Saturday, November 8, 2014

Implementing The Core: B lives twice as far from A as from C. Draw that!

From twitter.com/dmarain 11-8-14...

A,B,C live on a straight road. B lives 5 times as far from A as from C. If AC=12 draw,determine all possible distances!

COREFLECTIONS

1. 140 characters make the writing and interpretation of the problem challenging. But within each group of students there will usually be a few who will make more sense of it and they should be allowed to convince others in their group. When the inevitable hands go up and they ask "Do you mean...?" it's tempting to clarify but don't! Unless everyone is lost of course. The confusion will resolve itself in the class discussion and, yes, this consumes ("wastes"?) valuable time!

2. Of course I know that the phrase "5 times as far from A as from C" is the Waterloo of most students not to mention most humans! Can you guess which of my thousand or so blog posts have the most views over the past 8 years?  That's right -- the one that says ,"There are twice as many girls as boys..."!!
http://mathnotations.blogspot.com/2008/08/there-are-twice-as-many-girls-as-boys.html
Why are these phrases so troublesome? Many possibilities but the comments under that post are illuminating.

3. Do you believe this question is most appropriate in middle school? Geometry? Algebra?
OR of inappropriate difficulty for your groups?
My sense is that it's worth visiting it in ALL three!

4. So you're thinking your most capable students will rip right through this question. No problem. Then you or they explain it to the group and the rest will get it, right? Uh, try it out and let me know...

My experience tells me otherwise. Some of the strongest students will set it up incorrectly and get segments of lengths 60 and 12 for example. Or not recognize why there have to be TWO solutions depending on the relative location of, say, point C.

If you value a problem like this (and you may feel it's not worth the effort) and you anticipate the obstacles students will encounter, you may be tempted to provide a hint rather than see them struggle and "waste" time. I strongly urge you to let them work through it. You'll know when they need a hint. After a few minutes some will arrive at an incorrect result like 60 and 12. Invite them to share it. Discuss - explore--edit--revise. Learning can be messy.

After it's over what will the outcome be? They'll get it right on the assessment (as if it would show up on PARCC!)? Well if education is all about outcome-based performance then this has all been a grand waste of your time and mine...

Monday, November 3, 2014

Implementing The Core: Draining A Tank - A Real-World (?) Quadratic Model Problem

From twitter.com/dmarain today (of course the wording of the problem will exceed 140 characters!)...
Water is flowing out of a tank. The number of gallons after t min is given by the function
V(t) = k-2t-t^2. [Assume t≥0 and other suitable restrictions]
If 153 gallons remain after 3 min, in how many additional min will the tank empty?
I'll even provide an answer: 9 min
COREFLECTIONS
Problems like these which *artificially* model the real world are common these days on standardized tests but let's go beyond assessment issues.
Before throwing this problem out to the class I usually began with some thought-provoking questions to deepen understanding. For example:
(1) How do we know if the water is flowing out at a constant rate or not? Explain this to your partner.
[Suggested Answer: Constant rate implies a linear model]
(2) Draw a rough sketch before determining k. How can we do this if we don't have a value for k?
(3) Why is the quadratic model given more reasonable than say t^2-2t+k?
[Suggested Answer: The coefficient of the quadratic term should be negative since the quantity of water is decreasing. Note that students most often reply "'Because we want graph to open down!" This is insufficient IMO.
(4) What is the meaning of k both graphically and in the context of the problem?
[Suggested Answer: Graphically, k is the V-intercept; in the application, k = quantity of water at start or t=0]
(5) What strategy do we typically employ when working with function problems?
[Suggested Answers: Make a t,V(t) table; sketch a graph]
FURTHER COREFLECTIONS FOR INSTRUCTOR
(a)  Using a parameter like k makes it harder to just punch it into the graphing calculator. Common assessment technique these days. Students should be encouraged to also solve the problem with technology afterwards but that's teacher preference.
(b) Like most standardized test questions the quadratic doesn't require the quadratic formula, but for classroom discussion it certainly doesn't have to unless you're reinforcing factoring skills.
(c) Is asking for the "additional" number of minutes overkill here? A 'gotcha' ploy? Or does it discriminate as a difficult item should? If strong students, i.e , those who score high, do poorly then the question may be invalid. Serious issue here. What do you think?

Friday, October 31, 2014

Implement The Core: No *Mean* Tricks!

Halloween Twitter Problem
(@dmarain)

   Treats__Kids
          1 ___4
          2____5
          3____4
          4____4
M��an treats/child? M��dian?

COREFLECTIONS
(1) This question fits where in Common Core? Grade levels?
(2) What questions could you ask before calculation to develop number sense/conceptual thinking?
Some ideas...
Why is this sometimes referred to as a frequency table?
OR
Which is easier to determine -- mean or median? OR
If  frequency = 4 kids for all # of treats, mean = ? Mental Math!!
OR
Explain to your partner why mean > median.

HAPPY HALLOWEEN!!

Wednesday, October 22, 2014

A Dose of Reality -- My Latest Common Core Rant

I'm reproducing my comment to the post, "Who Needs Algebra?"on Mr. Honner's outstanding blog...
http://mrhonner.com/archives/14291#comment-10579.
I strongly recommend you  read all of his excellent pieces. The current one is compelling for all math educators not to mention the public...

MY COMMENTS...

First of all requiring an in-depth conceptual understanding of algebra for all students shows complete insensitivity to special needs students and their longsuffering teachers and parents. Sure just modify the curriculum for them. Go ahead. Show me exactly what that looks like and those who are pontificating the loudest come with me on the front lines of these classrooms and put your money where your mouth is.

Now for the rest…
Students should be expected to struggle much more than has been required of them for the past 3 decades. I've supported Common Core long before that name was coined because I believed not having uniform standards across the states was unethical and promotes inequalities for children. That belief is unwavering. However I've never believed all children should be subjected to a deluge of high-stakes assessments from the age of 8 or 9. Particularly when it takes 5-10 years for any new curriculum to "set". Particularly when teachers need extensive preservice and inservice training. Particularly when full released versions of these assessments have not yet been made public by PARCC or SBAC.

IMO, the rush to assess is purely politically driven and our leaders should be ashamed of themselves. In the name of accountability our children are needless guinea pigs. That is unconscionable. Sone of our best teachers are frustrated to the point that they might walk away from the profession they love. And that would be a real tragedy. The efficacy of the Core is dependent on our classroom leaders. If we lose the best of the best, we will all lose. Wake up before it's too late. Sadly that time may have passed…

Just How Common is our Core?

Borrowing a problem from the comments in the excellent blog CorkboardConnections. Hope that's ok...
http://corkboardconnections.blogspot.com/2014/08/common-core-math.html?showComment=1413889358852#c7893872002833512194

THE PROBLEM

Mdm Shanti bought 1/3 as many chocolates as sweets. She gave each of her neighbours' children 4 chocolates and 3 sweets, after which she had 6 chocolates and 180 sweets left.

(a) How many children received the chocolates and sweets?
(b) how many sweets did she buy?

ans: 18 children; 234 sweets.

FROM THE COMMENTER ON THE BLOG ABOVE

This is the questions our 12 year old do for their National exams.. is this type of questions easier or tougher than your core maths ?

MY REPLY

Dave MarainOctober 21, 2014 at 7:02 AM
My thoughts...
1. Unless Singapore Math materials are being used, US students could only solve this with algebra. For example, let y=# of children,etc. Students trained in Singapore Math might consider a "bar model" approach.
2. Problems of this level of complexity are unusual in US texts. Most 7th graders here are in prealgebra. This type of question would fit into 1st year algebra but I haven't yet seen many problems requiring this level of reasoning.
3. My instinct is that many of our **secondary** students would struggle with this! That's easy enough for teachers to verify.
4. Yes, Common Core has raised the bar but the proof will be in the difficulty of the problems students are expected to solve. If 12 year olds in your country are expected to solve this question on a National Exam then they must have been exposed to similar questions in their classes. In my opinion, we are not there yet...

OTHER COREFLECTIONS...

1. I hope you'll take exception to my comments above and prove me wrong by copying a page from a current COMMON CORE 7th-10th grade text. A page of problems similar to this one. Similar not only in content but in **difficulty**. An algebra problem tied to ratio concepts. In yesteryear, Dolciani would have problems like:

Determine a fraction in lowest terms with the property that that when the numerator and denominator are each increased by 2 the result is 4/5 (this one is easy; Mary P. Dolciani had harder ones!)

2. Some of my faithful readers are far more proficient with Singapore bar model methods). I tried it, it worked but I personally felt it wasn't worth the effort for me. Algebra seemed more natural. If you see a straightforward model solution, pls share!

3. What do you see as the complications in the problem above. The stumbling blocks for  some of your students? Remember the commenter is talking about a 12 year old, a 7th grader...

I asked myself if my 11 yr old grandson will be ready to tackle this next year? I think so if he's exposed to similar problems.

And that's the whole point of this post. Higher expectations are necessary but are they sufficient?

Monday, October 20, 2014

Round your answer to nearest cent: $1.29 or $1.30?

Tweeted (@dmarain) the above a couple of days ago. Moderate reaction so far which I find fascinating since I've done my own "random" survey...

SCENARIO
6th gr student calculates an *exact* answer of $1.29. Directions read "round ans to nearest cent." Student writes $1.30 in the answer box on the test. Teacher notes $1.29  was correct but the answer in box was wrong. No credit for problem...

COREFLECTIONS

Making too big a deal of this? After all "rounded to nearest cent" means "round to nearest hundredth". So $1.29 is already rounded to the nearest cent whereas $1.30 is rounded to the nearest tenths or dime, right? Adults know that, right? Students should know, right? Certainly higher-achieving HS students know that, right? Hmm...

Maybe you should try your own informal survey. Let me know...

Saturday, October 18, 2014

Implement The Core -- Opposite Corners of a Square

Twitter Problem 10-18-14
If (a,b),(-a,-b) are opposite vertices of a square, show that its area=2(a^2+b^2)
EXTENSION: What if (a,b),(-a,-b) are adjacent?
COREFLECTIONS
(1) What do you believe will challenge your geometry students here? The abstraction? "Show that"?
(2) Predict how many of your students would "complete the rectangle" by  incorrectly drawing sides || to the axes?
(3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy of more dialog IMO...
(4) How many of your students would question the lack of restrictions on a&b? Would most place (a,b) in 1st quadrant without thinking? So why doesn't it matter!
(5) Is it worth asking students to learn the formula "one-half diagonal squared" for the area of a square?  I generally don't promote a lot of memorization but this one is useful!
(6) EXTENSION
Answer to extra question: 4(a^2+b^2).
Ask your students to explain visually why this area is TWICE the area of the original square!

Thursday, October 16, 2014

Implement The Core: Arithmetic Patterns & Generalizations in Middle School Math

As tweeted on 10-16-14...

Pattern #1

Explore on calculator...
352×11=3872
527×11=5797
365×11=4015
Keep going!

Discuss!
Explain!
Generalize!

Pattern #2

18=9×2,81=9×9
27=9×3,72=9×8
36=9×4,63=9×7
Keep going!

Describe, extend,generalize!

Is 407×9=3663 unrelated?

COREFLECTIONS...

(1) But these are just math curiosities, Dave. They don't really tie into the Common Core, do they? Well, doesn't multiplying by 11 connect nicely to the Distributive Property:
352×11=352×(10+1)=3520+352 etc.
How about 9?

(2) My goal has always been to expose our students to engaging and meaningful mathematics. But deeper conceptual understanding results from going beyond the "Oh, I get the pattern!" response. That's where the "describe, extend, generalize" and group dialog come into play. Not to mention our guidance!

(3) Students are always intrigued by the mystery and wonder of 9 and 11. How ironic that these 2 numbers put together will forever have a negative connotation for our society. It's important for our  students to understand that many of the "tricks" involving these numbers are directly linked to their juxtaposition to 10, the base of our number system. In base 8, for example, 7 would display many of these properties!

(4) The more inquisitive students can research palindromes like 3663.

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