Tag Archives: elementary

Unanticipated Proof Before Algebra

I was talking with one of our 5th graders, S,  last week about the difference between showing a few examples of numerical computations and developing a way to know something was true no matter what numbers were chosen.  I hadn’t started our conversation thinking about introducing proof.  Once we turned in that direction, I anticipated scaffolding him in a completely different direction, but S went his own way and reinforced for me the importance of listening and giving students the encouragement and room to build their own reasoning.

SETUP:  S had been telling me that he “knew” the product of an even number with any other number would always be even, while the product of any two odds was always odd.  He demonstrated this by showing lots of particular products, but I asked him if he was sure that it was still true if I were to pick some numbers he hadn’t used yet.  He was.

Then I asked him how many numbers were possible to use.  He promptly replied “infinite” at which point he finally started to see the difficulty with demonstrating that every product worked.  “We don’t have enough time” to do all that, he said.  Finally, I had maneuvered him to perhaps his first ever realization for the need for proof.

ANTICIPATION:  But S knew nothing of formal algebra.  From my experiences with younger students sans algebra, I thought I would eventually need to help him translate his numerical problem into a geometric one.  But this story is about S’s reasoning, not mine.

INSIGHT:  I asked S how he would handle any numbers I asked him to multiply to prove his claims, even if I gave him some ridiculously large ones.  “It’s really not as hard as that,” S told me.  He quickly scribbled

s1

on his paper and covered up all but the one’s digit.  “You see,” he said, “all that matters is the units.  You can make the number as big as you want and I just need to look at the last digit.”  Without using this language, S was venturing into an even-odd proof via modular arithmetic.

With some more thought, he reasoned that he would focus on just the units digit through repeated multiples and see what happened.

FIFTH GRADE PROOF:  S’s math class is currently working through a multiplication unit in our 5th grade Bridges curriculum, so he was already in the mindset of multiples.  Since he said only the units digit mattered, he decided he could start with any even number and look at all of its multiples.  That is, he could keep adding the number to itself and see what happened.  As shown below, he first chose 32 and found the next four multiples, 64, 96, 128, and 160.  After that, S said the very next number in the list would end in a 2 and the loop would start all over again.

s2

He stopped talking for several seconds, and then he smiled.  “I don’t have to look at every multiple of 32.  Any multiple will end up somewhere in my cycle and I’ve already shown that every number in this cycle is even.  Every multiple of 32 must be even!”  It was a pretty powerful moment.  Since he only needed to see the last digit, and any number ending in 2 would just add 2s to the units, this cycle now represented every number ending in 2 in the universe.  The last line above was S’s use of 1002 to show that the same cycling happened for another “2 number.”

DIFFERENT KINDS OF CYCLES:  So could he use this for all multiples of even numbers?  His next try was an “8 number.”

s3

After five multiples of 18, he achieved the same cycling.  Even cooler, he noticed that the cycle for “8 numbers” was the 2 number” cycle backwards.

Also note that after S completed his 2s and 8s lists, he used only single digit seed numbers as the bigger starting numbers only complicated his examples.  He was on a roll now.

s4

I asked him how the “4 number” cycle was related.  He noticed that the 4s used every other number in the “2 number” cycle.  It was like skip counting, he said.  Another lightbulb went off.

“And that’s because 4 is twice 2, so I just take every 2nd multiple in the first cycle!”  He quickly scratched out a “6 number” example.

s5

This, too, cycled, but more importantly, because 6 is thrice 2, he said that was why this list used every 3rd number in the “2 number” cycle.  In that way, every even number multiple list was the same as the “2 number” list, you just skip-counted by different steps on your way through the list.

When I asked how he could get all the numbers in such a short list when he was counting by 3s, S said it wasn’t a problem at all.  Since it cycled, whenever you got to the end of a list, just go back to the beginning and keep counting.  We didn’t touch it last week, but he had opened the door to modular arithmetic.

I won’t show them here, but his “0 number” list always ended in 0s.  “This one isn’t very interesting,” he said.  I smiled.

ODDS:  It took a little more thought to start his odd number proof, because every other multiple was even.  After he recognized these as even numbers, S decided to list every other multiple as shown with his “1 number” and “3 number” lists.

s7

As with the evens, the odd number lists could all be seen as skip-counted versions of each other.  Also, the 1s and 9s were written backwards from each other, and so were the 3s and 7s.  “5 number” lists were declared to be as boring as “0 numbers”.  Not only did the odds ultimately end up cycling essentially the same as the evens, but they had the same sort of underlying relationships.

CONCLUSION:  At this point, S declared that since he had shown every possible case for evens and odds, then he had shown that any multiple of an even number was always even, and any odd multiple of an odd number was odd.  And he knew this because no matter how far down the list he went, eventually any multiple had to end up someplace in his cycles.  At that point I reminded S of his earlier claim that there was an infinite number of even and odd numbers.  When he realized that he had just shown a case-by-case reason for more numbers than he could ever demonstrate by hand, he sat back in his chair, exclaiming, “Whoa!  That’s cool!”

It’s not a formal mathematical proof, and when S learns some algebra, he’ll be able to accomplish his cases far more efficiently, but this was an unexpectedly nice and perfectly legitimate numerical proof of even and odd multiples for an elementary student.

 

Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.

Boaler

I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.

Robinsons1

Robinsons2

“Math Play” Presentation for Early Childhood teachers

Even though my teaching experiences are all middle and high school, as a PreSchool-12 math chair and father of 3 young children, I’m intensely interested in how math is presented to very young people.

As a result, I’m presenting ideas for teaching math through fun and exploration to about 55 Cleveland area pre-school through kindergarten teachers this morning.  My handout is on Scribd and should show below.  Math is  about Play and Curiosity.  Teach it that way.

 

Gender and Expectations Lessons from Research

A couple reports from NPR yesterday have me thinking about some of the articles I’ve compiling in my Diigo library about what I’ve learned about gender and expectations differences in parenting and teaching.  I don’t have anything particular to tie together here, but I thought these four resources were more than I could comfortably tie together in a coherent Tweet thread, so I thought I’d gather them into an impromptu ‘blog post.

Girls May Get More ‘Teaching Time’ From Parents Than Boys Do via @NPR.  Excerpts:
… ” ‘How often do you read with your child?’ or ‘Do you teach them the alphabet or numbers?’ … Systematically parents spent more time doing these activities with girls.”
… “Since parents say they spend the same amount of time overall with boys and girls, Baker’s analysis suggests that if parents are spending more time with girls on cognitive activities, they must be spending more time with boys on other kinds of activities.”
… “The costs of investing in cognitive activities is different when it comes to boys and girls. As an economist, he isn’t referring to cost in the sense of cash; he means cost in the sense of effort.”

Gender Gap Disappears in School Math Competitions via sciencedaily.com .  Excerpt:
… “Most school math contests are one-shot events where girls underperform relative to their male classmates. But a new study by a Brigham Young University economist presents a different picture.  Twenty-four local elementary schools changed the format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.”
… “It’s really encouraging that seemingly large gaps disappear just by keeping [girls] in the game longer.”

A broader look at school expectations leading to enhanced math performance:  What Distinguishes a Superschool From the Rest via ideas.time.com .
… “The difference seems to lie in whether a school focuses on basic competence or encourages exceptional achievement. While almost all the schools saw it as their responsibility to cover the math knowledge necessary to do well on the SATs, the authors noted that “there is much less uniformity in whether schools encourage gifted students to develop more advanced problem solving skills and reach the higher level of mastery of high school mathematics.”
… “The fact that the highest achieving girls in the U.S. are concentrated in a very small set of schools, the authors write, indicates ‘that almost all girls with the ability to reach high math achievement levels are not doing so.’ ”

Girls, Boys And Toys: Rethinking Stereotypes In What Kids Play With via @NPR.  Excerpts:
… Some toy companies are re-thinking gender-specific marketing and branding.
… “I think what they were worried about was causing gender identification needlessly — to turn off passive learning, passive expression down the road, even passive economic opportunity for girls or boys if they felt they couldn’t do something because of societal norms,”
… “It’ll be interesting to see how this changes the attitudes of parents and of kids over time or whether or not it does. There may be some hard-wired differences,”

 

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.

4s_1

4s_3

4s_2

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.

ADDENDA:

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.
Subtraction1

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.

33333You 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?

33If 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

333

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

Multiplication Puzzle for the Very Young

I just read a recent post on NRICH Mathematics that asked readers or students to list four consecutive whole numbers and compare the products of the outer pair of numbers in the list to the product of the inner pair.  For example, if you used the list {4, 5, 6, 7}, you would have 4\cdot 7=28 and 5\cdot 6=30.  Nothing particularly exciting seems to be here, but try another list of four consecutive whole numbers.  Grab a calculator if you want to be particularly daring or obnoxious with the members in your list.  Do you notice anything now?

I argue the beauty of mathematics as the “science of patterns” kicks in after you find these products for a few different lists.

LEVEL 1:  For the very young who are just learning to multiply, I think this is a GRAND problem.  No proof required.  It’s just crazy cool that those two products always have the same relationship.  Allowing calculators to permit young explorers to try lists beyond their ability to hand or mentally compute enhances the mystery, in my opinion.  

I just played this with my eldest daughter.  She first wrote {19, 20, 21, 22} when I asked her for a list of consecutive numbers.  When I then asked her for the products, she asked if she could use a smaller list.  She opted for {3, 4, 5, 6} and {1, 2, 3, 4} without seeing the pattern.  When I offered a calculator for her original list, she got 418 & 420.  Surprised that they were so close, she said, “Wow, they’re only 2 apart!” I asked if that happened other times.  She looked at her simpler two lists and exclaimed, “Cool!”  I asked if that always happened.  She said, “No.  It couldn’t.”  When I asked for a list where it wouldn’t, she suggested {401, 402, 403, 404}.  The outer product was 162004.  You should have seen her face after she pressed enter on the inner product to get 162006.  “Maybe it does always work!”  Then she asked if she could move on to clean her desk.  Game over … for now.

Part of the power and beauty of mathematics lies in showing that patterns are universal and aren’t limited to numbers we can manipulate quickly in our heads.  I think calculators added to my daughter’s wonder.  I’d love to see my daughter going up to one of her teachers, posing the problem, and predicting the answer without ever knowing the numbers the teacher (or anyone else) had picked.  I think I’d smile even bigger if she had a calculator at hand to offer the adult some “help” if needed!  Math is magical.  Play it up!

LEVEL 2a:  Extend to all integers.  NRICH suggests that the lists need to be whole numbers.  That just isn’t true.  You can start with any integer.  My eldest has been playing with adding negative numbers lately, so I may see if she’s interested in multiplication of negatives.  I’ll think about how to make that idea make sense to her.  At some point in the future, I’ll bring this problem up again and she’ll get an even bigger kick out of seeing that it doesn’t just apply to ordinary and ridiculously large numbers, but negatives, too.

LEVEL 2b:  Proof for the very young.  The NRICH site offers two solutions from “students”.  Whether she’s real or fictional, the approach “Alison” uses is one that I think some sophisticated young learners could grasp long before they learn what a variable is.  Granted, the geometric understanding of multiplication technically works only for specific (not generic) products, but if you set up a few of these, your young one might start to see how the areas grow as the list numbers grow, but the differences in the areas remain constant.

NOTE:  LEVELS 2a and 2b, in my mind, are pretty interchangeable, depending on the readiness and interest of your young learners.  As with all things for young people, throw out the line.  If the interest isn’t there, save the idea for another day.  If you get a nibble, prepare to play!!!

LEVEL 3:  Extend to any arithmetic sequence.  The suggestions NRICH makes for extending the problem all dance around the idea that this property works for any list of four consecutive elements of any arithmetic sequence.  The difference between the two products depends solely on the common difference of the sequence and is completely independent of the initial term in the sequence.  Try {1.1, 1.2, 1.3, 1.4}.  The difference in the outer and inner pair products will be the same as for {98.8, 98.9, 99.0, 99.1} simply because both lists increase by 0.1.

LEVEL 4:  Algebra.  Those who remember their algebra classes may have jumped right to an algebraic justification.  That’s what I did, and that’s the solution “Charlie” gives on the original NRICH post.  In a way, I think I cheated myself out of seeking the pattern as my daughter discovered it.  Whenever your young ones are ready to deal with the magic and power of variables, try out proving this for integers.  When they’re ready for more, prove it for all arithmetic sequences with any initial term.  You’ll know they’re strong when they can argue on their own why the initial term is irrelevant.

LEVEL 5:  More Algebra.  This “trick” extends to to any arithmetic sequence of any length.  With algebra, one can determine a formula for the difference between the products of the last terms and the next-to-last terms.  I think a talented middle school student or young high school student who knows how to handle very generic cases could find that formula.

And it all starts with playing with some little numbers.

Creative Subtraction for the Very Young

OK, this is far from a revolutionary idea, but I was impressed with my eldest daughter’s approach to a subtraction problem. What I present may already be used by others, but it was new to me.

She was interested a few months ago in learning what negative numbers were and what caused them.  Of course, I dove right in and explained.  From time-to-time she’s asked for more, but mostly I thought it was an idea that had gone fallow.  Then she wrote the following last night.

\begin{array}{ccc} &7&0 \\ -&4&2 \\\hline &3&0 \\ &-&2 \\\hline &2&8 \end{array}

I had learned column subtraction from right to left with “borrowing.”  It was an algorithm that I understood; many classmates struggled.  My daughter worked left to right by column/place value.  So, she computed 70-40 and wrote 30.  Then she used her knowledge of negative numbers and asked me, “So 0 minus 2 is -2, right?”.  She then wrote -2.  The rest was easy.  What impressed me was her adaptation of an algorithm she had been using for partial sums. If a student understands negative numbers, I wonder if this might be a cleaner approach.  The thinking is also similar to what one might use when calculating using an abacus.  Cool.

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.

SOLUTION ALERT: 
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?

Blokus

I mentioned in a post from earlier this month that I thought Blokus was one of the greatest games around for young and old.  Blokus is particularly phenomenal not because of the game’s published rules, but because its simplicity and flexibility allow the creation of many games-within-a-game.  In short, Blokus encourages creativity and allows the space for that to happen.

Here are some of the ways I’ve used/played Blokus successfully with my young daughters, my high school students, and with adult friends and colleagues.  I’ve ranked my suggested extensions according to how I think young learners would be able to handle them.  Just because I list something as challenging certainly doesn’t mean it is beyond any learner.  PLAY and maybe you’ll accidentally learn something!

  • First, the game as described in the Blokus rules is just phenomenal.  If you never try any of my extensions, the game is worth it.

ELEMENTARY EXTENSIONS

  • I started Blokus with my oldest daughter when she was 4 or 5 years old.  My philosophy of learning-play has been to give my children as much of the “rules” as they can understand and then let them decide how they want to play. My eldest wasn’t interested in the game I described, but she loved the brightly colored pieces and the freedom it gave her to create geometric designs.  No problem.  The limited number of pieces and shapes restricted her in a way that coloring and drawing on paper never did.  She had to plan her designs to get what she wanted.  Learning happened.
  • When she finally wanted to start playing a game, she didn’t like the competitive “block your opponent” rule, so she created a rule that you placed pieces on the board so that everyone could try to play all of their pieces.  She called it “Nice Blokus”.  We each played two colors and alternated laying down our pieces.  It can be just as (or even more) challenging trying to be nice to your opponents as it is competing against them.  (Perhaps there’s a broader lesson here!)
  • When we did start playing competitively, it was interesting to help her learn to strategize.  Don’t just play, talk about better ways to play.  What pieces do you want to play first and which do you want to save for late in the game?  She doesn’t always want to talk about strategies, sometimes she simply wants to play–so I adapt.  When she does want to talk, we definitely engage in serious, thoughtful, directed, and intentional play.

MODERATE EXTENSION

  • When you glance at the various pieces listed in the Blokus game, I bet many see not much more than different shapes, but there’s some very nice geometry involved.  How many shapes can you make from n squares, all connected by their edges, if you were permitted only 1 square?  2 squares?  3?  4?  5?  This could be an early exposure to the logic of mathematics of multiple cases.
    1. Obviously there is only one way to have one square.
    2. There is also only one way to have two squares.  While the orientation may change, ultimately all connections of two squares are the same.
    3. Three squares?  I don’t think it’s that hard to figure out that there are just 2 ways to do this:  three in a row–or–in an “L” shape.  One way to establish this would be to take a 2-square Blokus piece and a 1-square piece and try arranging them in different patterns until you are satisfied you have all the arrangements.
    4. What about 4-square arrangements?  Again you could use the 1-, 2-, and 3-square Blokus pieces to discover this answer on your own.  I’ll hold this answer to the end of this post in case you want to explore.
    5. What about 5-square arrangements?  These are called pentominoes, and there are many cool extensions using these. There are 12 unique pentominoes–a factoid I must give away now because it leads to many other games.  Whether or not you tell someone there are 12, can you prove that there are only 12?

NOTE:  Each color in Blokus contains every single possible 1-, 2-, 3-, 4-square, and pentomino arrangement.  Cool.

A math colleague, LG, once described the goal of the math problem-solving process this way:  FIRST, show your answer works.  SECOND, show no other answers work My experiences with most mathematics resources and teaching is that students always deal with LG’s first criterion, but rarely deal with the necessity of the second.  The problem of determining the number of unique pieces you can make with a given number of squares requires careful logic to make sure that one has found all solutions and much more importantly, that none are missing.

CHALLENGING EXTENSIONS

  • Because there are 12 unique pentominoes, obviously their combined area is 60 units.  If you create an 8×8 square grid and exclude any 2×2 square grid from its interior, it is always possible to fill in the remaining 60 units using the 12 pentominoes.  The short video below shows a solution to one of these problems.
  • While it seems like there might be many of these “omit a 2×2 square from an 8×8 square and fill in the rest with 12 pentominoes” puzzles, in reality, there are surprisingly few.  Using rotational and reflection symmetries, I’m convinced there are only 10 unique such puzzles.  Can you or one of your learners prove this?
  • Given that there are only 10 of these fill-in puzzles, how many unique solutions are there to each?  I don’t know the answer to this question.
  • In researching for this post, I encountered another interesting puzzle (link in the next bullet). In short, the total area of the pentominoes, 60, is equivalent to 6×10, 5×12, 4×15, and 3×20.  Use the 12 unique pentominoes to fill a rectangle defined by each of these four dimensions.  Even though 60=2×30, explain why it isn’t worth trying to use pentominoes to fill a 2×30 rectangle.
  • The last puzzle is nice, but going back to LG’s admonition, how do you know you’ve found all possible solutions?  This is a very hard problem to do and likely requires a computer search according to the Tiling Rectangles section in this article.

CONCLUSION

I’m hoping these examples have convinced you that Blokus has the potential to be so much more than just a box game.  Be creative.  Explore.  Maybe you’ll accidentally learn something.  Most of all, HAVE FUN!!!

SOLUTION ALERT

This next image shows the singleton solutions for the 1- and 2-square arrangements (yellow) and the two 3-square (green) arrangements described above.  There are five 3-square arrangements.  There are several ways to create convincing logical arguments that all arrangements have been found.  My personal approach was to imagine arrangements (from left to right in the red) with all four in a row, then with three in a row, and finally with no more than two in a row.

I mentioned earlier that there are 12 pentominoes.  The image below shows them all.  How can you extend my logic from the 4-square pieces to account for all of the pentominoes?

All of this makes me wonder if there is a numerical pattern lurking here.  If there is 1 1-square piece, 1 2-square piece, 2 3-square pieces, 5 4-square pieces, and 12 5-square pieces, how many n-square arrangements would there be?  Right now, I don’t know the answer.