Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW

SEASONS GREETINGS!

While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...

The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.

The extension at the bottom goes beyond what might be expected from the beginning of this exploration.

Math educators can adapt this for Algebra 1 through AP Calculus students...

THE PROBLEMS

What are the number(s) described in the following?

1. A number equals its reciprocal.

2. A number equals 25% of its reciprocal.

3.  A number equals twice its reciprocal.

4.  A number equals the opposite of its reciprocal.

5.  A number equals k times it's reciprocal. Restrictions on k? Cases?

Answers:
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined

OVERVIEW and much more...

• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:

(*) New = (Old + k/Old)/2

Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]

Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!

Two overlapping circles of radius r... - A Common Core Geometry Problem

OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.

The objectives of the problem below include:

• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)

THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.

Answer: (Pi-2)/2
Note: Please verify!

REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]

• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?

The Myth of Developing Math Skills Without Effort and Practice - A Rant

Every research study I read reinforces my belief in the children's fable, "The Emperor's New Clothes".

Why is the truth about the need for practicing math skills so evident to everyone EXCEPT those who actually develop and implement education policies in this country, the so-called 'experts'? Is it arrogance, short-sightedness or simply a reflection of a society which has lost its way? Perish the thought that there could be a profit motive in promoting new approaches to learning math...

Think of your most "talented" students/children for a moment. They may think more quickly, display more insight, have greater abstract or spatial ability. But do they ever make mechanical arithmetic errors? No? Then they are truly the exception. Because that's not what I observe. I see a generation of youngsters who are now better at problem-solving yet lack proficiency with the, should I utter the word, BASICS. Why? You all know why!

THIS IS NOT A CALL TO BASICS. DON'T ANYONE MISINTERPRET THESE WORDS. IT IS THE SAME CALL I'VE ECHOED FOR DECADES. BALANCE CONCEPTUAL DEVELOPMENT/APPLICATION WITH SKILL/PROCEDURAL PRACTICE TO BUILD THE FOUNDATION REQUIRED FOR SUCCESS IN MATHEMATICS. BALANCE...BALANCE... BALANCE.

Of course we want students to inquire. Of course we want out students to use tools to analyze the vast amounts of data they now have at their disposal. Of course we want students to understand the WHY as well as the HOW. Of course we want to reach a variety of learning styles. Of course we want to use multiple representations in class. Of course we want mathematics to be interesting and useful.

BUT THERE IS NO SUBSTITUTE FOR PRACTICE. FOR EFFORT. FOR TIME. FOR APPLYING ONESELF. FOR SELF-DISCIPLINE. IN FACT, MATHEMATICS HELPS TO DEVELOP THESE QUALITIES!

So why isn't the obvious visible to the  researchers and policy makers? I'm sure they'll tell you...

Continuation of A Very Inconvenient Math Truth

Pay 2 of my response to Prof Willingham...
Here is the link to his original post:

http://www.danielwillingham.com/1/post/2013/12/what-the-ny-times-doesnt-know-about-math-instruction.html

My reply (unfortunately my reply was duplicated several times. My bad. ..)

(a) 9+9+9+9+9+9+9+9+9+9 vs
(b) 10+10+10+10+10+10+10+10+10
Is their equality a coincidence?

On your grid paper, make 10 rows of
* * * * * * * * *
Which addition problem does this represent?

Should how you could represent the other addition problem without drawing any more stars. [Rotate paper 90°]

Now write both as multiplication sentences...

We can pontificate about all of this ad nauseam but in the end teachers have to be trained to provide an environment which BLENDS explicit and implicit instruction. I learned much about arithmetic and number sense from playing Monopoly but I didn't learn everything that way! Some concepts/skills/procedures had to be clearly demonstrated to me. I was observing my precocious 6-yr old grandson learning to play Monopoly. From playing a couple of times he decided to buy every property he landed on. When he ran out of money I told him he'd have to wait until her could collect $200. "No problem PopPop. Just let me be the banker!" Will he improve his understanding of the game without formal instruction? Of course. Will he also develop some misconceptions if not corrected and given a clear explicit explanation? Of course. INFORMAL LEARNING CAN GO ONLY SO FAR IN MATHEMATICS. This must be balanced with the child developing proficiency with skills/algorithms, attention to detail and recognizing the appropriateness of approximate vs exact results. I'm only scratching the surface here. But I do know that none of this happens by accident. CCSS are necessary to raise the bar but without the"heavy lifting" required to train/prepare teachers, it will be futile. But nothing substantive will occur until the education of our children is genuinely seen as an investment instead of an expense. When we truly put our money where our mouths are...

Another Very Inconvenient Truth About Math Education

I just responded to Daniel below. He was commenting to a NYT editorial on math education. I've included some of his response.

Daniel Willingham
What the NY Times Doesn't Know About Math Instruction
12/09/20131 Comment

"A New York Times editorial on December 6 called for improved math instruction, calling the current system "broken." Although I agree we could be doing a better job of teaching math, the suggestions in the editorial showed a striking naiveté about what it will take to improve."

Now for my reply...

COMMENTS

Dave Marainlink12/09/2013 8:38am

Some excellent points made here but sadly one could read editorials and letters on this topic from every decade for the past 40 years and I have and little has changed. Experienced math educators have forever recognized the problems identified here but what substantive change has occurred other than more testing and expecting more accountability from teachers who are expected to change water into wine.

Helping young children develop conceptual understanding of numbers, operations, relationships (spatial as well) requires specialized training that is not currently the norm in teacher preparation. We are very good at appointing commissions to draft world class standards and creating more ambitious testing but not very good at providing prospective and current teachers with the training necessary to implement these ambitious changes. THERE ARE NO SHORTCUTS HERE. It requires a sea change in teacher preparation and an investment of time and money that no one up to now has been willing to make. The money is out there to make this happen but saying we want to be the best is very different from preparing to be the best. There will always be fads and theories about how to improve the education of our children. But it's not rocket science to figure out that changing standards and assessments is putting the cart before the horse. Teaching children HOW to think isn't easy but it is doable. A young NFL player who happens to have majored in math was asked how math could help him with football. He replied, "It's all about problem-solving." Perhaps we should be listening more to young people like this...

The square root of x+1 equals x+1... A Common Core Investigation

OVERVIEW
Fairly straightforward radical equation in the title but there is so much hidden potential here for students in Alg 2/Precalculus.

REFLECTIONS
• The solutions to the equation above are -1 and 0. No big deal, right? The usual algorithm --- just square both sides and solve the resulting quadratic by any one of several methods. Done. Cheerio. But wait...

• We can encourage students to "make it simpler" by substituting 'a' for x+1 obtaining a^(1/2)=a, square both sides yielding a=a^2 which gives 2 easy solutions 0,1 and then x+1=0,1 producing the final result. Not that big a deal though except...

• A graphical interpretation of these equations is illuminating and illustrates multiple representations/The Rule of 4. You could demo this with the graphing calculator displayed on your smart board or have the students graph by hand or on their device. The graphs of y=x^(1/2) and y=x intersect at x=0 and x=1 then, by translation, the graphs of y=(x+1)^(1/2) and y=x+1 will intersect at x=-1 and x=0. Students should be asked for this conclusion BEFORE checking the graphs to verify!

• Is that all there is? Hardly! The current trend on assessments and hopefully in texts is to have students analyze a family of equations using a parameter. But first we can generalize numerically:
Solve
(i) (x+4)^(1/2)=x+2
(ii) (x+9)^(1/2)=x+3

Are there still 2 solutions for each of these? Solving just a couple of these and recognizing extraneous or apparent solutions would traditionally have been the WHOLE lesson! Not any more...

By the way -  why the "4" and "9"? Did I change the pattern from the original equation?

• Now for the parametric form:

(x+k^2)^0.5=x+k

What questions should STUDENTS be asking themselves BEFORE WE ASK THEM?

• Students can certainly be asked to solve the latter equation for x in terms of k. Some will struggle with the procedure/algorithm. Hopefully someone in each group (or the whole class!) will obtain x=0 and x=1-2k. BUT WILL THEY CHECK THE 2nd SOLUTION! The use of a  parameter goes beyond making a better standardized test question. Now the student has to recognize that, in order for there to be 2 solutions, k must be less than or equal to 1 which was suggested by the numerical examples above.

• Of course I'm anticipating most teachers' reactions to an exploration like this. I've provided much more than can reasonably fit in a 40 min lesson. Use it as you see fit or just ignore it. It will go away or will it? 

Turkey Day stuffing recipe...Have to double 3/4 cup...Three coins...

Happy Thanksgiving and Happy Hanukkah to all out there in cyberspace! The confluence of these two holidays provides plenty of grist for a math blogger but I'm not going there today...

Silly title, perhaps, but to all the experts who will tell us that money is the root of all evil when teaching fraction concepts/skills, uh whatever...

Are there any adults out there who don't want to mess up that Turkey Day recipe and will ask the 'math expert' in the family to verify what doubling three-fourths is?

Three-fourths cup...three quarters...75 cents...double...$1.50...1.5...1 1/2 cups

But, Dave, this won't work for multiplying 5/7 by 2 so this is worthless...

To quote a certain ad,
"Hey, if it works, it's not crazy!"

Seriously, from my perspective, money is as "real" as it gets. A child can be guided to discover the algorithm for multiplying a fraction by a whole number in an endless variety of ways ---
Concrete objects
Fraction pieces and all those other manipulatives
Bar models
Pictures
Simple numerical patterns

But I wouldn't be afraid to use money! Imagine making connections...

How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.

THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b)  algebraically

2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).

3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.

4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.

Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?

4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.

How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?

REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:

The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.

Note: In 3 dim, we can replace 'area' by what?

• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!

• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...

'.

Today's MathNotations # is 972...

OVERVIEW AND THE PROBLEM
As posted on Twitter (@dmarain), student's task is to list as many interesting properties/facts about 972 as possible, at least 10.  I posted a couple of possibilities but you may not want to give both of these away...

1) Prod of a perfect sq and a perf cube
2) a perf 5th power × a perf sq
...

REFLECTIONS
• Appropriate grade level? 3rd? 4th? Middle grades?
• 10+ properties way too much? Why do you think I chose that number? Depends on grade level? Abilities?
• What do you see as the principal
learning goals/benefits here? How highly would you rank "Expressing mathematical ideas/number properties in words?"
• We might naturally have students work in teams but I tended to respect that individual student who chose to work alone. Would you make this competitive to stimulate interest or that's not desirable?
• Beyond making a list of properties, what else should be expected here? How important is it for students to check their results? How about independent verification from another team member?

• Allow use of calculator throughout or after a little while?
• How is this different from MAA's "Today's Number is"?
• Anyone recall the last number I asked about a long time ago? [97 and 153!] You may want to click on the link...

Six parking spots were assigned randomly. What is the probability that...

OVERVIEW
No matter how many of these appear on standardized tests, a large per cent of test takers continue to get these wrong. Teachers will teach that unit on combinations, permutations, Multiplication Principle and rules of probability. But learners will still struggle and even if they survive the chapter test, they will probably get the following problem wrong. But some of you may have overcome this...

THE PROBLEM

Six parking spots are assigned randomly to six employees. If Jake and Alex are 2 of these employees, what is the probability they will be assigned to the first 2 spaces?
(A) 1/60 (B) 1/30 (C) 1/15 (D) 1/6 (E) 1/3

Answer: (C)

REFLECTIONS

• It's so simple: (2/6) (1/5) = 1/15  Next...
Of course that would never happen in a classroom!
How much understanding of probability principles and practice is required to feel comfortable with this efficient approach? More importantly when would this approach fail or need to be revised?

• So is it combinations? Permutations! Do we also need the multiplication principle?
How much experience do students need before being able to write
(2P2) (4P4) ÷ (6P6)

• Those who have experience teaching these will have developed their favorite instructional strategies. Please share!

"LEAST number to select to be CERTAIN of getting at LEAST..." -- Those annoying logic problems!

OVERVIEW

You've seen these on SATs, GREs, etc..
Where exactly does this fit into the Common Core?

Should the underlying Pigeonhole Principle and/or specific strategies/methods be taught to middle or high schoolers?

You can guess my feelings about this...

THE PROBLEM
There are thirty solid-colored scarves in 30 identical unmarked closed boxes, one per box. The 30 scarves consist of:
6 red
7 blue
8 yellow
9 green

What is the LEAST number of boxes that need to be opened to be CERTAIN of getting at LEAST

(A) 2 of the same color [5]
(B) 4 of the same color [9]
(C) one of each color [25]
(D) 4 of each color [28]
(E) 2 green and 2 blue [25]

REFLECTIONS

• Oh those nasty but critical keywords which I wrote in uppercase. Initial instruction must focus on developing meaning for these.

• Those who have had experience teaching this may want to share their strategies. Here are some I've used...

* Act it out with coins in a bag or any concrete objects. Let's say the goal is to get at least one of each color and the student or you selects 4 boxes or objects. I might say, "Do you want to stop? This is surely the LEAST number!" Were hoping students will respond with, "But we can't be sure!"
OR I might grab all of the boxes and say, "Ok, now we are CERTAIN!" Hopefully someone will respond...

* Without getting into the Pigeon Hole Principle I have used what I call the WORST POSSIBLE LUCK (CASE) approach. For example, if the goal is to get one of each color the WORST CASE would be getting the same color repeatedly. Don't like that approach? This is subtle and requires patience, time and several examples. Some learners will grasp it way before others but acting it out with objects and making a game out of it really has made a difference for my students.

Your thoughts?

When Mom was 40, Son was twice as old as Daughter... Age Problems and Singapore Bar Models

THE PROBLEM
When mom was 40, her son was twice as old as her daughter. Now her daughter is 28 and mom is twice as old as her son. How old is mom now?

Answer: 64

REFLECTIONS
Not exactly an authentic real world assessment but there's probably a reason why these are "Problems for the Ages"!! I don't believe harm is done by having students develop these kinds of relationships...

Of course for years I used to teach this using a chart and setting it up algebraically.

For younger students I would encourage a Guess-Test-Revise approach. Say, girls was 3 so boy was 6. Now it's 28-3 = 25 years later, so boy would be 6+25=31 and mom would be 40+25=65 which is a little more than twice 31. Revise to girl was 4, son was 8. 24 years later, girl would be 28, boy would be 32 and mom would be 40+24=64, Bingo!

Ah but children in Singapore perhaps as early as Grade 4 are representing these relationships using unit lengths and bar models. I simply don't have enough experience doing this so those out there with more knowledge please improve upon this so I can learn!

Then
Mom |-----40------|
Son |----||----|
Daughter |----| (some unit length)

Note: [//////] below represents the additional years from Then to Now

Now
Daughter |----| [//////] = [--28--]
Son |----||----| [//////]
Mom |----40---| [//////] = |----||----| [//////] |----||----| [//////] = |----||----||----| [//////] [--28--]
So 40 = 3u + 28 or u = 4 yrs, etc.

I'm limited by my graphics but I believe there are different,  better and more efficient models used in Singapore Math. If you're thinking I retrofitted an algebraic solution to make it look like a bar model you're probably right! I need wiser and more experienced 'bar modelers' to help me!

Mean Equals Median Problem

THE PROBLEM

5, 11, 19, 22, x
If the mean and median of these 5 numbers are equal, determine all possible values of x.

ANSWER
-2, 14.25, 38

REFLECTIONS

• If the question asked for one possible value for x, this could be an SAT Grid-In question of above-average difficulty. I chose to ask for ALL possible values not only to make it more challenging but also to encourage students to probe more deeply.

• Thinking of many strategies how YOU would solve this? But that's not the focus of this blog. After WE figure out how to solve the problem ourselves and, yes, we should try to find more than one method, we should be asking ourselves as educators:

What questions need to be asked to enable our students to solve it for themselves?

What learning environment best facilitates this?

• Do you believe some of your students would reason that the median could only be 11, 19, or x itself? Listen to their discussions. Give them time but if no one is thinking that way we might ask:

Which numbers could NOT be the median? Why can't 5 be the median? 22?

• Beyond solving this problem is an even bigger question:
What must be true about a set of data values if their mean (avg) equals their median? We know there are sufficient conditions like:
The data is normally distributed about the mean, in which case, the mean, median and mode coincide.
But this is NOT NECESSARY!

In the above example, you might want to ask students to examine the case where x=14.25:

"What is the mean of the numbers if x is NOT in the set? Why does this make sense?

[If you have an 80 average and one of your scores was exactly 80, what would happen to the mean if you removed that score?]

So what does it 'mean' if the mean and median of a set of data are equal?

Now that's an open-ended assessment question!

'Half as many as' vs 'Twice as many as' Revisited

OVERVIEW
Hundreds of math posts over the last 6+ years and my most popular post is still by far "There are twice as many girls as boys...".
http://mathnotations.blogspot.com/2008/08/there-are-twice-as-many-girls-as-boys.html

Still hard to do word problems without coping with the vagaries of language and grammar in particular.

Contrast phrases such as

'Half OF the girls' and 'Half AS MANY Girls as' OR

'Twice as many X's as Y's'

These are confusing enough for native English speakers. Imagine the terror if you're not! Phrases like these don't respect anyone!

THE PROBLEM

Attendance at a stadium broke down as follows...
Of the children, half as many girls as boys
Of the adults, half as many women as men
The number of females was what fractional part of the total attendance?

(A) 1/2  (B) 1/3  (C) 1/4  (D) 1/6  (E) cannot be determined from info given

Answer: B

REFLECTIONS

1. Although the title of the post suggests the focus was on interpreting the phrase "as many as", there are some significant underlying ratio concepts here.

2.  As I discussed in my earlier post, I've observed that students do better with the semantics if they first decide which is the larger of two quantities. Thus "half as many girls as boys" hopefully suggests that there are fewer girls but my experience is that some are simply blocked by the sentence structure and will guess randomly or say nothing. Remember we can always go back to concrete numerical relationships:

"Ok say there are 12 boys.
If there are half as many girls, then how many girls?"

Note how I not only used numbers but I also inverted the problem by giving the number of boys first. Yes we are also teachers of reading! Will students do this on their own? If trained!!

3. From this point there are many solution paths and I would definitely allow upper  elementary or middle school students to play with this for a few minutes in their groups. There's no rushing this process.

We can simply explain our method to them but this is only a part of their learning. Of course that's my opinion and there are many out there who would cringe at this. The eternal battle between The Direct Instructionists and the Constructivists! Those are just labels about which I care little. Whatever works... Since I can speak from 40 yrs of experience I know what worked for my students...
Besides I've already debated this ad infinitum and ad nauseam with some of the best. No one ever changes their mind!

4.  There are some subtle part:whole ratio concepts embedded here. Isn't it tempting to pick choice (E) here because we're not given the Adult:Child ratio. But the question asked for the ratio of the combined female to the total. It is very instructive to see this algebraically.

5. 'Plugging in' convenient numbers for the subgroups in this problem makes it accessible to 4th-5th graders. Organization is very helpful. I use a tree model to represent this kind of data but most do not do this.  Say there are 2 girls, 4 boys; 5 women, 10 men. Then the number of females (7) is still one-third of the total (21)!

6. Hopefully my readers will suggest more efficient ratio methods, Singapore models, other algebraic representations, etc... OR no comments at all!

Divide 3 Pizzas Equally Among 4 Children -- Developing Division Series Common Core

OVERVIEW

Note: You may want to first read #7 below in Reflections to clarify the suggested approach.

Some of you may be averse to using pizzas for concept development of division, fractions, ratios,... I've seen criticism of pizzas for awhile!

Similarly some educators don't like using money to develop fractions and decimals.

Here's my philosophy --
Whatever works...
I don't believe children's minds will be permanently damaged from pizzas and money!

THE PROBLEM
[Appropriate Grade Levels: 2+]

Directions to groups: Show at least 4 ways to share the 3 paper pizzas as described in the title.

Then write in words the part of one whole pizza each child gets.

REFLECTIONS...
1) Anything missing here in building division/fraction concepts?
I believe so!

BEFORE THE CHILDREN START SHARING PIECES,
WHAT QUESTION COULD WE ASK TO BUILD FRACTION SENSE, ESTIMATION, etc.,??

One possibility...
Teacher: Before you start drawing lines, choose from one of these:
(Write your choice on your paper in the next 5 secs)
(A) Each child will get more than one whole pie.
(B) Each child will get less than one whole pie.
OK, now defend your answer to your partners in the next minute.
Anyone change their first answer?
So the majority chose (B). Why?

2) Do you believe for young children it is important to give them the time for these kinds of activities?

3) What and how many prior division experiences do you think children need before tackling 3 divided by 4? Do you believe some children are ready from the beginning?

4) I always find an original approach from listening to children. If you choose to use this activity, what 2-3 methods do you think most children would use?

Do you think a few children will immediately divide each pie into quarters? What might they say if you ask them why they did that?

5) How would you enable the TRANSFER of learning from concrete manipulatives to WORDS (verbal representation) to the SYMBOLIC?
Sample...
So you have been dividing 3 pies equally among 4 children.
Let's say that together...
Now let's write 3 pies ÷ 4 children =
Three-_______ of ------ for ----- child =
3/4 pie/child

There's so much more to this. These are just a couple of suggestions which you could modify and improve upon!

6) How would you assess their understanding during the lesson? At the end of the lesson? The next day? On a written assessment or would you consider an individual performance assessment (Show me how you would...)
How do you think PARCC would assess the important ideas here?

7) Of course many of you are thinking:

Why not simply show them 3×8÷4=6 since most pizzas are cut into 8 slices? 

Well, I'm suggesting we introduce this to 7-8 yr olds before they learn multiplication and division. Secondly, the children are given BLANK circles to encourage creative open-ended approaches. I want them to experiment with dividing the circles into halves, quarters, eighths, etc...

Remember, as always, I'm writing these for my fellow educators to reflect on their practice, now in the context of the Mathematical Practices of the Common Core.

Been doing this now for almost 7 years on this blog and invariably these kinds of posts are viewed but not commented upon. Why do you think that is? Obviously a rhetorical question!

TEACHING DIVISION: I said to make 2 equal groups not groups of 2!

Ever experience this in the classroom working with elementary/middle school students, particularly with LLD students?  I believe to some degree most of us have some language processing issues, often connected to auditory processing issues and complicated by attentional deficits.

I recently observed this with a few students and my thought was that there is a developmental stage which precedes this question. Next time you're teaching division and you encounter this you might want to step back and assess the child's understanding of
NUMBER OF GROUPS vs
NUMBER IN EACH GROUP

A misunderstanding here could be a barrier to conceptual development of multiplication and division.

One strategy is to show the child 4 groups of 3 counters either with objects (preferably) or on paper. Keep the groups separated. On paper one could simply loop them.  Physically, you could put the counters in separate containers or put some kind of ring around them.

"So, how many groups do you see?
How many in each group?"

Most should get this but, if not, you know there's a  language barrier which must be addressed.

Now what? Ready for division questions? Not quite...
Have them now CONSTRUCT groups physically, then on paper.
For example have them "make" 3 groups of 4 from the original setup.

Some youngsters need considerable practice and reinforcement.

What do you think? Have you had similar experiences? Found another way to cope? A different theory about the underlying problem? Remember I'm just sharing my observations and conclusions. They're just mine...

International call costs x¢ per min for 1st n min, then y¢ for...

I can't believe I'm actually posting again. This won't last!

OVERVIEW

Developing abstract reasoning needs to start early and often and it's founded on a strong foundation of arithmetic/quantitative reasoning. That is, children normally learn to generalize from several concrete numerical examples before patterning takes place. Seems too obvious? Well, at what point would you expect a majority of your algebra students to do this successfully?

THE PROBLEM

Phone carrier charges for an international call are x¢ per minute for first n minutes, then y¢ per min for each additional minute or part thereof. Write an expression for the cost, in dollars, of a call lasting z minutes, z>n. Assume n,z are positive integers.

REFLECTIONS
Appropriate level of difficulty for Algebra 1? Algebra 2? Only for the 'honors' kids?

Too much expense of time to get students to be able to do this? Not worth the effort?

How many of this type are your algebra students normally exposed to? From the text? From teacher-constructed materials?

How many of these would most students need to practice to be proficient?

What % of your 'average' middle schoolers could solve this quantitatively, i.e., if all variables were replaced by numbers?

What are some of your favorite instructional strategies for these kinds of 'literal' word problems? By the way, this is the primary reason for my posting this!

What do you see as the main challenges to student performance on these kinds of assessment questions? Do you place 'understanding the question' high on this list?

Which of the 8 Common Core Mathematical Practices come into play here? By the way, do you have your own 'laminated' copy of these practices in front of you at all times! Here's the link:
http://www.corestandards.org/Math/Practice

Sorry, the 'answer' will have to come from someone who comments! I always assumed one of my students or a group would solve the problem...

Happy (31)(13) Day! No tricks here!

Wow, am I actually publishing a blog post for the first time in a long time? It took Halloween to motivate me! Of course by now I'm sure I've lost my readership but if you happen to come across this...

As posted on twitter. com/dmarain a few minutes ago, here is a challenge for your 4th graders and above...

We may choose to call 13 & 31 a '2-digit distinct prime reversal pair'. (This means I invented that name because it's my blog!)

The challenge is to list the other 3 such pairs in 31 sec or less! Of course you could argue for a 4th pair...

And, yes, I realize this is not a brand-new original problem. In fact I published a question like this before.

REFLECTIONS...
Do you think there is there any benefit to a problem like this?
Could a variation appear on a standardized test or SAT/PSAT?
Instead of racking your brain to invent more like these for your students why not ask each group of students to come up with a couple to stump their classmates! You could also look up 'primes' in my topics list in the sidebar and you'll find a few more...

NOTE: I don't usually post answers in my blog figuring that one of my commenters will do so.

Free Cup of Java! PerCent Word Problems, Strategies, Common Core...

So on Twitter.com/dmarain this morning I posted the following (I modified it slightly):

"Java Coffee Co." gives you a free cup after you purchase 12.  The % discount is 7.7% rounded.
EXPLAIN!

Common Core? 4th grade? 5th grade? Singapore Math? Blended learning? Flipped classrooms? Authentic assessments?

If you have difficulty viewing this video, watch it on YouTube on my channel MathNotationsVids. Here is the link. 

Wonderful concise conceptual explanation from @MrLeiss on Twitter this morning. This is what sharing of ideas is all about!
See it on my account at

twitter.com/dmarain

So where's the formula for % discount? % change in general? Why is it NOT in that video? Uh, that's rhetorical...


As I explained in my video --
THIS IS NOT INTENDED TO BE PRESCRIPTIVE OR SCRIPTED!
It is just one old retired math teacher's way of keeping the dialog going and we all know that sharing of ideas is the only way our students will understand math better. We do believe that, yes??

SO SHARE YOUR THOUGHTS BUT PLS REMEMBER THE RAISON D'ETRE OF THIS BLOG. THE PROBLEM AND THE "ANSWER" IS FAR LESS IMPORTANT THAN THE METHODS WE USE TO DEVELOP CONCEPTUAL UNDERSTANDING.



My Core Beliefs

1.  THERE IS NO ONE BEST WAY TO TEACH THIS!
2.  BELIEF #1, NOTWITHSTANDING, SOME WAYS ARE BETTER THAN OTHERS.
3. IT IS NEVER TOO EARLY IN A CHILD'S DEVELOPMENT TO INTRODUCE MATH CONCEPTS. IT'S JUST A MATTER OF HOW WE SAY IT AND HOW WE SHOW IT AND HOW WE ASK THE QUESTIONS AND WHAT ACTIONS WE ASK THEM TO TAKE!
4. CONCRETE (PHYSICAL OBJECTS) TO SYMBOLIC TO ABSTRACT ALWAYS MAKES SENSE REGARDLESS OF STUDENT DEVELOPMENT. 




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.

Where have all the problems gone...

I reminded my faithful that, after posting numerous times for 2 months, I would crash and burn! But I hope some of you are following me at twitter.com/dmarain.

I have been tweeting many SAT practice problems under my trademark ®SATMATH 800++++.

For example, here's my latest...

What is the probability that a number chosen at random from the first ten positive odd integers is prime?
[45 seconds...]

Here's another not yet tweeted...

In how many ways can 36 be written as the sum of 2 primes, p and q, p ≤ q?

These are NOT of a high order of item difficulty. They are intended to provide practice for this category of arithmetic problems on SATs and other standardized tests, such as the upcoming PARCC assessments.

Just as importantly, as in ALL problems I compose, they are intended to be used as discussion points in class to review fundamental ideas and help students improve their READING COMPREHENSION of math words/phrases.

AS ANY MATH EDUCATOR WILL READILY ACKNOWLEDGE:

LACK OF KNOWLEDGE OF KEY MATH TERMS AND LANGUAGE ISSUES IN GENERAL ARE MAJOR FACTORS IN STUDENTS NOT PERFORMING UP TO THEIR POTENTIAL.

INSCRIBING RECTANGLES IN AN EQUILATERAL TRIANGLE - A COMMON CORE INVESTIGATION

I'll let the video speak for itself...

I would really appreciate dialog here, focusing more on instructional methods -- balanced vs blended, conceptual development AND procedural understanding, etc.
Hope this helpful to someone...

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.

A brief respite...

I will not be posting for the next few days as my family and I observe the one year passing of my wife. Thank you for your understanding.

An SAT quiz to sharpen your brain for March 9

Click on the image to enlarge. Good luck trying to read my scrawl!
This is one I wrote from the 20th century! Feel free to use with your students but observe the copyright please.

No answers yet but you can share your thoughts...



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.

So is 75 the avg of the pos integers from 50 to 100 inclusive?

This very common type of question appears so straightforward. So why do variations of it recur so often on SATs and other standardized tests and math contests?

Why not test it out with your students and ask them to explain their reasoning. I am still surprised by the creativity of our students when given the opportunity to display it!

Again, my boring disclaimer...
This is not a conundrum for the math problem-solvers out there. It is intended as a discussion point for helping students develop some important ideas in mathematics.






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.

A RADICAL DEPARTURE - AN ALGEBRA 2 /CCSSM/MATH 2/MATH CLUB CHALLENGE

A Radical Departure...

(Inspired by Ramanujan and an excellent Wikipedia article on Nested Radicals)

Suggestion: Assign as a 2-day team or individual project after demonstrating a similar but simpler example such as the square root of 3+2√2 = 1+√2.

NOTE: The method below DOES NOT show a detailed algebraic solution, using substitutions and solution of resulting quadratic equations.  Rather, I suggested some reasonable educated guessing,  aka number sense. I would recommend both approaches. 

There is considerable more theory than is suggested by this example, e.g., justification of uniqueness of roots, conditions for roots to be of the form suggested in the solution, etc.  Encourage students to investigate further! 

PROBLEM: Demonstrate the following identity by simplification of the left-hand side only. No calculators permitted for derivation although numerical (decimal) verification that the left side equals the right is recommended prior to starting the 'proof'.

(SOLUTION GIVEN BELOW STATEMENT OF IDENTITY)

NOTE: Illegibility of next to last line of 3rd image!  Should be (Square root of 3 + Square of 2) not 'Square root of 4'.


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.

SAT/CCSSM: How many 3-digit positive integers satisfy...

Disclaimer/Reminder
Don't forget to comply with the Creative Commons License in sidebar. Thanks!

An acronym I just thought of for improving your students' performance on SATs or other standardized tests:


S: SPEED
A:ACCURACY
T:TERMINOLOGY

(For training purposes only)
TIME LIMIT: 45 sec
NO CALCULATOR

HOW MANY 3-DIGIT POSITIVE INTEGERS SATISFY BOTH OF THE FOLLOWING CONDITIONS?

• THE PRODUCT OF THE DIGITS IS 72
• THE 3-DIGIT INTEGER IS A PALINDROME (an integer that is the same when its digits are reversed)

Let me know how many of your students can do this within the time limit and no calculator.  And for those who could not? Guess that means they need more of these to practice! Why not ask each student to write a similar problem for hw! They learn more from writing their own and we give up control --- perfect!!


Answer: Read below shameless ad for my book...


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.

Answer (send yours if you believe I erred!):
2 (namely 383,626)

The Quintessential SAT Problem: If h hens eat p pounds of feed a day...

If one was to categorize every SAT question from the very first SAT ever published, I believe we would find the following type of algebraic ratio problem one of the most common type. Even with all the exposure students now have to SAT problems, my direct experience is that many students still struggle with these types of questions.
WHY?

More importantly, are these types of problems important enough in the CCSSM to justify the time investment to introduce them in middle school and reinforce in secondary algebra classes? IMO, ABSOLUTELY!

If h hens consume a total of p pounds of feed per day, then, at this rate, how many pounds of feed would c hens consume in x days?

Not only was a similar question the recent SAT Question of the Day on the College Board web  site, the statistics were also published:
35620 responded (up to the time I checked)
31% correct
So, about 7 out of 10 students attempting this question online got it wrong.

Note: The actual question was followed by 5 choices, allowing students to plug in numbers and test each choice, but I chose to focus on the question here rather than test-taking strategies.

IMO, the College Board hires highly competent math people who write succinct, accurate and helpful online solutions but this only scratches the surface. It only suggests one particular approach and has little to do with Instructional Strategies and the various ways children develop these important ideas.

REFLECTIONS...

1.  Where are ratio concepts introduced for the first time in the CCSSM? K? 1st? 4th 5th?

2. By your own estimate,  how many of these kinds of questions appear as sample problems or homework exercises in your elementary/prealgebra/algebra texts?

3.  Do you believe ALL your students receive adequate exposure to and review of these?

4. Would you be willing to share some of your favorite methods of laying the groundwork for and developing the skills and concepts needed for your students to be successful with ratio problems and ultimately algebraic types? If I take a risk, would you?

Putting myself out there...

The simplest and most instinctive approach usually makes the most sense, doesn't it? We know how we learn best and the same is true of all students.  Do you accept the following as a truism, an essential tenet of teaching and learning mathematics?

EVERYONE LEARNS BETTER WHEN PRESENTED WITH CONCRETE NUMERICAL RELATIONSHIPS BEFORE TACKLING ABSTRACTIONS. FURTHER, THE COMPLEXITY OF LANGUAGE SHOULD BE GRADUALLY INCREASED, STARTING WITH THE MOST ACCESSIBLE INFORMAL PHRASES.

For example,

If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would one hen eat in one day?

When first introduced, should our focus be on which operation to perform? In my view, our goal should be to develop number sense, in this case, ratio sense. 

We all know that a powerful construct for developing ratio/proportion sense is the idea of first reducing the information to a UNIT.
Many of us were taught this way and most children tend to think like this at first.

Scaffolding...
If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would nine hens eat in one day?

Working from one hen consumes 2 pounds per day, the child can usually move on to 9 hens eat 9x2 or 18 pounds per day.

Two points here...

First, I believe it is important to routinely use a variety of equivalent phrases:
"in one day" vs. "each day" vs. "per day."

Secondly, I would encourage students who can reason proportionally to share this with the group:

"Well, if 6 hens eat 12 pounds, then 3 hens will eat half as much or 6 pounds, so 9 hens will eat 12+6 or 18 pounds."

Teaching conceptually means NOT SETTING UP A PROPORTION initially. Procedures and algorithms turn off the child's sense-making and stifle intuition and number sense. You can fight me on this all you want, folks, but you will not win here on my blog!


So when do we introduce proportion problems involving variables and what are some good ways to solve the original problem??  I'll allow my readers to figure that out for themselves...




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.

What is the smallest positive odd integer which has exactly 10 factors?

PLS NOTE CORRECTION TO THE ANSWER TO THE PROBLEM. MY ERROR WAS CAUGHT BY NOVOTNY!!

Occasionally I like to respond to the topics in the Google searches which bring my readers to MathNotations.

Today's problem in the title of this post
(Humorous speech-to-text aside: "this post" was interpreted" as "disposed")
is a classic math challenge question, difficult SAT- or CCSSM-type question which is appropriate for grades 5-11.


What is the smallest positive odd integer which has exactly 10 factors?
Explain your method.


Answer at bottom of post after shameless promotion...
(Aside: Wouldn't Shameless be a a cool title for a premium cable TV show about a deadbeat dad mathematician!)

REFLECTIONS for my colleagues...
(Wouldn't it be awesome if someone actually read these!)

1.  How many of these types of questions have you seen in textbooks, math contests, SAT's, standardized tests or on other blogs?

2. How different would this question be if the word "odd" were removed? An easier or harder question in your opinion?
(Humorous speech-to-text aside: "the word odd" was interpreted as "The Word of God")

3. Would you like to share some math strategies you have used for this type of problem? Are there instructional strategies you prefer for this? Do you see these 2 questions as equivalent?

4.  Do we have to be the ones to devise variations on this?
My feeling is that all learners, including us, become more proficient at problem-solving and develop deeper understanding when we are asked to pose our own problems!

Do you think your students would, in groups or alone, arrive at variations like "use even in place of odd or drop the word completely? There's only one way to find out!  Perhaps you can share your experiences here...



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 first 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.

Answer to today's problem: 3^4 x 5 = 405  [Correction thanks to Novotny!!]

Do Parabolas Have Centers? Another PairoDucks?

Well, your chance to win a free copy of my Math Challenge book has come and gone but I thought I would post an original problem/investigation about graphs of quadratic functions.

In Alg2/Precalculus students learn about parabolas:
Vertex, axis, symmetry, intercepts then, perhaps, further into other defining properties involving focus & directrix. It's also fun to touch on other important and fascinating applications such as the reflecting properties of 3-dim parabolic surfaces. OK, enough overview.

So, is the focus of a parabola the closest analog to a "center"? It's my blog so I say no!

Consider the following sloppily drawn sketch...

Can you make sense out of this graph of y = (1/12)x^2?

Without referring to the focus-directrix form of a parabola (e.g., x^2=4py), determine the values of p and q.

Reflections...

1.  Is the "p" in this problem the same as the parameter p which defines the focus?

2.  Does the diagram suggest another way to define the focus? Explain.

3.  Of course, we do not refer to the point Q as the center. I just felt  like calling it that. Can you guess how the Circle Paradox posts led me to this? Hey, I may not be as creative as some of you but I am persistent!

4. OK, fellow (gender-free) colleagues. How might you extend this investigation? We need to share ideas, right?

5.  So what does "4p" represent geometrically? Refer to the diagram.

As always, feel free to share this but don't forget proper attribution.





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.

Making a Challenge Math Problem the Springboard for Concept Development

My problem-posing creativity peaks in the hours between 6-9 am so I'd better publish this fast!

Here's a sample of an SAT-type problem although it's verging on a math contest offering...

Disclaimer: Remember, the problem or its solution is never the objective of this blog. It's merely a framework for helping our students learn to think mathematically while developing concepts in a collaborative setting that builds self-esteem. Wow, where did that rhetoric come from? Uh, me...

If the sum of 2 positive integers is 2^16 and their greatest possible product is 2^k, then k = ?

Answer: 30

Reflections...

1.  What are some strategies you want your students to use for these types of questions? And what type is this!

2.  What do you see are the "big" mathematical ideas embedded in this problem? Are "exponents" a big idea?

3.  What are the prerequisite skills needed for success with this type of question? Would you review these first?

4.  How would you utilize this problem with middle schoolers? High schoolers? Algebra 2 vs precalculus?

5.  Would you begin with an easier problem first, then build up to this or let them struggle with it as is?

6.  How would you assess that students grasped the ideas here?  Make up 10 similar questions for homework? Give them another one to try in class? Include one of these on the next test?

7.  Would you ask students to generalize this problem? First demonstrate what "generalize" means?
[One possibility: If the sum is 2^n, then the greatest possible product is 2^(2n-2).]

8. How do you create an environment of making connections in and/or applying mathematics?
[One possibility: Relate this question to the problem of finding the rectangle of maximum area for a given perimeter!]

So, have I successfully killed off all potential comments!




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.

SAT QUADRATIC FUNCTION PARABOLA PROBLEM -- Level 4/5

This type of coordinate problem is occurring more frequently. Students need exposure to these...

The graph of the quadratic function f(x) = bx^2 + ax + c intersects the x-axis at 3 and 4 and the y-axis at 5.
b = ?

Answer: 5/12

Reflections...

1.  What in the question do you think  might cause students to struggle?
2. Do you use a standard approach to these types of coordinate problems, e.g., an x-y table?
3.  Do you usually discuss at least 2 methods for these? You know how I feel!

REMINDER
All the problems I post are original and are the property of MathNotations. Feel free to use them for classroom purposes according to the Creative Commons License in the sidebar.





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/@ 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.

WHAT IS MathNotations -- my annual rant...

The Circle Paradox Revisited seems to be provoking some interest however I have to remind my readers that the main purpose of this blog continues to be

HOW TO USE PROBLEM-SOLVING IN THE CLASSROOM TO DEVELOP CONCEPTUAL UNDERSTANDING AND STRENGTHEN STUDENT REASONING

I know that we all enjoy "solving the problem" but the Circle Paradox is not all that challenging and should prove straightforward for math teachers and mathphiles in general.

To clarify: My intent is to 

Generate dialog about how to effectively use non-routine problems in the classroom to enhance student thinking! 

WHY is it empowering for students when we encourage several approaches and not force feed our method or way of thinking?

IMO, the surest way to turn off students' minds is to "do it for them" or not allowing them the time to struggle.

Judicious guidance and applying "less is more" is, for me, the hallmark of the master teacher. I'm retired and I'm still learning how to do this with my grandkids!

For some reason, this intent has not taken hold in the six-year existence of this blog. When I try to shift the focus to how to use these problems with students, it is generally ignored. "Instructional strategies, Dave? Who cares!" 
Uh, I care...

Enough of my rant for now...

Oh, yes, hypocritically, I am reminding my readers that the deadline for submitting a solution is Fri 2-8-13, 12 noon EST. Of course, I'm hoping that your solutions are coming from your students!





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.

Circle Paradox Pt 2

"Proof" without words? There must be a dozen other ways...

No takers yet? My free offer will expire by Fri 2-8-13!






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.

The Circle Paradox Revisited

WHAT'S WRONG WITH THIS PICTURE OF A CIRCLE?

CHALLENGE YOUR STUDENTS TO EXPLAIN THE PARADOX!

FIND AS MANY WAYS AS POSSIBLE!

BEST EXPLANATIONS WIN A COPY OF MY CHALLENGE MATH WORKBOOK! OPEN TO ANYONE OVER THE AGE OF 4...




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.

The Super Bowl and the ratio 16 to 9

NOTE: The following classroom scenario requires a reimagining of how we normally present a math lesson and I already have heard all the negatives: not enough time for this, too much material to cover, this won't be on the test so why bother...
I hope you will be open-minded.


The 16 to 9 ratio of course refers to the aspect ratio of  high definition LCD or LED TV screens today.

Let's say you just purchased a 55 inch LED HDTV. We all know the 55 inches refers to the diagonal of the screen and, in fact it's slightly less than 55 inches. So what would the width of the screen actually be?

How does our method of presentation and the questions posed affect concept development? 

We know how to solve this, no nonsense. Just apply the Pythagorean Theorem with some algebra and voila. "Tradition" as the song title goes from Fiddler on the Roof.

A Non-Traditional Classroom Scenario

Let's try some estimates, boys and girls...
54" 52" 50" 48"?? Hmm, most 'guesstimated' 50?

Can anyone guess why there are sheets of paper, scissors and rulers on the table? Right, we will first "construct" a solution! Oh, so a 16" width is too big for standard 8.5×11 paper. Any ideas? Oh, it's a ratio so we don't have to use 16" and 9". Ok, we'll use 8"×4.5". OK, go to it...

Each member of your team should measure the diagonal to the nearest 1/8". Oh, that's right we could have used cm instead to make measurements more precise...

Alright, so most of you got around 9 1/8" for the diagonal.
(Aside: This is definitely an imaginary scenario!).
Guess, metric measurement would have been better.  Let's verify this using the Pythagorean Thm. Ok, 9.18" to nearest hundredth. So how will we apply this to a 55" diagonal? Oh, make a proportion, and we obtain 55×8/9.18 ≈ 47.9"!

In your groups, solve the ratio problem algebraically and compare results...

So a 55" screen is less than 48" in width. Wonder why they use diagonal measurements in the ads...

Yes, Alex. You found another way to estimate this mentally?

The closest Pythagorean triple to 16 and 9 is 15 and 8. The hypotenuse would be 17 and 17×3 is 51, sorta' close to 55. So we can triple the dimensions to get 16×3=48" for the width.  Hmm...



What non-traditional approaches for this kind of problem have my colleagues used? Share!

This imaginary lesson would consume the entire period, yes? Do you think it's worth it?




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.

MENTAL MATH TRICKS OR SOMETHING DEEPER

Hey folks, I do appreciate the increased readership and renewed interest in MathNotations.  I'm definitely a "binge" blogger, prolific for a few weeks, then burning out like a SuperNova, 'retweeting'  into oblivion again. I miss doing video solutions of some problems but I don't know if that will happen. Since I get lots of views but zero comments, I really don't know if I should continue...



EXPLAIN USING MENTAL MATH IN 10 SEC OR LESS...

72÷25=2.88
72 × 25 = 1800

Just curiosities to grab students and then fade away OR do you see these as worthwhile for developing deeper understanding of number operations?

For example, dividing by  25 is equivalent to multiplying by 4, then ÷ by 100.

How would you develop or extend this investigation?




If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175  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 multiplechoice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

A CCSSM GEOMETRY ACTIVITY -- IS IT NECESSARY OR SUFFICIENT

IF THE ALTITUDE ON THE HYPOTENUSE OF A RIGHT TRIANGLE DIVIDES THE HYP INTO A 1:3 RATIO, PROVE THAT THE TRIANGLE IS 30-60-90. 

ASK YOUR STUDENTS TO

FIRST VERIFY THE CONCLUSION BY CONSTRUCTING A TRIANGLE WITH GIVEN CONDITIONS
  (1)  by construction with mechanical tools
  (2) using electronic tools (e.g., Geogebra or Geom SketchPad)

THEN HAVE STUDENTS
(A) FiND AT LEAST TWO SYNTHETIC (DEDUCTIVE) METHODS WHICH DO NOT INVOLVE TRIG.


Reflections...
(1) So are given conditions both necessary and sufficient? Of course in geometry we usually write "if and only if" or "iff".
(2) I know most educators are annoyed that I ask lots of questions but don't ever answer them. Like, "who has time for this!" My children, grandkids and my students have always thought I was annoying too. I do pride myself on consistency!
(3) So who's going to call my bluff and ask me to show at least TWO methods of proof?
[Alt on hyp theorems same as similar triangle methods?]


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.

SAT QUADRATIC FUNCTIONS APPLICATION

C is a point on the portion of the graph of f(x) = 6x-x^2 in the 1st quadrant. If points A and B are the points of intersection of the graph of f with the x-axis, what is the greatest possible area of ΔACB?


Answer: 27



Questions:
 1) Do you think the College Board would provide the graph? Would you provide it or would you expect this from the student?

 2) Can you find examples like this in an Alg or Precalc text? Do students need more exposure to applied problems like these?

3) How would you rate the difficulty level of this question --- 3 is medium, 5 is hard.

4) EXTENSION
Change to f(x) = kx-x^2

Ans: k^3/8
Note: College Board is moving toward use of parameters. Can you guess why?



If interested in purchasing my NEW 2012 Math Challenge ç/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.

SAT TRIANGULAR PRISM PROBLEM

E

Animated gif showing one way to "draw" the prism above

gif make


SAT Geom questions often assess 3-D spatial visualization. Do you think your students will find the following trivial?

I'm still playing with my Handraw app so the diagrams will continue to be sloppy!

FIND THE VOLUME OF THE RIGHT ISOS TRIANGULAR RIGHT PRISM ABOVE.

Note: Markings are intended to indicate that the height of the prism equals the equal legs of the bases.

[NOTE: Error in labeling choices has been corrected.]

(A) 108 (B) 108√2  (C) 54√2 (D) 27√2 (E) (27√2)/2


Ans: D [Corrected]

How would you explain this?
What methods do you think students who solve this will use?
Apart from the poorly drawn diagram, how many students do you think would quickly recognize that the prism is half of a cube.





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.

A Variation on Using Powers of 2 to Eliminate Debt

Are you tired of the powers of 2 problems you've seen? Well, most students are not familiar with them, so why not impress another generation while developing their understanding of exponential growth and sharpening their mental math and exponent skills.

The title suggests there is a practical application for this activity, either for our national debt or our personal credit card or mortgage obligations. How you bring this in is all about your personal preference.

We'll start the usual way:

You receive a penny on January 1, 2013; 2 pennies on January 2nd, 4 on January 3rd, 8 on January fourth and so on.
How many dollars will you have accumulated by the end of the month?

To make this a little different why not have your students or children at home use 1 significant digit estimates and do this without pencil and paper!

For example we know 2^5 = 32, therefore 2^ 10 ≈ 1000 or 10^3, so 2^20 ≈ 1,000,000 or 10^6. Continuing, 2^30 ≈ (10^6)•(10^3) = 10^9.
But this is in pennies, so you'd have around 10^7 dollars or $10M!

So is this a fair estimate of how much would have accumulated by the end of Jan? If not, correct my error(s)!

As a teaching strategy, you could demonstrate one of these procedures first, then ask students to devise another path to 2^30 or
2^31.

So what is our approximate national debt today? First one to find it on your smartphone receives a penny!






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.

SAT GEOM: A special regular square pyramid

Consider the hand-drawn sketch of a regular square pyramid in which all 8 edges, both base and lateral, are congruent:

(1) Explain why angle PVR is a right angle.

(2) If edges each have length e, explain why the height PT has length e/√2.

(3) Any other interesting observations?


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.

Soln to SAT Quadr Problem 1-19-13

Sketch of solution to SAT quadratic problem

Note: I'm experimenting with a new Draw app on my Nexus 7 so pls

let me know if you can make any sense of it.





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.

Quadratic Function SAT Practice (Grid-in)

Student Constructed Response ("grid-in") type


If the graph of the quadratic function f(x) = -x^2 + kx - 25 intersects the graph of y = 0 in exactly one point, what is the value of k?

Answer: 10

NOTE: This is similar to but a notch above the College Board Problem of the Day of 1-19-13






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.

Tuneup for Jan 2013 SAT MATH -- TEST YOURSELF

The following questions reflect the medium to more difficult questions on a typical SAT math section. Both multiple choice and "grid-ins" are included.

ALLOW THREE MINUTES.


1.  The base of a parallelogram is 3 times the base of a triangle; the height of the parallelogram is twice the height of the triangle. What is the ratio of the area of the parallelogram to the area of the triangle?
(A) 24 (B) 12 (C) 9 (D) 6 (E) 3

2.  If (8x+8)^2 = 64 and x≠0, then x^2 = ?

3.  What is a value of x for which 1/(10x-20) > 10?

4.  The first 2 terms of a sequence are 4 and 6 and each term after that is the average of all the preceding terms. What is the sum of the first 22 terms?



Answers (formatting error corrected, hopefully! Thanks, Sue)
1.  B.
2.  4
3.  2.01>x>2
4.  110



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.

So WHY is 10 raised to the zero equal to 1 again?

This is a topic that recurs annually as we move traditional skills/concepts to lower grades according to the Common Core standards. I believe there will be a temptation to give rules without motivating them or explaining them. 

So perhaps raising a number to the power of 0 could be a fourth grade topic these days. I did publish a post several years ago on developing exponent concepts in the middle grades but I guess it can't hurt to revisit it now for even younger students.

One approach I have found useful is to have students construct a table in which the left column is the exponent and the right column is the corresponding power of 10.

I will start the table, suggest some questions but leave the rest for you to bring your own ideas and expertise to the  "table"!


EXPONENT  POWER OF TEN
          3           10^3 = 10x10x10 = 1000
          2           10^2 = 10x10 = 100
          1           10^1 = 10


Teacher: So what do you notice?
Remember --- a mathematician must first be very observant like a good detective!

You and your partner have 1 min to write as many observations as possible...

I HAVEN'T ADDRESSED THE TITLE OF THIS POST YET!
PLS COMMENT IF YOU WANT ME TO PROVIDE MORE DETAILS OR MAKE A SHORT VIDEO.






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.

How many positive integers less than 101 are not div by 2 and not div by 3?

Instead of posting this innocent little combinatorial problem to Twitter like I've been doing for the past 2 years, I decided to post it here so my readers will know MathNotations is not extinct!

Comments about this problem:

1) What would be an easier variation you might first pose? Oct 2009 SAT, Prob 17, "grid-in"!

2) To those who  have been teaching these types for awhile, SHARE your strategies!

3) A Black Box Soln:
100-(50+33-16)=33



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.

Math Teachers at Play 58 Up and Running -- DON'T MISS IT

Head on over to Denise's latest masterpiece collection of outstanding recent math blog posts. Something for everyone!




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.

RAISING AN EDUCATED CONSUMER - A CCSSM MIDDLE SCHOOL ACTIVITY

Ever wonder about the extra 9 in the cost of a gallon of gasoline at the pump? Here's a real-world activity for middle schoolers which might lead to a deeper understanding of decimals and fractions not to mention producing more educated consumers!

So what does the "raised" 9 mean at the end of $2.99? Some retailers display the extra 9 by writing it as 9/10. Nine-tenths of what? Hey, the cost is still less than $3.00 so that 9 is pretty insignificant, right, kids?

We would hope our youngsters would guess the extra 9 represents nine-tenths of a penny or nine thousandths of a dollar. We would hope...

First I would ask the students or my child or grandchild at home what she thinks the 9 means.  In the classroom, we ask the questions but we are not always obligated to confirm or reject students' replies! Research shows that the consensus of the group is usually correct so trust them and help them to develop more confidence!

So how would you develop this lesson to reinforce decimal/fraction skills and develop deeper understanding of decimals in a practical consumer setting? What are some questions you would ask?

Hey, it's my blog and I don't have to show my way first!






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.