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?

Friday, October 31, 2014

Implement The Core: No *Mean* Tricks!

(@dmarain)

Treats__Kids
1 ___4
2____5
3____4
4____4
MðŸŽƒan treats/child? MðŸŽƒdian?

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...

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 ?

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

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

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
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.

(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

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.

Examples...

(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.

COREFLECTIONS
(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!
(5) ***WARNING--RANT
Does anyone find the recycled arguments against Common Core Math as ironic and sad as I do? 1960? 1990? 2014? 'DejÃ¡ 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

f is a linear function with f(3)=5 and f(5)=3. f(0)=?
COREFLECTIONS...
(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

(ax+b)(cx+d)
List the different trinomials which result from assigning 1,2,3,5 to a,b,c,d in all possible ways.
List as follows:,
(3x+1)(5x+2)=15x^2+11x+2

Explain why there are 12 possibilities!

COREFLECTIONS...

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

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

Sample Student Solutions/Approaches

Student1:
40×9=360;360×.25=90

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

COREFLECTIONS
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

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

P=5x-12,Q=ax+17
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

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)

COREFLECTIONS

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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Sunday, October 5, 2014

Implement The Core with RE2PECT!

Avg × At Bats = Hits
0.310 × 11177 = 3465
Note: To make this more accurate we would need to ____ the number of walks, sacrifices???

1) Avg=0.310, At Bats=11177, Hits=x
Write equation, solve.

2) Write 2 more problems, solve!

Thursday, October 2, 2014

A Triangle Classic -- So many congruent parts but not enough...

∆ABC: AC=6,BC=9,AB=4
∆DEF: DE=6,DF=9
If angle BAC is congruent to angle EDF, EF=?

COREFLECTIONS...

1. Not that easy to find examples where two NONCONGRUENT triangles have 5 pairs of congruent parts!

2. This might drive home the meaning of important terms like corresponding parts, included vs non-included angles, etc

3. I used this example in the classroom to help students avoid jumping to conclusions! Geometry teachers love the play on words with "ASS-U-ME" but this example may be more about SSA and its reversal!

4. SAS is both a congruence and a similarity theorem/postulate. Once the student draws the diagram and labels corresponding parts carefully the similarity should become clearer. But do you think some students would match up the congruent sides before looking for proportional parts? Let me know if you use this at some point.

5. You might challenge your students to devise other pairs of similar triangles that have 2 pairs of congruent sides and three pairs of congruent angles. Maybe they'll notice the 2:3 ratio *within* each triangle as well as the 2:3 ratio between the triangles! Lots of interesting relationships there. Like 6 is the mean promotional (aka geometric mean) between 4 and 9!

Sunday, September 28, 2014

Implement The Core: A Variation on a Classic Rate Problem

Trip to work took 90 min including stopping b/c of accident for x min. Avg'd 30 mph overall; apart from delay, avg'd 50 mph. x=?

Solution??

REFLECTIONS
Do rate-time-distance problems still appear on SAT and other standardized tests? Yes!

Do our students get enough practice with these? In Prealgebra? Algebra 1?

Do you view these as applied problems?

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Friday, September 26, 2014

Implement The Core: Mean of 3 scores=90%,Range=30%,Median??

A little more detail from the Twitter Math Problem 9-26-14

Mean of 3 tests:90
Range:30
Explain why median must be 100.

Note: Assume all tests are based on 100 pts. The % info could be misleading, aka wrong!

REFLECTIONS...
1) Emphasis here is on explanation/reasoning rather than giving a numerical answer. That's why the problem is different from the title. This is at the "core" of the Mathematical Practices of the Common Core.
2) As any dedicated professional knows:
Finding challenging problems to promote collaboration and maximize participation is a daunting task. But isn't that what the Common Core is all about?
3) As educators would you promote an algebraic explanation or feel equally comfortable with one that uses a number-sense approach like "the lowest score has to be 70% or less because...", etc???
4) I've given away over a thousand original higher-order problems over 7 years on this blog and, more recently, on Twitter. And we know everyone is looking for freebies on the web. But writing detailed solutions/strategies/Common Core Implementation is labor-intensive. Creating new nonroutine problems every day is my passion but all good things must come to an end. Hope you RE2PECT that! Pls note the special offer in the sidebar which ends on 9-30-14.

Thursday, September 25, 2014

Least positive integer with 2014 factors - Detailed Solution

Explain why (3^52)(7^18)(11) is the smallest positive integer with 2014 factors and which doesn't end in 5 or an even digit.

Before the solution, a few
REFLECTIONS...
1) This is not an SAT or typical Common Core Problem. It's more challenging than that. But it does apply a fundamental principle of arithmetic which is often overlooked.
2) The solution below is more detailed than most but these are the kinds of solutions I will be emailing to you when you subscribe. See details at top of sidebar to the right.

Very Very Very Detailed Solution:
There's a fundamental rule about the number of factors of any positive integer > 1. I'll demo it with 12...
Step 1. Prime factorization of 12 is (2^2)•(3^1)
Step 2. Each factor of 12 is then of the form (2^a)(3^b) where a=0,1,2 and b=0,1
Step 3. Using the multiplication principle of counting there are (3)(2)=6 possible combinations of the exponents, each one producing a unique factor of 12:
1=(2^0)(3^0) (0,0) pair
2=(2^1)(3^0) (1,0) pair
3=(2^0)(3^1) (0,1) pair etc...
4=(2^2)(3^0)
6=(2^1)(3^1)
12=(2^2)(3^1)
So think of this as the
"Add 1 to the exponents and multiply" Rule!
Back to 2014 factors...
From Wolfram Alpha (enter "factor 2014")
2014=53•19•2
To construct an integer with this many factors we reverse the previous procedure, I.e., we SUBTRACT 1:
If p1,p2,p3 are different primes then
((p1)^52)•((p2)^18)•((p3)^1) will have
53•19•2 =2014 factors!
From the conditions we want to use the 3 smallest primes excluding 2 and 5, namely 3,7,11:
(3^52)(7^18)(11^1).
QED
(Mathematician's way of saying "I'm done!")

Wednesday, September 24, 2014

The sum of 2 pos int is 216, gcf=24 -- PUFM and Common Core

The sum of 2 positive integers is 216 and their gcf is 24. Find all possibilities.
Parents/Teachers/Students...
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REFLECTIONS
To solve the problem above without Guess-Test-Revise requires a more (P)rofound (U)nderstanding of (F)undamental (M)athematics - thus the acronym in the title. (Research Liping Ma for more info).
Students may find solutions by playing around with multiples of 24 on their calculator and that is a good thing. That's how we learn. But...
How many  will discover without our guidance a systematic approach to finding the 3 pairs of numbers. A method which makes sense to them and can be applied to more complex problems...
I believe there are BIG IDEAS, aka Fundamental Arithmetic Principles, embedded in this innocent question. I've said enough...

Which one of these might be on your child's math HW tonight? IMPLEMENTING THE CORE...

And I'm not talking about that so-called Challenge Problem at the bottom of the worksheet. The one where your child says, "Oh, we don't have to do that one!"

1) Remainder when 999 is ÷ by 30?

2) Largest multiple of 30  less than 1000?

3) Largest 3-digit integer div by 2,3 &5?

Which of these require more reasoning and conceptual understanding?

Mathematical Practices and Core Reflections...

1) How often do we just throw a challenge problem at a class knowing that only a couple will actually try it. You know, the "smart" ones. Not really for everyone else...

2) If we  don't seriously value the importance of such a question, WHY ASK IT? Because it's on the worksheet? Really? Are you going to review it carefully or is there no time for that?

3) What are the BIG IDEAS OF DIVISIBILITY underlying these questions? Are they identified in the Common Core? Where?

Oh yes...
The answer to #2 & #3 above is 990. See, that was easy. Guess that's all way can say about this problem, right?
Case closed...

Actually NOT...
The Common Core will not raise the bar by itself. Only we can do that. Teachers, parents and everyone in our society...

Do you sense an "edge" to these remarks? Then my message is getting through...

Tuesday, September 23, 2014

ImplementTheCore...It's 11:15 am. In 4hr 55min it will be?

Common Core Considerations...

The question in the title is appropriate for which grade levels?

To teachers/parents...

Think fast--get it right? wrong?
Need to write it out?
Need a clearly taught method?
Need less repetition? Extensive reps?

How do you make a variety of thinking/learning styles work in a collaborative setting when one of the children  in a group thinks
12-1-2-3
15+55=70=1 hr + 10 min
4:10

Do YOU show them that adding 55 min can be done by adding 60 then "backing off" 5? OR
Do you ask THEM who found another way?

Yup, teaching is the easiest job....

Monday, September 22, 2014

Implementing The Core: This is not a parenthetical remark...

-3a^2+4b+c; a=-3,b=-2,c=-1
Step One:  -3(  )^2+4(  )+(  )

Note that I'm recommending this BEFORE the numbers go in!

Do you share my belief in the critical role of (  ) in evaluating algebraic expressions?

OR

Are you thinking this is too much detail and most students don't need to do this?

And I haven't even gotten to replacing -7-3 by (-7)+(-3)!

Sunday, September 21, 2014

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Tuesday, September 16, 2014

CCSS: One Less Than a Million - How Many Nines? Grade 2? 4? 6? 8?

Implementing The Core - Raising The Bar
One less than a million. How many 9's?
Developmentally inappropriate for 7 year olds?
***What questions should we be asking to develop this kind of arithmetic reasoning?***
***WHAT ARE THE BIG IDEAS HERE?***
What should children be writing on their paper to make conjectures about numerical patterns?
[1] less than 10: 9 [1 nine, 1 zero]
[1] less than 100: 99 [2 nines, 2 zeros]
etc...
How can this be EXTENDED to challenge the child who's ready for higher-order thinking?
Note that I didn't say *OLDER* children!
EXTENSIONS/ASSESSMENT SUGGESTIONS
One less than a trillion? How many nines?
One less than a googol?
One less than 10^n?
One MORE than a billion? How many 1's?
One MORE than 10^10? What is the SUM OF THE DIGITS?
Discover a general rule and have them memorize it?
***STATE A RULE - YES!***
***MEMORIZE? NO! NO! NO!***
It's OK. I'm not expecting comments. I'm just planting seeds. Up to you to consider, modify, plant, add nutrients and illumination, watch growth
OR ignore all of this!

Monday, September 15, 2014

Typical 2nd Gr Assessment Questions and Your Thoughts...

Imagine that. I'm not promoting my problems/solutions!

Here are a couple of typical questions your 2nd grade child/student may be working on...

1.
(Clock shows 1:00)
In 1/2 hour, it will be ___.

Mathematical Practices Reflections...
Why do you believe some children would struggle with this?
Possible teacher/parent interventions?

2.
Write arrow rule. Fill in missing frames..
5---?---15---?---25---?

Mathematical Practices Reflections...
After child demonstrates proficiency, what can teacher/parent do to raise the bar? OR
Are you thinking this is ambitious enough for a 7 yr old?

Plugging in to avoid the algebra? Today's CCSS/SAT Twitter Problem

1/|8x-4| > 1
Possible value for x?
(SAT-type grid-in question)

Strategies...
"Plug in" - Easy?
Graphing calculator?
Algebra?

How do you think I devised this problem?

Want solution? Uh, you know what to do...

Sunday, September 14, 2014

More Solutions to Twitter CCSS/SAT Questions

The following is part of what everyone on my Twitter Problems mailing list has been receiving every day or two for the past 3 weeks. For free... You have 2 weeks left to sign up. Free...

As you can see I go beyond the answers. Way way beyond...
Just as the Math Practices of CCSS suggest we do...

1. I walked my daily path 25% slower than usual and took 5 min longer. How  many min does it usually take?

Solution:
Let R=usual rate (mi/min); T=usual time (min)
One can infer that distances are equal from the phrase "daily path". Using D=RT and equating:
((3/4)R)•(T+5)=R•T
R's " cancel" leaving
15/4=1/4•T or T=15.

Generalization:
If slower rate is k•R (0<k<1) and extra time is m min then
kR(T+m)=RT --->
km=T(1-k) --->
T=m(k/(1-k))
Test it: k=3/4,m=5 ---> T=5((3/4)/(1/4))=5•3=15
Special case: If k=1/2 then T=m or if one walks half as fast trip will take double the usual time!

2. Test this "rule":
3 more than the square of an odd integer is a mult of 4.
Now prove it!
Devise a rule if "more" is repl'd by "less!"

Solution:
An odd integer can be expressed as 1 more than even or 2n+1.
"3 more than the square of an odd integer" translates to
3+(2n+1)^2 = 3+4n^2+4n+1=4(n^2+n+1), a mult of 4.

Three less than the square of an odd becomes
(4n^2+4n+1)-3 = 4n^2+4n-2 which represents 2 less than a multiple of 4.
Thus "three less than the square of an odd" cannot be a multiple of 4 and in fact will always leave a remainder of 2. Why?

Saturday, September 13, 2014

Sample Solutions to Recent Twitter CCSS/SAT Problems

The following is copied from solutions I sent today to my mail list of those who have opted for free solutions for the rest of September...

Yes, I've been giving these away for weeks now. Hard to believe anyone would do this? There must be a catch, right?

I will continue this until the 30th then am considering a low fee subscription for the rest of the school year.

Subscribers will get detailed solutions which include strategies, big ideas, extensions, etc. Further I may include additional problems which will not appear on Twitter or this blog.

To sign up, provide all pertinent info in the Blogger Contact Form in the sidebar.

1. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

(a) 6
(b) Greatest perim:194; Least:40

Solution: 96 has 6 pairs of factors ---
1,96:2,48;3,32;4,24;6,16;8,12
Each pair has a different sum so there are 6 possible perimeters.
The greatest and least possible occur in the extreme cases, i.e., when the factors are farthest and closest apart. This is generally true.

Note: If integer condition is removed there would be no greatest perimeter and the least would be 16√6, a square!

2. Data:4,6;mean:5
(a) Avg diff from mean= ((4-5)+(6-5))/2=?
(b) v=((4-5)^2+(6-5)^2)/2=?
(c) √v=?
(d)Repeat for 3,7
Obs,Conj?
Common name for √v?

(a) 0
Note: This is always true -- the avg difference or deviation from the mean is zero! This is why we square the differences to measure deviation!
(b) v=1
Note: The avg of the squared "deviations" from the mean is called the variance.
(c) √v=1
Note: The square root of the variance is called the standard deviation!
(d) For 3,7 ---
Mean is still 5
v=(4+4)/2=4
√v=2, the stand dev.
√v gives a measure of how dispersed the data is from the mean...

Friday, September 12, 2014

Six more Free Twitter CCSS/SAT questions...

Free Twitter CCSS/SAT Problems with complete solutions sent to your inbox ending in 18 days on 9-30-14. You know what you have to do...

1. List in order the ten 5-digit pos int containing 3 nines & 2 eights?
Explain connection to 5C2.
Is this open-ended?

2. The median of 100 different integers is 100. If the numbers are in increasing order and the 50th # is 83 what is the 51st #? Explain/show...

3. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

4. Data:4,6;mean:5
Avg diff from mean= ((4-5)+(6-5))/2=?
v=((4-5)^2+(6-5)^2)/2=?
√v=?
Repeat for 3,7
Obs,Conj?
Common name for √v?

5. From the brilliant #xkcd...

Sneeze droplet: 200 million germs. Hand sanitizer kills 99.99%. How many live? No calculator-15 seconds!

6. Least positive odd integer with 8 factors?
Strategy: Make it ____
2 factors:3
4: 15 or 3×5
Generalize!

Wednesday, September 10, 2014

Seven More CCSS/SAT Twitter Math Challenges

Free solutions sent to your inbox?
Only 3 wks left...
http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

1. Circle has center (0,0), radius 2. Line y=1 intersects circle at P and Q. PQ=?
(A)1 (B)√3 (C)2 (D)2√2 (E)2√3

2.
(a) On a number line, how many positive integers are "closer" to 500 than to 1000?
b) Express the condition in (a) as an inequality using absolute values.

3. If the avg of 2 #'s is n, n>0, and the smaller # is 25% of the larger, the smaller # is what % of n?

4. For how many numbers x is the square of x equal to the opposite of x?

5. For how many integer values of x is |50-x| < 10?

6. A set of integers, T, has the property that if x is in T then x^2-1 is also in T. What is the least possible number of #'s in T?

7. Three-eighths of a candle's material remains after burning for 8 hrs. At this rate how many hrs for the rest to melt?

Monday, September 8, 2014

MathNotations Passionate View of the Common Core Controversy

Despite all the backlash against the Common Core, I am deeply committed to raising the bar for all students. The Common Core is an important step in this direction. I may have deep concerns about implementation and assessment, but I have no significant concerns about the quality, appropriateness and importance of the Core math standards and, in particular, the Eight Mathematical Practices.

Those states that are now defending their pre-CCSS standards are forgetting that the Common Core was developed by these states and approved by these states, not the Federal Government. The Common Core was a response to the fact that our students have been falling behind a majority of industrialized nations, particularly in math and science. The future of our children is at stake.

We need to stop the political rhetoric and do what's right for our children. Raise concerns about assessments and the timetable. You should. But remember this. On every international comparison our students have been performing at a mediocre level. That's reality. If there were no problem we wouldn't need to fix it. But there is a problem...

My commitment to our children and our teachers has been demonstrated by giving away hundreds of free higher-order math problems on this blog and on twitter for years now.

But implementation of CCSS will take the commitment of everyone. Or like every other educational initiative it will fail. But this time, if this fails, we will be failing our children. Hyperbole? Just check our nation's performance on international comparisons...

IMPLEMENTING THE CORE: Twitter SAT/CCSS/COORDINATE GEOMETRY Problem

Circle has center (0,0), radius 2. Line y=1 intersects circle at P and Q. PQ=?

(A)1 (B)√3 (C)2 (D)2√2 (E)2√3

I'm sorry to keep reminding my readers that free detailed solutions to all current Twitter problems can be sent to your inbox if you request it via Blogger Contact Form.

As any of my followers know, I've posted hundred of free problems on my blog and on Twitter for a very long time. However I have omitted answers and solutions to many of these.

For the past month I have offered to send free detailed solutions of these to your inbox if requested.  I will continue to do so until the end of this month. After that I will continue to post problems but solutions will be available only via paid subscription.

NOTE WELL:
In addition to solutions those on the mailing list can confirm that I'm also including alternate methods and specific references to the Mathematical Practices of the Common Core and strategies for implementing CCSS. This is my commitment to our teachers and our children.

Saturday, September 6, 2014

Free Twitter CCSS/SAT Problems/Solutions ending on 9-30

Right now it's still free to have these sent to your inbox. Blogger Contact Form must be filled out with all requested info. See http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

After that date by subscription only...

1. Right circ cylinder: base circumf 16Ï€. If greatest distance between bases=20, volume =kÏ€, k=?
Ans:768
Soln?

2. #ImplementTheCore

Median of 1st 1000 pos even int's?
Name of strategy one could use?

Free soln?

3. If area between 2 concentric circles=area of smaller circle then Larger radius:Smaller radius=?
Ans: √2:1
Free soln to inbox?

4.
2^100-1 must be div by
I. 2^10-1
II. 2^50-1
III. 2^50+1
(A)I (B)II (C)I,II (D)II,III (E)I,II,III
Soln?
http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

Sometimes "free" actually means free!

Thursday, September 4, 2014

Update on Free Solutions to Problems

Based on response thus far I am restricting emails to Solutions of the Twitter problems I post @dmarain.

The opportunity to be added to the mail list and have free solutions sent to you is limited.  Pls include all requested info - see link below...

http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

IMPLEMENT THE CORE-SIX Twitter/CCSS/SATPREP Math Questions to Start the Year

http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

1. Reg price:\$N
40% discount \$32 better than successive disc of 20%,20%
N=?
Want ans/soln/details/discussion sent to inbox?

2.
2^100-1 must be div by
I. 2^10-1
II. 2^50-1
III. 2^50+1
(A)I (B)II (C)I,II (D)II,III (E)I,II,III
Soln?

3.
Need  quarters&dimes for 35¢ bus fare. Asked bank for most I could get from \$10. How many dimes did I get?
Soln?

4.
(3/a)•(3/(a-12)) = -1
a=?

Relate this equation to a coord problem involving a right angle!

5.
Mork's age
O-yrs  46   90
E-yrs   1     10
When O-age:E-age=5:1, E-age=? (Assume linear)
Soln sent to inbox?

6. Inscribe a circle of rad=1 in a reg hexagon and inscribe a 2nd reg hex in the circle. Show that the avg of the areas of the hex's = (7√3)/4.
What does this have to do with Ï€?

Saturday, August 30, 2014

A Classic Algebra/Geometry Inequality Proof Without Words

In case one forgets that math can be beautiful...

Diagram is far from perfect. Hopefully you can make sense of it.

If you want more details let me know via Blogger Contact Form.

Use new contact form at top of right sidebar to contact me directly! If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Friday, August 29, 2014

Explore Inscribed Square in Rt Triangle - CCSS Activity

Just request this via Blogger Contact Form.

Incude
(c) State & Name of school district in which you are teaching (or a student)
(d) Grade Level(s) or Math subjects currently teaching or indicate grade level/current math courses if a student. Also indicate if you're teaching LD/Reg/Honors/AP Levels
(e) Preferred math problem types (E.g/, Middle School, CCSS, Algebra, Geometry, SAT, Precalculus, Explorations, etc)
(f) Indicate if you already subscribe to my blog feed (& which feed you use) and if you already follow me on Twitter.

Thursday, August 28, 2014

A Crazy Ages Problem - Is there a place for this in CCSS

A variation on a classic for your algebra group!

J is as old as K was when J was half the age he'll be in 10 years. If K is y yrs old, express J's current age in terms of y.

Want answer and detailed solution sent to your inbox? By now you should know what you have to do. Join the other teachers who have figured it out...

Reflections...
So you're probably thinking this question is either too hard or irrelevant in the Common Core. I beg to differ!

Before having students attempt to solve this algebraically I would encourage my students to experiment with numerical values. For example suggest a value for y, say y=40 and have them guess some values for x, J's current age. I'm sure you'd agree that this is still a challenging problem but someone will probably guess the correct value, x=30. In fact just verifying that 30 is correct is formidable enough!

Is making a table still a good idea for organizing information? I'll let you decide...

Present         10 yrs from now

J           x                          x+10

K           y                          y+10

So is there still a value to these "un-real" types of puzzle problems? Do you see the benefits that I do?

Tuesday, August 26, 2014

f(x)=(25--x^2)(16-x^2) Algebra 2 CCSS Challenge

f(x)=(25--x^2)(16-x^2)
P=product of x-int, Q=y-int of graph of f
|P| - |Q| =?
Generalize!

Want free ans/soln sent to your inbox? Just go to
http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

Back to School Challenge SAT/CCSS Problem

(100+99+98+...+51)-(1+2+3+...+50)=?

NO calculator/Noformulas!
Mental math - 30 seconds!

Are you getting your FREE answers and detailed solutions with strategies sent to your inbox?  For details go to http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

Think there's a catch here because there's no such thing as free? Why not ask the teachers who have been receiving these?

Sunday, August 24, 2014

Free CCSS/SAT/Challenge Math Problems/Solns sent to your inbox?

I'm considering a trial run depending on response. Many of these will be similar to the hundreds of Twitter Problems I've posted but more developed and with Answers/Solutions.

I will update my progress with this venture and let my readers here and on Twitter know if I will continue.

Since it's free at this juncture, I request that you

(2) Send me request via Blogger Contact Form with the following info:

(c) State & Name of school district in which you are teaching (or a student)
(d) Grade Level(s) or Math subjects currently teaching or indicate grade level/current math courses if a student. Also indicate if you're teaching LD/Reg/Honors/AP Levels
(e) Preferred math problem types (E.g/, Middle School, CCSS, Algebra, Geometry, SAT, Precalculus, Explorations, etc)
(f) Indicate if you already subscribe to my blog feed (& which feed you use) and if you already follow me on Twitter.

As you may know my problems are almost all conceptually based and require strong skill.

As always you are free to use these problems for personal use or teaching purposes but they cannot be reproduced for commercial use.

I have no idea what the response will be or whether I can maintain this on a daily or weekly basis. I may be able to customize it for your needs by category but no promises!

Friday, August 22, 2014

An Algebra Puzzle to Start The Year! Grades 6-10

As I posted on Twitter today (@dmarain)...

If I were to give you \$50 we'd have the same amount. If you were to give me \$50 I'd have 9 times as much as you. How much do we each have?

Submit your answer and solution via the Blogger Contact Form in right sidebar.

Reflections...

1. This was not intended to be highly challenging. It might engage students early in the year and I designed it to be accessible to most.

2. The language is open to interpretation by design! We want students to feel some disequilibrium. But we don't have to resolve ambiguities. I let my students do that among themselves.

3. Before jumping into an algebraic solution, I would allow my students to experiment with numbers - call it "plug in" or Guess-Test-Revise. After all on standardized tests this is what many will do in spite of all the algebra we teach!

Friday, August 15, 2014

Never ASS-U-ME in Geometry: A Triangle Problem to Get Them Thinking!

Not quite back to school for most but the problem above might prove interesting to review some geometric/deductive reasoning.

For new geometry students, replace 'a' by a value, say 40, and ask them to fill in all the missing angles. Most should deduce that angle 5 = 50, but my educated guess is that many will assume b = 40, so
angle 5= angle 6 = 50 and angle 3 = angle 4 = 40. From there to angle 1 = angle 2 = 50, so
angle 2 + angle 3 = 90. QED!  Not quite...

Well, the '90' is correct but the reasoning is another story! So this is all about justifying, checking validity of mathematical arguments, sorta' like some of the Eight Mathematical Practices of the Common Core!

In fact, you might ask them to redraw the diagram, keeping the given conditions but making it clear that b does not have to be 40 and that Angles 3&4 also do not have to be 40!

Monday, August 11, 2014

My Rant on MathShare 8 yrs ago

Haven't looked at the Yahoo group, MathShare I moderated several years ago. Phased it out when I started this blog.

Best way to describe what appears below? How about, " The more things change, the more they stay the same." Sad, but this could've been written today, except I'm retired...

From MathShare

Here's a novel thought...
I'd like to start a thread to explore what seems obvious to all of my teachers and probably to you and all of your colleagues as well. Not one of the teachers in my department teaches skills in a vacuum as in 'kill and drill.' They demand that students take notes and practice with many exercises until the concepts/skills are set in place and then come back to it in another context later on. They ask many questions in class to explore a topic, deepen conceptual understanding and assess: 'Show me why the absolute value of (x-1) is not x+1! How can you demonstrate this is not always true. Turn to your partner and convince each other!"
Technology enhances all of this but technology, standards, standardized tests and a plethora of new reports from various curriculum groups and governmental agencies WILL NEVER LEAD TO IMPROVEMENT of learning in the classroom and you all know why!
It's time to stop the political rhetoric and address what is really going on in the classroom. Effective teachers have always been effective. These are teachers who ravenously explore new instructional strategies, read everything they can get their hands on and then decide what will work best for their students. They are not mired in the the past nor are they easily swayed by buzz words or glitz. They are open to change but they will never abandon FIRST PRINCIPLES of learning. They will always be here after all the experts are long gone. ISN'T IT TIME WE CELEBRATE THEM AND USE THEM AS THE TRUE MODEL OF EDUCATION. THEY ARE THE GOLD STANDARD! Their students certainly know the truth: "She was the hardest and most demanding teacher I ever had. But boy did I learn!"

C'mon – don't be afraid to share 'self-evident' truths!! After all, someone has to tell the emperor he is naked! Ok, my rant is done!
Dave Marain

Tuesday, July 29, 2014

45 Free SAT MATH Problems Tweeted in 21 Days! Enjoy!

Pls remember that all of these are based on actual SAT questions but are completely original and therefore subject to the Creative Commons License in sidebar. Essentially, use them for your students or for yourserlf but no commercial reproduction. Thanks...

Perhaps my favorite is one I tweeted today:

How many combos of 2 pizza toppings can be selected from 10 choices?
Note: S,P is same as P,S; P,P is allowed

(A)100 (B)90 (C)55 (D)54 (E)45

Respond with answer and solution via Contact Form near top of sidebar.

Monday, July 21, 2014

THREE MORE SAT/COMMON CORE MATH PROBLEMS - FEED YOUR BRAIN!

See them on my twitter feed at

Did you submit your solution to the Geometry Puzzle from 7-20-14 via the Contact Form?

Sunday, July 20, 2014

Best Geometry Puzzle Ever?

Sorry. To see it, go to my twitter account:
Click on the image and magnify  or print as needed. If any angle measures are unclear let me know in Comments.
1. Submit answer and solution using Contact Form in right sidebar of my blog. PLS DO NOT SUBMIT SOLUTIONS IN COMMENTS!
3. Please include your full name, email, state/country, your connection to math (student,teacher,etc) and how you found my blog.

Sunday, July 13, 2014

Dozens of Free Common Core, SAT Practice Problems

I've been posting numerous challenge problems on Twitter of late, some with answers.
You can see these at twitter.com/dmarain.

Teachers: Feel free to use these to irritate your students.
Students: Feel free to use these to irritate your teachers. Lol

If you want to check your answers to these or you disagree with my answers or if you have any questions, use the new Contact Form in the sidebar.

Tuesday, June 3, 2014

And the winner is...

Our winning submission for the June 1st Challenge came from

KALEB LABBE

Kaleb is a freshman at Fort Kent Community High School in Fort Kent, Maine.

Congratulations Kaleb!

A)64
B)64, 16, 8, 4, 2

Sunday, June 1, 2014

First of June Quickie Math Challenge

Update-- One correct solution submitted thus far.
Reminder--
Deadline for submission is 6-2-14 Mon nite 9 pm (EDT).
This problem is appropriate for your Algebra 1/2 students.

For you or your Algebra students...
If a,b are positive integers greater than 1 and b^a=2^12 then
(a) what is the largest possible value of b?
(b) list all possible values of b
You or your students can respond with answers using the new Blogger Contact Form. Pls include names, school, state if applicable. I'll acknowledge all replies but only publish names of correct respondents who post by 9 pm (EDT) tomorrow 6-2. (up to the first million names!)

Wednesday, May 21, 2014

NEW Desmos Right Riemann Sum Calculus Investigation of Areas

UPDATED VERSION WITH BORDERS DRAWN BETWEEN RECTANGLES AND DETAILED EXPLANATIONS! The above is an extensive exploration of rectangular approximations to ares under a parabola utilizing the outstanding online graphing calculator, Desmos. CLICK on it to activate. In addition to a detailed overview and suggestions for the instructor, I've included a fleshed-out step-by-step activity for students which reflects my usual balance among the visual, symbolic, verbal and numerical- the Rule of Four... Please comment or use the new Contact Form in the sidebar to let me know if it's working properly and if you find it helpful. Your feedback is crucial to me. Use new contact form at top of right sidebar to contact me directly! If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Monday, May 12, 2014

Desmos Advanced Algebra Exploration -- Power Functions, Inverses and Graphs

NOTE: CLICKING ON THE GRAPH ABOVE SHOULD NOW LOAD THE DESMOS activity...

Hope you enjoy this new Common Core Investigation...

Students will examine the relationships among the graphs of y = kx^n, the inverse x = ky^n and their graphs. Beginning with particular values k=2, n=2 students will observe how the graphs are reflection images of each other over the line y=x. They will be asked to observe how the number of points of intersection vary over positive integer values of n, according to whether n is even or odd.

They will then determine the coordinates of these points first by estimating from the graphs, then by obtaining exact values using a system of equations.

Finally, they will use more advanced algebra tools to solve in terms of k and n. Some students will recognize the BIG IDEA that the points of intersection must lie on the graph of y=x, therefore the algebra is simplified by using y=x to replace x=ky^n when solving. This is crucial.

Thus, there is a blend of discovery and application of exponent skills. This is to me is the best use of technology - to enhance not replace instruction.

Desmos empowers the student to probe deep beneath the surface but the teacher must carefully plan and guide this process, otherwise many students will make pretty graphs and not get beyond moving sliders left and right. My opinion of course...

Desmos is a powerful teaching/learning tool because it enables students to discover important mathematical relationships and formulate key concepts for themselves. However, it is the expertise of the instructor which will determine WHAT they are learning. This is GUIDED self-discovery!

In this activity I included a detailed overview and guide for the instructor but I left it to the professional to tailor the investigation to the students and the curriculum. In other words, I did not provide a student worksheet. I encourage the professional to modify as he/she sees fit.

Your feedback is very important to me as I continue to develop these. Feel free to comment below or contact me directly using the new Blogger contact form. Also follow me on Twitter @dmarain.

Use new contact form at top of right sidebar to contact me directly!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Monday, May 5, 2014

Detailed investigation with extensive background notes for instructor and step by step outline for students to follow. Students will be asked to use a slider to approximate the position of a tangent line of slope -1 to a circle centered at (0,0). The tangent line, x+y=k, requires use of a parameter.

Students will begin with a particular radius, 3, then solve a linear-quadratic system to determine the exact equation of one of the tangent lines. They will also be asked to enter an expression for the other tangent line of slope -1 using the same parameter k. After approximating the locations of similar tangent lines for other radii, they will be asked to solve a general system using radius r.

There are different systems offered to the instructor, depending on the sophistication of the student. Finally, a geometric solution is suggested using 45-45-90 triangles.

Use new contact form at top of right sidebar to contact me directly!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Tuesday, April 29, 2014

An easier challenge?

NOTE: Don't forget to submit solutions and/or questions via the NEW Contact form at top of right sidebar...

And the winner is...
I'm happy to report that no one submitted an incorrect proof for my Challenge. Then again no one submitted anything! Disappointed but not surprised. Of course I've learned not to take this personally since it was a last minute announcement but I will leave the Challenge open for now for anyone who still wants to submit their proof or a counterexample to show I'm wrong!
A comment from anyone who tried the Challenge would be appreciated.  I saw the large number of views so it's hard to determine why no responses...
I don't give up easily so here's another Algebra Challenge which might also elicit zero responses!
Note: These are really designed for students...
1^3 + 2^3 < 3^3
2^3 + 3^3 < 4^3
Show algebraically that if a,b,c are consecutive positive integers with a<b<c,  then
a^3 + b^3 > c^3 for a> 5.

Monday, April 28, 2014

Deadline for Contest Submission is Midnight Tonight!!

Time's running out on the MathNotations Contest. I'm guessing one of 3 possibilities"

1) There's an error in the statement and you think I'm asking you to reprove Fermat's Last Theorem!
2) It's too easy to be bothered!
3) It's harder than I thought??

Remember I corrected it to positive integer values for n greater than 2.  Just a start here...

The first step is to show that c^3>a^3+b^3 and that may be a little sticky but I know many can do it

Use new contact form at top of right sidebar to contact me directly! If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Saturday, April 26, 2014

MathNotations Math Induction Challenge

Correction/Update: n is a positive integer

CHALLENGE
If a,b,c are the sides of a right triangle, c>a, c>b
prove by Mathematical Induction that
a^n + b^n < c^n, for n> 2

Submit detailed solution using the new contact form in the sidebar no later than Mon 5-28-14 11:59 pm EDT.

If your solution is too long for the contact form, indicate that that and I will contact you.

IMPORTANT NOTES:
1)Include your full name and state.
2)If you are a student, pls indicate grade level (or if in an Undergrad or Grad program).
3) If an educator,  pls indicate elementary,  middle, secondary or college.

The 3 best solutions will receive a free emailed copy of my Math Challenge Quiz Book and, of course,  recognition on this blog.

Note: If you feel there is an error in the problem or if the new contact form is not working, pls leave a comment.

Thursday, April 24, 2014

Parametric/Projectile Motion Simulated in Desmos - A Common Core Activity for Algebra/Precalculus

[Updated using folders to reduce amount of visible text. Click on the arrow next to the Folder icon to see the frames below. Thanks to Desmos team for this helpful hint!]

CLICK ON GRAPH TO ACTIVATE DESMOS...

The Desmos activity above is both an investigation of parametric representation and a tutorial for more advanced use of this remarkable WebApp. The The text in the side frames begins with a detailed background of the activity for the instructor and how Desmos can be used to demonstrate projectile motion using both parametric and rectangular coordinates. Some of the uses of slider 'variables' are demonstrated including animation, a powerful feature of Desmos.

In addition to showing how to use parameters in Desmos, the activity itself asks students to compare two different trajectories, representing an object dropped from some initial height, then a 2nd object two seconds later. The horizontal translation of the first graph is juxtaposed against the algebraic representations of these graphs using both system of coordinates.

The student activity starts about halfway down. There is a series of questions and actions the student needs to take in Desmos.

I'm hoping this will prove useful for both the instructor and the student.  Desmos is powerful but, in my opinion, some of the illustrative examples provided by Desmos do not flesh out the ideas behind the various uses of slider 'variables'. I'm hoping this will fill in some of those gaps.  I'm still a novice here so I'm sure more advanced users will be able to improve upon this...

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Sunday, April 20, 2014

A Common Core Alg Activity Using Desmos - Piecewise Linear Functions and Squares

This is my first attempt to use Desmos, the outstanding free online graphing calculator (and a free iPad app). I'm sure many of you have been utilizing this powerful resource. There are already many available teacher samples you can use.

CLICK ON THE GRAPH TO LOAD THE APP.

Let me know if you can view the graph (you may have to adjust the window slightly). More importantly, what do you think of the activity and do you see its potential for deepening understanding of algebra?

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Tuesday, April 15, 2014

Another Common Core Rant - not a video...

While I agree with the concerns of many parents that over-testing is damaging to children and subverts the purpose of education, I don't believe that the Common Core has set the bar too high, at least in math, my area of expertise. I know from direct experience with children that we can expect far more thinking of them than is commonly held. That is the Core Belief of my blog.

The problem is that teachers have not received the necessary preparation and the testing has been rushed and lacking in quality control.

We're trying to set the bar higher for children without raising the bar for those responsible for implementing these changes. That is irresponsible at best and criminal at worst.

Saturday, April 5, 2014

Developing Fraction Sense Using Egyptian Fractions

This MathCast describes a procedure for writing proper fractions as a sum of unit fractions, e.g., 5/6=1/2+1/3. This exploration is appropriate for Gr5 on up.A simplified form of the Greedy Algorithm.
Watch "Developing Fraction Sense Using Egyptian Fractions" on YouTube: http://youtu.be/ulb6E4OcQ8g
I'm experimenting with embedding these MathCasts directly into my blog post, however, the audio may not come out right. May work on your computer but not your tablets or smartphones. Please let me know. If there is a problem you can just click on the link to YouTube.

Sunday, March 30, 2014

A Data-Based Percent Problem for the New SAT and Common Core

The MathCast below is a more sophisticated % problem which requires processing several pieces of data expressed in percent form. Students have to make sense of the information and  develop a model to represent the data. Both a tree model and a table/spreadsheet approach are demonstrated.

These are the kinds of data-based applications which will become a central theme of the newer assessments from PARCC, SBAC and the College Board.

Sorry, you'll have to watch the video to see the problem but I'll give you a glimpse:

70% of the left-handed students surveyed were boys...

Watch "SAT/Common Core Data-based Percent Challenge" on YouTube - SAT/Common Core Data-based Percent Challenge: http://youtu.be/mhQN8imj-Pg

Also, pls let me know if these MathCasts are blurry.

Monday, March 24, 2014

40% of 9th graders play sports...Math Challenges for the Old/New SAT and the Common Core

David Coleman of the College Board has stated that ratios, percents, etc., will be an important focus of the new SAT, particularly in an applied setting.The MathCast below demonstrates a classic medium-hard percent word problem which should continue on all of our "new" assessments.

Watch "40% of the 9th grade... Percents,New SAT and…" on YouTube-40% of the 9th grade... Percents, the New SAT and…: http://youtu.be/WjNQ8VHhD4U

I encourage all my readers to offer constructive criticism re both the content and the technology of these videos and suggest topics/problems for future MathCasts.

Wednesday, March 19, 2014

SHOW ONE-THIRD OF THREE-FOURTHS-- TEACHING IN THE COMMON CORE

A hard adjustment for parents/educators who were raised as I was on traditional algorithms. Of course, understanding the rule for multiplying fractions DOES NOT mean students should use this to get the result.

Teaching conceptually is fundamental to the Mathematical Practice Standards of the Common Core. BUT this should be BALANCED with traditional methods. Current thinking is that the way we learned may be far more efficient but it leads one to think of math as just a set of meaningless rules. Kind of like a "black box." I see the need for both.

After watching this MathCast you'll probably conclude this is way too confusing for students not to mention adults! Just remember, some of that confusion results from sloppy drawings and carelessly worded explanations. I'm definitely guilty of both!

Sunday, March 16, 2014

5^(2x)=4; Find value of 5^(3x-1) Non-Calculator SAT-Type

These types of more challenging exponent problems requiring strong manipulative skills will probably not go away on the New SAT!
And on the NON-CALCULATOR section!

You have to watch the video to see the solution!

Watch "5^(2x)=4; Non-calculator exponent SAT-Type Problem" on YouTube - 5^(2x)=4; Non-calculator SAT-Type Problem: http://youtu.be/yaxx31L75hM

Saturday, March 8, 2014

4% of 8% of 16x is what % of 2x? New SAT, Common Core and Percents, Pt.1

A classic SAT math challenge that will probably not go away on the new SAT coming in 2016. And just imagine if calculators are not allowed for this!

Watch "4% of 8% of 16x is what % of 2x? New SAT, Common Core and Percents Pt 1" on YouTube - https://www.youtube.com/watch?v=WY58ebnZvCI&feature=youtube_gdata_player

Tuesday, March 4, 2014

Similar Figures MathCast - The Big Ideas and the Common Core

This 20 min video develops the Basic Proportionality Theorems re perimeters and areas of similar figures. Multiple representations and teaching conceptually are demonstrated with a strong emphasis on visualization.

Feedback would be helpful and appreciated...

Thursday, February 20, 2014

70a+19b=2838 SATs, Algebra, Number Sense and the Common Core

Click on link below to view video on YouTube. At this time I'm not embedding these videos in my blog.

Pls let me know if these screencasts are viewable on your desktop, laptop, tablet or phone. For now they have to be viewed in Portrait mode. The video quality is limited by my hardware and software at this time so the resolution is probably only fair.

Also any suggestions for improvement would be appreciated. If there's a particular problem or topic for which you'd like me to make a screencast let me know that as well.

Tuesday, February 18, 2014

Common Core SAT-type Quadratic Function Challenge Screencast

Tired of this yet? Well, the novelty of this screencast technology hasn't worn off yet for me!

One of my SAT students requested more practice with Quadratic Functions and coordinate problems so here it is...

Monday, February 17, 2014

Algebra and the Common Core - Teaching Abs Value Inequalities - Another Screencast

A screencast a day keeps everyone away!

Sunday, February 16, 2014

30-60-90 Video Tutorial and Challenge Problem

I'll soon tire of these screencasts but I'm having fun doing these for now!

Watch 30- 60-90 video uploaded today to my You Tube channel MathNotationsVids.

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

Saturday, February 15, 2014

Haven't posted in awhile and haven't uploaded videos to YouTube recently. I now have the technology to make short screencasts. Here's a link to the latest video I posted, a typical medium- level standardized test question relating to a non- convex quadrilateral. As aways these tutorials are intended for both students and educators.
Your feedback is important to me. You can support my efforts simply by subscribing to my channel. Thanks...

Wednesday, January 15, 2014

Cutting Corners -- The Square Transformed Into An Octagon Problem

A regular octagon is formed from a square by making 45° cuts from each corner.

(a) Draw a diagram or construct a model. See figure below.
(b) Since the octagon is "inscribed" in the square, its area is less than that of the square.
Explain using only "Euclidean" methods why the perimeter of the octagon is less than the perimeter of the square.

(c) If the perimeter of the square is 4 show that the perimeter of the octagon is 8(√2 - 1).

Note that the perimeter of the octagon is roughly 20% less than the perimeter of the square. Reasonable?

Saturday, January 4, 2014

Three congruent isosceles right triangles walked into a bar...

OVERVIEW
Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.

THE PROBLEM
Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.

(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.

REFLECTIONS
•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the  Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...

I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...