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!