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Yak Shaving and Typography

Posted on February 2, 2019 by psu

There are few things you know for sure in this world. But here is one that has been true for the last 30 years or so, and probably will be true for as many years as I can see into the future.

Fact: every student all over the world who decides to do an advanced degree in math, science, engineering and sometimes even related fields in the humanities will all have one thing in common no matter their background, their interests, or their future lives. What they will have in common is that they’ll all have to learn \TeX to write their papers and books, and they’ll have one of the greatest yak shaving exercises in history to thank for it.

If you’ve used \rm\TeX or done any computer programming in the last 40 years, you probably know what I’m going to say. But for the younger people in the audience (and because I like the story) here is the story.

Part 1: Prehistory

In 1968, at the tender age of thirty, Donald Knuth published the first volume of a book that was supposed to be about compilers. Volume 1 of The Art of Computer Programming instead formed the basis for what would become the first systematic investigation of techniques for designing and analyzing algorithms 1 in the future field of academic computer science.

Over the next ten years he would publish three volumes (1. Fundamental Algorithms, 2. Seminumerical Algorithms, and 3. Sorting and Searching) and also updated editions of volumes 1 and 3.

It’s hard to overstate the influence of these books, especially at the time. Over the years, at least in my opinion, they have become more of an extremely well written semi-historical survey than something a working programmer, or even working academic, would look at on a day to day basis. But still, it is an incredible work.

Then in 1976 he started to update volume 2. And here is where our story really starts.

Part 2: The Shaving Begins

The original printings of TAOCP were done on semi-automatic mechanical typesetting devices built by a company called Monotype beginning in the late 19th century. Someone would actually type all the text on a keyboard where it was recorded on paper tape. Then a second machine would read the tape and cast type in real time from molten hot metal 2 into galleys, one for each page. Then you make paper prints from the galleys and offset plates from the master pages and then you use the plates to print the books. For complicated technical books the typographer would also have to do a lot of hand-setting of type for the mathematical formulas and programs and whatnot. It was exactly as tedious and time consuming as it sounds.

In the late 70s the mechanical machines were starting to be phased out and replaced with machines that made the offset plates directly using a photographic system to place the text on the plates. In 1976 Knuth made enough changes to his second volume that all the text had to be reset … but the old mechanical Monotype machines at his publisher had been retired and the new photo-based machines did not generate text that looked the same. Knuth was left with the following choices:

1. Have the new edition of Volume 2 look different and weird.
2. Have the new edition of Volume 2 printed in England, where the mechanical machines still existed, at much higher cost.
3. Give up.

He was in fact thinking about giving up when he saw some output, apparently generated by a computer program, from a new high resolution laser-based digital typesetting machine. “Computer program? I can write computer programs!”, he thought to himself. And the idea for \rm\TeX was born.

Over the next few years Knuth (and his graduate students) built what would become the first versions of two programs: \rm\TeX for typesetting technical papers and books, and METAFONT (I can’t do the real logo in HTML, apparently, oh well) for creating fonts. The goal was to make digital versions of the type used in the original books (a font called Monotype Modern 8 3) and also a new markup language to allow the text of the books to be in digital form.

The first large scale test of this new system was the 1981 second edition of Volume 2. While Knuth was satisfied with proofs generated by his new system, the printed and bound versions of the new Volume 2 did not make him happy. I can say from personal experience that the second edition of Volume 2 does not really look like the originals. The fonts, especially the sans serif, are just not quite there yet.

That said, even if the fonts did not look right the typesetting was a triumph. The first final version of the \rm\TeX as we know it was finished in 1982 and released out to the world. While Knuth worked to improve his amateur font designs (with the help of many people who knew a lot more than he did) \rm\TeX (mostly in the form of \rm\LaTeX) became the standard way to write and publish technical text, especially in academic circles.

Part 3: Knuth and Me

I first looked at the Knuth books in high school just as I was getting interested in computer programming. What confused me at the time was why all that math was in there. My feeling was that math did not have much to do with computer programming. I was young, ignorant, and very much overestimated how much I understood about computer programming at the time. If I had been in high school in 2018 I would have posted a lot of stupid shit on twitter about this. Happily it was the 80s instead and I could not do this. By the time I graduated from college and went to study computer science in graduate school I had a better understanding of what Knuth was going on about and I had my own copy of the books.

I first came across \rm\TeX in college, but didn’t really do battle with it in earnest until graduate school. Like most graduate students I ended up fighting with \rm\TeX to get my thesis done. When I got out of the academic game I did my best to forget about Knuth and his software. \rm\TeX is a precisely designed and functional tool for what it does, but from a software engineering perspective it’s kind of a disaster. The fact that the source code is mashed up inside a giant \rm\TeX document so that you can print it as a book does not help. 4 Through the 80s and 90s publishing of course evolved from phototypesetting to fully digital printing workflows to publishing based on no printing at all. There are now tools by the hundreds to help you do this work, so you would have thought that \rm\TeX would have died the slow death. But this has not been the case.

But I do not print things or write books, so for twenty five happy years I did not think about \rm\TeX. Knuth, meanwhile, finished METAFONT and his fonts got back to writing his books in 1986, after shaving the yak for almost ten years. Just kidding the new editions of his books did not appear until the late 90s. I lost track of why. He is now about a third of the way through the material for Volume 4, but he has a long way to go before even getting to the section on NP … so don’t hold your breath. The shaving will continue for the foreseeable future.

Part 4: Modern \rm\TeX

These days \rm\TeX has become almost a standard wire format, if you will, for the expression of mathematical formulas in printed or more recently in electronically printed form. You can even put \rm\TeX commands into web pages now. But the \rm\TeX engine and user model remain mostly the same: you edit a plain text file with markup in it and \rm\TeX compiles that and spits out something you can preview on your screen, or print to your printer. These days you can edit your text on the Internet if you want, and you can format it on your iPhone, if you want, but the basic system remains the same. It turns out there isn’t a huge audience for anything else.

I came back to \rm\TeX for the dumbest of all possible reasons. I like to read papers about physics (and math, and astronomy, and sometimes computers). While these papers are now easy to find and store electronically (you go here. Give them money, they do great work), they are sometimes not the most readable artifacts. My complaints include:

1. Two column layouts are the standard for journals, but are a pain to read on screens.
2. Most older papers do not have hyperlinks for various things like references. Most newer papers do have hyperlinks but they use the default formatting for them, which is hideous.
3. Most papers use teeny tiny little letters and are better to read on screen if changed to use larger typefaces.
4. Lot’s of little picky things like using the right kinds of quotes, or using the \cite macro correctly.
5. I’ve never really liked Knuth’s typefaces (more on this later). I did my dissertation in Times. More modern \rm\TeX lets you use my favorite font of all time, Palatino, for both text and math. It is glorious.

So if I find a paper, and if the source code for the paper is available (and it usually is at the arXiv, which is amazing), then I will noodle with it for 10 minutes to try and make it better. Or give up. What is remarkable is that about 90% of the time you can take a ten or twenty year old paper that was originally written in some journal format and redo it without much trouble. The fact that \rm\TeX and its surrounding tools (esp. the \rm\LaTeX macros) have been stable enough over such a long period of time to make this possible turns out to be one of the most important parts of the Knuth legacy, IMHO. For better or worse, he froze the software and its formats and they have become a semi-accidentally archival in a way that no one would have considered possible.

Part 5: Knuth and Me, Revisited

Inevitably using \rm\TeX even casually brought my brain back around to checking in on Knuth. You can’t really avoid it. In the twenty five years since I looked last he had apparently:

1. Finished off his software, mostly.
2. He published 5 books about \rm\TeX and the fonts.
3. He co-wrote a textbook on math for Computer Science..
4. He re-published the five books about \rm\TeX and the fonts in 2000.
5. He finally redid Volumes 1 to 3, and started Volume 4.
6. He published a large series of retrospective collections, mostly about his academic work, and one is just about the \rm\TeX project and its history. That’s where most of my story came from.
7. He rebuilt the fake computer and assembly language used in his books to study “real” code.
8. Who knows what else.

In a fit of nostalgia over Christmas (and because my boss told me to do “something stupid”) I bought a new set of the Knuth books to look at again, along with a new copy of the \TeX book. I also got the Digital Typography volume, which has a lot of neat old historical material in it.

The Art of Computer Programming is as good as ever. In addition, even though I’ve never been a fan of the Computer Modern typefaces for whatever reason in those specific books they look good. They look especially good printed, but are also excellent in PDF form. Which is a rare thing. There is something about the entire page design of the books that holds them together and makes you forget about how the type still looks a bit off compared to what is in the classic books that Knuth so admired.

Digital Typography also looks fine. It has that more generic “I was formatted with \rm\TeX” look to it. But it’s fine.

In contrast the new editions of the \TeX book and its companions look awful. They are clearly not printed by the same machines and on the same paper used for TAOCP. The result looks like something I could pull off my inkjet fax machine if I wanted to. A disappointment.

Having come full circle, my only regret now is that I lost the source code to my PhD dissertation twenty years ago in a bad laptop move. It would have been cool to see how it looks in Palatino, with modern outline fonts.

Part 6: \rm\TeX and Me, Redux

The rest of the story is mostly about fonts. In 1990 when I did my PhD thesis the main set of typefaces available for \rm\TeX that would support both text and mathematics was Computer Modern. There were a few sets of fonts that you could buy for money, but no graduate student has enough money to do that. It turns out that creating a set of typefaces that work well for both text and math in \rm \TeX is a lot of hard detail work. Very few font sets even today get everything right. It’s long been my opinion that Computer Modern, especially before the later improvements, has never had the nicest set of letters to look at. But you have to admire all of the meticulous work that Knuth (and the folks at the AMS) did do get its support for \rm\TeX’s mathematical alphabets and layout right.

I compromised by using Postscript fonts for text but Computer Modern for math, which a lot of people have done, but in fact is not a great choice. I still see this done in some books published in two-thousand-and-nineteen and I just frown inside.

These days there are about a dozen different free sets of typefaces that also have good support for the wide range of math characters that are available Computer Modern, but are based on text faces that you might like better. As I said twice above my favorite is an old package that is based on Palatino, but you can also get your Times, Garamond, Baskerville, Charter and many more. Check them out here: http://www.tug.dk/FontCatalogue/.

There are also two well known and widely used commercial font sets: Lucida and the various versions of Minion. Both of these also look great, and I got interested in Lucida because its text face is perhaps a bit influenced by Palatino, since the designers were students of Zapf. It turns out that the Lucida people also have had a long relationship with Knuth so you can buy a package of their fonts specifically for \rm\TeX from the \rm\TeX Users Group and even get half off if you are a member. I’ve never been much for Users Groups but this seemed like a win so I grabbed a discounted first year membership and the fonts.

I like the Lucida fonts a lot. Maybe I’ll write a book to use them (that would be dumb). But I might still like Palatino more. Fiddling with fonts is a hobby that I could spend altogether too much time indulging in. Luckily there is a guy on the Internet who has done it all for me. Look at the print samples here and you never have to fiddle with fonts yourself again.

Surprisingly, I also like the back issues of the TUG newsletter. There are old stories about running \rm\TeX on old mainframes that took 10 seconds to process each page, catalogs of the pain and suffering some people put themselves through to try and modernize the code in the \rm\TeX engine itself (mostly to no avail), and old letters to Brian Reid (an inside joke). You can get these stories without paying the membership fee (back issues are free to download), but tossing a bit of monetary support towards the most niche of niche publications is a good, if useless, way to feel good about yourself. Since its inception more than forty years ago \rm\TeX has also become something that history will study, and these TUGboat publications will form a large art of the primary source material for those studies.

To sum up, I have realized that it’s not really fair to call \rm\TeX a Yak Shave. While it certainly started out as a classic yak shaving project: a small side project in support of a much larger magnum opus, it ended up being arguably the more important artifact. Knuth’s books will always have an audience that is primarily limited to people studying programming and computer science. \rm\TeX, on the other hand, has had a much wider impact. This quote from Charles Bigelow, one of the people who designed the Lucida fonts, supports this anecdotal conclusion:

A personal anecdote to support that claim: My neighbor is a retired mathematician, Norman Alling. He wrote a book on real elliptic curves and taught himself \rm\TeX in his 50s, so he could compose his book and papers himself, and he still uses \rm\TeX now, in his 80s. He says \rm\TeX liberated math journals and authors from dependence on commercial math typesetting, which was slow, expensive, and fraught with typographical errors needing proofreading and correction. When I told him that some people suggested that Knuth could have better spent his time finishing the Art of Computer Programming books instead of spending a decade developing \rm\TeX, he replied: Oh no, \rm\TeX liberated so many mathematicians and scientists from the bottleneck of typesetting that it was a great boon to all of math and science, more important for the world-wide science and technical professions than Knuth’s unpublished books on computing, however excellent they might be.5

You can read more about various aspects of the history of \rm\TeX from the early days until now in the PDFs linked below:

Notes

1. By algorithms here I do not mean the current usage where what we mean by an “algorithm” is some intern-built glorified gradient following engine driven by terabytes of data collected by automatically surveilling the actions of millions of people while they innocently use their computers on a world-wide information network. No, I mean specific programs written to do specific tasks, which are usually specified in some semi-formal mathematical language. You can then ask the question “how fast will these programs run?” … and this is question that the Knuth books explore in incredible detail and with incredible precision.↩

2. Here are two cool youtube videos about how the Monotype machine works.


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# How do I apply boldface to multiple Khmer words in tabular environment?

I create documents that include both Latin and Khmer (Cambodian) characters. To type in Unicode, I use the fontspec package and typeset using XeLaTeX. To move between the two scripts I use the package ucharclasses.

My problem is this: when I put two or more Khmer words into a tabular environment, any text style that ought to target the entire string instead alters only the first word in the string. Here's a MWE:

\documentclass{article}
\usepackage{fontspec}
\setromanfont[Mapping=tex-text]{Times}
\usepackage[Khmer]{ucharclasses}
\setTransitionTo{Khmer}{\fontspec[Scale=0.8,Script=Khmer]{Khmer Mondulkiri}}
\setTransitionFrom{Khmer}{\normalfont}
\begin{document}
\begin{tabular}{l l}
ខុសហើយ & \quad not bolded\\
\textbf{នេះ ខុស ដែ} & \quad \textbf{three bolded words}\\
\bf នេះ ខុស ដែ & \quad \bf three bolded words\\
\end{tabular}
\end{document}

Once typeset, it looks like this:

Notice that neither \boldface nor \bf works for the Khmer – except the first word.

(To get the MWE example to work, you'll need a decent Khmer font: many pre-installed stock Khmer fonts don't have boldface at all. I've used Khmer Mondulkiri, an open font available from SIL.)

You are using \normalfont for the transition, and as obviously the space is a transition places this resets the series. Use only \rmfamily. It is also better to define a dedicated font family for khmer instead of loading it with fontspec::

\documentclass{article}
\usepackage{fontspec}
\setmainfont[Mapping=tex-text]{Times New Roman}
\newfontfamily\khmerfamily[Scale=0.8,Script=Khmer]{Khmer Mondulkiri}
\usepackage[Khmer]{ucharclasses}
\setTransitionTo{Khmer}{\khmerfamily}
\setTransitionFrom{Khmer}{\rmfamily}
\begin{document}
\begin{tabular}{l l}
ខុសហើយ & \quad not bolded\\
\textbf{នេះ ខុស ដែ} & \quad \textbf{three bolded words}\\
\bfseries នេះ ខុស ដែ & \quad \bfseries three bolded words\\
\end{tabular}
\end{document}

• Thanks, Ulrike. Worked perfectly. Honestly, I don't understand what \normalfont is (I copied the code from somewhere!), or why a transition would be different with \normalfont rather than \rmfamily. It seems as though a space is a transition either way. – viverechristus Mar 15 '17 at 12:47
• \normalfont is defined as \usefont\encodingdefault \familydefault\seriesdefault\shapedefault so it resets everything to "normal" and so e.g. instead of the bold series your get the normal mdseries. \rmfamily only changes the family. – Ulrike Fischer Mar 15 '17 at 12:56

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                   sir, what mark should i get for computer science & engineering in old iit
smit mehta

5 years ago

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                                        Dear smit
You need to score more than 270 marks to get computer science in old IITs
best of luck

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5 years ago

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# Inverting data for Zero-inflated mixed effects models

I am looking for some advice on my analyses, I have been going back and forth between co-authors about the validity of my approach and would appreciate some external input.

My data are derived from mutli-country household surveys of rural farmers living under different landuse configurations. We are focusing on frequency of food consumption to examine dietary patterns. In order to do so we asked how many days did you consume a particular food group in the last 7 days? (7day recall). I am applying mixed effects models run using the glmmTMB package in R to account for the nested nature of the experimental design. I am happy with this approach but am being questioned as to my logic for using Zero-inflated models. I will explain below and would appreciate any comments, clarifications, suggested alternatives or even a "you cant do that".

There are two main issues I am having with my coauthors.

1. The "inverting" of the data in order to model with a zero-inflated model
2. The assumptions I am making about there being "real" and "structural" zeros and two different processes determining them.

So my data looks like this for meat and vegetable consumption.

there are obviously lots of people who are eating both food groups 7 days a week and this is potentially inflating the data, especially with the vegetable distribution of counts.

My approach was to "invert" the data so instead it would represent a count of the number of days a respondent went without consuming that food group.

In my mind this makes the data more manageable from a modelling perspective and was my starting point to examining which models would be appropriate.

My first question is: Is it OK to do this from a modelling and statistics point of view?

From here I ran a bunch of models and examined the model fits, the best fits were either zero-inflated Poisson or zero-inflated neg binomial.

The biggest blowback I have been getting (apart from the inverting which I think is fine) is that my coauthors do not agree that there could be two sources of the zeros (which is actually two sources of 7days consumption in its original form before I inverted the data).

I have suggested that there are respondents we surveyed that would always answer 7 days (structural zeros) and respondents that answered 7 days who have a variable weekly intake but just happend to eat 7 days that week (real zeros).

I assume that if people have the the opportunity to eat these food groups daily then they do. So because these people are farmers and have access to these food groups either because they grow them themselves or can readily collect them from the forest or purchase/trade for them by being wealthy enough (all these factors are in controlled for in the model). It seems reasonable to me to assume that some of the people who answered '7 days' are from a group of people that would only ever answer '7 days' (structural zeros) and therefore the zero-inflated models are appropriate, remembering that because I inverted the data a structural zero actually means being from the group of people who could only ever answer zero days without consumption of that particular food group. So there is one set of processes that determines the odds of one someone only being able (close enough to only) to answer 7days and another set of processes that determine the frequency of consumption for those who cannot only answer 7days.

My second question is: Is this approach appropriate and does it justify the use of zero inflated models?

I would really appreciate people to point out any flaws in my approach or my logic and suggest alternatives or point me in the direction of alternative approaches. I have looked into quite a few different approaches and I always end up back here.

It should be noted that the outputs from the models make perfect sense in terms of the significant factors driving each process for real and structural zeros(I would be worried if they were throwing weird results).

• Model the outcome and reversed outcome while ignoring the zero-inflation using the count model. You will find that changing the direction affects the estimated coefficients, not just affecting the signs, but the magnitudes change. The answer below from @DimitrisRizopoulos is a reasonable approach that will not be affected by the value you choose as success since it is binomial, it should only reverse the sign. glmmTMB permits cbind(success, failure) outcome type which is what you'll need to implement. See its manual. Nov 17 '18 at 5:04
• this would be a beta binomial model right? if so how would I deal with the 1's and 0's? I have looked into the zoib package in R but it cannot handle nested random effects. Nov 20 '18 at 3:46
• Not a beta-binomial, a regular multilevel binomial as suggested in the single answer you have. Was only pointing out that using a model with a log link and reversing the outcome has consequential effects on the results you find. So the choice to reverse should not be arbitrary. But you can arbitrarily reverse if using binomial with logit link. Nov 20 '18 at 3:58

The negative binomial model is typically used for count data with no upper limit, i.e., the outcome variable $$Y$$ can take values in the set $$[0, 1, 2, \ldots, \infty)$$. However, in your case you can only have a maximum count of 7.
A more appropriate distribution for your data is the binomial distribution, which describes the number of “successes” (i.e., in your case success = eating meat & vegetables) in $$N$$ trials (i.e., in your case $$N = 7$$ is the number of days).


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# Engaging students: Slope-intercept form of a line

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jessica Williams. Her topic, from Algebra I: the point-slope intercept form of a line.

A.2 How could you as a teacher create an activity or project that involves your topic?

In order to teach a lesson regarding slope intercept form of a line, I believe it is crucial to use visual learning to really open the student’s minds to the concept. Prior to this lesson, students should know how to find the slope of a line. I would provide each student with a piece of graph paper and small square deli sheet paper. I would have them fold their deli sheet paper into half corner to corner/triangle way). I would ask each student to put the triangle anywhere on the graph so that it passes through the x and the y-axis. Then I will ask the students to trace the side of the triangle and to find two points that are on that line. For the next step, each student will find the slope of the line they created. Once the students have discovered their slope, I will ask each of them to continue their line further using the slope they found. I will ask a few students to show theirs as an example (picking the one who went through the origin and one who did not). I will scaffold the students into asking what the difference would look like in a formula if you go through the origin or if you go through (0,4) or (0,-3) and so on. Eventually the students will come to the conclusion how the place where their line crosses the y-axis is their y intercept. Lastly, each student will be able to write their equation of the line they specifically created. I will then introduce the y=mx+b formula to them and show how the discovery they found is that exact formula. This is a great way to allow the students to work hands on with the material and have their own individual accountability for the concept. They will have the pride of knowing that they learned the slope intercept formula of a line on their own.

E.1 How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Graphing calculators are a very important aspect of teaching slope-intercept form of a line. It allows the students to visually see where the y-intercept is and what the slope is. Also, another good program to use is desmos. It allows the students to see the graph on the big screen and you can put multiple graphs on the screen at one time to see the affects that the different slopes and y intercept have on the graph. This leads students into learning about transformations of linear functions. Also, the teacher can provide the students with a graph, with no points labeled, and ask them to find the equation of the line on the screen. This could lead into a fun group activity/relay race of who can write the formula of the graph in the quickest time. Also, khan academy has a graphing program where the students are asked to create the graph for a specific equation. This allows the students to practice their graphing abilities and truly master the concept at home. To engage the students, you could also use Kahoot to practice vocabulary. For Kahoot quizzes, you can set the time for any amount up to 2 minutes, so you could throw a few formula questions in their as well. It is an engaging way to have each student actively involved and practicing his or her vocabulary.

B1. How can this topic be used in your students’ future courses in mathematics or science?

Learning slope intercept form is very important for the success of their future courses and real world problems. Linear equations are found all over the world in different jobs, art, etc. By mastering this concept, it is easier for students to visualize what the graph of a specific equation will look like, without actually having to graph it. The students will understand that the b in y=mx+b is the y-intercept and they will know how steep the graph will be depending on the value of m. Mastering this concept will better prepare them to lead into quadratic equations and eventually cubic. Slope intercept form is the beginning of what is to come in the graphing world. Once you grasp the concept of how to identify what the graph will look like, it is easier to introduce the students to a graph with a higher degree. It will be easier to explain how y=mx+b is for linear graphs because it is increases or decreases at a constant rate. You could start by asking,
1.What about if we raise the degree of the graph to x^2?
2.What will happen to the graph?
3.Why do you think this will happen, can you explain?
4.What does squaring the x value mean?
It really just prepares the students for real world applications as well. When they are presented a problem in real life, for example, the student is throwing a bday party and has $100 dollars to go to the skating rink. If they have to spend$20 on pizza and each friend costs $10 to take, how many friends can you take? Linear equations are used every day, and it truly helps each one of the students. # Engaging students: Using the point-slope equation of a line In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Rachel Delflache. Her topic, from Algebra: using the point-slope equation of a line. A2: How could you as a teacher create an activity that involves the topic? An adaptation of the stained-glass window project could be used to practice the point-slope formula (picture beside). Start by giving the students a piece of graph paper that is shaped like a traditional stained-glass window and then let they students create a window of their choosing using straight lines only. Once they are done creating their window, ask them to solve for and label the equations of the lines used in their design. While this project involves the point slope formula in a rather obvious way, giving the students the freedom to create a stained-glass window that they like helps to engage the students more than a normal worksheet. Also, by having them solve for the equations of the lines they created it is very probable that the numbers they must use for the equation will not be “pretty numbers” which would add an addition level of difficulty to the assignment. B2: How does this topic extend what your students should have learned in previous courses? The point-slope formula extends from the students’ knowledge of the slope formula m = (y2-y1)/(x2-x1) (x2-x1)m = y2-y1 y-y1 = m(x-x1). This means that the students could solve for the point-slope formula given the proper information and prompts. By allowing students to solve for the point-slope formula given the previous knowledge of the formula for slope, it gives the students a deeper understanding of how and why the point-slope formula works the way it does. Allowing the students to solve for the point-slope formula also increases the retention rate among the students. C1&3: How has this topic appeared in pop culture and the news? Graphs are everywhere in the news, like the first graph below. While they are often time line charts, each section of the line has its own equation that could be solved for given the information found on the graph. One of the simplest way to solve for each section of the line graph would be to use point slope formula. The benefit of using point slope formula to solve for the equations of these graphs is that there is very minimal information needed—assuming that two coordinates can be located on the graph, the linear equation can be solved for. Another place where graphs appear is in pop culture. It is becoming more common to find graphs like the second one below. These graphs are often time linear equation for which the formula could be solved for using the point slope formula. These kinds of graphs could be used to create an activity where the students use the point slope formula to solve to the equations shown in either the real world or comical graph. References: Stained glass window- Stained Glass Window Graphing Project # Engaging students: Finding the slope of a line In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Deanna Cravens. Her topic, from Algebra: finding the slope of a line. C3. How has this topic appeared in high culture (art/sports)? While one might not think of ski jumping as an art but more of a sport, there is definitely an artistic way about doing the jumping. The winter Olympics is one of the most popular sporting events, besides the summer Olympics that the world watches. This is a perfect engage for the beginning of class, not only is it extremely humorous but it is extremely engaging. It will instantly get a class interested in the topic of the day. I would first ask the students what the hill the skiers going down is called. Of course the answer that I would be looking for is the “ski slope.” This draws on prior knowledge to help students make a meaningful connection to the mathematical term of slope. Then I would ask students to interpret the meaning of slope in the context of the skiers. This allows for an easy transition into the topic for finding the slope of a line. C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)? Look at this scene from Transformers, it shows a perfect example of a linear line on the edge of the pyramid that the Decepticon is destroying. This video easily catches the attention of students because it is from the very popular Transformer movie. I would play the short twenty second clip and then have some student discussion at the beginning of class. This could be done as an introduction to the topic where students could be asked “how can we find the steepness of that edge of the pyrmaid?” Then the students can discuss with a partner and then group discussion can ensue. It could also be done as a quick review, where students are asked to recall how to find the slope of a line and what it determines. The students would be asked to draw on their knowledge of slope and produce a formula that would calculate it. How can this topic be used in your students’ future courses in mathematics or science? Finding the slope of a line is an essential part of mathematics. It is used in statistics, algebra, calculus, and so much more. One could say it is an integral part of calculus (pun intended). Not only is it used in mathematics classes, but it is also very relevant to science. One specific example is chemistry. There are specific reaction rates of solutions. These rates are expressed in terms of change in concentration divided by the change in time. This is exactly the formula that is used in math classes to find the slope. However, it is usually expressed in terms of change in y divided by change in x. Slope is also used in physics when working with velocity and acceleration of objects. While one could think of slope in the standard way of ‘rise over run,’ in these advanced classes whether math or science, it usually better thought of as ∆y/∆x. References: # Engaging students: Graphs of linear equations In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Anna Park. Her topic, from Algebra: graphs of linear equations. How could you as a teacher create an activity or project that involves your topic? • Have the students enter the room with all of the desks and chairs to the wall, to create a clear floor. On the floor, put 2 long pieces of duct tape that represent the x and y-axis. Have the students get into groups of 3 or 4 and on the board put up a linear equation. One of the students will stand on the Y-axis and will represent the point of the Y-Intercept. The rest of the students have to represent the slope of the line. The students will be able to see if they are graphing the equation right based on how they form the line. This way the students will be able to participate with each other and get immediate feedback. Have the remaining groups of students, those not participating in the current equation, graph the line on a piece of paper that the other group is representing for them. By the end of the engage, students will have a full paper of linear equation examples. The teacher can make it harder by telling the students to make adjustments like changing the y intercept but keeping the slope the same. Or have two groups race at once to see who can physically graph the equation the fastest. Because there is only one “graph” on the floor, have each group go separately and time each group. • Have the students put their desks into rows of even numbers. Each group should have between 4 and 5 students. On the wall or white board the teacher has an empty, laminated graph. The teacher will have one group go at a time. The teacher will give the group a linear equation and the student’s have to finish graphing the equation as fast as possible. Each group is given one marker, once the equation is given the first student runs up to the graph and will graph ONLY ONE point. The first student runs back to the second student and hands the marker off to them. That student runs up to the board and marks another point for that graph. The graph is completed once all points are on the graph, the x and y intercepts being the most important. If there are two laminated graphs on the board two groups can go at one time to compete against the other. Similar to the first engage, students will have multiple empty graphs on a sheet of paper that they need to fill out during the whole engage. This activity also gives the students immediate feedback. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? Sir William Rowan Hamilton was an Irish mathematician who lived to be 60 years old. Hamilton invented linear equations in 1843. At age 13 he could already speak 13 languages and at the age of 22 he was a professor at the University of Dublin. He also invented quaternions, which are equations that help extend complex numbers. A complex number of the form w + xi + yj + zk, where wxyz are real numbers and ijk are imaginary units that satisfy certain conditions. Hamilton was an Irish physicist, mathematician and astronomer. Hamilton has a paper written over fluctuating functions and solving equations of the 5th degree. He is celebrated in Ireland for being their leading scientist, and through the years he has been celebrated even more because of Ireland’s appreciation of their scientific heritage. Culture: How has this topic appeared in pop culture? An online video game called “Rescue the Zogs” is a fun game for anyone to play. In order for the player to rescue the zogs, they have to identify the linear equation that the zogs are on. This video game is found on mathplayground.com. References https://www.teachingchannel.org/videos/graphing-linear-equations-lesson https://www.reference.com/math/invented-linear-equations-ad360b1f0e2b43b8# https://en.wikipedia.org/wiki/William_Rowan_Hamilton http://www.mathplayground.com/SaveTheZogs/SaveTheZogs.html # Engaging students: Solving systems of linear inequalities In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Heidee Nicoll. Her topic, from Algebra: solving linear systems of inequalities. How could you as a teacher create an activity or project that involves your topic? I found a fun activity on a high school math teacher’s blog that makes solving systems of linear inequalities rather exciting. The students are given a map of the U.S. with a grid and axes over the top, and their goal is to find where the treasure is hidden. At the bottom of the page there are six possible places the treasure has been buried, marked by points on the map. The students identify the six coordinate points, and then use the given system of inequalities to find the buried treasure. This teacher’s worksheet has six equations, and once the students have graphed all of them, the solution contains only one of the six possible burial points. I think this activity would be very engaging and interesting for the students. Using the map of the U.S. is a good idea, since it gives them a bit of geography as well, but you could also create a map of a fictional island or continent, and use that as well. To make it even more interesting, you could have each student create their own map and system of equations, and then trade with a partner to solve. How does this topic extend what your students should have learned in previous courses? If students have a firm understanding of inequalities as well as linear systems of equations, then they have all the pieces they need to understand linear systems of inequalities quite easily and effectively. They know how to write an inequality, how to graph it on the coordinate plane, and how to shade in the correct region. They also know the different processes whereby they can solve linear systems of equations, whether by graphing or by algebra. The main difference they would need to see is that when solving a linear system of equations, their solution is a point, whereas with a linear system of inequalities, it is a region with many, possibly infinitely many, points that fit the parameters of the system. It would be very easy to remind them of what they have learned before, possibly do a little review if need be, and then make the connection to systems of inequalities and show them that it is not something completely different, but is simply an extension of what they have learned before. How can technology be used effectively to engage students with this topic? Graphing calculators are sufficiently effective when working with linear systems of equations, but when working with inequalities, they are rather limited in what they can help students visualize. They can only do ≥, not just >, and have the same problem with <. It is also difficult to see the regions if you have multiple inequalities because the screen has no color. This link is an online graphing calculator that has several options for inequalities: https://www.desmos.com/calculator. You can choose any inequality, <, >, ≤, or ≥, type in several equations or inequalities, and the regions show up on the graph in different colors, making it easier to find the solution region. Another feature of the graphing calculator is that the equations or inequalities do not have to be in the form of y=. You can type in something like 3x+2y<7 or solve for y and then type it in. I would use this graphing calculator to help students visualize the systems of inequalities, and see the solution. When working with more than two inequalities, I would add just one region at a time to the graph, which you can do in this graphing calculator by clicking the equation on or off, so the students could keep track of what was going on. References Live.Love.Laugh.Teach. Blog by Mrs. Graves. https://livelovelaughteach.wordpress.com/category/linear-inequalities/ Graphing calculator https://www.desmos.com/calculator # Engaging students: Finding the slope of a line In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Brianna Horwedel. Her topic, from Algebra: finding the slope of a line. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Algebra vs. the Cockroaches is a great way to get students engaged in learning about slopes. The object of the game is to kill the cockroaches by figuring out what the equation of the line that they are walking on is. It progresses from simple lines such as y=5 to more complicated equations such as y=(-2/3)x+7. It allows the students to quickly recognize y-intercepts and slopes. Once finished, you can print out a “report” that tells you how many the student got correct and how many tries it took them to complete a level. This game could even be used as a formative assessment for the teacher. http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf How could you as a teacher create an activity or project that involves your topic? Last year, I was placed in an eighth grade classroom that was learning about slope. One of the things that really stuck out to me was that the teacher gave a ski illustration to get the students talking about slope. The illustration starts off with the teacher going skiing. She talks about how when she is going up the ski lift she is really excited and having a “positive” experience which correlates to the slope being positive. Once she gets off of the ski lift, she isn’t going up or down, but in a straight line. She talks about how she doesn’t really feel either excited or nervous because she is on flat ground. This corresponds to lines that have a slope of 0. She then proceeds to talk about how when she starts actually going down the ski slope, she hates it! This relates to the negative slope of a line. She also mentions how she went over the side of a cliff and fell straight down. She was so scared she couldn’t even think or “define” her thoughts. This is tied to slopes that are undefined. I thought that this illustration was a great way of explaining the concept of slope from a real world example. After sharing the illustration, the students could work on problems involving calculating the slope of ski hills. How can this topic be used in your students’ future courses in mathematics or science? Understanding how to find the slope of a line is crucial for mathematics courses beyond Algebra I and Algebra II. Particularly, knowing how to find the slope of a line is essential for finding tangent lines of curves. This comes in handy for Calculus when you have to use limits to determine the slope. If a student does not have a strong grasp of what slope means and what its relationship is with the graph and the equation in Algebra I, then they will have a difficult time understanding slopes of lines that are not straight. # Engaging students: Finding the slope of a line In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Jason Trejo. His topic, from Algebra: finding the slope of a line. A2) How could you as a teacher create an activity or project that involves your topic? I have to start off by giving some credit to my 5th grade math teacher for giving me the idea on how I could create an activity involving this topic. You see, back in my 5th grade math class, we were to plot points given to us on a Cartesian plane and then connect the dots to create a picture (which turned out to be a caveman). Once we created the picture, we were to add more to it and the best drawing would win a prize. My idea is to split the class up into groups and give them an assortment of lines on separate pieces of transparent graphing sheets. They would then find the slopes and trace over the line in a predetermined color (e.g. all lines with m=2 will be blue, when m=1/3 then red, etc.). Next they stack each line with matching slopes above the other to create pictures like this: Of course, what I have them create would be more intricate and colorful, but this is the idea for now. It is also possible to have the students fine the slope of lines at certain points to create a picture like I did back in 5th grade and then have them color their drawing. They would end up with pictures such as: C1) How has this topic appeared in pop culture (movies, TV, current music, videogames, etc.)? Sure there aren’t many places where finding the slope of a line will be the topic that everyone goes on and one about on TV or on the hottest blog or all over Vine (whatever that is), but take a look around and you will be able to see a slope maybe on a building or from the top of Tom Hank’s head to the end of his shadow. Think about it, with enough effort, anyone could imagine a coordinate plane “behind” anything and try to find the slop from one point to another. The example I came up with goes along with this picture I edited: *Picture not accurately to scale This is the infamous, first double backflip ever landed in a major competition. The athlete: Travis Pastrana; the competition: the 2006 X-Games. I would first show the video (found here: https://www.youtube.com/watch?v=rLKERGvwBQ8), then show them the picture above to have them solve for each of the different slopes seen. In reality this is a parabola, but we can break up his motion to certain points in the trick (like when Travis is on the ground or when Travis is upside down for the first backflip). When the students go over parabolas at a later time, we could then come back to this picture. B2) How does this topic extend what your students should have learned in previous courses? It has been many years since I was first introduced to finding the slope of the line so I’m not sure exactly when I learned it, but I do know that I at least saw what a line was in 5th grade based on the drawing project I stated earlier. At that point, all I knew was to plot points on a graph and “connect the dots”, so this builds on that by actually being able to give a formula for those lines that connected the dots. Other than that, finding slopes on a Cartesian plane can give more insight on what negative numbers are and how they relate to positive numbers. Finally, students should have already learned about speed and time, so by creating a representation how those two relate, a line can be drawn. The students would see the rate of change based on speed and time. References: Minimalistic Landscape: http://imgur.com/a/44DNn Minimalistic Flowers: http://imgur.com/Kwk0tW0 Double Backflip Image: http://cdn.motocross.transworld.net/files/2010/03/tp_doubleback_final.jpg Double Backflip Video: : https://www.youtube.com/watch?v=rLKERGvwBQ8 # Engaging students: Graphs of linear equations In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Nada Al-Ghussain. Her topic, from Algebra: graphs of linear equations. How could you as a teacher create an activity or project that involves your topic? Positive slope, negative slope, no slope, and undefined, are four lines that cross over the coordinate plane. Boring. So how can I engage my students during the topic of graphs of linear equations, when all they can think of is the four images of slope? Simple, I assign a project that brings out the Individuality and creativity of each student. Something to wake up their minds! An individualized image-graphing project. I would give each student a large coordinate plane, where they will graph their picture using straight lines only. I would ask them to use only points at intersections, but this can change to half points if needed. Then each student will receive an Equation sheet where they will find and write 2 equations for each different type of slope. So a student will have equations for two horizontal lines, vertical lines, positive slope, and negative slope. The best part is the project can be tailored to each class weakness or strength. I can also ask them to write the slop-intercept form, point slope form, or to even compare slopes that are parallel or perpendicular. When they are done, students would have practiced graphing and writing linear equations many times using their drawn images. Some students would be able to recognize slopes easier when they recall this project and their specific work on it. Example of a project template: Examples of student work: How has this topic appeared in the news? Millions of people tune in to watch the news daily. Information is poured into our ears and images through our eyes. We cannot absorb it all, so the news makes it easy for us to understand and uses graphs of linear equations. Plus, the Whoa! Factor of the slopping lines is really the attention grabber. News comes in many forms either through, TV, Internet, or newspaper. Students can learn to quickly understand the meaning of graphs with the different slopes the few seconds they are exposed to them. On television, FOX news shows a positive slope of increasing number of job losses through a few years. (Beware for misrepresented data!) A journal article contains the cost of college increase between public and private colleges showing the negative slope of private costs decreasing. Most importantly line graphs can help muggles, half bloods, witches, and wizards to better understand the rise and decline of attractive characters through the Harry Potter series. How can this topic be used in your students’ future courses in mathematics or science? Students are introduced to simple graphs of linear equations where they should be able to name and find the equation of the slope. In a student’s future course with computers or tablets, I would use the Desmos graphing calculator online. This tool gives the students the ability to work backwards. I would ask a class to make certain lines, and they will have to come up with the equation with only their knowledge from previous class. It would really help the students understand the reason behind a negative slope and positive slope plus the difference between zero slope and undefined. After checking their previous knowledge, students can make visual representations of graphing linear inequalities and apply them to real-world problems. References: http://www.hoppeninjamath.com/teacherblog/?p=217 http://walkinginmathland.weebly.com/teaching-math-blog/animal-project-graphing-linear-lines-and-stating-equations http://mediamatters.org/research/2012/10/01/a-history-of-dishonest-fox-charts/190225 http://money.cnn.com/2010/10/28/pf/college/college_tuition/ http://dailyfig.figment.com/2011/07/13/harry-potter-in-charts/ https://www.desmos.com/calculator # Engaging students: Slope-intercept form of a line In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Kelley Nguyen. Her topic, from Algebra: slope-intercept form of a line. How has this topic appeared in high culture (art, classical music, theatre, etc.)? The slope-intercept form of a line is a linear function. Linear functions are dealt with in many ways in everyday life, some of which you probably don’t even notice. One example where the slope-intercept form of a line appears in high culture is through music and arts. Suppose a band wants to book an auditorium for their upcoming concert. As most bands do, they meet with the manager of the location, book a date, and determine a payment. Let’s say it costs$1,500 to rent the building for 2 hours. In addition to this fee, the band earns 20% of each $30 ticket sold. Write an equation that determines whether the band made profit or lost money due to the number of tickets sold – the equation would be y = 0.2(30)x – 1500, where y is the amount gained or lost and x is the number of tickets sold that night. This can also help the band determine their goal on how many tickets to sell. If they want to make a profit of$2,000, they would have to sell x-many tickets to accomplish that.

In reality, most arts performances make a profit from their shows or concerts. Not only do mathematicians and scientists use slope-intercept of a line, but with this example, it shows up in many types of arts and real-world situations. Not only does the form work for calculating cost or profit, it can relate to the number of seats in a theatre, such as x rows of 30 seats and a VIP section of 20 seats. The equation to find how many seats are available in the theatre is y = 30x + 20, where x is the number of rows.

How can technology be used to effectively engage students with this topic?

A great way to engage students when learning about slope-intercept form of a line is to use Geometer’s Sketchpad. After opening a graph with an x- and y-axis, use the tools to create a line. From there, you can drag the line up or down and notice that the slope increases as you move upward and decreases as you move downward. Students can also find the equation of the line by selecting the line, clicking “Measure” in the menu bar, and selecting “Equation” in the drop-down list. This gives the students an accurate equation of the line they selected in slope-intercept form. Geometer’s Sketchpad allows students to experiment and explore directions of lines, determine whether or not it has an increasing slope, and help create a visual image for positive and negative slopes.

Also, with this program, students can play a matching game with slope-intercept equations and lines. You will instruct the student to create five random lines that move in any direction. Next, they will select all of the lines, go to “Measure” in the menu bar, and click “Equation.” From there, it’ll give them the equation of each line. Then, the student will go back and select the lines once again, go to “Edit” on the menu bar, hover over “Action Buttons,” and select “Hide/Show.” Once a box comes up, they will click the “Label” tab and type Scramble Lines in the text line. Next, the lines will scramble and stop when clicked on. Once the lines are done scrambling, the student could then match the equations with their lines. This activity gives the students the chance to look at equations and determine whether the slope is increasing and decreasing and where the line hits the y-axis.

How could you as a teacher create an activity or project that involves your topic?

With this topic, I could definitely do a project that consists of slope-intercept equations, their graphs, and word problems that involve computations. For example, growing up, some students had to earn money by doing chores around the house. Parents give allowance on daily duties that their children did.

The project will give the daily amount of allowance that each student earned. With that, say the student needed to reach a certain amount of money before purchasing the iPad Air. In part one of the project, the student will create an equation that reflects their daily allowing of $5 and the amount of money they have at the moment. In part two, the student will construct a graph that shows the rate of their earnings, supposing that they don’t skip a day of chores. In part three, the students will answer a series of questions, such as, • What will you earn after a week? • What is your total amount of money after that week? • When will you have enough money to buy that iPad Air at$540 after tax?

This would be a short project, but it’s definitely something that the students can do outside of class as a fun activity. It can also help them reach their goals of owning something they want and making a financial plan on how to accomplish that.

References

# Finding the equation of a line between two points

Here’s a standard problem that could be found in any Algebra I textbook.

Find the equation of the line between $(-1,-2)$ and $(4,2)$.

The first step is clear: the slope of the line is

$m = \displaystyle \frac{2-(-2)}{4-(-1)} = \frac{4}{5}$

At this point, there are two reasonable approaches for finding the equation of the line.

Method #1. This is the method that was hammered into my head when I took Algebra I. We use the point-slope form of the line:

$y - y_1 = m (x - x_1)$

$y - 2 = \displaystyle \frac{4}{5} (x-4)$

$y - 2 = \displaystyle \frac{4}{5}x - \frac{16}{5}$

$y = \displaystyle \frac{4}{5}x - \frac{6}{5}$

For what it’s worth, the point-slope form of the line relies on the fact that the slope between $(x,y)$ and $(x_1,y_1)$ is also equal to $m$.

Method #2. I can honestly say that I never saw this second method until I became a college professor and I saw it on my students’ homework. In fact, I was so taken aback that I almost marked the solution incorrect until I took a minute to think through the logic of my students’ solution. Let’s set up the slope-intercept form of a line:

$y= \displaystyle \frac{4}{5}x + b$

Then we plug in one of the points for $x$ and $y$ to solve for $b$.

$2 = \displaystyle \frac{4}{5}(4) + b$

$\displaystyle -\frac{6}{5} = b$

Therefore, the line is $y = \displaystyle \frac{4}{5}x - \frac{6}{5}$.

My experience is that most college students prefer Method #2, and I can’t say that I blame them. The slope-intercept form of a line is far easier to use than the point-slope form, and it’s one less formula to memorize.

Still, I’d like to point out that there are instances in courses above Algebra I that the point-slope form is really helpful, and so the point-slope form should continue to be taught in Algebra I so that students are prepared for these applications later in life.

Topic #1. In calculus, if $f$ is differentiable, then the tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ has slope $f'(a)$. Therefore, the equation of the tangent line (or the linearization) has the form

$y = f(a) + f'(a) \cdot (x-a)$

This linearization is immediately obtained from the point-slope form of a line. It also can be obtained using Method #2 above, so it takes a little bit of extra work.

This linearization is used to derive Newton’s method for approximating the roots of functions, and it is a precursor to Taylor series.

Topic #2. In statistics, a common topic is finding the least-squares fit to a set of points $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$. The solution is called the regression line, which has the form

$y - \overline{y} = r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$

In this equation,

• $\overline{x}$ and $\overline{y}$ are the means of the $x-$ and $y-$values, respectively.
• $s_x$ and $s_y$ are the sample standard deviations of the $x-$ and $y-$values, respectively.
• $r$ is the correlation coefficient between the $x-$ and $y-$values.

The formula of the regression line is decidedly easier to write in point-slope form than in slope-intercept form. Also, the point-slope form makes the interpretation of the regression line clear: it must pass through the point of averages $(\overline{x}, \overline{y})$.


text:
# Redshift of mass-dominant universe

What was the value of redshift $z$ when matter started to dominate the universe?

Is there any way to calculate it without knowing the time?

• $z_{eq} \sim 3000$. It should be easy to calculate if you know the present amount of radiation and matter energy density. – Ihle Nov 1 '15 at 21:17

We can calculate the redshift of matter-radiation equality by using the scaling of the energy density of radiation and matter:

$$\rho_{matter} = \rho_{matter}^0(1+z)^3,$$ $$\rho_{radiation} = \rho_{radiation}^0(1+z)^4,$$ where $\rho^0$ denotes the current enegy density.

This gives us $$z_{eq} = \frac{\rho_{matter}^0}{\rho_{radiation}^0} -1 \sim 3400.$$

The exact current value can be found in: http://planck.caltech.edu/pub/2015results/Planck_2015_Results_XIII_Cosmological_Parameters.pdf


text:
Optics series, lecture – IX

This lecture was delivered on 14th February 2017 ( 1.5 hours ) to honors students.

( all optics series lectures ) go to other available optics lectures

## “Young’s double slit experiment: coherent sources and conditions of interference”

### Background

The concurrent lecture is particularly divided into two parts.

a. The first part pertains to what are coherent sources and what are the sustainable conditions for interference, for such to be observed.

b. The second part leads us to describe in requisite detail the phenomenon of Young’s double slit interference.

Recall that we have already discussed the phenomenon of interference in our lecture — VII.

We will only passively mention that there are two kinds of interference, the so called wave-front splitting and the amplitude splitting interference. Later on we will discuss the required details of both kinds. Before we do so we will have several instances of lecture on interference phenomena of both types.

Young’s double slit interference is an example of the wave-front splitting interference. What happens here is there are two primary or secondary coherent sources and two distinct waves interfere at a given observation vantage.

Another example of wave-front splitting interference is Fresnel’s bi-prism set-up which we will study soon, in an imminent lecture.

( lecture XI: Fresnel’s bi-prism ) wave front splitting interference in Fresnel’s bi-prism

For amplitude splitting interference only one wave ( i.e. one source ) produces the interference patterns because the wave amplitude is partially reflected and partially transmitted ( i.e. refracted ) and both channels meet up somewhere.

Just to mention it for the time being, Newton’s ring interference patterns is an example of amplitude splitting interference. Later we will study the details of all sorts of interference phenomena such as the ones just mentioned. ( see link below for the same )

( lecture XIV, XV, XVI: several interference phenomena ) Read about Newton’s ring and many other interference phenomena.

### Coherent sources and conditions for interference

#### Coherence

Let us now discuss the first part of our lecture. Let us for the time being define coherence as the attribute of a light source such that there is no arbitrary and unexpected changes in the phases of different light waves such that when these waves meet at an observation point, we can apply the results of our interference analysis that we discussed in lecture — VII. ( lecture — VII has been linked above )

Accordingly the coherence of light sources exists for certain amount of time, after which the coherence is lost. This is known as temporal coherence. If this value of time is denoted as Δt then light wave moves a distance of Δs = c × Δt and Δs is known as spatial coherence, as this is the distance for which coherence exists in the light wave. Coherence is necessary for light-waves, for us to observe the resultant interference

#### Conditions of interference

Stable interference pattern
Stable interference pattern is produced when two wave sources have same frequency. Any large difference in the frequency destroys the coherence and therefore spoils the interference pattern. Large frequency difference results in rapid time – dependent phase difference, I12 averages to zero. Recall that I12 is the interference term, in our description of interference in lecture — VII.
For white light
When the light source produces white light there is interference among each color component, red interferes with red, blue interferes with blue and so on. An overall white interference pattern is observable.
Clearest interference pattern
The clearest interference pattern is produced when waves have nearly equal amplitude.
Sources need to be in phase
Sources of light need to be in phase with each other for observation of fringe pattern.
Coherent sources
Coherent sources are necessary for formation of interference pattern, as long as the initial phase difference between the two sources is constant, an interference pattern ( which is shifted ) is produced.
Visibility of interference pattern
The visibility of interference is given by $v=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$ where the maximum and minimum irradiance of the two sources are related to the irradiance of the individual sources by the following, as discussed in lecture — VII: a. $I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2$, b. $I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2$, c. $v=\bigg[ \frac{\,2\,(I_1+I_2\,)}{4\sqrt{I_1\,I_2}} \bigg]^{-1}=\frac{2\,\sqrt{I_1\,I_2} }{I_1+I_2}$ when both sources have the same irradiance, that is I1 = I2 this would mean Imin = 0 and v = 1. The visibility is perfect or 100 %. Therefore the amplitudes need to be equal as we stated in point 3 above.

#### Difference between coherent and incoherent sources

Let us now discuss the differences between coherent and incoherent sources. Let us also define formally, the coherence length and coherence time, that we just discussed above.

Coherence time: coherence time is the time interval during which light wave has sinusoidal purity and there are no random phase changes. Its given by: Δtc.

Coherence length: coherence length is the distance along the direction of propagation of light wave, over which temporal phase coherence exists in the wave, given by: L= c × Δtc.

Coherent source
a. equal frequencies and constant phase difference b. produces interference c. resultant intensity varies between maximum and minimum, zero intensity is possible d. LASER and secondary sources from primary light source are examples of coherent sources e. coherence is produced when there is a constant phase difference between two sources
Incoherent source
a. unequal frequencies and varying phase difference b. do not produce interference c. resultant intensity is sum of individual intensities, can’t produce zero intensity d. ordinary light sources producing visible light such as incandescent lamp filaments, candle light, sun light, head lights etc are examples of incoherent sources e. produced by random and rapid generation of light waves at atomic level

### Young’s double slit experiment

Now we are in a position to discuss the interesting results of interference in a Young’s double slit experiment set-up. Grimaldi was a famous physicist and astronomer who was deceased shortly before Galileo was born. Grimaldi had hoped to produce interference of light waves emanating from sun by letting the light pass through a small hole.

That way it would have produced two different wave – fronts emanating from the secondary source ( the hole ) acting like a primary source, a premonition about the wave – front splitting interference effect, so to say. But the set up resulted in a null effect.

He did not observe any interference. Later Young modified his set up slightly to produce the expected interference patterns of light waves. The small change that Young, so inquisitive though, adopted is because of one of the primary reason for production of interference pattern as an observable. Namely coherence.

The sun is a vast region in the cosmos as far as we are concerned. This naturally means it does not have the required coherence, the primary condition for interference patterns, as light rays come from extremely long and uncorrelated points. Thus the small hole in the purported interference set up does not guarantee the desirable coherence.

Thomas Young, more than 100 years later, recognized the shortcoming that had clad Grimaldi’s understanding of the situation and brought a small change to the system. He introduced two small pinholes after a short distance to the primary hole. The first hole therefore acted like a primary source and acted like a coherent source for the two secondary sources.

This way while sun remained as an incoherent source to satisfy its own hubris, the secondary ( holes ) sources acted in tandem with the primary hole and provided a reasonable source of coherence, for the interference pattern to be observed.

Let us now draw a diagram to depict this experiment famously known as the Young’s double slit experiment.

Let us consider two secondary coherent sources S1 and S2 which are distances r1 and r2 apart from the point of observation P. The path difference of these two rays is given by S1B = S1P – S2P.

Now S1B = r– r= a sin θ. This implies: r– r= a θ when angle θ is small. This also means θ = y / S. So: r– r= ( a / S ) y.

Constructive interference occurs when path difference is an integral multiple of λ. Thus: r– r= m λ, where m is any integer; 0, ± 1, ± 2, ± 3 …. So the height of observation point y can be given in terms of S: the distance of plane of observation Σo from screen of the sources Σa, here a: width between sources S1 and S2, m: order of interference, this is any integer and λ: the wavelength of the light sources.

$\boxed{y_{\,m} \simeq \frac{S}{a}\,m\,\lambda }$.

Thus ym gives the position of the m-th order maxima or bright fringe, on the vertical axis. Since y = 0 for m = 0, we see a central bright fringe. The angular position of the m-th order bright fringe is given by: $\theta_m \simeq \frac{m\,\lambda }{a}$.

Two consecutive maxima are given by the integers m + 1 and m. So, we can easily determine the spacing between bright fringes. $y_{m+1}-y_m \simeq\frac{S}{a}(\,m+1\,)\,\lambda-\frac{S}{a}\,m\,\lambda$.

$\boxed{\Delta y =\frac{S}{a}\lambda \,\,\,\,or\,\,\,\, \beta_b=\frac{D\lambda}{d}}$,

where we have given the fringe spacing an alternative symbol used in some textbooks: βb is fringe spacing for bright fringes, D is distance of separation between sources and the screen of observation, d is the slit separation.

Similarly dark fringes or minima are formed when the path difference is odd integer multiple of λ/2 or phase difference is odd integer multiple of π.

We can thus write: $y_m \simeq \frac{S}{a}\,\Big(\,m+\frac{1}{2}\,\Big)\,\lambda$, $y_{m+1} \simeq \frac{S}{a}\,\Big(\,m+\frac{3}{2}\,\Big)\,\lambda$.

So $\boxed{\Delta\, y _{\,dark} =\frac{S}{a}\,\lambda \,\,\,or \, \,\,\,\beta_d=\frac{D\,\lambda}{d}}$

We see that the spacing between dark and bright fringes is the same and it remains constant even if we change the order of the interference: m.

When we studied interference due to two coherent plane harmonic waves, in lecture VII, we derived the intensity distribution for two beams having equal irradiance;  $I=4\,I_0\,\cos^2{\,\Big(\frac{\delta}{2}}\Big)$. The phase difference is δ = k (r– r2).

By using r– r= ( a / S ) y we have: $I=4\,I_0\,\cos^2{\,\Big(\frac{y\,a\,\pi}{S\,\lambda}}\Big)$. In alternative symbols that we mentioned above: $I=4\,I_0\,\cos^2{\,\Big(\frac{y\,d\,\pi}{D\,\lambda}}\Big)$.

Let us plot this distribution as a function of y.


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# Cos - cos identity

Feb 22, 2018 · We will prove the cosine of the sum of two angles identity first, and then show that this result can be extended to all the other identities given. cos (α+β) = cos α cos β − sin α sin β We draw a circle with radius 1 unit, with point P on the circumference at (1, 0).

To do this we use the Pythagorean identity sin 2 (A) + cos 2 (A) = 1. In this case, we find: cos 2 (A) = 1 − sin 2 (A) = 1 − (3/5) 2 = 1 − (9/25) = 16/25. The cosine itself will be plus What should cos 𝑥𝑥+ 𝑦𝑦and sin 𝑥𝑥+ 𝑦𝑦be? Do these trigonometric functions behave linearly? Is cos 𝑥𝑥+ 𝑦𝑦= cos 𝑥𝑥+ cos 𝑦𝑦and sin 𝑥𝑥+ 𝑦𝑦= sin 𝑦𝑦+ sin 𝑦𝑦? Try with some known values: cos 𝜋𝜋 6 + 𝜋𝜋 3 = cos 𝜋𝜋 6 + cos 𝜋𝜋 3 cos 3𝜋𝜋 6 = cos 𝜋𝜋 6 The sum-to-product trigonometric identities are similar to the product-to-sum trigonometric identities.

s i n (2 θ) = 2 s i n (θ) c o s (θ) From these relationships, the cofunction identities are formed. Notice also that sinθ = cos(π 2 − θ): opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of θ equals the cofunction of the complement of θ. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions. CORE BY COS Wardrobe foundations, for all facets of life. Made from the finest fabrics and sustainably sourced materials, explore our edits of essentials. The following (particularly the first of the three below) are called "Pythagorean" identities.

## Apr 15, 2015 The problem here is "how far back do we need to go?" when we try to explain " why?" Assuming that the following identities are known:

Note that an identity holds true for  sin 2x = 2 sin x cos x. Double-angle identity for sine.

### 1 + cot2 θ = cosec2θ. (2) tan2 θ + 1 = sec2 θ. (3). Note that (2) = (1)/ sin2 θ and (3 ) = (1)/ cos2 θ. Compound-angle formulae cos(A + B) = cos A cos B − sin A sin B.

Trigonometry Trigonometric Identities and Equations Sum and Difference Identities.

The following (particularly the first of the three below) are called "Pythagorean" identities. sin2(t) + cos2(t) = 1. tan2(t) + 1 = sec2(t). 1 + cot2(t) = csc2(t).

Second result. Radians. First result. This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides many practice problems on identifying the sides  (23) Once again, we replace β with (−β), and the identity in (22) becomes: cos (α − β) = cos α cos β + sin α sin β. Next, we re-group the angles inside the cosine

Identities for negative angles. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions. Ptolemy’s identities, the sum and difference formulas for sine and cosine. Sine, tangent, cotangent and cosecant in mathematics an identity is an equation that is always true. Meanwhile trigonometric identities are equations that involve trigonometric functions that are always true. This identities mostly refer to one angle labelled θ. Defining Tangent, Cotangent, Secant and Cosecant from Sine and Cosine 2 - The cosine laws a 2 = b 2 + c 2 - 2 b c cos A b 2 = a 2 + c 2 - 2 a c cos B c 2 = a 2 + b 2 - 2 a b cos C Relations Between Trigonometric Functions cscX = 1 / sinX sinX = 1 / cscX secX = 1 / cosX cosX = 1 / secX tanX = 1 / cotX cotX = 1 / tanX tanX = sinX / cosX cotX = cosX / sinX Pythagorean Identities sin 2 X + cos 2 X = 1 1 + tan 2 X Trigonometric Identities Sum and Di erence Formulas sin(x+ y) = sinxcosy+ cosxsiny sin(x y) = sinxcosy cosxsiny 1 cos 2 cos 2 = q 1+cos 2 tan 2 = q 1+cos tan 2 But in the cosine formulas, + on the left becomes − on the right; and vice-versa.

They are just the length of one side divided by another. For a right triangle with an angle θ   Formulas and Identities. Tangent and Cotangent Identities sin cos tan cot cos sin θ θ θ θ θ θ. = = Reciprocal Identities. 1.

2 The complex plane A complex number cis given as a sum c= a+ ib cos(4x) in terms of cos(x), write cos(4x) in terms of cos(x), using the angle sum formula and the double angle formulas, prove trig identities, verify trig i Odd/Even Identities. sin (–x) = –sin x cos (–x) = cos x tan (–x) = –tan x csc (–x) = –csc x sec (–x) = sec x cot (–x) = –cot x I know that there is a trig identity for $\cos(a+b)$ and an identity for $\cos(2a)$, but is there an identity for $\cos(ab)$? $\cos(a+b)=\cos a \cos b -\sin a \sin b$ $\cos(2a)=\cos^2a-\sin^2a$ 0)) = cos( 0 0), and we get the identity in this case, too. To get the sum identity for cosine, we use the di erence formula along with the Even/Odd Identities cos( + ) = cos( ( )) = cos( )cos( ) + sin( )sin( ) = cos( )cos( ) sin( )sin( ) We put these newfound identities to good … Use sum and difference formulas for cosine. Use sum and difference formulas to verify identities.

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### See (Figure). Sum formula for cosine, \mathrm{cos}\left(\alpha +\beta \right). Difference formula

cos –t = cos t. tan –t = –tan t. Sum formulas for sine and cosine sin (s + t) = sin s cos t + cos s sin t.


text:
# interpreted scripting language

S

I would like to put in my $.02 against using an interpreted language. I have limited experience with SCADA systems, but the one thing that I have found is that using an interpreted language makes it more possible for errors to become prevalent at inopportune times. Yea I know, test, test, test - but.... Nathan Stowe Systems Engineer A Finkl & Sons 2011 North Southport Chicago, IL 60614-4079 [email protected] http://www.finkl.com _______________________________________________ LinuxPLC mailing list [email protected] http://linuxplc.org/mailman/listinfo/linuxplc B #### Bill Sturm You have a great point. A compiler will scan your complete program for syntax errors and report them at compile time. An interpreter will generally only test code when it executes. I have been burned in VB by a line of code that I never tested which caused the program to abort. For some reason, this happens most often right after you leave for the day Some interpreters do have syntax checking programs. I have never tried them, I assume that they help. VB has an option to test all code at the beginning of runtime, I recommend checking that option. Bill _______________________________________________ LinuxPLC mailing list [email protected] http://linuxplc.org/mailman/listinfo/linuxplc K #### Ken Irving Perl (for one) does compile the source, so will catch syntax errors, but valid code can still have bugs and application errors. I'm not convinced that compiled code is intrinsically any less prone to bugs and errors than interpreted. Some programming styles can help prevent hidden bugs, e.g,. by ensuring that (nearly) all code is processed on each pass. This tends to rule out conditional jumps, and increases the amount of code to run, but sometimes it's worth it. This can apply regardless of whether the language is compiled or interpreted. Just another$.02.

--
Ken Irving <[email protected]>

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LinuxPLC mailing list
[email protected]
http://linuxplc.org/mailman/listinfo/linuxplc

P

J

#### Jocelyn Boily

One of the main advantages of PLC languages is that you can change the program while it is running.

It happen to me very often to maintain the codes of robots on machine that can not be stopped for monetary reasons (production lines, ... ) and had to change the program "On-Line". This is done very easily with PLC like Allen Bradley, GEFanuc and others.

I would hate to see Our LinuxPLC to shy away from this feature. Interpretative languages would allows to do this quite easily, but would it
be possible with a compiling language?

jb

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T

#### Tim

> You have a great point. A compiler will scan your complete program for
> syntax errors and report them at compile time. An interpreter will
> generally only test code when it executes. I have been burned in VB by a
> line of code that I never tested which caused the program to abort. For
> some reason, this happens most often right after you leave for the day

Python "compiles" from .py to .pyc, and catches errors very nicely. It does it on the fly if there's no pre-generated .pyc files handy. It
also has syntax checking tools. I typically make a lot fewer errors (and far less serious ones) in Python than in most other languages, compiled or otherwise.

I think it's worth considering. Of course, if you go with some standard way of exchanging data on the backend and the SCADA is totally separate, it won't much matter what's used. I'd like to play
with a python front end if it goes that way, but I'm not a SCADA expert by any devious twist of the imagination.

Tim

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G

#### Greg Goodman

> I would like to put in my \$.02 against using an interpreted language. I have limited experience with SCADA systems, but the one thing that I have found is that using an interpreted language makes it more possible for errors to become prevalent at inopportune times.

here's my 2 cents.

i have a lot of experience with SCADA systems, and i know that all sorts of errors make themselves known at inopportune times. in fact, any error that doesn't get caught during design, code or lab test is going to be inopportune when it does show up. and compiled code demonstrates this just as readily as does interpreted code.

compiled code can suffer from memory leaks, buffer overflows, crashes due to unexpected values or characters in the data stream, read/write failures that produce garbage data because the coder didn't check the error returns... lots of stuff that doesn't show up even in rigorous testing, because it's dependent on system load, or the operating environment, or a real-life data stream with gunk in it that nobody knew about or thought to test for.

i write a lot of interpreted code, and pretty much every line gets tested to some degree; every routine has a unit tester that exercises all of the cases that i coded for... and (for obvious reasons) nothing that i didn't think to handle. that verifies that the code works the way i think it should (which isn't necessarily the way it *should* work) and catches the syntax errors that aren't the result of building variable names on the fly.

it's worth noting that all data or event processor code is subject to the behavior of the data/event stream which, in the automation world, is often (usually?) different from the test data streams we handle in the lab. even a correct program operating beyond its limits will fail, which is as likely to be the fault of the programmer making assumptions as it is of the user abusing the program.


text:
[SOLUTION]-BIBL 450 assignment- Begin by choosing 1 narrative passage and 1 apocalyptic passage from the Book of Daniel (2 passages total). Both the narrative text and the apo
5 (100%) 5 vote[s]

Essay 1 Instructions
Begin by choosing 1 narrative passage and 1 apocalyptic passage from the Book of Daniel (2 passages total). Both the narrative text and the apocalyptic text must be chosen because they are considered by you as especially significant to the overall message of the Book of Daniel. Then, in a 1000–1500-word essay, in current Turabian format, complete the following:
1. Write an interpretive exposition of each text (the narrative passage and the apocalyptic passage).
2. State how the mixture of narrative and apocalyptic literature contributes to the overall message of Daniel.
3. State how these 2 passages, together, contribute to the overall message of the Book of Daniel.
Step 1 must be 800–1000 words, step 2 must be 100–250 words, and step 3 100–250 words for a total of 1000–1500 words.


text:
Structure of Blastwaves in a Magnetized Medium
Session 41 -- Computational Astrophysics II
Display presentation, Wednesday, 1, 1994, 9:20-6:30

[41.16] Structure of Blastwaves in a Magnetized Medium

Dongsu Ryu (Chungnam National University, Korea), T. W.~Jones (U.~of Minnesota), Adam Frank (U.~of Minnesota)

We present multidimensional simulations of adiabatic blastwaves propagating through a magnetized ideal medium. Simulations have been done with a multidimensional magnetohydrodynamic (MHD) code, which was built by extending a one-dimensional code described in Ryu and Jones (1994) using a Strang-type operator splitting method. The constraint of a divergence-free field was enforced exactly by adding a necessary correction calculated in a separate step. Extensive tests including two-dimensional MHD shock tubes proved the ability to capture discontinuities in all three MHD waves families as well as contact and tangential discontinuities

In this poster, we address specifically the question on the existence of intermediate shocks suggested to form in concave fronts around the magnetic pole (e.~g. , Ferri\'ere, Mac Low, and Zweibel, 1992). In our simulations with two-dimensional magnetic field, where the third component of field has been suppressed, we have confirmed the reversal of the transverse magnetic field in the concave fronts but the structures formed resemble fast compound structures, which are composed of intermediate shocks (so called the 1-3'' intermediate shocks) preceded by fast rarefaction waves. However, we suggest that those intermediate shocks or compound structures are artifacts of the two-dimensional nature of the calculations. In simulations where all the three components of magnetic field are allowed, the transverse magnetic field should be reversed only in rotational discontinuities or tangential discontinuities.

Support for this work at the University of Minnesota is provided through the NSF, NASA and the University of Minnesota Supercomputer Institute. DR also acknowledges the support by the Non-Directed Research Fund of the Korea Research Foundation 1993.

{\noindent Reference:\hfil\break Ryu, D.~and Jones, T.~W., 1994, in preparation.\hfil\break Ferri\'ere, K.~M., Mac Low, M.-M., and Zweibel, E.~G., 1992, Ap.~J., 375, 239}


text:
UK Secondary (7-11)
Fully factorise after completing the square

## Interactive practice questions

Factorise the quadratic using the method of completing the square to get it into the form $y=\left(x+a\right)\left(x+b\right)$y=(x+a)(x+b).

$y=x^2+4x+3$y=x2+4x+3

Easy
Approx 5 minutes

Factorise the quadratic using the method of completing the square or otherwise to get it into the form $y=\left(x-a\right)\left(x-b\right)$y=(xa)(xb).

$y=x^2-6x+8$y=x26x+8

Factorise the quadratic using the method of completing the square to get it into the form $y=\left(x+a\right)\left(x+b\right)$y=(x+a)(x+b).

$y=x^2+56x+159$y=x2+56x+159

Factorise the quadratic using the method of completing the square to get it into the form $y=\left(x-a\right)\left(x-b\right)$y=(xa)(xb).

$y=x^2-24x+63$y=x224x+63