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?


(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 if 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!
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...
Keep going!


Pattern #2

Keep going!

Describe, extend,generalize!

Is 407×9=3663 unrelated?


(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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Monday, October 13, 2014

Implement The Core: 'Dates' and 'Figs' - Middle School Investigation

Twitter Problem @dmarain...

Yesterday's date here in the US was 10-12-14: an arithmetic sequence.

(a) List the other 5 such dates this year

(b) List them for 2015 & 2016

(c) Observations & Explanations

In your group make at least 5 observations and/or conjectures. Explain/prove or show they are false.


(1) Observation: There are fewer such dates in 2016 than in 2015.
Possible Explanation: In 2015, the months are the 7 odd numbers  from 1 through 13; in 2016, the months are the 6 evens from 2-12 . There is no 14-15-16.
Note:Would the same be true for all even years from 2014 on?

(2) Conjecture: The middle number in the date is the average (arithmetic mean) of the other 2.
Possible Explanation for Algebra Students: The 3 numbers can be expressed as n,n+k,n+2k. The average of n and n+2k is (2n+2k)/2 or n+k.

(1) Ask students about the phrase "here in the US".
(2) Arithmetic sequences a middle school topic in the Common Core? What about patterns and linear relationships?
(3) Rich discussion of odds and evens
(4) Connections to Geometry: Find a "Pythagorean Triple" date!
Does anyone find the recycled arguments against Common Core Math as ironic and sad as I do? 1960? 1990? 2014? 'Dej√° vu all over again' as Yogi would say. Meanwhile students in other countries are bemused as they pass us in the fast lane...

Saturday, October 11, 2014

Implement The Core: f(3)=5,f(5)=5 and much more

Twitter Problem 10-11-14
f is a linear function with f(3)=5 and f(5)=3. f(0)=?
(1) The title has an error and omits the critical linear condition. Note that f(5)=3 not f(5)=5.
(2) The Mathematics Practice Standards ask us to extend student thinking, make connections and go beyond the superficial qualities of a problem.
My hope is that you will see the Twitter problem as a  door marked ENTER not EXIT...
How do we do this?
One possibility is to ask our students to generalize. Note that the responsibility is shifted from us to them. We can guide this by prompting with: "Suppose f(a)=b..."
Here's one possible generalization:
f is linear, f(a)=b,f(b)=a. Show that f(0)=a+b
Since the answer to the Twitter problem is 8, do you believe some of your students will make the connection from 8=3+5?
Yes, this is time-consuming. Some of the best food requires slow cooking! (Sorry for all the metaphors...)
(3) Would you also want your students to relate f(a)=b and f(b)=a to the reflection relationship between the points (a,b) and (b,a)?

Friday, October 10, 2014

Implement the Core-- A binomial activity with connections

Twitter Problem 10-13-14

List the different trinomials which result from assigning 1,2,3,5 to a,b,c,d in all possible ways.
List as follows:,

Explain why there are 12 possibilities!


1) Do you think this type of activity will facilitate factoring? OR factoring involves different skills/reasoning?

2) Activities which connect algebra to other content areas like discrete math (combinations, multiplication principle,etc) are fundamental to the Common Core. While students are practicing multiplication of binomials ( a lower-level algorithm) they are also exercising higher-order reasoning. Do you feel this is overly ambitious for students who struggle with distributive property?

3) Students need to understand that listing the 12 possibilities is not the same as **EXPLAINING WHY** there are 12!

You might challenge them to explain the flaw in the following reasoning:

There are 4 choices for 'a'.
Then 3 remaining choices for 'b, so there are 12 assignments for a,b. For each of these there are two assignments for c, etc. Thus there are (4)(3)(2)(1) = 24 outcomes.

Tuesday, October 7, 2014

Implement The Core:9 rolls of quarters,40 quarters per roll


#CCSS #Grade3-6

9 rolls of quarters
40 quarters in a roll
Total Value?

Sample Student Solutions/Approaches


Student 2
9×40 quarters=9×(10×4) quarters=
90×(4 quarters)=90×$1=$90

1) Which approach is more likely to be used by a 4th grader? 5th? 6th?

2) Which method is more commonly demonstrated in the text or by the teacher?

3) How do you get Student 1 to include UNITS/LABELS like
9 rolls × 40 quarters/roll × $0.25 etc???

4) How many children in Grades 3-6 intuitively use 2nd approach and can solve it mentally?

Of course no one out there is thinking:

"Well those are the 'smart' kids. We don't even have to teach them a method. Besides, their way of thinking is just not accessible to the rest of the group and would only confuse them. So I wouldn't even bring it up..."


Monday, October 6, 2014

Twitter CCSS Algebra Challenge Problem

Higher level of difficulty here. You may want to give this as a team challenge...

If P,Q are consecutive integer values, P<Q, for some integer value of x, what is the greatest possible integer value of a?

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Implement The Core: Delving Deeper into Quadratic Functions

Twitter Problem 10-6-14

The graphs of f(x)=x^2 and g(x)=k-f(x), k>0, intersect at pts A and B. Show that AB=√(2k)


1) To me, Implementing the Core means challenging our students with problems that go beyond the "standard" textbook exercise.

2) In your opinion, what makes this question difficult?

The definition of g(x) in terms of f?
The coordinate concepts?
The algebra (simplifying √, etc)?

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