Another Fraction Question

OK, following up on our previous post, we have another fraction question. This one is not new, has appeared in the comments of a previous post, and many readers will have heard us bang on about it. Nonetheless, given the discussion on the previous question, and given the possibility that some new readers might not have yet read Marty’s Collected Sermons, it seems worthwhile giving the question its own post. Here it is:

    \[\color{Mulberry}\boldsymbol{(-1)^\frac26 \overset{?}{=} (-1)^\frac13}\]

Continue reading “Another Fraction Question”

Mr. McRae’s Triple Gift

This is a story from long, long ago. It is about Mr. McRae, who was our grade 4 teacher, at Macleod State School. We have written about Macleod before, and we have written, briefly, about Mr. McRae before, in regard to the moon landing:

I still have vivid-grainy memories of watching Armstrong’s first steps. A random few students from each class in Macleod State School were selected to go to the library to watch the event on the school’s one TV. I was not one of the lucky few. But Mr. Macrae, our wonderful Grade 4 teacher, just declared “Bugger it!”, determined which student in our class lived closest to the school, and sent out a posse to haul back the kid’s 2-ton TV. We then all watched the moon landing, enthralled and eternally grateful to Mr. Macrae.

He was that kind of guy. No-nonsense and intelligent and cultured.

The year he taught us, Mr. McRae was new to Macleod. He had just appeared on the playground before the first class of the year, tall and commanding. Rumour had it that he had played Under 19s for the Richmond Football Club, making Mr. McRae just shy of a Greek god. (The actual Greek god was, of course, Carl Ditterich.) He was a standard and excellent teacher. Firm, disciplined and disciplining, but kind, and with a calm and intelligent air of bemusement. He was the boss, but a thoughtful and unpredictable boss. Hence, our class getting to watch the moon landing. And, how else to explain the boxing match?

One day, Mr. McRae inadvertently started a harmless play-scuffle between two students. He then decided the dispute should be settled by a proper boxing match in front of the class. Once, of course, a kid had been sent home to fetch a couple pairs of boxing gloves. We can’t remember whether we lost, although we remember we didn’t win. In any case, neither of us had a clue how to box, and so the match was followed by Mr. McRae giving the class an impromptu lesson on technique. This was, to explain it a little, the era of Lionel Rose and Johnny Famechon and TV Ringside.

That’s all by way of background. The story we want to tell is of a mathematics lesson.

One Friday afternoon, Mr. McRae introduced his grade 4 class to Pythagoras’s theorem. Or, at least, to Pythagorean triples; we can’t specifically remember the triangles, or anything, but undoubtedly \boldsymbol{3^2 + 4^2 = 5^2} made an appearance. Why he showed us this, God only knows, but Mr. McRae ended the class with a challenge: find more triples. Our memory is that the specific challenge was to find a certain number of triples, maybe three, maybe five.

We have no idea what Mr. McRae hoped to achieve with this challenge, but we remember pondering, aimlessly, hoping to find triples. Eventually, by smart persistence and dumb luck, we stumbled upon the trick: doubling a triple gives a new triple. So, \boldsymbol{6^2 + 8^2 = 10^2}, and so on. With this kid-Eureka insight, we then happily spent the week-end doubling away.

Come Monday morning, Mr. McRae asked for the class’s triples. We proudly went to the blackboard and wrote up our largest creation. By memory, it was something in the millions. So,

    \[\boldsymbol{1572864^2 + 2096152^2 = 2621440^2}\]

or thereabouts. And then Mr. McRae uttered the fateful words:

“Let’s check it!”

There were the inevitable groans from the class, and the little Archimedes hero of the story was more popular than ever. But, Mr. McRae was the boss, and so we all set down to multiplying, including Mr. McRae himself. And, ten or so minutes later, the class collectively started to conclude … the equation was wrong. Yep, Little Archimedes had stuffed up. Which led to more fateful words:

“Let’s find the mistake!”

More groans, more multiplying, and eventually the error was found. By memory, after quite a few doubles, somewhere in the mid thousands. And, satisfied, Mr. McRae led the class on to whatever he had been planned for that day.

What is the moral? We have a reason for telling the story, beyond a simple tribute to a great, memorable teacher. We think there are morals there. We’ll leave it for the reader to ponder.

NotCH 2: A Digits Puzzle

OK, hands up who thought there was ever gonna be a second NotCH?

We’re not really a puzzle sort of guy, and base ten puzzles in particular tend to bore us. So, this is unlikely to be a regular thing. Still, the following question came up in some non-puzzle reading (upon which we plan to post very soon), and it struck us as interesting, for a couple reasons. And, a request to you smart loudmouths who comment frequently:

Please don’t give the game away until non-regular commenters have had time to think and/or comment. 

Start by writing out a few terms of the standard doubling sequence:

1, 2, 4, 8, 16, 32, 64, etc.  Continue reading “NotCH 2: A Digits Puzzle”

NotCH 1: Maths Masters Quizzes

More or less by accident, this post is the beginning of a new series: Not Crap Here.

A couple of people have suggested that we could occasionally include Dr. Jekyll material on this blog. You know, helpful stuff. It’s a decent idea, if our current thoughts weren’t so influenced by misanthropic disgust and murderous rage. Still, we’ve received two specific requests for the same old Jekyll material,* and which entailed some digging. Having finally dug, we’ve decided to post the material here, for whomever is interested. Whether or not there will ever be a NotCH 2 is anybody’s guess.

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ACARA CRASH 16: Unlevel Playing Field

Each Year’s content in the draft curriculum begins with a Level description, and each of the thirteen Level description begins with the exact same sentences:

The Australian Curriculum: Mathematics focuses on the development of a deep knowledge and conceptual understanding of mathematical structures and fluency with procedures. Students learn through the approaches for working mathematically, including modelling, investigation, experimentation and problem solving, all underpinned by the different forms of mathematical reasoning. [emphasis added]

Yep, “an equal focus on building fluency“, no doubt about it.

 

ACARA CRASH 15: Digital Insertion

Continuing to try to rid ourselves of ACARA irritants, the following are the “calculator” elaborations from Year 1 – Year 6 Number and Algebra (sic):

YEAR 1

using the constant function on a calculator to add ten to single digit numbers, recording the numbers to make, show and explore the patterns in a 0 – 100 chart

with the use of a calculator, exploring skip-counting sequences that start from different numbers, discussing patterns

modeling skip counting sequences using the constant function on a calculator, while saying, reading and recording the numbers as they go

Continue reading “ACARA CRASH 15: Digital Insertion”