Another Quadratic Function SAT Problem

Have you noticed the SAT Tips of the Week in the sidebar? These are intended both for math teachers and students.

The "new" SAT has a few 2nd year algebra questions and, typically, there is at least one 'parabola' problem usually expressed in function form. Here are two different versions - one multiple choice and one "grid-in" (student-generated response).
Can you predict which one might give most students more difficulty?
It might be interesting to list all of the skills, knowledge and concepts being tested here. Are all of these typically included in your Algebra 2 course? Do students get enough exposure to these kinds of problems?


Version I
For some constant r, the graph of the quadratic function f(x) = -x² + 2rx is a parabola with x-intercepts at P and Q and vertex V. What is the area of ΔPQV, in terms of r?

(A) r² (B) r³ (C) 2r²

(D) 2r³ (E) 4r³


Version II

For some constant r, the graph of the quadratic function f(x) = -x² + 2rx is a parabola with x-intercepts at P and Q and vertex V. If the area of ΔPQV equals 27, what is the value of r?

Answers, solutions, strategies and comments will appear below the Read more...



Answers/Solutions/Comments/...

Version I
Answer: (B) r³

Possible Solution (no frills):
Factoring, we have f(x) = -x(x-2r); x-intercepts are 0 and 2r. Therefore base of triangle has length 2r.
The x-coordinate of the vertex is r (why?), so y-coord = f(r) = -r(r-2r) = r².
Area of triangle = (1/2)(2r)(r²) = r³.

Version II
Answer: r = 3

Possible Solution:
From Version I, we obtain r³ = 27, so r = 3.

Comments

• These kinds of questions typically appear among the last 3-4 problems on a section, meaning they are of above-average difficulty. Students who are in Algebra 2 or beyond should definitely attempt it. After reviewing it, most students may conclude it's not very hard at all!
  
• Testmakers are more frequently using a parameter like 'r' to make it more difficult to merely punch it into the graphing calculator and read off the intercepts and vertex.
• This kind of question could also appear on standardized tests like the Algebra 2 End of Course Exam from Achieve/ADP or other state tests.
• Pick up a copy of 10 Real SATs from the College Board to find several other practice problems like this.
• One could modify and extend these problems in many ways. For example inscribe the parabolic region (bounded by the cruve and the x-axis) in a rectangle and determine its area, a simple variation. More interestingly is to note that the ratio of the area enclosed by the parabola to the area of the rectangle is 2:3, a famous result proved in Calculus.
• Skills, knowledge required for this question? Worth enumerating in my opinion...
  
• I also believe strongly that our students should be tackling these kinds of problems on a regular basis to deepen their understanding of the relationship among the function, the coordinates of key points and the geometry. This used to be known as Analytic Geometry.
  

...Read more

Algebra For All - Another Report...

A recent article in Education Week, entitled Algebra-for-All Policy Found to Raise Rates Of Failure in Chicago, is generating some provocative online comments. Although access is usually restricted to subscribers to Education Week, there may be limited access to this article.

Meanwhile, California currently is mired in legal action regarding implementation of Algebra for all eighth graders.

What does all of this mean? IMO, mandating that all students take Algebra at the same point in time, ready or not, reflects a lack of understanding of the prerequisites for student success in learning algebra.

I've been advocating for some time that the content of an algebra course be standardized. As long as the course contains a common body of knowledge, I would argue that the name of the text, the approach, the instructional strategies, and the extent of integration of technology are of less importance.

Teachers should also be provided with samples of the kinds of assessment questions students should be able to handle. If these exemplars reflect a variety of question-types, balancing skill, conceptual understanding and problem-solving we can be reasonably certain that students are getting an authentic algebra course.

After reading some of the excellent comments on this article, I decided to share my own thoughts. Click on Read more if you would like to see these comments...


First Comment
Considering that this debate has been ongoing for years, one would hope that our current administration will listen to the voices of reason, many of whom have already submitted excellent comments to this article. If you bring together 100 parents and educators and sit them down in a room to discuss this issue, a consensus could be reached that would probably be far more reasonable and helpful to our children than all the research studies and commission reports that have been published.

Here's my best guess of what this group would recommend (much of which was stated above by some of the commenters:

(1) Algebra for All makes sense only if we have Arithmetic for All, i.e., a STANDARDIZED body of content/core knowledge of skills AND concepts, K-7 or K-8. Yes, it is possible to balance UNDERSTANDING AND SKILL and, yes, the preparation of our K-8 teachers must similarly be upgraded to deliver this!

(2) Students must be expected to demonstrate conceptual understanding of and proficiency in basic arithmetic skills including fractions, decimals, percents and ratios. Does this sound impossible given the current levels of student performance? What seems far more absurd to me is expecting proficiency in algebraic reasoning and skill UNLESS this foundation is in place.

(3) We also know that mandating ALL children to demonstrate proficiency AT THE SAME TIME time is unreasonable, however, there must still be strong EXPECTATIONS THAT ALL WILL GET THERE if we provide enough support and demand the needed effort and commitment from each child. Why should parents (and most do not have the means) have to pay thousands of dollars to private after-school companies to supplement their children's learning? This administration should provide whatever funds are needed to provide extra tutorial time in mathematics DURING THE SCHOOL DAY or before or afterwards or on Saturdays or during the summer or whatever is needed to bring children up to level. In return, students must be expected to work hard - NO EXCUSES. Coming for this extra help should not be optional! IMO, that would truly be a 'stimulus' for success.

Other countries have shown that most children can be ready for more algebra at an earlier age provided the necessary foundation is laid.

Prof. Escalante demonstrated that, through superhuman effort, it may even be possible to have CALCULUS FOR ALL! I'm not advocating this nor am I convinced that this is feasible unless one can clone this remarkable educator, however, the main lesson to take from him is the POWER OF HIGH EXPECTATIONS.

Until our society believes that EDUCATION OF OUR CHILDREN IS AN INVESTMENT NOT AN EXPENSE, all the recommendations from all the experts will fail. Listen to the voices of reason, please, before another generation is lost...
Dave Marain

Second Comment

How about some real specifics of what it means to develop algebra sense...

Here is one of many possible ways of developing the distributive property (the basis of 'combining like terms').

Visual:
[{&&&&}{&&&&}{&&&&}] combined with [{&&&&}{&&&&}] =
[{&&&&}{&&&&}{&&&&}{&&&&}{&&&&}]

Verbal:
Three groups of 4 added to two groups of 4 equals how many groups of 4?

Numerical:
(3x4) + (2x4) = 5x4

Symbolic:
3a + 2a = 5a

The language of algebra is the generalization of the language of numbers and arithmetic.

Other countries introduce the symbolic form early on as children are learning their arithmetic facts. Do we? In fact, children can develop both number sense and symbol sense if these are presented in a systematic organized manner. BTW, the use of multiple representations I've shown above is not just to get at different learning styles; it also deepens the child's understanding of numbers and relationships.

BUT, in the end, children also need to KNOW that 3x4 = 12 without hesitation. Knowledge of fact and skills can only be achieved through repetition and practice. Educators know this self-evident truth and they also know that one can accomplish this while students are gaining insight from solving problems and communicating their thoughts. Until we all make a commitment to this BALANCED VIEW of learning, it won't make any difference what curriculum a district purchases.
Dave Marain
MathNotations


Your thoughts...

...Read more

What Does A Teacher Make? A Message for Everyone...

Update: As several of my readers have pointed out, the following is adapted from a poem, "What Teachers Make", written by former teacher and poet Taylor Mali. His words have inspired thousands of teachers. I strongly urge you to view his powerful performance of this poem on YouTube or visit his website.

One of my favorite quotes of his is:
MALI: ...it’s more important for me to love my students than it is for them to like me.
...

The dinner guests were sitting around the table discussing life.

One man, a CEO, decided to explain the problem with education. He argued,
"What's a kid going to learn from someone who decided his best option in
life was to become a teacher?"

He reminded the other dinner guests what they say about teachers: "Those
who can, do.. Those who can't, teach."

To stress his point he said to another guest; "You're a teacher, Bonnie.
Be honest. What do you make?"

Bonnie, who had a reputation for honesty and frankness replied, "You want
to know what I make? (She paused for a second, then began...)

"Well, I make kids work harder than they ever thought they could.

I make a C+ feel like the Congressional Medal of Honor winner.

I make kids sit through 40 minutes of class time when their parents can't
make them sit for 5 without an I Pod, Game Cube or movie rental.

You want to know what I make?" (She paused again and looked at each and
every person at the table.)

I make kids wonder.

I make them question.

I make them apologize and mean it.

I make them have respect and take responsibility for their actions.

I teach them to write and then I make them write. Keyboarding isn't
everything.

I make them read, read, read.

I make them show all their work in math. They use their God given brain,
not the man-made calculator.

I make my students from other countries learn everything they need to know
about English while preserving their unique cultural identity.

I make my classroom a place where all my students feel safe.

I make my students stand, placing their hand over their heart to say the
Pledge of Allegiance to the Flag, One Nation Under God, because we live in
the United States of America.

Finally, I make them understand that if they use the gifts they were
given, work hard, and follow their hearts, they can succeed in life.

(Bonnie paused one last time and then continued.)

"Then, when people try to judge me by what I make, with me knowing money
isn't everything, I can hold my head up high and pay no attention because
they are ignorant.... You want to know what I make?

I MAKE A DIFFERENCE. What do you make, Mr. CEO?"

His jaw dropped, he went silent.

An "Average" Looking SAT-Type Problem for Middle Schoolers Too!

With juniors preparing for the May or June SAT (many already took it for the first time in March), I plan on having several posts dedicated to the kinds of questions one often encounters along with a discussion of math and test-taking strategies. The math content of many SAT questions is middle school level although the level of reasoning, the wording and the symbolism raise the bar higher.

Why so much focus on this standardized test? My contention has always been that these questions provide our math teachers with endless material for asking more higher-order questions, promoting reasoning and thinking 'outside the box'. They should not be thought of as 'taking away from the curriculum', rather they enhance the curriculum. Most importantly, questions like these should be integrated into regular textbook assignments. They need to be inside the assignment not placed at the end of the section or end of a chapter in a separate section (aka, Standardized Test Practice) or in a supplementary book.

The other central point about using these kinds of questions is to think of them as more than a warmup or SAT review before the test. Each of these questions can help students develop a deeper understanding of fundamental mathematics and therefore is integrally connected to the curriculum. Students often view these questions as something different from what they learn in school. Instead of applying the knowledge they've gained from the classroom, they abandon what they know. Test-taking strategies are fine but these should complement actual mathematics, not replace it.



The table below shows the relative population of students and average GPA by grade level at Standardized High School.

GRADE	%	Avg. GPA
FRESHMEN	28%	2.85
SOPHOMORES	24%	2.74
JUNIORS	22%	3.34
SENIORS	26%	3.21

What was the average GPA for all students in the school?

Please click Read more to see the answer, suggested solution and more discussion.


Answer: Students must "grid in" 3.02 or 3.03

Suggested Solution:

"Weighted average" method: (0.28 x 2.85) + (0.24 x 2.74) + (0.22 x 3.34) +(0.26 x 3.21)

Comments

• You might ask students why the process of adding the four GPAs in the table and dividing by four is incorrect here. Unfortunately, this incorrect method produces 3.035 and if the student doesn't round they would grid in 3.03 and receive credit! Hopefully, the testmaker would catch this and adjust the numbers slightly to catch this student error.
• Would students have less difficulty with this question if the actual number of students in each grade level were given, rather than percents? Should this non-percent version be presented first when teaching this topic? I would think so. The problem in this post is not intended to be introductory. The instructor could deal with the percents by assuming a total school population of 100 students and proceed from there. The weighted average method shown above is more sophisticated. By the way, does it remind anyone of the concept of "expected value"? Make those connections! (when appropriate of course).
  
• IMO, middle school students should be introduced to the ideas of weighted averages early on. Is this standard curriculum in 6th, 7th or 8th?
• On the actual test the student might see many variations of this problem. For example, the data could be given in two separate displays: The % distribution of students by grade level could be presented in pie chart form and the other data in table form.
• Can you think of variations on this problem? Do you have a good source of these? Are these questions designed primarily for the accelerated students? The Honors students? The Math Contest crowd?
  

...Read more

The String of 100 Saturdays Problem -- READ MORE!!

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.

...Read more

Updates Squared...

Update: My RSS feed seems to have been repaired. I want to thank Denise for contacting me with helpful info...

• I think there may be a problem with my RSS Feed. Please email me (dmarain at geeemail dot com) or post a comment here to let me know if you're still getting my feed or if there has been any interruption. Also indicate which feed reader/aggregator you are using (Live Bookmarks, Google reader, Bloglines, etc.). Since Google purchased Feedburner I've noted some quirky behaviors of late...

• I decided to place an SAT Math Tip of the Day in the sidebar. If any of your students (or students who happen to visit) find it helpful, let me know. I'm also working on an SAT Problem of the Day (different from the College Board), but that will require some work to make the math symbols appear correctly.

• Blogger, unlike Wordpress, doesn't make it easy to have expandable posts (aka, Read More...). There are some solutions out there from techies so I hope to have this feature up soon. This would be particularly helpful for challenge problems where I can "hide" the solution until you click on something. This is done very well in the Math Problems of the Day in the sidebar but the solution is not easily transferable to the layout in Blogger.

• Have you looked at the Math Problem of the Day in the sidebar recently? Some excellent algebra and geometry problems which are quite challenging. They're along the lines of math contest problems, some close to AIME questions or even higher. Some require careful derivation or justification and the solutions are done well. I have noticed a few repetitions in the problems but, overall, the quality is good. I'll have more to say about the source of these later on...
  
  
• Achieve/American Diploma Project/Algebra I
  As soon as released items/practice test for Algebra I is posted, I will let my readers know. My understanding is that this should be coming very soon, considering that the test will be operational in May! Anyone in your state participating? I would strongly recommend looking into this for your district as the benefits will be great IMO:
  
  • Helps to standardize your Algebra curriculum and insure consistency of content and instruction; helpful for programmatic review
    
  • Aligned with most state standards and may well serve as a basis for national algebra standards
    
  • May become a mandated test in the future in some states
    
  • Individual report forms can identify students with potential weaknesses who may require additional remediation or can even be used for placement purposes
  • My understanding is that levels of proficiency will be set this summer and that reporting will be more timely the following year
  • Excellent practice for students in taking standardized tests
  • Early exposure to this type of testing usually results in better performance later on
    
  • Contains both calculator and non-calculator sections consistent with current recommendations for assessment
  • Contains objective, short-answer and open-ended questions, again consistent with most recommendations for assessments
    

Updates: Math Teachers at Play #3, Recognition for MathNotations,...

• Be sure to stop by at f(t), Kate's excellent math teacher blog. Kate graciously agreed to host the 3rd Edition of the new carnival, Math Teachers at Play, originated by Denise at Let's Play Math. This edition has some fascinating posts, particularly, "When does the sum of three numbers equal their product?" The solution is surprising! I wrote a comment to this on John Cook's blog. Kate has broken the contributions into 4 well-defined categories: Secondary, Primary, Pi Day Roundup and Unclassifiable. Also, don't miss the hysterical cartoon Kate posted at the bottom. It's precious...
  

• dy/dan, Let's Play Math, 360 and MathNotations were featured in Mathematics Teaching, one of the journals of the UK's Association of Teachers of Mathematics. The authors took the time to capture the uniqueness and essence of each of these blogs and the reviews are objective and fair. I made a connection with one of the authors and I'm hoping to do an interview focusing on comparisons between math education in our two countries. Here's the link to the article which will be downloaded as a pdf document.
  

• MathNotations is currently ranked as the top Math ed blog on Alltop Math.There are a few other sites rated higher but they are not math ed (teacher) blogs.
  

• Plenty of new features in the sidebar including the first "ads" I've run on this site. No apologies here -- any support you can give to keep this blog rolling along would be gratefully appreciated...

• Pi Day may be over but it never really terminates, does it? Well, at least, us math bloggers have twelve monthhs to come up with some new ideas for the occasion!

• Finally, a 'quickie' for your middle schoolers or for those taking the SATs. Might make a nice review of divisibility, remainders, patterns, problem-solving strategies, etc:
  
  What is the greatest possble number of Saturdays in a string of 100 consecutive days?
  

Analysis of a Series: An Investigation before the AP Calculus BC Exam

The remarkable identity above could be the subject of many math blog posts but we will look at a variation, one that is accessible to precalculus and calculus students. With the AP Calculus BC Exam looming, the following investigation can be used to introduce or to review the topic.

I'm not sure if I have ever made it really clear on this blog that I routinely used these kinds of investigations in the classroom. For those who wonder how I could possibly have completed the required coursework for the AP Calculus BC syllabus or who might question my sanity, a couple of points here:

(1) Of course I didn't do this every day. I might have done an extensive investigation once per unit.
(2) Imagine my surprise when I first saw the Finney, Demana, Waits and Kennedy text, a book that has these kinds of explorations in every chapter! I thought they had found my old lesson plans.
(3) Most of the extensive investigations were assigned for work outside the classroom. In fact, for a while, the first investigation of the year was posted on my web site and emailed to students at the end of August before they arrived in school (I met them in June before they left for the summer or I got their phone numbers from guidance and called each of them to tell them to look for the assignment online, and to download and print it.)
(4) Even if I didn't prepare an exploration every day, most every lesson plan which introduced a new topic included a series of leading questions like these. My intent was always to have them think more deeply about a topic, i.e., to understand

• the historical origins of the topic
  
• how it was connected to their prior learning
• its usefulness and application
• why a method or theorem works (derivation, justification)

Developing these lessons initially was labor-intensive but a work of love. Perhaps no more laborious than what another of my colleagues did for his students: developing a PowerPoint presentation for every lesson for the entire year. As time-consuming as all of that sounds, once you've done a few of these, they start to flow naturally and the following year you only have to revise!

A Series Investigation

Consider the following finite series:



(a) Write the series using summation notation.

(b) Verify the following identity for n > 1:



(c) Use the identity in (b) to show that the value of the series above is

Hint: What was Galileo's most famous invention?

(d) Using a method similar to (c) verify the following for n, even:



Note: If n = 2, the right side would be accurate however the left side would consist of only one term. I could have used summation notation for the left side but I didn't want to give away the answer to part (a).

(e) If n is odd, show that the series on the left of part (d) can be written:



(f) Show that the expression on the right side of the equation in (d) and the expression in (e) are algebraically equivalent.

(g) Use the expressions from (d) and (e) to show that the sum of the following infinite series is 3/4:




(h) There are many ways (p-series, integral test, etc.) to prove that the series

converges. However, for this exploration, we will use the convergence of the series in (g) to do this:
Demonstrate that this series converges using both the Comparison Test and the Limit Comparison Test by using the series in (g).

Notes:

• More commonly, the convergence of the series in (g) is demonstrated by comparing it to the p-series. We're doing the reverse here.
• Another important aspect for precalculus and calculus students is to have them compare the partial sums to the sum of the infinite series. Thus, it's worth taking the time to have them see how close the sum is to 0.75 when adding the first 100 terms, the first 1000 terms etc. Also, indicate that the difference can be thought of as the "error" in the approximation. All of this is needed for further study and it deepens their understanding of infinite series.
  
• As indicated above, this investigation may be too time-consuming for a regular period of 40-45 minutes. I would recommend doing parts (a)-(c) (or (d)) in class and assigning the rest for homework to be collected after 2-3 days.
• Teachers of precalculus can use parts of this investigation when developing the concepts of series. Much of the groundwork for infinite series can be laid before students get to calculus!
  

Those "Function" Questions on the SATs - Practice, Tips

PLS NOTE THE EDIT TO THE PROBLEM BELOW. THE ORIGINAL WORDING WAS INACCURATE.

The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.

If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:

A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.
A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.

What is the ratio of the number of elements in F(20) to the number of elements in P(20)?

Click Read more below to see answer (suggested solution will be posted later).

Answer: 1
Explanation: Not yet...

...Read more

Pi Day, SAT Day - Last Minute Reminders and a Problem to Try...

It's fitting that many juniors will be taking their SAT on 3-14! The following are some thoughts for students to take with them on Saturday morning.

Each question on the Math section is worth about 11 points. Six to seven careless errors (which is common) can cost you up to 90 points! How many students would like to increase their score 90 points without that high-powered SAT Course? It is within you! So what can you do to minimize the damage?

• HIGHLIGHT (circle, underline) KEY WORDS AS YOU READ THE QUESTION.
  
• LOOK FOR WORDS/PHRASES LIKE EVEN, POSITIVE INTEGERS VS. INTEGERS, NOT, LEAST POSSIBLE, ...
• SLOW DOWN! WRITE OUT DETAILS - DO NOT SKIP STEPS; DON'T WORRY ABOUT FINISHING EVERY QUESTION. IT'S THE 6-7 CARELESS ERRORS THAT HURT MOST!
• IF YOU HAVE ANY DOUBT ABOUT DOING THE PROBLEM ALGEBRAICALLY, PLUG IN SIMPLE WHOLE NUMBERS LIKE 1, 2 OR 3. See Example below.
  
• REMEMBER:
  INTEGERS: ...,-3,-2,-1,0,1,2,3,...
  EVEN INTEGERS: ...,-4,-2,0,2,4,..
  PRIMES: 2,3,5,7,11,... NOTE THAT ONE IS NOT PRIME!!
  
• ZERO IS THE MOST IMPORTANT NUMBER ON THE SATS AND MATH IN GENERAL! LEARN THE TRUTH ABOUT ZERO:
  WHOLE, EVEN, INTEGER, NOT POSITIVE, NOT NEGATIVE, 0/5 = 0, 5/0 IS UNDEFINED!

Example 1: If m > 0, the average (arithmetic mean) of 2⋅3^m and 2⋅3^m+2 can be written as c⋅3^m+1. What is the value of c?

Possible solution: Unless you're a math team whiz, you should immediately substitute a simple whole number for m and not worry about c. Also, ignore the phrase "arithmetic mean" which is math terminology for the common average.

"Plug in" m = 1: We want the average of 2⋅3¹ and 2⋅3¹⁺². This translates to the average of 6 and 54 which equals 30. If you're prone to any careless arithmetic errors (order of operations, exponent issues), do this on the calculator even though your math teacher would cringe!
We want to express 30 as c⋅3¹⁺¹ or 9c. Thus 9c = 30 or c = 30/9 = 10/3 = 3.33 if you're gridding in.

Additional Comments

• The above example would normally be among the last 5 questions of a section, therefore, considered to be more difficult. Students should not give up too quickly on these. They often can be solved by straightforward methods like the one described above.
• For the mathematical purists out there who are offended by "plug-in" methods (btw, I'm one of those purists!), this post was about providing test-taking strategies or 'survival' techniques. For the classroom development of algebraic skills, I would certainly have demonstrated an algebraic method:
  2⋅3^m+2 = 2⋅3²⋅3^m = 18(3^m). The average of 2(3^m) and 18(3^m) = 10(3^m) = 10(3^-1)(3^m+1) =
  (10/3)(3^m+1) ,etc. Wonderful review of exponents and operations but not necessarily for everyone taking this test...
  

A Middle School Mental Math 'Practical' Problem

Look here for links to MathNotations π-Day Resources, Activities and Investigations!


You're on a superhighway in the middle of nowhere at 10 PM, your fuel tank shows about a quarter of a tank remaining, your trip odometer shows you've gone about 180 miles since the last fill-up and you just passed a fuel rest stop. The sign reads "Next Rest Area and Gas - 58 miles." Are you in trouble or will you make it?

Comments
(1) Unrealistic scenario? Rest areas rarely that far apart on the Interstate? Why didn't the driver realize that he was low on gas and just stop! Why not just get off the Interstate and look for a local station or call AAA or Onstar if needed or use a navigation system to locate another gas station? (Do all of us have access to these technologies?). Does the student have to know that one still has some fuel remaining even when the gauge points to empty? Should a discussion of these kinds of practical issues be an integral part of 'real-world' problem-solving?

(2) Is this type of question fairly common in the texts you're using?

(3) What prerequisite skills and conceptual understandings are needed for this problem? Should most 7th or 8th graders be able to do this?

(4) Why do you think I suggested a mental math approach? Should we eliminate the mental math/estimation aspect altogether or use it as a starting point for further discussion?

(5) How do you think your students would fare on this problem? How many students would recognize the 1:3 ratio between gas remaining to gas consumed? Is this part:part construct something stressed in texts and in our classrooms? Should it be? Is it worthwhile discussing several approaches to this problem?

(6) Do you have a favorite visual model for this kind of ratio problem? Your thoughts?

Updates, Math Teachers at Play, ADP Algebra Links, More Features,...

Math Teachers at Play #2
Denise has done another wonderful job of gathering some excellent posts for K-12 educators. Puzzles, articles on creative ways to develop arithmetic skills, bouncing balls, Pythagorean Day, Math Contests, vectors, integrals, a link to the 50th Carnival of Math, more Monday Math Madness and I'm just scratching the surface! And with all of this, Denise sprinkles in engaging images, graphics and intriguing quotes including one from Woody Allen! Denise, you're the best. If I can get my act together, I'll try to submit something for next time.

Recent Comments Widget in Sidebar
It's about time that this feature appeared! Wordpress and Typepad make this easy for bloggers but I had to figure out a way to do this for Blogger. Actually, it's not that hard and long overdue...

π and π-Day Links!
Over the past couple of years, I've posted several articles on π Day activities and other π factoids and investigations. The π-Day Scavenger Hunt post is definitely the most popular but you can locate all of these easily by going to the labels section in the sidebar and looking for π or π-Day. Better yet, this link will get you to all 7 of my π-Day posts!


Personal Thoughts or a Silly Math Riddle in the Sidebar?
Well, for now, I put up the latter in the sidebar. Please don't groan too much...

ADP/ACHIEVE LINKS ALGEBRA I & II
Many visitors seem interested in all of the updates I've posted about the End of Course Exams, so I've decided to keep relevant links in the sidebar. I don't think Algebra I released test items have yet been posted (as of this date) but keep checking the Achieve site or, better yet, check here! What is particularly interesting are the test specifications for Algebra I.

NATIONAL STANDARDS MOVEMENT MOVING INEXORABLY FORWARD...
I've been keeping a very low profile here as one education leader after another comes out with their latest support for this novel idea! Those of you who have been following this blog for the past two years already know my position but it is gratifying to see one's views validated even if it took 15 years! I will have more to say about how this movement is growing exponentially but what is critical here is the idea that we're talking National Standards in Math, not Federal! Subtle but critical distinction...

Another New Feature - KENKEN is Mind-Bending and Instructive!!

KENKEN® Puzzles

Invented by Japanese math teacher, Tetsuya Miyamoto, KENKEN® allows you to test your puzzle acumen and improve your math skills at the same time.
Exclusively on NYTimes.com, updated with 6 new puzzles daily.

This extraordinary new math-logic puzzle started appearing in the NYTimes about a month ago. Link here to start playing. I also placed a link in the sidebar so that you can play every day. It will take you a few minutes to catch on to the rules and then you will give up Sudoku, Kakuro, your daily crossword puzzle and Jumble! I just started playing and I'm hooked. Most importantly, it will reinforce and develop basic arithmetic skills for your students and/or children!

I also recommend that you read the Times article which introduced this new feature. The creator is a talented and unique math teacher. Here is some fascinating background from the article:

KenKen was invented in 2004 by the Japanese educator Tetsuya Miyamoto, who founded and teaches at the Miyamoto Math Classroom in Tokyo. Students attend his class on weekends to improve their math and thinking skills. Mr. Miyamoto said he believes in “the art of teaching without teaching.”

He provides the tools for students to learn at their own pace using their own trial-and-error methods. If these tools are engaging enough, he said, students are more motivated and learn better than they would through formal instruction.

About 90 minutes of class time each week is set aside for solving puzzles, usually designed by Mr. Miyamoto. The most popular one has been KenKen.

NOTE: The Math Problem of the Day for Wed 3-4-09 was deleted because I deemed that the context was inappropriate for younger readers who may visit. I'm assuming this was an aberration and this feature will resume shortly.

"Multiple" Pi Challenges for MS and HS

Consider the following table displaying the first four positive integer multiples of π rounded to 4 places:


So what's the challenge here? You will be a π-multiple investigator!

(1) Determine the EIGHT positive integer multiples of π, up to 1000π, which, when rounded to THREE places, produce an integer result. For example, the decimal 17.9996 when rounded to 3 places results in the integer 18.000, which we will regard as an 'integer'. Unfortunately, this decimal is not an integer multiple of π so have fun searching! Do you notice any pattern in your results? Describe it! Can you explain this pattern? [This requires more than a Yes/No response!]

(2) In fact, the "pattern" you may have found in (1) continues beyond 1000π. Show that you can go up to SIXTEEN multiples of π before the pattern breaks down. Why do you think the pattern eventually ends?

(3) Which of your results in (1) would produce an integer when rounded to FOUR places?

(4) Can you think of any practical application for finding multiples of π which are very close to an integer? Be creative! Responses may depend on how advanced your math background is.

Comments:

• Do your students know how to program their graphing calculator to do the search? OR in Java or C++ or Python or Perl?? This would certainly facilitate the search! I may display the code for the TI-83 or -84. However, students may also find a creative way to use the TABLE feature on their graphing calculators to save time without programming. Have fun!
• Please post feedback if you use this in your classes. I will not post answers yet in case students find this post in their 'searching'!
• For most students, the full significance of this innocent looking search problem will not be apparent. You might want to give them a hint. Perhaps we should call this post:
  
  IS π ALMOST RATIONAL?
  

NEW FEATURES!

Update: I decided to delete the Math Problem of the Day for Wed 3-4-09 as I felt it was inappropriate for our younger viewers. Hopefully this was an aberration. Now you're probably all wondering what it said (except for those in other countries who already viewed it!!).

Check out the sidebar as I experiment with some new features...

• My Personal Thoughts on Education
  My own reflections - I'll change these every few days
  
  
• Math Problem of the Day
  Many of you have been commenting on these. Exceptional open-ended problems with solutions for the secondary student. Most require extensive work to be shown and/or justification.
  
  
• Amazing Fraction-Decimal Calculator
  You and your students will be mesmerized by the capabilities of this software. Wonderful for discoveries of patterns, development of conceptual understanding of repeating decimals, etc.
  
  
• Free Daily Kakuro Puzzles - a link
  For me, these are far more addictive than Sudoku. Further your students will be practicing both their addition and logic skills!
  
  
• Aristotle Quote of the Day - from the consummate logician/philosopher
  
  
• MathNEXUS Portal
  As I've mentioned often, this website is one of the best compilations of resources for math teachers and is updated weekly - an online journal!

Subscribers may need to visit MathNotations to see these new features in the sidebar. Enjoy!

Stuck on a Calculus Problem? Ask St. Patrick!!

With existing technologies and all the help available on the web I am incredulous that students still feel lost at sea the night before a major exam or just getting through an assignment. I've asked so many students what they do when they're frustrated by some math problem at 10 PM: "So who do you call? CalcBusters!"Seriously, they often just look at me as if I have two heads. Well, what are their options?

(a) Call their teacher/professor? Uh, not likely...

(b) Email their teacher/professor? Assuming one has this email address, what are the odds you will receive an immediate reply which will illuminate everything...

(c) Go into a Calc help chat room enabled by your teacher or one set up by some student. A good option if anyone is actually online at that moment. Your teacher or another student can establish guidelines for this so that students know a help session will always be available from 9 PM to 11 PM for example. See (e) below for a similar idea.

(d) Call or email a friend (remember you will then have only 2 lifelines left!). Of course this presumes you have a friend who understands it better than you and can communicate a solution over the phone or via email. Remember what time it is...

(e) Go to a Calculus forum/discussion group in which you can post your question and someone with the knowledge will reply in short order. This is a viable option as there are now many such help groups out there and I will review these and provide links in another post.

OR...

Go to YouTube and find a free video tutorial demonstrating a similar problem in detail.
For example, suppose you're floundering with "integrating by partial fractions." You can just Google "YouTube partial fractions calc video" or something like that and, presto, you are transported here. Ok, the video covers a more sophisticated problem involving a rationalizing substitution as well as partial fractions, but you immediately see dozens of related videos in the sidebar. There are many excellent free videos online from many talented teachers/professors but on this post I will feature one of the best.

If you're interested in seeing an exceptionally clear presentation of calculus or other math topics you cannot do better than Patrick's videos which he offers at no cost. Patrick does not know that I am writing this review so rest assured I am not getting any commission here! The link above is one of his lessons.

He tries to limit his lessons to 10 minutes to make the file size manageable. His writing on the whiteboard is crystal clear, his organizational skills are exemplary and his speaking voice is soooo calming. Further, his explanations are mathematically precise and include just enough rigor to make the purists out there happy without sacrificing clarity. Compare his videos to the amateurish attempts I have posted on this blog - uh, there is no comparison.

Patrick also has a website where you can find all of his free videos. Look here first on YouTube or go directly to his site.
Patrick's background (from his site)
About me: I have been teaching mathematics for over 8 years at the college/university level and tutoring for over 15 years. Currently I teach part time at Austin Community College, but have also taught at Vanderbilt University (a top 20 ranked university) and at the University of Louisville.

He runs a tutoring service in Austin and I'm quite sure he is doing well considering the quality of what he is offering for free. And his videos are not exclusively calculus. Enjoy!

Updates: RSS Feed, Preparing for SATs on Pi Day, A Ratio Problem,...

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


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


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

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

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

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

Notes, Comments, Solutions, Strategies,...

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

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

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

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

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

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

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


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