the next person will have to give the correct answer and so on. You may want to start them off by giving them the first number (judgment call here). I suggest you allow a maximum of 5 minutes for this activity.

Here's the problem I gave to a group of high schoolers but it is highly suitable for middle schoolers as well:

Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave! You will have 3 seconds to respond. If incorrect or time runs out, I will ask you to sit down and the next person will need to give the correct number. We will continue until there is only one star shining or time runs out and we have co-champions!

Learning Objectives:
(a) Reviews primes (How many of your students do you think would be eliminated early by starting with 1? You can always start them off with 3 if you feel this will help. Some students will begin with 11, assuming that you meant 2-digit numbers. By the way, 51 and 91, in particular, typically knock out many, if they get that far! Finally, are you thinking this question is not particularly relevant for high schoolers? Count how many questions relate to primes on the SATs!)
(b) Improves listening skills and concentration (How many of your students do you think will forget the last number given either because their minds are wandering or from trying to think ahead to their turn?)
(c) Learning how to think under pressure. (Although we know some students will 'freeze up' or be embarrassed if they are eliminated, they will not be alone! Typically, about half of the students in an above-average class will be sitting down on the first pass through the class! With a high-achieving class of very strong students, you may need to make several passes to reach a winner. If there are a couple left after 3-4 minutes, proclaim co-champions.

Let me know how this goes if you try it after the Thanksgiving break. What variations would you use to make this more effective for your students? What other kinds of problems are suitable for this 'Bee?' Were my predicted statistics way off? Did you predict that students would get past 100?
Again, please share some of your favorite warmups!

Posted by Dave Marain at 6:50 AM 2 comments

Labels: math challenge, primes, warmup

Here's a warm-up you can give to your Algebra 2 (and beyond) students to welcome them back to math class after a summer of brain drain. NO CALCULATOR ALLOWED! This oughta' set the tone...

A total of 9991 M&M's were eaten by a group of Freshmen. Here are the facts:
(1) Each freshman ate the same number of M&M's.
(2) There was more than one freshman.
(3) Each freshman ate more than one M&M.
(4) The number of freshman was less than the number of M&M's each freshman ate.

How many M&M's did each freshman eat?

Work in your groups of 3-4. You have 3 minutes. First team to arrive at the correct number AND explain their method, gets to eat ______________.


Let me know how many groups solved it or your thoughts about the appropriateness of this question.

Posted by Dave Marain at 6:51 PM 14 comments

Labels: algebra 2, factoring, warmup