Tag Archives: arithmetic

Measuring Calculator Speed

Two weeks ago, my summer school Algebra 2 students were exploring sequences and series.  A problem I thought would be a routine check on students’ ability to compute the sum of a finite arithmetic series morphed into an experimental measure of the computational speed of the TI-Nspire CX handheld calculator.  This experiment can be replicated on any calculator that can compute sums of arithmetic series.

PHILOSOPHY

Teaching this topic in prior years, I’ve found that sometimes students have found series sums by actually adding all of the individual sequence terms.  Some former students have solved problems involving  addition of more than 50 terms, in sequence order, to find their sums.  That’s a valid, but computationally painful approach. I wanted my students to practice less brute-force series manipulations.  Despite my intentions, we ended up measuring brute-force anyway!

Readers of this ‘blog hopefully know that I’m not at all a fan of memorizing formulas.  One of my class mantras is

“Memorize as little as possible.  Use what you know as broadly as possible.”

Formulas can be mis-remembered and typically apply only in very particular scenarios.  Learning WHY a procedure works allows you to apply or adapt it to any situation.

THE PROBLEM I POSED AND STUDENT RESPONSES

Not wanting students to add terms, I allowed use of their Nspire handheld calculators and asked a question that couldn’t feasibly be solved without technological assistance.

The first two terms of a sequence are $t_1=3$ and $t_2=6$.  Another term farther down the sequence is $t_k=25165824$.

A)  If the sequence is arithmetic, what is k?

B)  Compute $\sum_{n=1}^{k}t_n$ where $t_n$ is the arithmetic sequence defined above, and k is the number you computed in part A.

Part A was easy.  They quickly recognized the terms were multiples of 3, so $t_k=25165824=3\cdot k$, or $k=8388608$.

For Part B, I expected students to use the Gaussian approach to summing long arithmetic series that we had explored/discovered the day before.   For arithmetic series, rearrange the terms in pairs:  the first with last, the second with next-to-last, the third with next-to-next-to-last, etc..  Each such pair will have a constant sum, so the sum of any arithmetic series can be computed by multiplying that constant sum by the number of pairs.

Unfortunately, I think I led my students astray by phrasing part B in summation notation.  They were working in pairs and (unexpectedly for me) every partnership tried to answer part B by entering $\sum_{n=1}^{838860}(3n)$ into their calculators.  All became frustrated when their calculators appeared to freeze.  That’s when the fun began.

Multiple groups began reporting identical calculator “freezes”; it took me a few moments to realize what what happening.  That’s when I reminded students what I say at the start of every course:  Their graphing calculator will become their best, most loyal, hardworking, non-judgemental mathematical friend, but you should have some concept of what you are asking it to do.  Whatever you ask, the calculator will diligently attempt to answer until it finds a solution or runs out of energy, no matter how long it takes.  In this case, the students had asked their calculators to compute values of 8,388,608 terms and add them all up.  The machines hadn’t frozen; they were diligently computing and adding 8+ million terms, just as requested.  Nice calculator friends!

A few “Oh”s sounded around the room as they recognized the enormity of the task they had absentmindedly asked of their machines.  When I asked if there was another way to get the answer, most remembered what I had hoped they’d use in the first place.  Using a partner’s machine, they used Gauss’s approach to find $\sum_{n=1}^{8388608}(3n)=(3+25165824)\cdot (8388608/2)=105553128849408$ in an imperceptable fraction of a second.  Nice connections happened when, minutes later, the hard-working Nspires returned the same 15-digit result by the computationally painful approach.  My question phrasing hadn’t eliminated the term-by-term addition I’d hoped to avoid, but I did unintentionally create reinforcement of a concept.  Better yet, I got an idea for a data analysis lab.

LINEAR TIME

They had some fundamental understanding that their calculators were “fast”, but couldn’t quantify what “fast” meant.  The question I posed them the next day was to compute $\sum_{n=1}^k(3n)$ for various values of k, record the amount of time it took for the Nspire to return a solution, determine any pattern, and make predictions.

Recognizing the machine’s speed, one group said “K needs to be a large number, otherwise the calculator would be done before you even started to time.”  Here’s their data.

They graphed the first 5 values on a second Nspire and used the results to estimate how long it would take their first machine to compute the even more monumental task of adding up the first 50 million terms of the series–a task they had set their “loyal mathematical friend” to computing while they calculated their estimate.

Some claimed to be initially surprised that the data was so linear.  With some additional thought, they realized that every time k increased by 1, the Nspire had to do 2 additional computations:  one multiplication and one addition–a perfectly linear pattern.  They used a regression to find a quick linear model and checked residuals to make sure nothing strange was lurking in the background.

The lack of pattern and maximum residual magnitude of about 0.30 seconds over times as long as 390 seconds completely dispelled any remaining doubts of underlying linearity.  Using the linear regression, they estimated their first Nspire would be working for 32 minutes 29 seconds.

They looked at the calculator at 32 minutes, noted that it was still running, and unfortunately were briefly distracted.  When they looked back at 32 minutes, 48 seconds, the calculator had stopped.  It wasn’t worth it to them to re-run the experiment.  They were VERY IMPRESSED that even with the error, their estimate was off just 19 seconds (arguably up to 29 seconds off if the machine had stopped running right after their 32 minute observation).

The units of the linear regression slope (0.000039) were seconds per k.  Reciprocating gave approximately 25,657 computed and summed values of k per second.  As every increase in k required the calculator to multiply the next term number by 3 and add that new term value to the existing sum, each k represented 2 Nspire calculations.  Doubling the last result meant their Nspire was performing about 51,314 calculations per second when calculating the sum of an arithmetic series.

My students were impressed by the speed, the lurking linear function, and their ability to predict computation times within seconds for very long arithmetic series calculations.

Not a bad diversion from unexpected student work, I thought.

Multiplication Practice Plus Creativity

I hope this post is particularly helpful for parents and teachers of elementary school children.  Through my Twitter network last week I found via @Maths_Master‘s Great Maths Teaching Ideas ‘blog a 2010 post summarizing Dan Finkel’s Damult dice game. Recognizing that “practicing times tables can be unmotivated and boring for kids,” Damult is an attempt to make learning elementary multiplication facts more entertaining. I offer some game variations and strategies following a description of the game.

Here’s Dan’s game:

image via Wikipedia

Each player takes turns rolling 3 dice. First to break 200 (or 500, etc.) wins. On your turn, you get to choose two dice to add together, then you multiply the sum by the final die. That’s your score for that turn.

Simple; no bells, no whistles. For example, I roll a 3, a 4, and a 6 on my turn. I could either do (3+4) times 6 for 42 points, OR (3+6) times 4 for 36 points, OR (4+6) times 3 for 30 points. I’ll take the 42 points.

I spent some time playing this with kids the other day and I saw that (1) it was genuinely fun, and (2) it gives you almost all the multiplication practice you could ask for. In fact, it gives even more, because the choice of which dice to add and which to multiply reveals some interesting structure of numbers. Seriously, get a kid hooked on this game, and it’s the equivalent of dozens or hundreds of times table practice sheets.

It’s a fun activity idea by itself.  Damult combines a bit of luck and memory, and rewards the ability to recall multiplication facts.  As an added bonus, it requires players to be able to manipulate objects in their heads–how many different ways can the three given dice be manipulated in summation stage to create unique products? How can a player ensure that she has found the biggest product for her score?  Try the game!

CONNECTIONS AND EXTENSIONS:

This is a great opportunity for parents to engage with their  children as they learn multiplication facts.  Parents and teachers could play along, or the learner might be the only player, talking out loud so that the teacher or parent can “hear the thinking.”

I love that the game completely randomizes the multiplication tables.  This significantly enhances recall as memory is not tied to particular patterns or positions on fact pages. Players must adapt to each random roll.

In any variation, there obviously should be a discussion among all players about what products were found to confirm the results. Make the game more formative or more competitive, depending on the experience level of the players.  In more competitive variations with experienced learners, if a product was miscalculated and claimed, you might decide that no score should be recorded for that round.

If you’re guiding someone on this it is critical that you DO NOT give answers.  Students need to explore, hypothesize, discover errors, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  Experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

The first comments on Dan’s post noted that while one player was summing and multiplying, the other player(s) were largely disengaged. Also, the game could drag on as unconfident players tried to make sure  they had explored every possibility.  To address that and several other possibilities, I offer the following Damult variations.  Some more complex variations are toward the end. Read on!

Finally, if you’ve read my ‘blog much, you know that I’m a huge fan of leveraging technology for math learning, but this is one of those situations where I think you should 100% unplug. To learn multiplication facts is to learn some of the basic grammar and vocabulary that makes the language of mathematics work.  You simply can’t communicate mathematically with an underlying awareness of how the structure of the language works.

GAME VARIATIONS:

Variation 1: Adding a timer to the game could cure the slow-down issue. Depending on the age of the child and his/her familiarity with multiplication, the timer can be longer or shorter.  If the skill levels of the players are unequal, make the timer unequal.  (I love the adage, “Fair is seldom equal, and equal is seldom fair.”)

Variation 2: Why must only one player be active? The players could take turns rolling the dice while both record scores based on what they find.  If a particular combination was not noticed by one player, that player doesn’t get to consider it for his/her score.

Variation 3 – As an aside, notice that Dan implicitly claims there are only 3 possible sums from a 3-dice roll. Will that always be the case?  Can you convince someone why your solution is correct?

(For 3 dice the maximum number of possible sums is 3. When and why would there be fewer products?)

Variation 4 – How many multiplication facts are possible using only 3 dice?

This would be a great number sense exploration.  Some may try it by gathering lots of data, others may have more sophisticated reasoning.  I suggest that you or your students hypothesize an answer first along with some reason why you think your hypothesis is correct.  Different answers are OK, and you can always revise your hypotheses if you get evidence leaning in another direction. No matter what, have fun exploring and learning.

(Middle School extension: Damult creates products of axb where a can be any integer 1 – 6 and b can be any integer 2 – 12.  That gives 66 different products if you count different arrangements (3×4 and 4×3) as different products. Can you or your student see why? How many outcomes are possible if you look only at the product result and not at the factors which created it?)

Variation 5 – After discovering or just using the answer to the last variation, you could use a table of multiplication facts and see how quickly different facts and be “discovered” from rolls of the dice.  After rolling 3 dice, mark off all multiplication facts you can using the sum-then-multiply combination rules posed at the beginning.  This might be a fun way for early learners to familiarize themselves with multiplication patterns.

NOTE: If you play variations 4 or 5 as a game, you’ll likely want (or need) to stop before all possibilities are found.  Some (eg, 6×12 and 1×2) will be pretty uncommon from dice rolls.

Variation 6 – You could make a Bingo-like or a 4 or 5-in-a-row game.  The first person to mark off a certain number of facts or the first to get a certain number in a row would be a winner.

Variation 6 – If you try the last few variations, you’ll see that some products occur much more frequently from the dice rolls than others.  This could be used to introduce probability. Which products are more likely and why?

As an example, I suspect 3×7 could happen six times more often 1×2.  Can you convince yourself why 3×7 is so much more likely?  Can you see why 3×7 is exactly six times more likely than 1×2?

Variation 7 – Why restrict yourself to 3 dice? When just starting out, using more than 3 dice would definitely be a frustration factor, but once you’ve got a good grip on the game, consider rolling 4 dice and allow players to multiply the sum of any 2 or 3 of the dice by the sum of the remaining dice.

By my computation, using 4 dice means there are up to 7 possible combinations in a given roll.  Can you prove that? Being able to consistently find them all is likely to be a very difficult challenge, but it is a phenomenal and early opportunity to stretch a young person’s mind into considering multiple outcomes and reliable ways to guarantee that you’ve considered all possibilities.

Variation 8 – Why go for maximum products and being the first to get to 200 or 500 points?  Why not try for a low score (like golf), seeking minimum products  and being the last to exceed 100 or 200?

Variation 9 – Stealthy Calculus:  OK, my analysis on this one goes way deeper than is necessary to play the game, but sometimes knowing more than is necessary can give insights and can help you lead others toward developing “math sense”–a truly invaluable skill.

LOW LEVEL – After you’ve played this a few times, ask the player(s) if there is some strategy that could be used to guarantee the biggest (or smallest) possible product for any roll.  This could be a great mathematical experiment for which the solutions are not at all intuitive, I think.  Some might figure it out quickly and others might need to gather lots of data, comparing products from lots of rolls before distilling the relationship.

If you’re guiding someone on this it is critical that you DO NOT give the answer.  Students need to explore, hypothesize, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  If you don’t know the answer, stop reading now and figure it out for yourself. I provide an answer in the next paragraphs, but experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

MUCH HIGHER LEVEL – As a calculus teacher, the very first fact that struck me was Damult’s implied goal: Getting the largest possible product from any roll of three dice.  That’s an optimization problem, and I knew from calculus that the greatest possible product of two numbers whose sum was constant happens when the two numbers are as close as possible to being equal.  Likewise, the smallest possible product happens when the two factors are as far apart as possible.  (If you recall some calculus of derivatives, I encourage you to prove these for yourself.  If anyone asks, I could write a future post with the proof.)

In Dan’s initial example above in which 3, 4, and 6 were rolled, I stopped reading after the first sentence of paragraph 2 (pausing to think and draw your own conclusions is a great habit of the mind) for a few moments as I thought, “I know 3+4 and 6 are as close to equivalent as I can get, so 7*6=42 is the greatest possible product.”  I didn’t even look at the other possibilities, I knew they were less. This fact was established (unnecessarily for me) in the end of the paragraph.

Without calculus, I propose students try making tables of their data.  They’ll have up to three unique products (Variation 3) and will need to explore the data before hopefully discovering the relationship. If a young person doesn’t discover the relationship, Don’t tell him/her! it is far better to leave a question as unanswered to think on and answer another day than to have a relationship given unearned.  Value comes from effort and discovery. Don’t cheat young learners out of that experience or lesson.

Conclusion: Don’t just play a game. Be creative! Strategize! Encourage young ones not just to play, but to play well. Children are quite creative in free play as they continually make new and adapt old “rules”.  Why should intellectual play be any different?  I’d love to see what variations others discover or have to offer.

Teaching Creativity in Mathematics

This will be the first of two ‘blog posts on an activity that could promote creativity for elementary, middle school, and high school students.  A suggestion for parents and teachers is in the middle of this post.

ABOUT A DECADE AGO, I first discovered what I call the Four 4s activity.  In brief, the game says that using exactly four 4s (no more, no less, and no other digits) and any mathematical operation you want, you can create every integer from 1 to 100.  Two quick simple examples are $\displaystyle 3= \frac{4+4+4}{4}$ and $\displaystyle 16= 4\cdot 4+4-4$.

As for mathematical operations, anything goes!  The basic +, -, *, / along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals ($.\overline{4}$), and many more are legal.

At the time, I was teaching a 7th grade prealgebra course with several students who were struggling to master order of operations–that pesky, but critical mathematical grammar topic that bedevils some students through high school and beyond.  I thought it would be a good way to motivate some of my students to 1) be creative, and 2) improve their order of operations abilities to find numbers others hadn’t found or to find unique approaches to some numbers.

My students learned that even within the strict rules of mathematical grammar, there is lots of room for creativity.  Sometimes (often? usually?) there are multiple ways of thinking about a problem, some clever and some blunt but effective.  People deserve respect and congratulations for clever, simple, and elegant solutions.  Seeing how others solve one problem (or number) can often grant insights into how to find other nearby solutions.  Perhaps most importantly, they learned to a small degree how to deal with frustration and to not give up just because an answer didn’t immediately reveal itself.  It took us a few weeks, but we eventually completed with great communal satisfaction our 1-100 integer list.

PARENTS and TEACHERS:  Try this game with your young ones or pursue it just for the fun of a mental challenge.  See what variations you can create.  Compare your solutions with your child, children, or student(s).  From my experiences, this activity has led many younger students to ask how repeating decimals, factorials, and other mathematical operations work.  After all, now there’s a clear purpose to learning, even if only for a “game.”

I’ve created an easy page for you to record your solutions.

A FEW WEEKS AGO, I read a recent post from the always great MathMunch about the IntegerMania site and its additional restriction on the activity–an exquisiteness scale.  My interpretation of “exquisiteness” is that a ‘premium’ is awarded to solutions that express an integer in the simplest, cleanest way possible.  Just like a simple, elegant explanation that gets to the heart of a problem is often considered “better”, the exquisiteness scale rewards simple, elegant formulations of integers over more complex forms.  The scale also includes surcharges for functions which presume the presence of other numbers not required to be explicitly written in common notation (like the 1, 2, & 3 in 4!, the 0 in front of .4, and the infinite 4s in $.\overline{4}$.

In the past, I simply asked students to create solutions of any kind.  I recorded their variations on a class Web site.  Over the past three weeks, I renamed exquisiteness to “complexity” and re-ran Four 4s across all of my high school junior and senior classes, always accepting new formulations of numbers that hadn’t been found yet, and (paralleling Integermania’s example) allowed a maximum 3 submissions per student per week to prevent a few super-active students from dominating the board.  Also following Integermania’s lead, I allowed any new submission to remain on the board for at least a week before it could be “sniped” by a “less complex” formulation.  I used differently colored index cards to indicate the base level of each submission.

Here are a few images of my students’ progress.  I opted for the physical bulletin board to force the game and advancements visible.  In the latter two images, you can see that, unlike Integermania, I layered later snipes of numbers so that the names of earlier submissions were still on the board, preserving the “first found” credit of the earliest formulations.  The boxed number in the upper left of each card is the complexity rating.

The creativity output was strong, with contributions even from some who weren’t in my classes–friends of students curious about what their friends were so animatedly discussing.  Even my 3rd grade daughter offered some contributions, including a level 1.0 snipe, $\displaystyle 5=\frac{4\cdot 4+4}{4}$ of a senior’s level 3.0 $\displaystyle 5=4+\left( \frac{4}{4} \right)^4$.  The 4th grade son of a colleague added several other formulations.

When obviously complicated solutions were posted early in a week, I heard several discussing ways to snipe in less complex solutions.  Occasionally, students would find an integer using only three 4s and had to find ways to cleverly dispose of the extra digit.  One of my sometimes struggling regular calculus students did this by adding 4′, the derivative of a constant. Another had already used a repeating decimal ( $. \overline{4}$), and realized she could just bury the extra 4 there ( $.\overline{44}$).  Two juniors dove into the complexity scale and learned more mathematics so they could deliberately create some of the most complicated solutions possible, even if just for a week before they were sniped.  Their ventures are the topic of my next post.

AFTERTHOUGHTS:  When I next use Four 4s with elementary or middle school students, I’m not sure I’d want to use the complexity scale.  I think getting lots of solutions visible and discussing the pros, cons, and insights of different approaches for those learning the grammar of mathematical operations would be far more valuable for that age.

The addition of the complexity scale definitely changed the game for my high school students.  Mine is a pretty academically competitive school, so most of the early energy went into finding snipes rather than new numbers.  I also liked how this game drove several conversations about mathematical elegance.

One conversation was particularly insightful.  My colleague’s 4th grade son proposed $\displaystyle 1=\frac{44}{44}$ and argued that from his perspective, it was simpler than the level 1.0 $\displaystyle \frac{4+4}{4+4}$ already on the board because his solution required two fewer operations.    From the complexity scale established at the start of the activity, his solution was a level 2.0 because it used concatenated 4s, but his larger point is definitely hard to refute and taught me that the next time I use this activity, I should engage my students in defining the complexity levels.

1) IntegerMania’s collection has extended the Four 4s list from 1 to well past 2000.  I wouldn’t have thought it possible to extend the streak so far, but the collection there shows a potential arrangement of Four 4s for every single integer from 1 to up to 1137 before breaking.  Impressive.  Click here to see the list, but don’t look quite yet if you want to explore for yourself.

As a colleague noted, it would be cool for those involved in the contest to see how their potential solutions stacked up against those submitted from around the world.  Can you create solutions to rival those already posted?

2) IntegerMania has several other ongoing and semi-retired competitions along the same lines including one using Four 1s, Four 9s, and another using Ramanujan’s ‘famous’ taxi cab number, 1729.  I’ve convinced some of my students to make contributions.

Play these yourself or with colleagues, students, and/or your children.  Above all, have fun, be creative, and learn something new.

It’s amazing what can be built from the simplest of assumptions.  That, after all, is what mathematics is all about.

Arrangements Connections for Young Students

Mathematics is not arithmetic.

The latter is a set of symbol manipulation rules that dominates most of what we teach in school.  Mathematics, on the other hand, is a science of patterns.  It is a way of logical thinking, making sense of forms and arrangements–sometimes applied and sometimes purely imagined.  It involves looking at the implications of what we know and pushing that knowledge as far as we can to see what else can be learned based solely upon connections we can make from our assumptions.

Within the last few weeks, I’ve discovered a great daily ‘blog run by @Five_Triangles “for (but not limited to) school years 6-8.”  I’d argue that those posts are great for a broader range of ages. I gave my 3rd grade daughter one of the puzzles during breakfast.  We had some great conversations then and on the way to school.  I share those below.  Another offering extends that thinking in a way that may not be immediately obvious to young people.

Here’s the part of the post I used at breakfast.

For my daughter, I saw this problem presenting two different possibilities–the obvious arithmetic problem and a mathematics extension.  The arithmetic requires very basic subtraction facts and wee bit of trial-and-error (a GREAT mathematics skill!) to tease out a solution.  Part of the mathematics here, in my opinion, involves asking a “What if?” question.

I posed this problem to my 3rd grade daughter and after randomly dropping in some numbers at first and seeing some frustration, I said to her, “I wonder what sorts of numbers subtract to give 3.”  Her frustration evaporated as she started making a list of several possibilities for such digits. She noted that there were far more possibilities for these difference than space in the problem allowed.  I encouraged her to keep trying.  We never explicitly discussed the problem’s set up with a four-digit number subtracted from a five-digit number, but I saw her try a couple different first digits before realizing that the first character of the five-digit number clearly had to be “1”.  A little more experimentation and she had an answer.

She thought the puzzle was over–after all, school has trained her to think that once she had “an” answer, she must have found “the” answer.

That’s when I prompted some mathematics.  I asked if she could find another answer.  A few other prompts and she had found 6 different solutions.  I asked her how she found them.  “Easy,” she replied.  “You just put the number pairs in different orders.”  She found through trial-and-error that the five-digit number always started “12…” and therefore the four-digit number started “9…”.  Checking her list of differences leading to 3 left no other possibilities.  Everything else was flexible, thus her six different answers.

• Can you explain why the five-digit number must start “12…”?
• Once I had the “12…” and “9…”, I knew there were at least 6 solutions  before I had found even the first one.  My daughter wasn’t ready for this thought, but can you explain why this is true?
• Can you find all 6 answers?
• Better: Can you explain why there cannot be any more?

The second part of the problem (with the same rules and a different result) is definitely tougher.

You can quickly conclude that the first digit of the five-digit number must be 4 or 3, but it’s definitely more challenging to tease out the rest.  Rather than dealing with the entire problem at once, I suggest another great mathematics strategy:  Simplify the problem.  Using only the digits 1 to 9, can you find all possibilities that would result in the beginning of the problem?

If this is part of an answer, the six digits not used in those three boxes must have an arrangement that subtracts to 333.  Unfortunately, none of these actually pan out.  Convince yourself why this must be true.  Students need to learn that not finding an answer is OK.  Knowing that there’s not a solution is actually a solution–you’ve learned something.

Extending the beginning of the problem to

eventually shows that the five-digit number could start “412..” with the four-digit number starting “79..”.  That means the remaining four digits must have exactly two arrangements for precisely the same reasons that the first problem had six solutions.  Can you find the two arrangements that satisfy the 33333 problem?  In case you want to check, I list the answers at the end of this post.

The next week provided another puzzle using the arrangements idea.

The problem doesn’t yield a straightforward solution that can be solved.  Instead, laying out all possible finishing arrangements and testing the veracity of the claims leads to a solution.  Again, there are three entries, so this problem is (mathematically) just like the 3333 subtraction problem above–both have six possible arrangements.  Helping a young person see this connection would be a great thought achievement.

Start by listing the six possible 1st, 2nd, and 3rd place arrangements of the letters A, B, and C:  A-B-C, A-C-B, B-A-C, B-C-A, C-A-B, & C-B-A.   As an example, if the boys finished A-B-C, all three boys would have told the truth, so that finish doesn’t satisfy the problem requirement of one false statement.  Comparing each arrangement to the boys’ statements eventually shows that only one of these arrangements satisfies the problem’s requirement that exactly one of the three boys made a false statement.

A good mathematical extension would be to see if there are any other questions that could be asked from the boys’ statements.  Is it possible that all three told the truth?  Is it possible that only one was truthful?  Are there any other possible outcomes?  Do any of these have unique outcomes given the boys’ statements, or do some have multiple possibilities?

CONCLUSION:  I fear that too often school and students stop at a single answer and don’t explore other possibilities.  Asking “What if” is a critical question in all of science and mathematics.  It inspires creativity, wonder, and exploration.  It doesn’t always yield results, so it also helps motivate stamina.  Convincing yourself that there are no (more) solutions is itself an intellectual accomplishment.

We need more of this.

SOLUTIONS:

• 3333 solutions: 12678-9345, 12687-9354, 12768-9435, 12786-9453, 12867-9534, & 12876-9543.
• 33333 solutions: 41268-7935 & 41286-7953.
• Competition solution:  A-C-B

4×4 Grid and Extensions

Ben Vitale’s Fun with Num3ers ‘blog is a prolific source of all sorts of interesting number patterns.  He just posted a great problem that would be appropriate for students from elementary school through algebra.  Here it is:

Any students who understand nothing more two-digit addition could enjoy the magic that comes from getting the same answer every time.  Older students who are beginning to understand something about variables can handle the generalized question Ben asks.  Depending how one approaches the proof, a student might discover that this problem generalizes even a bit further than Ben suggests in his initial post.

Don’t read any further if you want to solve this problem on your own.

PROOF:  Let the number in the upper left of the grid be a.  One way to tackle this proof is to write the grid elements with the upper left number in parentheses, values added to that number along a row placed inside the parentheses, and values added to that number down a column placed outside the parentheses.  The revised grid looks like this:

Following the rules of selecting a number and then crossing out any other entries in that numbers row and column, every sum of four numbers selected this way will contain exactly one element from every row and every column making the overall sum contain an (a) from column 1, an (a + 1) from column 2, an (a + 2) from column 3, and an (a + 3) from column 4.  Also, every set of four numbers will have outside the parentheses nothing from row 1, a “+4” from row 2, a “+8” from row 3, and a “+12” from row 4.  That means the numbers you add for this sum will be some arrangement of $(a)+(a+1)+(a+2)+(a+3)+4+8+12=(4a+6)+24$.  Because $a=1$ for the given problem, the magic sum for this problem is 34. That solves an arithmetic problem.

EXTENSION 1:  Now think a bit more mathematically.  Notice that all my proof requires is that the upper left number be (a).  That means any consecutive integer run starting at any integer a in the upper left corner of a 4×4 grid would produce a constant sum of $4a+30$.  Encourage your mathematical explorers to start with or include all types of integers, including zero; include negative numbers if they’re ready for that.

EXTENSION 2:  How many different ways are there to pick numbers from a 4×4 grid in this manner, no matter what value (a) you place in the upper left corner?

EXTENSION 3:  Pushing just a little further, can you prove why any square grid of any size filled with any consecutive elements of any arithmetic sequence produces a constant sum?

Math for the very young

It’s title promotes unnecessary parental mania, but once you get past that bit of self-marketing, I think this Atlantic article offers five great ideas for helping parents encourage numeracy (my word, not the article’s) in very young children.  Here are its key points.

1. Talk about numbers.  Don’t just count to ten, refer to physical objects to help young ones make connections.
2. Talk about spatial relations.  Learn names for different shapes and make comparisons.  [My addition: Continued conversations about comparisons are the seeds for phenomenal connections later.  One example is my ‘blog post here.]
3. Talk about math with your hands.  Point to objects as you talk about them.  Connections between multiple representations is huge for brain development.
4. Engage your child in spatial play.  The article talks about puzzles (more below), but I’d add rolling balls, stacking blocks, and more.
5. Engage your child in number play.   Play games like Chutes & Ladders to encourage counting.

These are all great points and not particularly revolutionary if you think about it.  So, I’d like to add a few topics to the list from my play with my own children.

• Play with blocks, Legos, etc.  Get down on the floor and stack those blocks yourself alongside your child.  Sometimes build your own tower and sometimes add to your child’s creations.  The connection time is great, and your child learns greater creativity by trying to imitate some of your more sophisticated constructions.  Talk about what you’re doing, but if your child doesn’t seem to care or follow, no problem!  Over time, the ideas will sink in, and you got quality time anyway.  As they get older, play Jenga or–even better–make your own game of Jenga using blocks.  Don’t forget to laugh and have fun when the tower falls and you get the chance to rebuild the original tower.  For young children, play with a much smaller stack of bigger blocks.  Check out this sophisticated Jenga tower.  What else can you and your child make with this game that can be good for years of play?
• Play with jigsaw and similar puzzles.  I’m quite fond of the great creativity of most of the Melissa & Doug and other similar puzzles, but they can be pricey.  I save lots of money on puzzles at local consignment sales.  Early puzzles for my kids have knobs on them to enable easier handling.

Long before they figure out how to put them back in place, you can use the pictures for conversations about names, colors, etc.  So much room for general creativity!

Before they can assemble them, we put puzzles like the one below in their play space.  Straight-edged puzzles eventually give way to more traditional jigsaw puzzles.  As they got more sophisticated in their thinking, we encouraged them to assemble connected puzzles outside their frames.  As always, other games with the puzzles are great:  How many pieces does the puzzles have?  What colors are there?  What is Pooh doing?  Tell me a story.

I was about to put away some very simple puzzles when my oldest daughter created a new game.  She knew the early puzzles were too simple, so she turned all the pieces upside down and tried to reassemble them without the aid of pictures.

We sometimes work more complicated, increasingly difficult jigsaw puzzles together.  Talking about shapes, colors, and searching for where individual pieces might fit into the big picture of the final puzzle are all great activities.  Pointing to part of the picture on a puzzle box and then pointing to the corresponding location in the puzzle as it is assembled is a tremendous lesson that helps children make connections between multiple representations of ideas.

We also have a big foam floor puzzle of the alphabet (Thanks, N!).  In the earliest days, it was a nice floor pad.  It comes apart and can be assembled in different ways.  Actually, being able to disassemble is a great early skill for children (good to remember when you’re annoyed the 100+ times you put it back together yourself).  Some pieces are easier to put back than others, but cheer every time they accomplish a new task. Sometimes we sit in another room and ask them to retrieve a specific letter.  We’ve encouraged early literacy by connecting every letter with something familiar: “D is for Daddy”, “Y is for yogurt”.  They remember these special relationships long before they’ve memorized the alphabet.  Literacy and numeracy are not isolated skills.

Don’t forget to try something unusual.  I’ve had great fun making blocks and fences out of this floor puzzle.  My girls giggle as we build interlocking walls around them, creating and filling in windows, etc.

• When they’re older, draw pictures of their room or maps of your home.  It doesn’t matter if the scales are right or the pictures are accurate.  Show them on maps of your neighborhood, city, state, country, or world where they live and where special family and friends live.  Revisit this when you travel or walk with your child.  Connections between the real world and 2D maps can be tough, but are phenomenal skills for later mathematical abilities.
• Older games we’ve used that also happen to be great for visual-spatial development Connect 4 and the absolutely glorious Blokus.  I’ll post more on this game another time.

WARNING:  I’ve said this before, but I worry about parents who might succumb to the mania suggested by the Atlantic article’s title.  Education of children should not be about competition or creating math whizzes.  Play, model creative play yourself to encourage their creativity, cheer for your kids when they do something new to encourage out-of-the-box thinking, don’t worry if your kids don’t get it right away, be patient, be 100% willing to move to a different task/game if your child isn’t interested, and make connections.

Growth will happen if you keep them surrounded by challenges and point out how much fun it is to think, to create, and to solve.

Developing fractional understanding

Here’s another installment of my infrequent commentaries on my eldest daughter’s math development, this time on an unexpected result when we were talking about equivalent fractions.

I’ve been a firm believer in keeping challenging ideas, games, etc. around my children at all times, some of them intentionally beyond what they’re developmentally ready to handle.  Sometimes I’ll ask leading questions to see if there is any interest; sometimes they pick up an idea or toy again and create their own play rules or ask me to explain how it works.  My ground rule is that they should PLAY.  If my explanations ever bore them, they are welcome to drop it at any time and move on to something more interesting.  Not only does this keep with my mantra that learning should be fun (even if it involves work), but I believe it helps them see that mathematics (and anything else they learn) is about enjoyment and pushing yourself to discover more than what exists within the current boundaries of your understanding.

Several weeks ago

With that philosophy in mind, my 2nd grade daughter had been helping me make some fresh bread one afternoon when we needed a cup of one ingredient, but our cup measurer was dirty.  Whether from school or one of our earlier conversations, she responded something like, ‘No problem.  Just use 2 half-cups.‘  Maybe she actually said  ‘3 third-cups,’ but she clearly had the beginnings of fractions down, so I jumped.

So if I needed a half cup of something and my half-cup measurer was dirty, what else could I use?

It took some conversations, but eventually she drew a circle with a line through the center and shaded one side.  When she drew another line through the center roughly bisecting the original sectors, she declared, “Look, Dad.  Cutting each part in half doesn’t change what you have, it just cuts it into more pieces.  So, $\frac{1}{2}$ must be the same as $\frac{2}{4}$.”

That was a pretty cool moment of discovery for her, but then she upped the ante.  After some additional thought, she noticed that both parts of the fraction had been doubled, so she applied her rule again and asked if $\frac{1}{2}$ would also be the same as $\frac{4}{8}$.  Another drawing confirmed her discovery which led to gleeful proclamations that $\frac{1}{2}=\frac{2}{4}=\frac{4}{8}=\frac{8}{16}=\frac{16}{32}$. She knew she could go on, but what was the point?

I asked if doubling was the only way she could make equivalent fractions, which led to $\frac{1}{2}=\frac{3}{6}=\frac{9}{18}$. We didn’t go any further that day, but we had already “cooked up” far more than I had anticipated.  Off the top of my head, I don’t recall how elementary school curricula deal with the scope and sequence of teaching equivalent fractions, but it will be difficult for anyone to convince me that my daughter could have had a better experience or more fun learning.

Last weekend

I was putting my daughter to bed and for some reason she asked how much of a year 9 months was.  Being the teacher, I responded to her question with another question:  “Well, how many months are in a year?

So 9 months is $\frac{9}{12}$ of a year?” She asked.

Good job.  Can you think of any other smaller fractions that might be the same as $\frac{9}{12}$?” It was an innocent question, I thought, trying to get her to take our fraction doubling-trebling idea from earlier in reverse. If she didn’t get it, no big deal, but it was certainly worth asking.  That’s when I got surprised.

Almost immediately, she said, “Four and a half sixths.  Is that right?

Some purists out there might complain that $\frac{4.5}{6}$ isn’t “proper,” but I’ve seen far too many situations where rigid insistence on proper form served instead to stifle creativity far more than to enhance understanding or to encourage deeper exploration or creativity.  I praised the heck out of her solution, letting her know that she had just made a fraction “smaller” for the first time (that I knew of).”  I didn’t mention that her proposed numerator wasn’t whole.  It didn’t matter.

How did you do that?

Easy,” she said, as if her answer would have been obvious to anyone.  “If you can double the parts of a fraction, why can’t you halve them, too?

Why not, indeed?  No matter what they end up looking like.

I asked if there were any other smaller, equivalent fractions.  That took lots more thought and time than I expected, certainly more than her nearly instantaneous $\frac{4.5}{6}$ had required.  Eventually, she asked if I could hold out 2 fingers beside her 10 so that she could look at 12 fingers together.  A little more thought led to her grouping the 12 fingers into 4 equal groups and a claim that $\frac{9}{12}=\frac{3}{4}$.  I still don’t completely understand her long explanation, but it wasn’t as clear (to me, anyway) that she was saying that 3 of her sub-groups contained the original 9 fingers (or months), and so 9 of an original 12 individual months was the same as 3 of 4 equal sub-groups of months.

When she woke the next morning, she asked if we could play any more fraction games

Conclusion

As students of all ages are learning, we need to reserve space for them to think and be creative.  They should be allowed to give correct mathematical solutions, even if those answers aren’t the arithmetic solutions our “trained” minds expect.  For me, I expected my daughter to say $\frac{9}{12}=\frac{3}{4}$, if she answered at all.  I was far more delighted to hear $\frac{9}{12}=\frac{4.5}{6}$ than she will ever know, and I’m convinced that she’s becoming more confident and capable because she was allowed to do so.

Three people and a monkey

Personally, I’ve always thought of educated Trial-and-Error as a valid problem-solving technique, or at least a problem-insight technique.  The challenge with this approach is the difficulty of trial-and-error to distinguish between situations with multiple solutions and those with only one.  I take every opportunity to remind my students of the dual goals of every problem solution:

1. Show that your solution(s) is (are) correct, and
2. Show that no other solution(s) exist(s).

Following is a variation of a fun problem from a member of the CPAM list-serve.

PARENTS:  This problem has been around for centuries in many different forms and more importantly, is a great problem-solving opportunity for elementary and middle school children.  Change the roles, genders, and animal to make the problem interesting for your children/students.

Three people and their pet monkey spend a day gathering bananas and go to sleep.  During the night, one wakes up, splits the bananas into three equal piles with one banana left over.  She gives the extra banana to the monkey, hides one pile for herself, combines the other two piles and goes back to sleep.
A little later, another person wakes and splits the remaining pile of bananas into three equal piles with one left over.  She gives the extra to the monkey, hides one pile for herself, combines the other two piles and goes back to sleep.
The third person wakes later and repeats the process.
In the morning, the remaining pile is divided evenly among the three peo
ple with nothing left for the monkey.

What is the smallest number of bananas that could have been in the original pile?

Don’t read any further if you want to solve this problem for yourself.

Because the problem asks for the smallest size of the original pile, there is only one answer and the fundamental weakness of a trial-and-error solution has been eliminated.

The recent CPAM post asked:  “Please help me to solve the attached problem in the way I could explain it to grade 6-7 students. I came up with very complicated equations. Is there any other method to solve it?”

From experience, I knew that I could approach this with lots of equations involving fractions, but I could instantly hear the groans of so many middle school students if this approach was the first attempted.  That’s when I tried another great problem-solving approach:  Work Backwards!

Because the problem wanted the smallest possible pile, I wondered if there could have been 1 banana each in the divided piles the next morning.  If so, there would have been 3 bananas in the re-combined pile, an impossible situation when you remember that this pile was the result of combining the two piles that remained after the third person had split the piles and hid her “share.”

Because the sum of three odd numbers is always odd, the evenly divided piles the next morning must have an even number of bananas each to avoid the impossible situation from that of the paragraph above.

So what if there were 2 bananas each in the divided piles the next morning?

• That would give 6 bananas in the re-combined pile after the 3rd person went back to sleep.
• The 6 bananas would have been in piles of 3 & 3 after the 3rd person’s pile of 3 was hidden and the monkey had 1, giving 10 (3+3+3+1) bananas in the re-combined pile after the 2nd person went back to sleep.
• Those 10 bananas would have been in piles of 5 & 5 after the 2nd person’s pile of 5 was hidden and the monkey had 1, giving 16 (5+5+5+1) bananas in the re-combined pile after the 1st person went back to sleep.
• Those 16 bananas would have been in piles of 8 & 8 after the 1st person’s pile of 8 was hidden and the monkey had 1, giving 25 (8+8+8+1) bananas in the original pile.

GENERALIZATION:  The original problem suggests that there are lots of solutions, so algebra may be the best way from here.  The approach learned from the trial-and-error approach can guide the global solution.

• If there are $x$ bananas in each final pile the next morning, then there were $3x$ after the 3rd went to sleep.
• That means there were $\frac{3x}{2}$ in each of the 3rd person’s piles for a total of $3*\frac{3x}{2}+1=\frac{9x+2}{2}$ bananas after the 2nd went to sleep.
• So there were $\frac{(9x+2)/2}{2}=\frac{9x+2}{4}$ in each of the 2nd person’s piles for a total of $3*\frac{9x+2}{4}+1=\frac{27x+10}{4}$ bananas after the 1st went to sleep.
• Finally, there were $\frac{(27x+10)/4}{2}=\frac{27x+10}{8}$ in each of the 1st person’s piles for a total of $3*\frac{27x+10}{8}+1=\frac{81x+38}{8}$ original bananas.

Notice that $x=2$ gives $\frac{81*2+38}{8}=\frac{200}{8}=25$, confirming the trial-and-error solution from earlier.

CONCLUSIONS:

• Because the answer is an integer, $81x+38$ must be a multiple of 8 which can only happen for even values of $x$, confirming the earlier hypothesis.
• $81x+38=(80x+32)+(x+6)=8(10x+4)+(x+6)$, therefore the only solutions, $x$, to this problem are those which make $x+6$ a multiple of 8.  The smallest positive value for which this is true is $x=2$, again confirming the trial-and-error solution.  All such final pile values of $x$ form an arithmetic sequence:  $2, 10, 18, 26, ...$  which correspond to initial piles of size $25, 106, 187, 268, ...$.

In my opinion, the initial trial-and-error solution is attainable by any elementary school student who knows how to add, but the logic to get there might need to be scaffolded for younger students.  The arithmetic behind the generalization can be expected of middle school students, but the inclusion of a variable on top of the trial-and-error procedure would push it out of reach of most students who haven’t had a pre-algebra course.  There are many high school students who eventually could understand the factoring arguments in the conclusion, but that logic, again, seems to be absent from most curricula.

This is a good problem that can be approached again and again by students as their mathematical sophistication matures.  I’d love to hear how others tackle it or specific results of student attempts.

Area 10 Squares – Proof & Additional Musings

Additional musings on the problem of Area 10 Squares:

Thanks, again to Dave Gale‘s inspirations and comments on my initial post. For some initial clarifications, what I was asking in Question 3 was whether these square areas ultimately can all be found after a certain undetermined point, thereby creating a largest area that could not be drawn on a square grid. I’m now convinced that the answer to this is a resounding NO–there is no area after which all integral square areas can be constructed using square grid paper. This is because there is no largest un-constructable area (proof below). This opens a new question.

Question 4:
Is there some type of 2-dimensional grid paper which does allow the construction of all square areas?

The 3-dimensional version of this question has been asked previously, and this year in the College Math Journal, Rick Parris of Exeter has “proved that if a cube has all of its vertices in then the edge length is an integer.”

Dave’s proposition above about determining whether an area 112 (or any other) can be made is very interesting. (BTW, 112 cannot be made.) I don’t have any thoughts at present about how to approach the feasibility of a random area. As a result of my searches, I still suspect (but haven’t proven) that non-perfect square multiples of 3 that aren’t multiples of pre-existing squares seem to be completely absent. This feels like a number theory question … not my area of expertise.

Whether or not you decide to read the following proof for why there are an infinite number of impossible-to-draw square areas using square grids, I think one more very interesting question is now raised.

Question 5:
Like the prime numbers, there is an infinite number of impossible-to-draw square areas. Is there a pattern to these impossible areas? (Remember that the pattern of the primes is one of the great unanswered questions in all of mathematics.)

THE PROOF
My proof does not feel the most elegant to me. But I do like how it proves the infinite nature of these numbers without ever looking at the numbers themselves. It works by showing that there are far more integers than there are ways to arrange them on a square grid, basically establishing that there is simply not enough room for all of the integers forcing some to be impossible. I don’t know the formal mathematics name for this principle, but I think of it as a reverse Pigeonhole Principle. Rather than having more pigeons than holes (guaranteeing duplication), in this case, the number of holes (numbers available to be found) grows faster than the number of pigeons (the areas of squares that can actually be determined on a square grid), guaranteeing that there will always be open holes (areas of squares that cannot be determined on using a square grid).

This exploration and proof far exceeds most (all?) textbooks, but the individual steps require nothing more than the ability to write an equation for an exponential function and find the sum a finite arithmetic sequence. The mathematics used here is clearly within the realm of what high school students CAN do. So will we allow them to explore, discover, and prove mathematics outside our formal curricula? I’m not saying that students should do THIS problem (although they should be encouraged in this direction if interested), but they must be encouraged to do something real to them.

Now on to a proof for why there must be an infinite number of impossible-to-draw square areas on a square grid.

This chart shows all possible areas that can be formed on a square grid. The level 0 squares are the horizontal squares discussed earlier. It is lower left-upper right symmetric (as noted on Dave’s ‘blog), so only the upper triangle is shown.

From this, the following can be counted.
Level 1 – Areas 1-9: 6 of 9 possibilities found (yellow)
Level 2 – Areas 10-99: 40 of 90 possibilities found (orange)
Level 3 – Areas 100-999: 342 of 900 possibilities found (blue)

The percentage of possible numbers appears to be declining and is always less than the possible number of areas. But a scant handful of data points does not always definitively describe a pattern.

Determining the total number of possible areas:
Level 1 has 9 single-digit areas. Level 2 has 90 two-digit areas, and Level 3 has 900 three-digit areas. By this pattern, Level M has M-digit areas. This is the number of holes that need to be filled by the squares we can find on the square grid.

Determining an upper bound for the number of areas that can be accommodated on a square grid:
Notice that if a horizontally-oriented square has area of Level M, then every tilted square in its column has area AT LEAST of Level M. Also, the last column that contains any Level M areas is column where floor is the floor function.

In the chart, Column 1 contains 2 areas, and every Column N contains exactly (N+1) areas. The total number of areas represented for Columns 1 through N is an arithmetic sequence, so an upper bound for the number of distinct square areas represented in Columns 1 to N (assuming no duplication, which of course there is) is .

The last column that contains any Level M areas has column number . Assuming all of the entries in the data chart up to column are Level M (another overestimate if is not an integer), then there are

maximum area values to fill the Level M area holes. This is an extreme over-estimate as it ignores the fact that this chart also contains all square areas from Level 1 through Level (M-1), and it also contains a few squares which can be determined multiple ways (e.g., area 25 squares).

Conclusion:
Both of these are dominated by base-10 exponential functions, but the number of areas to be found has a coefficient of 9 and the number of squares that can be found has coefficient 1/2. Further, the number of squares that can be found is decreased by an exponential function of base , accounting in part for the decreasing percentage of found areas noted in the data chart. That is, the number of possible areas grows faster than the number of areas that actually can be created on square grid paper.

While this proof does not say WHICH areas are possible (a great source for further questions and investigation!), it does show that the number of areas of squares impossible to find using a square grid grows without bound. Therefore, there is no largest area possible.