Challenge Their Minds Day 1 - A 'Means to an End'

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.

Another 'Average' Problem for Standardized Tests and Conceptual Understanding

After 4 tests, Barry's average score was 5 points higher than Michelle's. After the 5th test, Michelle's overall average was 5 points higher than Barry's. Michelle's score on the 5th test was how many points higher than Barry's?

Can you find at least three methods for solving this?
Algebraic, "plug-in", conceptual, etc...

As teachers we need to have a deep understanding of these kinds of problems and familiarity with several approaches. Of course, our students will show us a variety of methods, both right and wrong, when we open up the dialog!


Comments
Students from middle school on see many problems relating to means. However, they need to see a variety of problems of increasing difficulty. This question is certainly not a highly challenging math contest problem but I believe it demonstrates some important principles of averages and can be used to review different problem-solving strategies. Middle schoolers would struggle with the algebraic approach (a system of two equations), however they should be thoroughly comfortable with the underlying ideas.

Since the focus is on concept and method, I will give the answer: 45

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

SOMETHING NEW! Instructional Strategy Series: Teaching Average Rates

The following is the first in a series of strategies for teaching concepts that often prove difficult for many students from middle school on. These are not based on carefully controlled research studies following clinical methodology for a dissertation. They are based on 30+ years of learning how to do it better!! I suspect that's why we refer to the practice of teaching. Our readers are encouraged to share their own favorite methods that have been helpful to their students or to themselves. These ideas are intended only as suggestions. Each teacher will, of course, bring her/his own ideas and style to bear on the lesson.

Most of you know the classic algebra word problem type that has appeared frequently on standardized tests and math contests:

THE BIG QUESTION
Jack averaged 40 mi/hr going to school and 60 mi/hr returning from school over the same route. What was his average speed in mi/hr for the round trip?

Since there has been a decrease over the past 25 years in the number of word problems to which our students are exposed, some youngsters may not get to see one of these until reviewing for SATs or in their physics class.

From watching how students approach this type of question, I'm getting a sense that we need to introduce the basic concepts earlier on in middle school, which I am sure already occurs in some programs. In planning to teach methods of solving these kinds of problems, I usually tried to return to basic principles of math pedagogy - keep it simple and start with concrete numerical exercises that built on prior knowledge. What does all this jargon mean?


Start with a review of averages, then move on to combined averages before attempting to explain the round-trip rate problem!

[Concerned that such development will take too much time? There won't be enough time to review homework and provide enough practice for the homework assignment? My supervisors never threatened to fire me if a lesson lasted for more than one day and if, heaven forbid, I did not assign homework that first evening! Some ideas just cannot be rushed.]

Suggested Question #1:
Jack had a 70 avg on some tests and a 90 average on some other tests. Can his overall average be determined?
More specifically: When do you think 80 will be the correct answer? When will it not?

Comment:
Question 1 is intended to provoke thought and encourage an intuitive response, not a calculated answer!

Suggested Question #2:
Jack had a 70 average on his first 4 tests and a 90 average on his next 6 tests. What was his overall average for the 10 tests?

Comments
Note that I am suggesting beginning with problems to which middle school students may better be able to relate than a rate-time-distance question. The first question above is fundamental in developing the concept of the original rate problem.

These questions should help many students focus on the essential idea that we need to know how many are in each sub-group!

Since most students connect average to dividing a TOTAL by some quantity, they should feel comfortable in solving the average grade question as follows:

(TOTAL PTS)/(TOTAL NUMBER OF TESTS) to arrive at an average of 82.

BUT DON'T STOP THERE! Stress the UNITS of this result to build the rate concept:

AVG PTS/TEST = (TOTAL PTS)/(TOTAL TESTS)

Since students generally do not attach units to the 82, stress that the combined average is 82 PTS PER TEST or 82 PTS/TEST! BTW, not a bad time to mention that PER MEANS DIVIDE!!

Suggested Question #3:
Jack averaged 40 mi/hr for 2 hours, then 60 mi/hr for the next 2 hours. What was his average speed (rate), in mi/hr, for the 4 hours?

[Note the incremental development (commonly termed scaffolding in today's world!). Rather than jump to the abstraction of the original problem, we move on to the next logical step - giving them both the rates and the times for each part of the trip. In this case, we use equal times to provoke their thinking about why the result is also the simple arithmetic mean of the two rates. Each of us needs to make decisions about how many of these examples are needed before moving on to the main question.

Depending on the background and ability level of the group, you may be able to skip one or more of these suggested questions. Further, you may already be thinking of placing these questions on a worksheet for students to try alone or in pairs, stopping and reviewing as needed.

Suggested Question #4:
Jack averaged 40 mi/hr for 4 hours, then 60 mi/hr for 2 hours. What was his average rate, in mi/hr, for the 6 hours?

Suggested Question #5:
Jack averaged 40 mi/hr for the first 120 miles of a trip, then 60 mi/hr for the remaining 120 miles. What was his average rate, in mi/hr, for the entire trip?
Key question: Why does it turn out that the answer is NOT 50 mi/hr?

Comments
Do you think your students would now be ready for the BIG QUESTION near the top of this post? OR do you think they would need at least one more interim problem? Again, could these questions have just as effectively been placed on a worksheet and given to students, working in pairs?

I'll leave the rest to our readers. Pls feel free to share your ideas, comments, thoughts and questions. There's no question in my mind that some of you would develop these ideas differently! Remember you can always email me personally at dmarain at geemail dot com (the last 4 words misspelled intentionally of course!). Unfortunately, I typically get little response from posts about instruction since most readers prefer to solve a challenging problem!

Final Comment: Note that I didn't once suggest that students use a short-cut for the original round-trip problem. Ok, so it is the harmonic mean of the two rates, and can be calculated
from the formula: 2R₁R₂/(R₁+R₂).
But who would want to use that (uh, SATs, GREs, GMATs,...)???