Wednesday, January 21, 2015

When is a rectangle an equilateral triangle?

As posted on ...

Diagonal of a rectangle has length 6 and makes a 30° angle with a side.
(a) Area of rectangle=?
(b) If diagonal has length d, area=?


(1) A moderate difficulty problem for SATs? Appropriate or too hard for a PARCC assessment with both parts?

(2) Should diagram be given or is drawing part of what's being assessed?

(3) Will some students recognize that the expression in terms of 'd' is the formula for the area of an equilateral triangle of side length d? If no one does then is it our responsibility to model and facilitate "connection-making"? Uh, yes ...

More interestingly, will some students realize that the rectangle divides into two 30-60-90 triangles which can be rearranged to form an equilateral triangle? Hence, the title of this post!

If we create this kind of environment in our classes it may happen. I think we're all conditioned to thinking that's only for the top honors groups,  and only for a few students. But, for me, helping ALL children discover and uncover the beauty of mathematics was my raison d'ĂȘtre for teaching. Idealistic perhaps but when that is lost, what's left?

Monday, January 12, 2015

How one 2nd grader knows his 7 Times Table!

As posted on twitter @dmarain today...

Question to 2nd grader: 7×6
How did you know that?
Easy --- 6 touchdowns! I know all my 7's!

1) So do you think this is about a real 7-yr old?

2) Would this be useful to many or just for girls/boys who watch a lot of football? OR

Is there a bigger issue here re the individual ways in which children learn? I think there are some HUGE implications here for teaching/learning in the Common Core and beyond...

3) All these "strategies" turning you off? Yearning for the good old days -- having children write their facts 10 times each or flash cards and memorization?

I have mixed emotions since I'm probably older than most of my readers but the anecdote above is real and it did work for this particular child! Further the child said, "I know some of my sixes too!" Missed PATs?? Maybe field goals will help with the 3s!!

Your thoughts?

Friday, January 9, 2015

Implement The Core: Quadratic Function SAT-Type Assessment

As posted on Twitter @dmarain...

The graph of f(x)=-(x-k)^2+h has one x-int and a y-int=-16. Coordinates of all possible vertices? Sketch graph(s).


(1) How do you feel about the "h,k switch" on an assessment? Would you revise it or leave it alone?
(2) Level of difficulty here? How do you think your students would perform? Let me know if you use it!
(3) Are you finding more of these types of questions in current texts? If not, what resources do you use to raise the bar?
(4) What if the question had asked for the PRODUCT of all possible x-intercepts? Better question for standardized assessments? Since the answer to this revised question is -16 do you think some of your students would ask if that's a coincidence? Why not ask them to check that -- Then GENERALIZE!

Thursday, January 8, 2015

BREAK THE CODE: 12-91-1305

As tweeted  on Twitter @dmarain today:

Break the code:12-91-1305
Then multiply these #'s by 4/3...
And you'll get my OBJective here!

Use contact form in sidebar to send me your answer/thoughts or leave a "hint" or question in Comments!

Saturday, November 8, 2014

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

From 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!


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..."!!
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 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
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]
(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

          1 ___4
M��an treats/child? M��dian?

(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?
Which is easier to determine -- mean or median? OR
If  frequency = 4 kids for all # of treats, mean = ? Mental Math!!
Explain to your partner why mean > median.


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...
I strongly recommend you  read all of his excellent pieces. The current one is compelling for all math educators not to mention the public...


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…