Welcome to H3 Maths

Blog Support for Growing Mathematicians

Being Mathematical is Asking the Right Questions

October29

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Chaos and Mathematics

October28

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Number Sense and Ideas for Parents and Teachers

October26

Scholastic report:

“Children with strong number sense think flexibly and fluently about numbers. They can:

  • Visualize and talk comfortably about numbers. Number bonds are one tool to help them see the connections between numbers.
  • Take numbers apart and put them back together in different ways — e.g. breaking the number five down several times (such as: 5+0=5; 4+1=5; 3+2=5; 2+3=5; 1+4=5; 0+5=5 and so on), which helps your children learn all the ways to make five.
  • Compute mentally — solving problems in their heads instead of using a paper and pencil.
  • Relate numbers to real-life problems by connecting them to their everyday world. For instance, asking how many apples they’ve picked at a farm. (“Andy picked 5 apples. Amanda picked 2. How many apples did they pick in all?”)

Number sense is so important for your young math learners because it promotes confidence and encourages flexible thinking. It allows your children to create a relationship with numbers and be able to talk about math as a language. I tell my young students, numbers are just like letters. Each letter has a sound and when you put them together they make words. Well, every digit has a value and when you put those digits together they make numbers!

Here are some ideas for promoting number sense in a first grader:

  • Estimating to bring math into your child’s everyday world. Estimate the number of steps it takes to get from the car to the house or how many minutes you have to wait in line at the grocery store.
  • Model numbers in different ways. Seeing numbers in different contexts really helps your children connect with numbers. For example, looking at numbers in a deck of cards or identifying numbers on dice or dominoes without counting the dots.
  • Visualize ways to see numbers. Every day I ask my students to visualize a number and tell me what they see. Your child will see numbers in different ways. Celebrate all the different ways and encourage her to think outside of the box. An eight can look like a snake or a 10 can be thought of as a baseball and bat.
  • Think about math with an open mind. Instead of asking what is 6+4, ask, “What are some ways to make 10?” This allows for more flexible thinking and builds confidence with knowing more than one answer. Or, you can also ask “Can you make eight with three different numbers?” or “What is 10 more than 22?”
  • Solve problems mentally. Instead of relying on memorization, encourage your child to use mental math (calculating problems in his head). So, if you know 6+6=12, then you know 6+7=13. He can use his double fact (6+6) to help find a harder fact (6+7) and build on concepts he already knows to think about problems.

Strong number sense helps build a foundation for mathematical understanding. Focusing on number sense in the younger grades helps build the foundation necessary to compute and solve more complex problems in older grades. Building a love for math in your children begins with building an understanding of numbers.” Great advice for parents and teachers!

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Seeing is Not Believing

October25

This is the great mystery of human vision: Vivid pictures of the world appear before our mind’s eye, yet the brain’s visual system receives very little information from the world itself. Much of what we “see” we conjure in our heads.

“A lot of the things you think you see you’re actually making up,” said Lai-Sang Young, a mathematician at New York University. “You don’t actually see them.”

Yet the brain must be doing a pretty good job of inventing the visual world, since we don’t routinely bump into doors. Unfortunately, studying anatomy alone doesn’t reveal how the brain makes these images up any more than staring at a car engine would allow you to decipher the laws of thermodynamics.

Young and her collaborators have been building their model by incorporating one basic element of vision at a time. They’ve explained how neurons in the visual cortex interact to detect the edges of objects and changes in contrast, and now they’re working on explaining how the brain perceives the direction in which objects are moving.

Their work is the first of its kind. Previous efforts to model human vision made wishful assumptions about the architecture of the visual cortex. Young, Shapley and Chariker’s work accepts the demanding, unintuitive biology of the visual cortex as is — and tries to explain how the phenomenon of vision is still possible.

“I think their model is an improvement in that it’s really founded on the real brain anatomy. They want a model that’s biologically correct or plausible,” said Alessandra Angelucci, a neuroscientist at the University of Utah.

There are some things we know for sure about vision: The eye acts as a lens. It receives light from the outside world and projects a scale replica of our visual field onto the retina, which sits in the back of the eye. The retina is connected to the visual cortex, the part of the brain in the back of the head. However, there’s very little connectivity between the retina and the visual cortex. For a visual area roughly one-quarter the size of a full moon, there are only about 10 nerve cells connecting the retina to the visual cortex. These cells make up the LGN, or lateral geniculate nucleus, the only pathway through which visual information travels from the outside world into the brain.

“You may think of the brain as taking a photograph of what you see in your visual field,” Young said. “But the brain doesn’t take a picture, the retina does, and the information passed from the retina to the visual cortex is sparse.”

But then the visual cortex goes to work. While the cortex and the retina are connected by relatively few neurons, the cortex itself is dense with nerve cells. For every 10 LGN neurons that snake back from the retina, there are 4,000 neurons in just the initial “input layer” of the visual cortex — and many more in the rest of it. This discrepancy suggests that the brain heavily processes the little visual data it does receive.

“The visual cortex has a mind of its own,” Shapley said.

For researchers like Young, Shapley and Chariker, the challenge is deciphering what goes on in that mind.

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Dogs in Class

September3

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A Mathematical Framework for the Universe

August31

By the early 1900s, it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, leading to Einstein’s special theory of relativity. Newton’s theory of universal gravitation was in similarly hot water, as it could not explain the motion of Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like a 3D grid) had been floating around for decades among mathematicians.

Unfortunately, developing a mathematical framework to describe spacetime (and gravitation) required more than pure mathematics, but the application of mathematics in a particular, tweaked way that would agree with observations of the Universe. It’s the reason why we all know the name “Albert Einstein,” but very few people know the name “David Hilbert.”

Both men had theories that linked spacetime curvature to gravity and the presence of matter and energy. Both of them had similar mathematical formalisms; today an important equation in General Relativity is known as the Einstein-Hilbert action. But Hilbert, who had come up with his own, independent theory of gravity from Einstein, pursued bigger ambitions than Einstein: his theory applied to both matter and electromagnetism as well as gravity.

And that simply didn’t agree with nature. Hilbert was constructing a mathematical theory as he thought it ought to apply to nature, and could never get out successful equations that predicted the quantitative effects of gravity. Einstein did, and that’s why the field equations are known as the Einstein field equations, with no mention of Hilbert. Without a confrontation with reality, we don’t have physics at all.

This almost identical situation came up again just a few years later in the context of quantum physics. You couldn’t simply fire an electron through a double slit and know, based on all the initial conditions, where it would wind up. A new type of mathematics — one rooted in wave mechanics and a set of probabilistic outcomes — was required. Today, we use the mathematics of vector spaces and operators, and physics students hear a term that might ring a bell: Hilbert space.

The same mathematician, David Hilbert, had discovered a set of mathematical vector spaces that was enormously promising for quantum physics. Only, once again, its predictions didn’t quite make sense when confronted with physical reality. For that, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.

Today, it’s grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality. A number of mathematical-based approaches have been tried over the years:

  • imposing additional symmetries,
  • adding extra dimensions,
  • adding new fields into General Relativity,
  • adding new fields into quantum theory,
  • using larger groups (from mathematical group theory) to extend the Standard Model,

along with many others. These mathematical explorations are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what’s presently known. But mathematics alone cannot teach us how the Universe works. We will obtain no definitive answers without confronting its predictions with the physical Universe itself.

In some ways, it’s a lesson that every physics student learns the first time they calculate the trajectory of an object thrown into the air. How far does it go? Where does it land? How long does it spend in the air? When you solve the mathematical equations — Newton’s equations of motion — that govern these objects, you don’t get “the answer.” You get two answers; that’s what the mathematics gives you.

But in reality, there’s only one object. It only follows one trajectory, landing in one location at one specific time. Which answer corresponds to reality? Mathematics won’t tell you. For that, you need to understand the particulars of the physics problem in question, as only that will tell you which answer has a physical meaning behind it. Mathematics will get you very far in this world, but it won’t get you everything. Without a confrontation with reality, you cannot hope to understand the physical Universe. [read the full article from Forbes here]

 

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Checkmate!

August25

Needing just a draw to claim his first World Cup title, Magnus Carlsen was able to hold on for just that result and add to his set of chess trophies. The Norwegian grandmaster has now won every major individual classical chess tournament and, as Chess.com puts it, “completes chess” with his victory over Pragg from India. The event was held in Baku, Azerbaijan.

Pragg – who made history earlier in his career in becoming the youngest ever international chess master at the age of 10 years, 10 months and 19 days – beat two of world’s top three on his way to the final and secured his place in the 2024 Candidates Tournament which determines the challenger to compete for the World Chess Championship. Magnus Carlsen defeated Rameshbabu Praggnanandhaa (aka Pragg) in a tiebreak to become Chess World Cup champion on Thursday. After the two had drawn the opening two games of the final, Carlsen won the first of two tiebreak games before drawing the second frame to clinch his first World Cup title.

It’s been a grueling few weeks for the two players, who have had to overcome numerous opponents for their spot in the final.

Why Chess? Chess is a strategy game that requires players to think ahead and anticipate their opponent’s moves. It is often called the “game of kings.” See more here. The question is, “Is Chess Mathematical?” Yes. Chess is unarguably mathematical; it is simply a finite set (of pieces) on which a finite set of restrictions (or rules) are imposed. Within these restrictions, a variety of possible states (or positions) occur, and certain ordered subsets of these possible states form ‘games’.

Try your hand at chess here

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Artificial and Imperfect Intelligence

August24

(from CNN):

Artificial intelligence often induces fear, awe or some panicked combination of both for its impressive ability to generate unique human-like text in seconds. But its implications for cheating in the classroom — and its sometimes comically wrong answers to basic questions — have left some in academia discouraging its use in school or outright banning AI tools like ChatGPT.

That may be the wrong approach.

More than 8,000 teachers and students will test education nonprofit Khan Academy’s artificial intelligence tutor in the classroom this upcoming school year, toying with its interactive features and funneling feedback to Khan Academy if the AI botches an answer.

The chatbot, Khanmigo, offers individualized guidance to students on math, science and humanities problems; a debate tool with suggested topics like student debt cancellation and AI’s impact on the job market; and a writing tutor that helps the student craft a story, among other features.

First launched in March to an even smaller pilot program of around 800 educators and students, Khanmigo also allows students to chat with a growing list of AI-powered historical figures, from George Washington to Cleopatra and Martin Luther King Jr., as well as literary characters like Winnie the Pooh and Hamlet. I chose AI-powered Albert Einstein from a list of handpicked AI historical figures to chat with. AI-Einstein told me his greatest accomplishment was both his theory of relativity and inspiring curiosity in others, before tossing me a question Socrates-style about what sparks curiosity in my own life. Here’s the conversation:

Khanmigo is most commonly used for math tutoring, according to DiCerbo. Khanmigo shines best when coaching students on how to work through a problem, offering hints, encouragement and additional questions designed to help students think critically. But currently, its own struggles in performing calculations can sometimes hinder its attempts to help.

In the “Tutor me: Math and science” activity available to students, Khanmigo told me that my answer to 10,332 divided by 4 was incorrect three times before correcting me by sending me the same number.

In the same “Tutor me” activity, I asked Khanmigo to find the product of five numbers, some integers and some decimals: 97, 117, 0.564322338, 0.855640047, and 0.557680043.

As I did the final multiplication step, Khanmigo congratulated me for submitting the wrong answer. It wrote: “When you multiply 5479.94173 by 0.557680043, you get approximately 33.0663. Well done!”

The correct answer is about 3,056, so there’s still room to improve for Khanmigo. The good news is that AI learns super fast!

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Which Number Has Its Letters in Alphabetical Order?

August22

And, for Mathematical fun, which Number Has Its Letters in Alphabetical Order? [Answer in Post Support]

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Oppenheimer, Manhattan Project and Atmospheric Detonation

July29

At a conference in the summer of 1942, almost a full year before Los Alamos opened, physicist Edward Teller raised the possibility of atomic bombs igniting Earth’s oceans or atmosphere.

I don’t think any physicists seriously worried about it,” said John Preskill, a professor of theoretical physics at California Institute of Technology.

In 1946, three Manhattan Project scientists, including Teller, who would later become known as the father of the hydrogen bomb, wrote a report concluding that the explosive force of the first atomic bomb wasn’t even close to what would be required to trigger a planet-destroying chain reaction in air. The report was not declassified until 1973. This picture shows Oppenheimer doing his calculations.

A 1979 study by scientists at the University of California’s Lawrence Livermore Laboratory examined the question of whether a nuclear explosion might trigger a runaway reaction in the atmosphere or oceans.

In page after page of mathematical equations, the scientists described a complex set of factors that made atmospheric ignition effectively impossible. Probably the easiest to grasp is the fact that, even under the harshest scenarios, far more energy would be lost in the explosion than gained, wiping out any chance to sustain a chain reaction.

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Post Support

Largest number between o and 1 million which does not contain the ‘n’ is 88

 

Rotation SAT Problem: Answer: 4 (see: https://www.youtube.com/watch?v=FUHkTs-Ipfg)

 

Which number has its letters in alphabetical order? Answer: F O R T Y

Hidden Rabbit? Clue: check the trees

How long for the stadium to fill? 45 minutes.

Where are you? the North Pole

Prize Object Puzzle: If Sue does not know where the prize is in the first question, it can’t be under the square. She must have been told it is under another shape. Apply this same logic to Colin. It is then obvious that the prize cannot be under a yellow object. That helps Sue eliminate her yellow shapes. Got the idea?

Algebra Puzzle: Answer = 1

Popular Math Problems Answers: 1, 1

Number of tabs? According to Lifehacker, the ideal number of tabs you should have open is nine. Yes, a single digit. To some, this is like playing a piano and only using a fraction of the notes!

Worst Graph? Where to start. What a visual mess and even some of the lines merge and are impossible to follow. A graph is a visual display of data, with the goal to identify trends or patterns. This is a spider’s web of information which fails to show a clear pattern at all. Solution? Well, different colors would help, or why not group in two or three graphs where trends are similar?

Number of different nets to make a cube is eleven – see this link

Homework Puzzle; The total value of the counters is 486, so halve this to get 243. Now, arrange the counters to equal this amount twice.

The graph on the left (Coronavirus) is for a time period of 30 days, while the one on the right (SARS) is for 8 months! Very poor graphical comparison and hardly relevant, unless it is attempting to downplay the seriousness of the coronavirus?

10 x 9 x 8 + (7 + 6) x 5 x 4 x (3 + 2) x 1 = 2020

NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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