SAT TWITTER PROBLEM

Students taking May SAT MAY want to try today's Twitter problem I just posted at twitter.com/dmarain

If n is a positive integer, then the expression n(n+3) + (n+3)(n+8) must be divisible by
I.   2
II.  4
III. 8
EXPLAIN!

This is a typical "cases" type but I omitted the usual choices like
(A) I only
etc...

Might be worth some discussion to consider more than the typical student's "plug-in"  approach. That's why I added "EXPLAIN! "
There is some rich mathematics to be unearthed here IMO...

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The Third Wheel...

Two wheels with  diameters 18 and 8 are touching and are on level ground. Show that the diameter of a 3rd wheel on the ground which touches the other 2 is 2.88.


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Any child struggling with mixed numerals=?UTF-8?B?Pw==?=

The answer may be "no" in some parallel universe but here on earth the title of this post is rhetorical.

So we show children a diagram of 4 identical pizzas each divided into 8 equal slices or for the younger set we have manipulatives. We would probably not use so many pieces when introducing this but I needed an example which could also appear on the next state test.

We cross out or shade all the slices in 3 of the pizzas and 5 of the slices in the 4th pizza, representing what a group of kids ate.

What are the questions we ask or might appear in the text or on the worksheets or on the state mandated tests?

What do you believe are the major stumbling blocks for most children and what can we as educators or parents or tutors do to help?

Here are some thoughts...
Is the issue more conceptual or procedural?

How would you rank the importance of how each question is worded?

You want the answer to be both the improper fraction 29/8 and the mixed numeral 3 5/8. How should the questions be worded?  Hey, there's no universal remedy here! Some children will misunderstand the questions no matter how they're expressed or simply have not yet made sense of the ideas. BUT on an assessment the wording must be mathematically correct and age-appropriate, right?

How would you react to the child who responds 29/32? Is (s)he wrong? How could the question be asked for which this correct? Is the child confused or was it the question itself?

Whether you're a 3rd grade teacher, a professor of math/math ed, a math staff developer or coordinator/administrator I hope you'll weigh in on this with your reflections and/or anecdotal experiences.

I consider this issue to be of vital importance in the development of the concepts and skills of fractions and part vs whole.

What do you think?
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If a hen and a half can lay...

Share your teaching methods for the classic problem in the title:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can 3 hens lay in 3 days?

Would you ask students for an immediate intuitive guess and expect many to say 3?
The answer is 6 so the purpose of this post is reflection on sharing instructional strategies.

What are the BIG IDEAS here? Do you use one basic strategy in teaching all ratio problems?  Does dimensional analysis work for middle schoolers?  Should students always reduce everything to a single unit ratio like 1 hen per day? How many ways could your students devise if we tell them the answer is not 3?

Not much sharing going on like in the old days of this blog but maybe it's time..


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Shameless self-promotion!
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State Testing Review Time In Your District=?UTF-8?B?Pw==?=

This is that special time of the year when some districts, particularly in elementary grades, hand out practice materials for the test and teachers are expected to devote the majority of class time to it.

Here's my question...
Does anyone out there feel that these materials seem to be somewhat different and of a higher level of difficulty than regular classroom materials/tests?  If this is the case then what are the implications for the child, the teacher and the district?  I do have strong feelings about this but I'll wait for your comments first. WHAT ARE YOUR THOUGHTS?

Also I have to repeat a tweet I just saw from the brilliant Timandra Harkness from the UK. She made my day...

"I'm decorating my bathroom with those new fractal tiles. I think it's going to take forever."

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PerCent "More" - Applying RATIOS DEEPLY

PROBLEM
Two thin cylindrical steel  disks have diameters of 35 in and 25 in. The area of the base of the larger is what % more than the smaller?

We would hope juniors in Alg 2 or Precalc would know the basic setup for % more, % increase/decrease or % change types, particularly since this is a middle school concept. Of course we know this is often not the case!

After having students work in small groups for a few minutes and watching them pushing calculator buttons you have someone come up and explain, asking questions and reviewing basics. As is typical, some students will use the diameters instead of the radii and get the right answer anyway. What are the "BIG IDEAS" here?

Write on the board (35/25)^2 = 49/25, then 24/25 = 96%.  No explanation. You give students in small groups 2 minutes to make sense of this and have 2 groups take turns explaining it to the class.A mental calculation?

Before you kneejerk reflexively react to this with " Even some of my honors students would struggle with that", I would like my readers to reflect on our obligation to stretch their minds and promote conceptual understanding.

IN NO WAY AM I SUGGESTING THAT IS THE METHOD MOST STUDENTS SHOULD USE!

SO WHAT ARE THE KEY MATH CONCEPTS USED IN MY EXPLANATION?

Of cou

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Investigation for "Squares"

CHALLENGE YOUR GEOM STUDENTS OR YOUR MATH TEAM

In square ABCD of side 1, E is the point on diagonal AC such that AE=1.

(a) Explain without numerical calculation why
√2 < BE + DE < 2
(b) Show that BE+DE = 2(√(2-√2)) ≈ 1.531 without using Law of Cosines
(c)  Be a math researcher! How might you generalize this?

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NEW MATH CHALLENGE QUIZ/PROBLEM BOOK AVAILABLE

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT I, SAT Math I/II, Math Contest practice or daily/weekly Problems of the Day. Questions include multiple choice, cases I/II/III type and constructed response. Price is $9.95 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT! (dmarain "at" "geemail dot com")