## Saturday, November 8, 2014

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

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?