The Parabola This was the first equation sheet that I created. There is something on this one for everyone. Unlike with the line, I cannot have students try to work with a single equation for parabolas. The reason is that the test questions are often designed to make it quite difficult to convert between the different forms. While conversion from factored form or vertex form to standard form is straightforward conversion from standard form to factored form (factoring) is a task which ranges from simple to nearly impossible depending on the student and the numbers involved. If factoring is needed to solve a particular problem, the College Board generally (nearly always) makes the factoring simple. Here is a link to an online interactive parabolic equation in standard form y = ax^2 + bx + c . Here is a link to an online interactive parabolic equation in factored form y = a(x - d)(x - f) . (I was not able to use e as one of the roots in the online graph since it is a numeric constant so I used f instead on the interactive graph.) Here is a link to an online interactive parabolic equation in vertex form y = a(x - h)^2 + k . While this equation sheet looks quite busy, it is much simpler once you realize that the entirety of the sheet is a mapping between those three forms of parabolic equations and four points on the curve: the vertex, the y-intercept, and the two x-intercepts. Those x-intercepts are also known as the roots or solutions of the equation. Once I explained the mapping to students, I taught them to only memorize what they had to and *understand* the rest. For instance, as I wrote in the notes for the line equation sheet, you can find the y-intercept by simply plugging in x = 0 into the equations. Similarly, there is no reason to memorize the equations for k: if you look at those equations you will see that they are simply the original equations with h substituted for x. Just find h and then plug that value in to get k. That leaves three points on the graph for which they need to memorize equations: - Equations for h (which is the x-value of the vertex) IMO, these are the most valuable equations on the entire sheet since textbooks do not teach how to find this value for the standard or factored forms. Note that most (all?) textbooks write the quadratic formula as: (-b +or- sqrt(b^2 - 4ac))/2a I suppose they write it like that because you don't write any term more than once. Unfortunately, it loses one of the most insightful aspects of the formula. As such, when we get to parabolas, I teach my students to write the quadratic formula "Mr. Guheert's way": -b/2a +or- sqrt(b^2 - 4ac)/2a Not much different, is it? But notice one thing: it is now clear that the two roots are simply some number above or below a central number. Since parabolas are symmetrical, that central number is the x-value of the vertex and is simply -b/2a !! That works regardless of whether the roots are real or complex. It is a gigantic time saver on these tests. h = -b/2a Similarly, in factored form there is a very simple way to find h: it is the number which is half way between the two roots. So I ask them, "How do you find a number which is half way between two other numbers?" Blank stare. It is simply the average of those two numbers or (d + e)/2 h = (d + e)/2 - Equations for d and e (the x-intercepts, which are the roots or solutions to the equation) Of course the solutions to a quadratic equation in standard form are given by the quadratic formula. On this sheet I wrote that formula as two *separate* equations. That makes it much more clear to students that those are two different values of x that will give you a y value of zero (if there are real roots). d = -b/2a - sqrt(b^2 - 4ac)/2a and e = -b/2a + sqrt(b^2 - 4ac)/2a (For the purists out there: notice that the above "simplification" only makes d < e if the parabola opens up and therefore a is a positive number. If a is negative, e will be to the left of d on the number line. I decided to represent the equations this way because of the insight it gave. I explained this slight discrepancy if/when needed by the student.) In factored form, d and e are provided directly by the equations. But this can be a bit tricky since the students often struggle with the sign. The best way that I found to clarify finding d and e was to ask this: "What value do you need to put in for x to make that particular factor equal to zero?" I found that I had to repeat that often until they got that it is THE way to determine d and e. I wanted that question playing in their ears as they took the test. I will leave the derivation of the equations for d and e for vertex form to you. To my knowledge, those particular equations have only proven to be useful on the SAT or PSAT for one question in the past five years. They are there for completeness and in case any student wants to memorize them. d = h - sqrt(-ak)/a and e = h + sqrt(-ak)/a (And again, purists, d and e swap when a is negative.) Note that when a is equal to 1, these simplify to: d = h - sqrt(-k) and e = h + sqrt(-k) Now for one comment about vertex form: Just like students get tripped up by the sign in factored form when finding d and e, they have a similar problem finding h. So I ask them the same question: "What value do you need to put in for x to make the base inside the parentheses that gets squared equal to zero?" I also tell them to note that the form is the same for d and e in factored form, for h in vertex form, and for h and k in the circle form given on the next sheet. There is a lot here to take in, but I can promise you that any student that fully understands these relationships will have a major leg up on the SAT and PSAT math sections. They simply do NOT have time to do everything the hard way on those tests. One thing which is not covered on this equation sheet that students need to know is a fact about standard form: The sign of the formula under the radical determines both how many roots there are and whether the roots are real or complex: If b^2 - 4ac > 0 then there are two real roots (i.e. d and e are separate and real) If b^2 - 4ac = 0 then there is only one real root (i.e. d and e are the same real number) If b^2 - 4ac < 0 then there are two complex roots (i.e. d and e are separate complex numbers) Finally, I must point out that I am a man with a fragile ego and that this is the piece of test prep material of which I am most proud. As such, feel free to ask questions about it or lavish praise upon it. But as far as criticisms go, I ask that you JAWM that it is a masterpiece and that no improvements are necessary! 😉

The Line This was the last equation sheet that I did. I put this off because I thought perhaps it was too basic. Maybe it was for some of my students, but it certainly was needed by some of them. In my prep sessions, I ONLY teach my students the slope-intercept form of the linear equation. "What is Mr. Guheert's favorite equation for the line?" was something my students heard OFTEN. I told them to forget about those other forms they learned in school like point-slope and two-point equations since they simply did not need to know them. Here is a link to an online interactive linear equation in slope-intercept form y = mx + b . Aside: One misgiving I have about ONLY teaching slope-intercept is that some of my students would be inclined to convert to that form when they were dealing with a system of equations in ax + by = c form. I had to make sure they did not automatically go down that path. Solving simultaneous equations is a topic all on its own for which I did not create a sheet. Perhaps I will write some notes in this thread later on regarding approaches that I thought were helpful in that area. Even seemingly simple concepts like point notation were confusing to some of my students, so that is something I tried to make clear on these sheets. Including variables within point notation is something that really threw off some of my students. Slope is tested in many ways on the SAT and PSAT and I tried to ensure that my students understood exactly what it was and how to determine the value and sign. By a few weeks into our prep sessions, I wanted each of my students to be able to look at the graph of *any* line and give me its equation. Conversely, they needed to be able to convert any linear equation into a graphical representation. There were a couple of *special* equations which I handled separately with my students. Here are some of the questions I found myself asking often: "What is the equation for a line on the X-axis?" Answer: y = 0 "What is the equation for a line on the Y-axis?" Answer: x = 0 (This one is particularly useful for teaching about how to find the y-intercept: plug in x = 0 into any equation! That works for *any* type of equation except those which do not cross the y axis.) If I saw any vertical or horizontal line not on the axis, I would always ask them to tell me what its equation was. Feel free to ask any questions about the line equation sheet. Why did I do such-and-such? Why didn't I include something else? How do you teach blah? etc.

I have done PSAT and SAT prep for 8 to 10 homeschool students for the past couple of years. The parents of many of the students who came to me were mostly interested in help with math, but I did prep for everything except the essay part of the test. After working with these tests for a while it became clear to me that the test writers of the math sections expect the students to be able to work fluidly between the equation-based representations of shapes and their graphical representations. I came to the realization that this was not how I was taught math and, with the exception of one student I had this year who used Singapore Math, that is not how the current curricula teach it either. Most of my student were unable to look at an equation on the test and tell me what shape it represented. They were very uncomfortable with me always asking "What is that?"...at least at first. I often got "It's an equation!" back. Sometimes "What is what?" What I see in textbooks are separate sections covering the graphs and the equations. What I also see is that the students have learned about these different things over a period of many years, so bringing the concepts together all in one place was helpful to them. So I decided to make graphical equation sheets that mapped the important features from the graph into the forms of the equations which appear on the test. The most basic of these is for the line and it shows up on the largest number of problems, but the one which I think is most valuable for students trying to get into higher echelon of test scores is the parabola. I will say that even my most mathy moms, and I had some moms who are *very* good at math (better than I am), found nuggets on those sheets which they had never realized before. I also learned a lot about these relationships myself! I will encourage you to have a look at these yourself if you are teaching high school math as there may be some tips that can help you better understand the relationships involved in a way that can help you teach your students. There are four equation sheets that I have created. I will attach them to four separate posts in this thread and will give a brief introduction to each to allow separate discussions for each one. I hope these are helpful to someone!

Please tell us where you are, as this is a very large storm (again). We are in Northern VA and today's storm seems to be passing us by to the north. It's actually quite warm here right now: 34F. That said, the forecast for Wednesday night and Thursday have not changed much over the past few days: 6 to 10 inches of snow and/or ice (whatever that means!).

I do not have any answers, so this is NOT a JAWM thread. If I compare Texas outages right now with Oklahoma outages I see the images attached below. Basically, Oklahoma is experiencing WAY fewer outages than Texas, even in adjacent counties directly across the border. This is something that the Texas energy regulators are going to have to answer for. So what are the possible reasons for this huge difference in results? I have a few ideas, but I suspect it is likely a complex combination of all of these and perhaps other factors of which I am unaware. Here is a brief list: Too many electric-only homes I know that a lot of the posters from Texas have indicated that their homes are all-electric. Many do not seem to have any other way to provide heat, so when it gets cold, more of the load of heating falls to the electricity grid than to other utilities like the natural gas pipelines. Houses are too poorly-insulated Since Texas is further south than Oklahoma, it makes sense that houses would not be as well-insulated there. That said, it also seems that the temperatures yesterday and today are colder in Oklahoma. Texas' electricity grid is not well-connected to neighboring states My understanding is that there are basically three electricity grids in the US: West of the Rockies, East of the Rockies, and Texas. I know that sounds like the beginning of a Texas joke, but I believe it is true. There are links between these three grids, but the links are not as strong as the interconnections within each of the grids. The bottom line is that a state like Oklahoma can more easily borrow from neighboring states if they have a shortfall than can Texas. Texans are more wasteful of electricity Everything is bigger in Texas and their electricity usage does not seem to be an exception. If you look at the last column in this list you will see that Texas used more than twice as much electricity as California in 2017. Of course the climates are not identical, but I doubt that accounts for the entire difference. It may just be historical, since a lot of the natural gas in the country originally came from Texas. (The same could be said about Oklahoma.) ERCOT's "energy-only" electricity market ERCOT stands for the Electricity Reliability Council of Texas. That is the name of the grid operator for the Texas electricity grid. They use something called an "energy-only" electricity market, which has come under fire during hot summers when electricity prices have spiked. Pundits have defended this market approach as a good thing: Summer price spikes are a feature of Texas' power market, not a bug. Price spikes are one thing, but outages during extreme weather is quite another! I expect this "feature" will get a long, hard look after the current debacle is over. Just have a look at electricity prices in Texas for last Thursday (mostly around $40/MWh or $0.04/kWh) versus yesterday's prices (often $9000/MWh or $9/kWh but never below $1000/MWh or $1/kWh). (My understanding is that those are the spot prices paid to the electricity generators on the grid. Those values are averaged over the month and over the entire Texas grid and then are passed on to the consumers.) Too much renewable energy with insufficient backup Texas has more installed renewable energy production than any other state. And they are growing that very rapidly. I have heard reports that the cold has caused perhaps half of the wind generators in Texas to shut down, which is one of the reasons for the current shortfall. As I said above, I suspect all of the above may be factors in what is going on right now, but perhaps there are other factors at play. Of course none of this discussion helps anyone who is suffering in the cold in Texas right now, but ultimately someone will need to figure these things out in order to minimize the risk of this type of problem in the future. What other factors do you think might be leading to the ongoing outages in Texas?

I Duck Duck Go'ed the Texas power outages just now and one of the things I found was recommendations from the power utility in Texas (called the Electricity Reliability Council of Texas or ERCOT) on how to save electricity during this emergency. The first item on the list was this: Really?! Around here, we would have to turn the thermostat UP to get to 68 degrees. It seems that there must be some (many?) Texans who run their houses cooler in summertime than they do in the wintertime! If ERCOT really wants to save electricity, why aren't they recommending setting thermostats to 58 degrees instead of 68? /rant

It's 2:00 AM and the Texas outage map still looks terrible. I hope everyone is managing to stay warm somehow.

Thanks! DD18 and DS18 are both in their second year at UAH. DD18 is in Computer Science while DS18 is in Computer Engineering. Both were remote last semester, but this semester DD18 is doing classes remotely from here in VA while DS18 is on campus. DD18 is a bit resentful about that, but there is NO WAY we were going to pay $4000 for her to live on campus for three months to take online classes. It's bad enough that they charge the same amount of tuition for online school as the did for in-person instruction. I'm sorry, but the two are NOT the same thing! DD18 did manage to tell me that they are closed tomorrow *after* I confronted her about having to hear about today's closing from people online.

Yeah, so does MomsintheGarden. Maybe she will come back to me once all of her clothing is photographed, catalogued, made into outfits, etc. ...but I'm not so sure...

Therein lies the problem. For you fashion-conscious women it is probably more addictive than ANY video game!

Hi RootAnn! That's very sweet of you! We miss you guys here, also. I am here today because MomsintheGarden pointed me to the "Texas" thread. Now I am "sucked in"! How do I get "sucked back out"? I need to tell you "Thanks!" because I only found out that UAH was closed today due to one of your posts! Far be it from our twins to tell us things like that! We sent down three freshmen to UAH last fall. The twins are doing great down there, but their older brother has now failed out. We are still trying to figure this out since he is an excellent student. Let's just say there is more to it than just school (and, no, it is not a problem with UAH as far as we know). I have had a couple of years of (mostly) successful SAT/PSAT prep here for homeschool students in the area. I create some materials during that time that I may post sometime if anyone is interested. Hi to everyone else, also! (Oh, and Quill: I have a bone to pick with you about an app called StyleBook.)

Dreamergirl: Does your emoticon on my last post mean you have a gas oven?

PSA: Baking or cooking to heat the house is fine if you have an ELECTRIC oven and/or an ELECTRIC cooktop. However, DO NOT try to keep your house warm by baking or cooking with a gas (propane or natural gas) oven or cooktop. Here are a couple of links regarding why you should not do this: Three reasons you should never use a natural gas oven to heat your home Quora: Is it O.K. to use a gas oven to heat my house? In fact our propane oven has a safety feature which prevents the gas from flowing when there is no electricity. That said, some older models do not have this feature. However, we could use a match to light up the burners on the cooktop if we were so inclined. If you need to heat your home, please use *something* that is designed for that purpose.

Oh, good! I read your post to mean that power had been off for a couple of hours and then you had no water when it came back on. I guess your pressure tank has simply lost pressure. Good news!

I hesitate to write this, but I will in case you or someone else does not already know this: Once your pipes freeze, they are likely already broken. Unfortunately, you do not find out about the damage until they are thawed and water sprays everywhere. I hope this is not true in your case, but it is usually the case.

Our daughter started her first professorship in the Fall of 2019. We were just commenting over the weekend that it was good that she did not start teaching THIS year. I have similar feelings about students starting out as college freshmen this year. What an awful way to start college!

Doesn't she have a cellular backup of some sort? If not, can she hop onto a friend's cellular hotspot?

That brings up a good question in my mind: Which is more traumatic to our college students these days: losing heat in 20F weather or losing the internet?

I posted a link to an outage map in the Texas thread, which looks much worse than Oklahoma, so far. Of course that is little consolation to those of you who are losing power in the cold. Here is your map: Oklahoma state power outage map As far as the amount of power used, heating at 3F is WAY harder than cooling at 115F (at least if both are done using electricity). It is true even if we assume that every home is heated and cooled by a heat pump. Why? First, 115F is 45 degrees away from 70F, while 3F is 67F degrees away from 70F. In other words, the rate of heat transfer through your walls is about 50% faster than on a 115F day. On top of that, heat pumps in warm climates are optimized for cooling rather than heating. Finally, even the best heat pumps DO NOT HEAT effectively below about 15F, so they revert to resistive backup heaters instead. The result is that at 115F your heat pump (or air conditioner) will be able to keep your house cool with a coefficient of performance of about 2, meaning that it takes about half as much electricity to cool the house as is being transferred through the walls while at 3F, every bit of heat loss has to be provided in the form of electricity. The bottom line is that a house in your area with a heat pump will be using about 3X as much electricity to heat at 3F as it would use to cool at 115F. Of course not all houses are heated with heat pumps. Some will be heated with fossil fuels like natural gas, so the electricity usage will be lower than the 3X I have calculated.

This link leads to a map which will show you how widespread the outages are in Texas right now (and going forward): https://poweroutage.us/area/state/texas A note in red letters at the top of the page at the above link says this:

Is she still your friend after running off with your husband? 🤪

The Texans have gone quiet. Hopefully it is because they are homeschooling their children right now.

I would look carefully at the label to see if they provide storage instructions. If those instructions are not there, then the manufacturer of the paints certainly knows. Check their website and if you do not find anything there, then I would contact customer support. I'm sure they want you to be satisfied with the product they manufactured. (Who knows, they might even be willing to replace the products for you.)

I think the word you are looking for is "bacronym". 🙂

This comment makes me wonder if you understand the risks associated with freezing rain. Yes, it makes the roads hazardous, but the bigger issue is that it is VERY HEAVY. Freezing rain can and does destroy power lines by making the wires so heavy that they fall down. The worst damage I am aware of from freezing rain was the damage done in Quebec, Canada, in late November of 2006. They got nearly TWO INCHES of freezing rain. That amount of ice was so heavy that even the very large towers that hold the high-voltage wires that deliver massive amounts of electricity buckled and crumbled to the ground. After that storm, some of the people in Montreal did not get their power back for over a month. The temperatures rarely got above 0 degrees Fahrenheit during that time. Here is a link to an article about that storm: https://en.wikipedia.org/wiki/Early_Winter_2006_North_American_storm_complex I visited the research arm of the Hydro Quebec a few months after that storm occurred. They had placed one of the crumpled power towers near the front door so that all of the engineers inside could see what freezing rain could do to their system. One of the engineers we met with told me that the government required them to design their system to hold at least 18 mm of ice. He said that they always designed for at least double that: 36 mm of ice. Unfortunately, that storm deposited over 45 mm of ice over a very wide area. The point is that critical infrastructure can and does get damaged during severe weather events. Just because the storm has passed does not mean that life will immediately return to normal. Hopefully you will not get enough ice to do serious damage, but it appears to me from the weather models that I have recemtly seen that some areas of the country will get very thick ice during this next week. (And many places have already had lots of damage from ice during the last week or so.)