TWITTERMATH @dmarain
12-31-18

5 executives are each assigned their own parking spot. On any given day 0 thru 5 of these spots may be filled. Total # of ways for these spaces to be occupied? Other math connections, examples?

#MathProbs_Pedagogy









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Twitter Problem 12-27-18


Compare, contrast
 0.1111... (base 2) and
0.9999... (base 10)


@dmarain @HarMath @johnjoy1966 @benjamin_leis @mrdardy @aranglancy @MrATeachesMath #MathProbs_Pedagogy





Use new contact form at top of right sidebar to contact me directly. Note: If you'd like detailed solutions and Common Core strategies for my Twitter problems emailed daily, see info at top of sidebar.

Use new contact form at top of right sidebar to contact me directly. Note: If you'd like detailed solutions and Common Core strategies for my Twitter problems emailed daily, see info at top of sidebar.

From my Twitter account 12-21-18
The 1st 4 terms of a sequence of integers are
1
23
456
78,910
1,112,131,415
(a) Write the 10th term
(b) Using algebra describe how to construct the nth term (its... https://t.co/pcqZSq6Q7l

XMASMATH TWITTER PROBLEM 12-24-18

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From my Twitter account @dmarain  12-24-18

#MERRYANDHAPPY
x²=1: 2 integer solns
x²+y²=2: 4 solns
x²+y²+z²=3: 8 solns
(1) x²+y²+z²+w²=4 How many?
(2) Write eqn for 8 vars. How many solns?
Note: For @benjamin_leis use 2019 vars!😊
@dmarain
@johnjoy1966
@mrdardy @HarMath  @aranglancy @MrATeachesMath
#MathProbs_Pedagogy

Tribute to a very special woman...

The following was written by one of my daughters. It expresses what each of us feels every day after six years...

A little boy walks by alone on the sidewalk. A woman stops gardening looks up and sees this little boy. She asks the little boy to help her plant seeds in the ground. The little boy puts down his bag and sits next to the woman to help her dig a hole in the ground. He is intent in digging the perfect hole in the ground to fit the seeds. The woman tells him to open his hand and she gives him one seed. He looks at her and asks if he should have more, so it could grow big and strong. The woman looks into the little boy’s dark brown eyes and explains that all you need is one seed for it to grow into the most beautiful, strong, and tall tree that will live a very long life. The boy didn’t believe her. She took the little boy for a walk around the yard and showed him all of the trees that were planted. He asked the woman who planted all these trees. The woman told him that they were each planted by someone special. The little boy looked at all the trees and couldn’t believe how many people she thought was special. This little boy asked if he could come back and visit this tree. The woman said this tree is yours and will always be here for you. The woman showed the little boy how to pat down the dirt to make sure the seed was completely covered and would be protected. Then the woman gave the little boy a watering can to begin the process of bringing the seed to life. She explained that taking care of the seed in the beginning of its life is the most crucial time, so that it grows strong roots. The roots will hold the tree nice and tall and make sure it doesn’t ever fall. She explained that when the sun shines down upon the tree, the tree will feel its warmth and as the branches grow so will the leaves. All of a sudden a little bird flew by and landed right next to where they were planting. The woman told the little boy when the tree grows, she will hang a bird feeder off of it, so the bird can have a place to eat. The little boy questioned what if many birds want to eat. The woman said we can put as many bird feeders on it as we need. Everyday, the woman worked in her garden and everyday the boy would pass and stop to check on his tree. As the tree grew, so did the little boy. Years later, a man was taking a walk and stopped and saw the biggest, strongest, most beautiful tree he has ever seen. The tree was covered with bird feeders hanging all over with bluejays, robins, and squirrels climbing on the branches. The tree had a blanket of sunshine upon it. He looked closer and the tree had a word carved in it. He looked even closer and it said… “Home”. He looked around for the woman but she was nowhere to be found. He put his hand on his tree and felt her spirit, which lifted his heart. As he walked through the garden, he now realized all the trees were different sizes and types, but they all said.. “Home”. It was at this moment, the man realized this was where he belonged all along. He was home. I am proud to say that this woman was my mother and the little boy represents all the children my mom loved and took care of her whole life.. The tree represents stability, strength, and life. My mom would always say it doesn’t take much to love someone if you just open your heart and let the sunshine in. I miss her so much. I try to carry on her spirit, but it is almost an impossible feat since she was superwoman, but all I can do is always be kind and open my heart. I know that would make her proud. I also know that home is where your heart lives. Love you mom today, tomorrow, and always. Use contact form at top of right sidebar to contact me directly. Note: If you'd like detailed solutions and Common Core strategies for my Twitter problems emailed daily, see info at top of sidebar.

Common Core Activity: Square in Right Triangle - One More Variation!

As posted on twitter.com/dmarain ...
Share your thoughts in Comments or email directly using Blogger Contact Form.

AN UNCOMMON? CORE GEOMETRY ACTIVITY

As posted on twitter.com/dmarain@gmail...
Feel free to Comment or email me using the Blogger Contact Form.

COMMON CORE ALGEBRA/GEOMETRY INVESTIGATION

Also posted on twitter.com/dmarain
Let me know how you use/modify this activity for your Algebra and Geometry students! Feel free to comment or send email via Blogger Contact Form in sidebar.

COMMON CORE: LINKING MEAN AND CENTER OF MASS

Common Core Desmos Activity for Algebra 2 and beyond

Click on graph to see the activity!
Set viewing window: -5<=x<=5
Also, it's better to manually animate the tangent lines by dragging the point or the slider!

Just a simple demo of the power of Desmos to build interactive Investigations for our math students.

My primary goal is to model the critical balance between technology and concept development. While students can freely explore the graphs, they are also guided with carefully crafted questions. Feel free to use it and modify as needed.

This activity can be improved upon with Desmos' New Activity Builder! As I learn it, I'll share more of these. See Desmos.com and on Twitter visit @Desmos and @mathhombre for more examples, many of which are far more creative than mine.

Use this link if graph does not display: https://www.desmos.com/calculator/ukpdfbmaf6

Balanced Learning is not Blended or Flipped

Haven't been up to one my passionate rants in a long time so buckle up...

Technology has enabled educators to reimagine the  traditional classroom, how students learn and how we facilitate this process, aka teach. Incredible new opportunities to empower students to take more control over their own learning in a "space-time continuum" sort of way. Not to mention providing powerful tools to analyze data to individualize and maximize learning. Are there any buzzwords I omitted!

BUT...

We have strayed from NCTM'S central message from over 25 years ago:
***BALANCING*** PROCEDURAL LEARNING and CONCEPTUAL UNDERSTANDING.

Technology changes the landscape in a fundamental way but the best source code cannot quite replace the critical dialog and face-to-face interaction that is needed to accomplish the above goal. The spontaneous give-and-take of questions and ideas. Interaction vs interactive...

Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....

The hexagon/triangle problem in the diagram above can be approached using dynamic software like Geogebra & Desmos. You could develop an extraordinary exploration with carefully crafted questions enabling the student to discover relationships in the figures. I love doing that. I used to do this in the classroom long before it was fashionable. Anyone who follows me knows I'm a techie geek at heart.

BUT I came to realize that there was something missing. If you believe I'm not knowledgeable enough of how these new tools can accomplish the BALANCING I speak of, then challenge my premise! Let the New Math Wars begin...

And I haven't even addressed the myriad of approaches to "solving" this multifaceted geometry problem. Most students/groups will find their own solution paths but it is human nature to CHOOSE THE METHOD THAT FITS YOUR OWN WAY OF THINKING.

To develop the deeper ideas of geometry - symmetry, transformations, dissecting, combining and rearranging pieces of a puzzle, students need to be TAKEN OUT OF THEIR COMFORT ZONE and experience others' ideas and we need to fill in the gaps. That is part of teaching, yes?

And, oh yes, there certainly are algebraic/geometric approaches here with lots of nice formulas like (x²√3)/4...

Modeling - What Algebra Looks Like on the New SAT/PSAT and the Common Core

On twitter.com/dmarain...

A bear population, P(t), after t yrs,  is modeled by
P(t)=M-k(t-20)², 0≤t≤20.
Initial population:356
Max pop'n:500
Estimated population after 10 yrs?
Answer: 464

COREFLECTIONS

Is this the "new" algebra? Students given a function with PARAMETERS which "models" real world data? Questions like this have appeared on SATs for a few years now and, based on the sample new SAT/PSATs released by the College Board, they will become even more common. Students will be asked to analyze the function and use it in application.

The Common Core also emphasizes algebra models - "using" algebra to solve applied problems.

Middle and secondary math educators are not surprised by any of this as these changes have been around for a while but textbooks may need to include even more examples and homework problems of this type.

The real challenge, IMO, is to find that proverbial BALANCE between traditional algebra skills and  applications.

How much knowledge of quadratic functions is needed for this question? Will most students relate the form of the model to f(x) = a(x-h)²+k? Will they immediately recognize that M must be 500 since (20,500) is the vertex or maximum point? Try it and let me know!

Students should be allowed to explore this function using powerful software like Desmos and Geogebra. Sliders in Desmos allow for considerable analysis when parameters like M and k are given. 

BUT they also need to develop a fundamental knowledge of quadratic functions.

A key question for me is:
Should some background be developed BEFORE exploring with technology or AFTER or something in between?

I included a screenshot from Desmos for the bear population problem but this does NOT show how to IMPLEMENT this powerful tool in the classroom. I'll leave that for the real experts like John Golden! (@mathhombre).

37 not 42 the Answer to The Meaning of Life? A Common Core Investigation

From twitter.com/dmarain...

A Middle School Common Core Investigation

Is 37 an "interesting" #?

37x4=148; 4-1="3",8-1="7"
37x13=481
37x22=814

37x5=185; 8-5="3",8-1="7"
37x14=518
37x23=851

How far can you extend the pattern?

And is 37 patriotic (apologies to AK&HI)??

37x48=1776

And my favorite ...

1/37=0.027027...
1/27=0.037037...

Is it all because 37x3=111?

First we engage, then illuminate...

Parabolas, NEW PSAT/SAT and the Common Core

As posted on twitter.com/dmarain...

SHOW: The line with slope 1 intersecting y=-(x-h)²+k at its vertex also intersects at (h-1,k-1).

COREFLECTIONS
How would you modify this to make a grid-in or multiple choice question? A question similar to this appears on the published practice NEW PSAT. It is one of the last 3-4 questions on the grid-in with calculator section and was rated "medium" difficulty. I would rate it as more difficult! I recently tweeted the link for this practice test but easy to find on the College Board website.

Do the parameters h,k discourage use of graphing software?

Does the student need the equation of the line to solve the linear-quadratic system? Why does (h-1,k-1) have to be on the line? Then what?

What will be your source of questions like this for your students?

Median = Geometric Mean? A Common Core Investigation

As tweeted on twitter.com/dmarain...

J noticed that for an arithmetic sequence like 3,7,11,15,19 the median equals the arithmetic mean. In this case, the median and "mean" are both 11. She found this was well-known and not too difficult to prove.

She wondered if there was an analogous rule for geometric sequences like 2,4,8,16,32. Instead of the arithmetic mean she tried the geometric mean:
(2•4•8•16•32)^(1/5) which equals 8, the median. VERIFY THIS WITHOUT A CALCULATOR!

Unfortunately her conjecture failed for a geometric sequence with an even number of terms like 2,4,8,16 in which the median equals 6 while the GM = 4√2.

(a) Test her conjectures with at least 4 other finite geometric sequences, some with an odd number of terms, some with an even #.

(b) PROVE her conjecture for an odd number of terms.
Hint: If n is odd then a,ar,ar²,...,ar^(n-1) would have an odd # of terms. Why?

(c) How would the definition of median have to be modified for an even # of terms?

COREFLECTIONS
How much arithmetic/algebraic background is needed here?

Arithmetic sequences more than enough for middle schoolers to explore? Geometric too ambitious?

PROOF too sophisticated for middle schoolers? How would you adapt it? We are trying to raise the bar, right?

Tangrams Forever...

Posted today on twitter.com/dmarain...

Math educators K-14 have used tangrams for creative activities and to make learning "fun" but the underlying mathematics is rich. Whether you cut out the 7 pieces and rearrange to re-form the original square or a cat or a swan it's all math! Enjoy!

Dys-Functional but Rational

As posted on twitter.com/dmarain...

f(x)=(x²-x-6)/(x²-5x+6)
Find x such that f(x)=
0 [-2]
5 [no soln]
1 [no soln]

Answers in brackets

COREFLECTIONS
Fairly traditional rational function question in precalculus? Normally we'd ask students to analyze the function. What was my focus here?

Of course graphing software and CAS systems can be used but students need to walk before they run. The technology helps visualization but, more importantly it allows DIFFERENT, MORE PROBING QUESTIONS to be asked! I'm sure many of you might provide the graph and some key points and ask students to construct a rational function that fits the graph!

So why didn't I simply ask for the zeros, asymptotes, "holes"??

0,1,2,3,x If mean=median, x=?

As posted on twitter.com/dmarain today...

Let's get the "answer" out of the way first.
x can = -1,1.5 or 4. Not much more to say about this, right?

COREFLECTIONS

If this were an SAT-type question, it might be a "grid-in" asking for a possible value of x.

So what is needed to be successful with this type of problem? A basic understanding of mean and median for sure but there are the intangibles of problem-solving here. This question requires clear thinking/reasoning. Confident risk-taking is very important also. When one seems blocked, not knowing how to start, some students just jump in anywhere and see where it goes. Insight enables a student to move in the right direction more quickly.

Many students intuitively suspect that the median could be 1 or 2 or something in between. Even if they can't precisely justify this, they should be encouraged to run with their ideas. "Guessing" the median first seems easier than guessing a mean! One can always test conjectures.

Recognizing that there are THREE cases to consider is critical here. In retrospect, this will make sense for most but they have to make that sense of it for themselves!

So why not just give a nice clean efficient solution here? Because problem-solving for most of us is not clean at all! When the student is GIVEN the solution it may help them to grasp the essence of the problem but more often it shuts down thinking and doesn't help the student learn to overcome frustration. Yes, we can provide a model solution but how will that lead to solving a similar but different problem. We learn when we construct a solution for ourselves or reconstruct other's solutions in our own way.

Annoyed yet? If you solved it, you're fine. If not, frustration sets in quickly for some. If everyone in the class is stumped we can always give a hint. I think I already did!

1,012×1,008=1,020,096 A Mental Math Shortcut for MS!

Calculators and other technology enable students to "see" possible patterns/relationships without being discouraged by arduous calculations. The above multiplication is a well-known type of example to engage students in the mystery, magic and beauty of our subject.

COREFLECTIONS
Would you expect groups of middle schoolers to devise a rule or observe and describe a pattern based on this one example?

Would you start with simpler 3-digit examples like 102x103=10506 first to make relationships easier to see and formulate or does that depend on the group?

What do you find are the greatest challenges when implementing these kinds of activities? Is helping them express ideas in verbal and symbolic form one of them?

How important is "testing hypotheses" in this discovery/problem-solving process. Some students are naturally more patient and careful about "jumping to conclusions", a quality we should cultivate. But the risk-takers are necessary to move forward. The " testers" and skeptics are cautious and equally necessary, n'est-ce pas?

I don't expect many comments but if you have the opportunity to share this with children, pls share your experiences!

If (a-3)x+(b+2)=0 for at least 2 values of x then...

Many conclusions here but would you want your students to know why 'a' MUST EQUAL 3 and 'b' MUST EQUAL -2.

COREFLECTIONS
So what's the BIG IDEA here? Is this really "Fundamental"? Where is it in the Common Core?

So if a polynomial equation of "degree" n has more than n solutions, what exactly does that imply? Any restrictions on the coefficients? And what does this have to do with an identity?

For me, it's critical that we don't see these problems as curiosities or challenges designed for only the accelerated groups or the mathletes.