#anatomy/physiology - Keto eating and measuring weight loss

Question

I started following a Keto way of eating (if you aren't familiar with this concept, you can look to the videos of Dr. Ken Berry, Dr. Anette Bosworth (she calls herself Dr. Boz), and Nurse Cindy (sorry - don't know her full name), among others.

One of the hard parts of researching this way of eating, there is a lot of testimonials concerning weight loss. There are a lot of claims about various amounts of weight lost, but it's hard to look at all these numbers and try to figure out what's what - someone who loses a little bit of weight vs someone who loses a lot of weight and different starting points. Is there a way to help make some sense of it? Help!

Answer

There is a way - use percentages. See below for how...

Analysis

First off, I totally hear you on having to deal with a sea of numbers and trying to find a way to make sense of what is being reported.

Also, my wife and I started eating Keto a couple of months ago and so I am familiar with Keto. Further, my wife runs a Facebook support group for Keto called Keto Living for Health and so you can access support through her group, if you are looking for a Facebook group that speaks to you.

Before I start the weight discussion, I would like to point out that weight is a very difficult metric - there is so much going on inside our 100 trillion cells of our body that weight will fluctuate, and sometimes wildly, with no apparent reason. (Keep in mind that only 10% of those cells are actually you - or in other words has DNA that is your human DNA, and the other 90% is all the flora and fauna that live within us that do all the work (such as the gut flora that actually break down food we consume into the constituent nutrients that get absorbed by our intestines)).

Ok - so weight is not a reliable indicator on a day-to-day basis to see what is happening inside your body. However, every house has a weight scale and it's easy to use, so oftentimes people will use day-to-day weight change as a way to track progress. However, as you lose size, weight will start to follow.

With all that understood about weight, let's now talk about measuring weight loss.

Let's talk about my weight loss. I started at 87.5 kg and I recently measured my progress at 78.7 kg. How much have I lost? Well... we can do the subtraction:

87.5 - 78.7 = 8.8 kg

And I'll note I lost that over the course of 10 or so weeks.

Ok... 8.8 kg isn't a lot compared to some of the reported weight losses online. But 8.8 kg is actually quite a bit compared to where I started. If I divide my weight loss by my starting weight and then multiply by 100, that will give a percentage weight loss:

8.8 / 87.5 x 100 = 10.0

This number is my percentage - I've lost 10.0%. To have lost 10% of my weight in 10 weeks - that's pretty amazing.

Ok - so let's do another person. Let's say they start at 350 pounds and they now weight 330 pounds. What percentage of their body weight have they lost?

First we look at how much weight was lost:

350 - 330 = 20 pounds.

For someone who is 330 pounds, they may think that they've "only lost" 20 pounds and might not be happy with that number. But let's look at the percentage they've lost:

Divide 20 by 350 and then multiply by 100 to get the percentage:

20 / 350 x 100 = 5.7%

A weight loss of 5.7% of their starting weight is amazing.

I hope this helps!

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And now time for shameless plugs - this blog has other "household math" topics and opinion pieces - I invite you to look around.

I have another resource that is an online inter-connected textbook. It's focused pretty much on math topics right now but will continue to expand outward called Fact-orials.

~~~~~

And as always, questions and comments are always welcome!

 

 

 

#Admin - Refocusing my efforts

Hi all,

Projects, like life, evolve and change. Goals are evaluated, re-evaluated. Reasons are examined, values tested. And then, after much thought and deliberation, actions are taken to bring that new evolution to fruition.

When I first started Math Fact-orials, I was seeking to re-create the experience I was having at Socratic.org - an experience of engaging with students, answering their questions, and finding within myself a feeling of satisfaction and accomplishment.

Then came the realization that where my heart really lay was in creating an online, inter-linked, education resource that would help show not just how things work, but why they work and where they came from. From this came the birth of Fact-orials. While that project envisions being able to explain everything at a level where a pre-teen can understand any given subject, it's also designed for curious adults who wonder "How is it that the Pythagorean Theorem, a^2 + b^2 = c^2, works?". (The answer is - I don't know. But I intend to write about it when I do find out...).

My heart also lies in creating content that helps adults live their lives. Helping people understand just what it means where a $100 shirt is 10% off (you pay $90 before any sales taxes), how to work that out on a calculator (1 - 0.1 x 100 = 90) and then to understand if it's actually a good deal (the same shirt in a different store for full retail of $85 is still a better deal than the one on sale) - these are topics that are of interest to me.

Also important is having a way to talk about things that aren't strictly financial, but more of opinion pieces where I can try to break down complicated (and often deliberately obfuscated) issues into what I hope are thought-provoking simple ideas that everyone can relate to.

As a result, while my goals for Fact-orials have't changed and that project will continue to steam along, Math Fact-orials will change signifcantly:

• Math Fact-orials will now be focused on what I like to call Household Math and also on opinion pieces,
• Math Fact-orials will no longer accept nor answer questions from students,
• The academic content currently on Math Fact-orials, as Fact-orials expands, may become part of that project, but will no longer be hosted on Math Fact-orials,
• Entries on Math Fact-orials will no longer be daily (not that they've been for the last little while, but now I'm officially comfortable with that state of affairs).

I want to thank you, one and all, for reading this and continuing to support me as I continue to explore my interests, my passions, and my heart's desire.

#PreAgebra - Which fraction is bigger? Smaller?

Questions:



We'll be doing 3c, 4c, 5c, 6b, 6c

Answers:

See below:

Analysis:

In all of these questions, we're dealing with what appears to be a bunch of random numbers dividing other rather numbers. It's all quite confusing! But if we work with these numbers, they'll tell us which of them is bigger and which is smaller.

There are a bunch of ways to think of fractions and to understand what they are telling us, but one of my favourites is a pizza pie analogy. All of these questions have three fractions, so let's imagine three pizza pies all the same size. We're going to cut those pies into some number of same sized pieces - it's the denominator, or the bottom number, that tells us how many. Keep in mind that as the number of pieces increases, the size of those pieces drops (if I cut a pizza in half, 1 slice is really big. If I cut that same pizza into 100 pieces, the pieces are really really small).

After we've cut the pizza into the number of pieces we want, then we'll pick out some number of those pieces. It's the numerator, or the top number, that tells us that. Let's start working 3C so you can see what I mean:



That first pizza has been cut into 7 pieces and we'll take 3, the second pizza has been cut into 2 pieces and we'll take 1, and the third pizza has been cut into 5 pieces and we get 3. Ok - so which option gives us the most amount of pizza? Right now it's a little hard to tell, isn't it? The slices are all different sizes and we're taking different amounts. What would really help is if the slices were all the same size - if we cut the pizza into slices that are all the same size - then all we'd need to do is pick the fraction with the largest numerator (the top number).

Now, as we cut the pizza into smaller and smaller slices, we still want to get the same amount of pizza! So as we increase the number of slices, we get more of them.

As an example, look at the  and think about a pizza where it's cut into 2 and you get 1 slice. If we cut the pizza into quarters, so 4 pieces, how many pieces of pizza do you get now? 2. Getting 2 slices when it's cut into 4 is the same as when you get 1 slice when the pizza is cut into 2 (image from chilimath.com):



This is where the idea of a Common Denominator comes into play - and all that means is that we want the bottom numbers to all be the same (so that the pizza slices are all the same size).

So how do we get to the Common Denominator? One way is to simply multiply all the denominators together (this may not get you to the Lowest Common Denominator, which you'll need to find in other questions, but it'll work well here):

7 x 2 x 5 = 70

Ok - so our denominator will be 70.

Now - we need to remember a couple of things at this point. The first is that we can always multiply by the number 1 and end up with the same number we started with. 3 x 1 = 3. I can multiply by 1 and not change the value of the fraction.

The other thing we need to remember is that 1 doesn't always look like 1. If I divide a number by itself, what do I get? 1. And so:



So now we can multiply the different fractions by versions of the number 1 so that we get our denominators to all be 70:







Remember, these are all equivalent to the fractions I had before - all I did is multiply by 1.

Ok - which amount of pizza is most? The one with 42 slices in it. And which fraction was that? 

All the other questions can be worked this same way:

4C



The Common Denominator is 8 x 6 x 3 = 144:





The one with 36 pieces is the smallest, and that is 

5C



The common denominator is 7 x 6 x 5 = 210



And now they can be arranged easily from largest to smallest:



6B

This question has a little twist to it - they include the number 1 as a fraction. Let's change it's look slightly so that it really does look like a fraction:



This says I cut the pizza into 1 piece (or in other words, it's the whole pizza) and I get that one piece (again, the whole pizza).

Now let's work this the same we have the others:



The common denominator is 5 x 1 x 8 = 40:



From smallest to largest, it's the one with 20, then 24, then 40, or:



6C



The common denominator is 7 x 9 x 3 = 189



From smallest to largest it's the one with 27, then 63, then 84, or:



~~~~~

Questions and comments always welcome!

#Admin - changes are coming...

Hi Everyone,

Over the course of the two or so months now that I've been doing this blog, two directions for where it would go developed:

1. A Q&A site that covers specific topics in a given post, and
2. An interlinked learning resource where posts support subsequent posts.

At first I tried doing both within the same blog, but that just made an organizational mess that was hard to navigate. 

And so I've decided to start another blog - one that will be that interlinked learning resource (and still maintain this one, where I do the Q&A). It'll be called Fact-orials and will take the initial posts here in Math Fact-orials (they'll move and get an edit in the process) and build upon them to explore topics as wide ranging as various fields of math, physics, other sciences, accounting, business, and so much more. 

In the end, if it seems like Math Fact-orials has gone rather quiet, that's because I'm hunkered down working on the other blog. 

I'll put in all the links and everything into both blogs and hopefully they'll work together to help everyone everywhere learn some cool stuff.

Many many thanks to both Stefan Velicu, a former Socratic.org moderator, Hero, contributor, and overall doer of all things, and my wife Aleesha, for helping to make Fact-orials something that can become reality, one entry at a time.

#Admin - Dealing with the moving goalposts...

Hi all,

After searching around for a bit to house this new brainchild, crazy idea of mine (to build a hyper-linked, non-linear website of a thing for all things... well... all things (but in particular math), I've finally decided on doing it right here on Blogger (mostly because I'm getting more familiar with it and it's free).

The idea is that there are going to be two kinds of posts - one that focuses on answering questions, and the other that focuses on building the online resource.

The ones that answer questions will follow a more traditional, blog-like scheme of being rather linear, dealing with a particular topic/question, utilizing the definitions pages and such when needed, but otherwise being stand-alone posts.

The ones that help to build the resource are going to be "living posts" - they are going to be edited and change, sometimes at great length, as content is added and the world of... well... everything (but in particular math) expands. As such, those posts won't remain static at all (or there are odds that they won't be. Perhaps I'll do a post about what those odds are???).

My hope is that this won't end up being a giant mess and instead will allow for integration of the resource, asking of questions, an ability to collaborate with readers, and overall be an enjoyable read for you and an enjoyable creation for me.

The place where the resource starts is the Index page - you can hyperlink from here or click on the link on the left side of the blog where it says "Start Here".

And with that, it's time to start writing!

(If there are areas that you feel have been missed or should be covered, please do drop me a line and I'll do what I can to make the needed content!)

#Admin - The goalposts just moved...

Hi all,

When I first started the blog back in late July, I had it in mind that I would answer questions, create content, and do the things I used to do at Socratic.org. I've been having fun doing it and plan on continuing to do so.

At the same time, however, it struck me today that through the use of definition pages and other supporting structures for the blog, something else can arise and it's that thing that I'm going to start building. It's a resource for the information age.

You see, I have often asserted that assembly line education is dumb, that teaching everyone exactly the same thing at the same time just because the child's age is x years old, just doesn't work at all  (echoing the thoughts of the speaker on a TedTalk I listened to raptly many years ago). I think that no matter what subject it is within mathematics (and probably within all of education itself), we can start with something that is of interest and move into domains that are related and help students (of all ages) be curious about math, science, and all that. For instance, if a student is really into triangles, that can lead to discussions about shape, line, area, perimeter, ratios, trig functions, graphing, and so much more. Any particular topic can lead to a whole heap of other topics.

And so that is the resource I want to build. I don't think I've seen its like in existence and perhaps I'm too naive to know that what I'm looking at building is crazy. Well... that's ok. A bit of crazy, applied long enough, can result in amazing things. Like the biggest ball of string in the world. Or a scientific breakthrough. Or perhaps, just perhaps, a useful online resource that will help index, categorize, and bring a flow to knowledge that doesn't exist yet.

I'll continue to write the blog as I start to build this new resource, so keep the questions coming!

#PreAlgebra - What are the different kinds of numbers (Natural, Whole, Integer,...)

Question

What are the different kinds of numbers (Natural, Whole, Integer,...) and where do they come from?

Answer

See below for a discussion:

Analysis

The groupings of numbers and the development of them is a fascinating walk through history. Let's take a brief walk down that path:

• We can start with the number 1

• One shows existence, it shows something I can point to - someone can hand something to someone else and say "here's this thing". You have to think really really basic representation stuff here (which operates below where essentially all of us do), but even if it never really happened, one serves as that bedrock number from which everything else is going to arise. As a side note, the development of the number 0 took much much longer to form (like thousands of years after the development of the number 1).

• Counting/Natural/Whole numbers

• Once we had the number 1, people would be able to count things with tick marks (essentially little number 1s) and add them up. If there's a tick mark and a tick mark... well that's tick tick, right? At some point, it'd be simpler to count, not by saying tick tick tick tick tick.... but by having a word that associated with the number of ticks. Thus came 2, 3, 4, etc... These are therefore called the Counting Numbers. And I believe they get the name Natural Numbers because they naturally arise from the number 1. Whole Numbers are the counting numbers with the number 0 included. The symbol for these kinds of number is .

•  Integers 

• Integers are a result of subtraction. The basic thought goes like this: I have 2 things. What happens if I subtract 3 things? In the world of "I have 2 chickens. And you want to take away 3?", 2 - 3 makes no sense. However, in many other applications, the use of negative numbers is very useful. And that is what the integers are - the counting numbers that are both above zero (positive) and below zero (negative). The symbol for these kinds of numbers is .

•  Rational numbers 

• Rational numbers are a result of division. We start with an integer (say 4) and divide it into a number of pieces (say 7). And in fact this is the definition of a rational number - any number that can be expressed as a fraction of integers. The symbol for these kinds of numbers is .

•  Real numbers 

• Real numbers are all the numbers that sit on a number line and are symbolized with . There are two types of numbers on the number line - Rational (discussed above) and Irrational numbers. 
• Irrational numbers are those numbers that can't be expressed in terms of a fraction using integers. These kinds of numbers have endless strings of digits, such as more famous ones like  and others like . Irrational numbers are symbolized by .

•  Complex numbers 

• Complex numbers consist of two parts - a real part and an imaginary number part that is a real number multiplied by the square root of -1. An example of a complex number is.

• Imaginary numbers came about from needing to take the square roots of negative numbers. Consider - the square root(s) of a number is/are the numbers we can multiply together to find the number under the square root. For instance, take . We can multiply 2 by itself, so 2 X 2 = 4, or we can multiply -2 by itself, so -2 X -2 = 4. We're good so far. So what happens if we have . There are no two numbers that we can multiply together that will get us to -4. So what to do? Well, if we were to rewrite it this way, , we know what the square root of 4 is - it's 2. So what to do with the square root of -1? Assign it a symbol and move on, and the symbol is the small letter i. 

 ~~~~~

Questions and comments always welcome!

#Algebra - the folding of A0 paper and the square root of 2...

Question

I was listening to a podcast (it's podcast 10, and here's the transcript on his blog) by the artist Danny Gregory (ok - more like I overheard my wife listening to it and my ears perked up when Danny started talking about the mathematical development of the A, B, and C paper sizes). He said that the paper sizes (A, B, and C) as they become smaller, a ratio of square root of 2 : 1. With the paper getting folded in half (the longer side) each time, how does that ratio hold? And are there any other ratios that would work just as well as square root of 2 : 1?

This question is related to this one which looks at finding the measures of the sides of the paper. 

Answer

See below on how it is that the ratio holds. There are no other ratios that will hold.

Analysis

Ok - so how does the ratio manage to stay the same?

Let's first consider a situation where the ratio does not stay the same. Let's set up a piece of paper where the ratio is 2:1 and we fold the long side over (the 2 side) and now we have a piece of paper that has a ratio of 1:1 - not the same at all. So how does the ratio keep maintained?

We start with a ratio of:



We fold the long side in half (that's the square root side) and now we have:



We started out with the "short side" being the "1" side, so let's scale up the numbers so that the short side is 1 again and we'll see where the long side is:





We're almost done. Let's rationalize the denominator of the left hand fraction:





And taa daa! It's the same ratio.

Is there any other ratio that will hold like this? To find out, we can set up a ratio:



We know that the "short pre-fold" will be 1 as will be the long post-fold (they're the same side). The long pre-fold will be some value, x, and the short post-fold will be one half of x:



We can cross-multiply:







And so the square root of 2 : 1 ratio is the only one that will work.

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As always, questions and comments always welcome!

 

#Algebra - rectangle area and side ratios...

Question

I was listening to a podcast (it's podcast 10, and here's the transcript on his blog) by the artist Danny Gregory (ok - more like I overheard my wife listening to it and my ears perked up when Danny started talking about the mathematical development of the A, B, and C paper sizes). He said that an A0 piece of paper is defined as being 1 metre squared with the sides of the paper having a ratio of square root 2 : 1 (rounded to the closest millimetre). What are the sides of the paper?

This question is related to the one regarding the folding of A, B, and C series paper and the holding of the square root of 2 : 1 ratio. 

Answer

1189 mm X 841 mm

Analysis

Let's first remember that the area of a rectangle can be found by multiplying the base times the height:

A = bh

We're told the area of the paper in metres. Let's convert that to millimetres:

1 m^2 = 1 m X 1 m = 1000 mm X 1000 mm = 1,000,000 mm^2

With the sides of the paper, we know the ratio is square root 2 : 1. We'll need a variable, x, to make sure we count the same amount of 1's as we do square root 2's:







The short side is 841 mm. The long side is:



These are both in millimetres.

And now let's see just how big the paper really is (in millimetre squared):



and so just a wee bit smaller than the defined size!

~~~~~

Questions and comments always welcome!