Sunday, April 22, 2012
SAT CHALLENGE : Counting Non -Multiples of 7
How many pos integers less than 1000 are not multiples of 7?
Middle school problem?
Strategies you teach your students?
Calculator appropriate?
"Big Ideas" here?
Ans: 857
Sketch of one possible method:
1000/7=142.857... ---> 142 multiples of 7 less than 1000 ---> 999-142 = 857 non-mult
The devil is in the details of course which I intentionally omitted! Why didn't I mention that the largest mult of 7 less than 1000 is 994? Would most solutions involve finding 994 first?
Someone out there is thinking about the repeating decimal expansion of 1/7 = 0.142857142857… and why the ans to our problem is 857. A coincidence?
Too bad we have no time in our classrooms to explore and go in depth. If we spend time doing that we'll never cover all the required topics in the Core Curriculum. Yes?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 10:22 AM 0 comments
Labels: Instructional Strategy Series, SAT strategies, SAT-type problems
Sunday, April 15, 2012
SAT Logic and Semantics Twitter Problem
Too easy for most secondary students?
Too ambiguous?
How would it be modified for SATs?
How would 3rd or 4th graders respond?
What do think my underlying purpose is?
What are the "Big Ideas" here?
Hiw would you present this in a 4th grade vs a 10th grade classroom?
After discussion how would you assess understanding?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 6:36 AM 0 comments
Labels: Instructional Strategy Series, logic, SAT-type problems
Tuesday, March 9, 2010
Counting, Multiplication Principle, Pigeonhole Principle and Reasoning for Middle Schoolers and Beyond
UPDATE: SEE THE NEW VIDEO BELOW EXPLAINING THE PROBLEMS IN THIS POST. PLS SUBSCRIBE TO THE NEW MathNotationsVids Channel and share your comments and ratings!
The following video is available on my new MathNotations Videos Channe l.
This particular video is a 10 minute discussion of developing the Multiplication Principle of Counting. It is designed more for the instructor than the student although it may be helpful in clarifying this important concept. The focus is on using multiple representations to reach the widest variety of learning styles. It is appropriate for any teacher of mathematics but particularly for the middle school teacher or those who work with students who struggle with math concepts.
After watching the video (or skip it if you wish) scroll down to the two problems below. These are more sophisticated than the one in the video and they require application of other concepts as well. I believe they are appropriate for 8th graders through high school. A full investigation with questions is provided for each problem. Feel free to edit them to your own tastes or as needed for your students.
[埋込みオブジェクト:http://www.youtube.com/v/IAhJHV-qmlM&hl=en_US&fs=1&rel=0]
Problem I
Mr. M told his Period I 8th grade math class about the following imaginary scenario...
Before the first day of school, Mr. Serling noticed that the names of the 26 students in his 1st period class had an unusual property. All of their initials (First Initial, Last Initial) came from the letters A, B, C, D and E. Furthermore, some had duplicate initials like B.B.
Part (a)
He now asked his actual class to make a conjecture:
Do you think it's possible that all 26 students in this imaginary class could have different initials (from each other)? Write down your "initial" prediction (Y or N) on a slip of paper and fold it over.
Part (b) Ok, now that you've made a conjecture, get into your learning groups of 4 and individually make a list of all possible sets of initials using the letters A, B, C, D and E with repetitions like "B.B." allowed as I explained before. Make sure your lists agree - edit as needed. Are your lists easy to compare? Why or why not?
HOW MANY DIFFERENT SETS OF INITIALS DID YOUR GROUP AGREE ON? ________
Part (c) Show your predictions to your partners and, in pairs, explain your reasoning why you would stay with your original prediction or change. Then write your reasoning as follows:
I believe that it is/is not possible for the 26 students to have different initials because ___________________________________.
At this point, Mr. M reviewed the Multiplication Principle of Counting (see the video above).
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The following problem may be assigned for classwork or homework after Problem I has been discussed in class. You could also use it as an assessment.
Problem II
Mr. M decides to assign to each student in his 5 classes a unique code consisting of up to 5 colors in sequence. He has a total of 129 students and the codes will use only the colors Red, Yellow, Green, Blue and Purple. Mr. M explains that codes may have repeated colors (like GGG or GYG) and RYG is a different code from YGR.
Comment: I haven't mentioned how the Pigeonhole Principle can be applied to these two problems. I'll leave it to my astute readers to comment on that!
Ok, here's another video explaining the two problems above. I hope you will subscribe to my new channel on YouTube, MathNotationsVids .
[埋込みオブジェクト:http://www.youtube.com/v/9_P_wyk7N4M&hl=en_US&fs=1&rel=0]
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Note: I've been asked why I'm using these signatures on my posts, particularly the 2nd one. Well...
"It's my party and I'll try what I want to!"
(Apologies to Lesley!)
I'm sure some of my devoted readers can figure out why I included Schopenhauer's quote and the 2nd one is really all about education, isn't it?
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)
You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific
Posted by Dave Marain at 7:38 PM 2 comments
Labels: counting problems, Instructional Strategy Series, middle school, multiplication principle
Thursday, December 11, 2008
Instructional Strategy Series for Middle School and Beyond: Developing Direct & Inverse Ratio Concepts
Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?
Standard Assumptions
(1) All beagles work at the same rate. (If you understand beagle behavior intimately, you might question this). Seriously, it's the underlying assumption of constant "rate of work" that is so fundamental here.
(2) All holes are the same size.
Instructional Commentary
Well, at least, I didn't ask the classic: "How many eggs can 1.5 hens lay in 1.5 days (my all-time favorite word problem)!
The focus of this post will be on the first two stages of concept development using a concrete numerical example. You may take strong exception to the approach below of combining both direct and inverse variation in the same lesson, but, remember, the goal here is concept development, not proficiency with an algorithm! The algebraic stage will be deferred or left to the reader. The algebraic relationships are extremely important and worthy of extended discussion but that needs to be a separate discussion.
Stage I: Building on Intuition
Before developing a strict mathematical procedure involving direct, inverse or joint variation I feel it is critical for students to trust their "math sense." Encourage this with comments like:
"Forget calculations here, boys and girls, just think about this problem, use commonsense, and you might be able to arrive at the answer in less than 10 seconds!"
Don't think they can? No harm in trying...
I believe that when we tell them to trust their intuition, some will arrive at the correct answer of 5 days. Encourage those who "see" it to share their reasoning: WHY will the number days not change! This will vary according to the ability level and confidence of individuals in the group but, even more importantly, according to the environment you create in the classroom (accepting non-judgmental climate leads to greater risk-taking).
When review of homework, content coverage and time for guided practice (before the assignment is given) are the highest priorities of our lessons, then it is natural to question the wisdom of the above strategy.
This is obvious from typical comments like:
"Very nice, Dave, but who has the time for that, it's not going to be tested on the State Test and, moreover, I'm not teaching gifted kids like you must have had."
I won't react to my own Devil's Advocate arguments. Those you who know my philosophy of education know what my response would be!
Stage II: Beyond Intuition - Developing Proportionality Concepts via a Systematic Approach
"Well, boys and girls, now that we believe the answer is still FIVE days, let's try to approach this more mathematically, that is, more logically and systematically, in case the answer cannot be 'guessed' so easily."
I have found over the years that the following TABLE or matrix approach is a powerful model for devleoping proportionality concepts before the student sees a single algebraic relationship:
| EVERY DOG HAS HIS DAY! | |||
|---|---|---|---|
Beagles | Holes | Days | |
| 3 | 4 | 5 | |
| 3 | 8 | ?? | |
| 6 | 8 | ??? | |
Note how this approach avoid changing both the number of holes and the number of dogs in the same step! By keeping one quantity fixed, the student may better be able to focus on the relationship between the other two. Thus, in the second row I kept the number of dogs constant, changing only the number of holes:
"Boys and girls, if the number of dogs stays the same and we double the number of holes, then what will happen to the number of days ?"(they will double).
[Note that I asked for the effect on the the number of days before I asked for the actual number of days, namely 10 days.]
This approach develops the idea of direct variation before we express the relationship algebraically: As one quantity increases, so does a second quantity proportionately.
Now that we have filled in the second row (replace the ?? with 10 days), we can move on to another relationship:
"Boys and girls, look at the 3rd row. What quantity (variable) did we not change (keep constant)? What quantity did change? If we double the number of dogs, what should happen to the number of days needed to dig the same number of holes?"
(Yes, some will think 'double', since direct variation is often the initial reaction of many students).
Thus we are literally constructing direct and inverse variation via numerical computation before we develop any general relationships. Yes, this is time-consuming, but hopefully you will see the payoff in comprehension.
Stage III: Expressing Relationships Algebraically
Not in this post!
Important Note:
Normally, we would be very reluctant to mix both types of variation in one lesson, choosing to develop mastery of just direct variation first, then inverse much later on. Yes? Therefore you might feel that combining these will lead to confusion on the part of most students in most classes. Remember, though, the intent here was to develop a strong intuitive base for different types of variations before attempting to formalize any of this! You may not agree, but I'm proposing it anyway. I have done this with good results. Once the concept foundation is laid, students can take off with all the formulas!
Posted by Dave Marain at 8:03 AM 7 comments
Labels: direct variation, instructional strategies, Instructional Strategy Series, inverse variation, variation
Tuesday, June 17, 2008
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 practiceof 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:
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,...)???
Posted by Dave Marain at 7:28 AM 9 comments
Labels: average rates, averages, instructional strategies, Instructional Strategy Series, pedagogy