RegGuheert Posted February 16, 2021 Share Posted February 16, 2021 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! 7 6 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 (edited) 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. Edited February 25, 2021 by RegGuheert Added link to online interactive linear graph. 4 2 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 (edited) 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! 😉 Edited February 25, 2021 by RegGuheert Added link to online interactive parabolic graphs. 1 4 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 (edited) The Circle The College Board only seems to use one form for the circle equation. Here is a link to an online interactive circle equation in standard form (x - h)^2 + (y - k)^2 = r^2 . Students have the same problem with the signs of h and k as they do with d, e, and h on the parabola sheet. I point out that all five of them have the same form and that in each case they need to ask themselves the following: "What value do I need to put in for x (or y) to make that particular factor equal to zero?" Also they need to remember that the radius on the right-hand side of the equation is squared. But the biggest challenge for students comes when the question requires them to complete the square -- twice. I have put in one approach to completing the square on the sheet, but students often have memorized other ways to get the squares that give the result quickly. FWIW, I learned two interesting things from the parabola sheet which come in handy when completing the square. First, since the x-value (or y-value) of the vertex is -b/2a, you only need to complete the square when b is NOT zero. Second, completing the square boils down to solving the following equation: b^2 - 4ac = 0 (or simply b^2 = 4ac) Since a typically (not always) equals 1 in these circle problems, that simplifies to: b^2 = 4c Solving for c gives: c = b^2 / 4 So, yeah, this sheet on the circle has room for improvement! Memorizing the equation immediately above is certainly easier than what I wrote on the sheet! One other thing I do not like about this sheet is that it looks like the circle is tangential to the x-axis. Of course that could be true, but that certainly is not the general case. So, please feel free to ask questions or make suggestions. Yes, even criticisms are fine here! 🙂 Edited February 25, 2021 by RegGuheert Added link to online interactive circle graph. 2 2 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 TTLFs TTLFs stands for "Things To Look For" This sheet contains a list of a few things which come up frequently on the SAT and PSAT. My goal is to get students to evaluate each problem for one of these items before trying to work them out "the hard way". They should expect to find a few TTLFs on EACH test. The main issue with this sheet is that you can certainly add to the list that is given. The issue is that students already have a lot of things to remember, so the list shouldn't be too long. Do you have other favorite things to look for on these tests? 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 (edited) Factoring I do not have a factoring sheet, but there are some tips that I wanted to capture on that subject. Avoiding Factoring I have discovered that there are a few problems which appear to require factoring that can be done more easily, more accurately, and more quickly without doing any factoring. Here is an example (and I am going to do this with symbolic equations to demonstrate the challenges you might encounter if you try to factor): Question: What is the sum of the solutions to the following equation? ax^2 + bx + c = 0 On a real test, such a question would typically have numerical coefficients and many (most?) students would be tempted to factor the polynomial to find d and e and then sum them together to get the answer. There are several problems with that approach including the fact that it includes many steps, there are quite a few places to make math errors, and also the fact that many combinations of a, b, and c will send you "down the primrose path", never to recover. On the parabola sheet, I use the symbols d and e to represent the two solutions to a quadratic equation. If I look at the sheet, I can quickly see that: h = (d + e) / 2 But the question is in standard form, so I only have the coefficients represented by a, b, and c. That said, I can easily relate those to h: h = -b / 2a So we have a simple, low-risk path to the answer: (d + e) / 2 = -b / 2a Solving for d + e gives: d + e = -b / a Ta-da! Note that the answer comes very quickly and there is greatly-reduced risk of making a math error. In fact, this equation is so simple and insightful, it might be useful to simply memorize it! One interesting observation from this new equation is that equations of the form ax^2 + c = 0 are squares, which means the x-value of the vertex falls on the y-axis since b = 0 and therefore h = d + e = -b / a = 0. Oh, and BTW, this falls under TTLF 3), but I am spelling this one out since it is a bit less obvious and more valuable than most. But I admit that most people will say (and have said!): "Is that really true?" Once you have used these relationships extensively, I can tell you that you learn to trust them, but that doesn't come right away. So let's check it! Since I posed the problem in symbolic form, you are drawn to solve the problem using the quadratic formula: d + e = -b / 2a - sqrt(b^2 - 4ac) / 2a - b / 2a + sqrt(b^2 - 4ac) / 2a Of course, the discriminants cancel out (and, yes, that seems obvious without even writing down the equation, but I can tell you that many, many students will not see that and will write the whole thing out). (At this point I will reiterate that if the student attempts to use the quadratic formula in the form given in the textbooks, some (many) students will get tripped up by the combination of the +/- and the common denominator. Did I mention that my formulation is better? 🙃) Now you get: d + e = - 2b / 2a = -b / a Finally, no, I am not suggesting that students avoid factoring at all costs. Most factoring on the PSAT and SAT is designed to be simple and quick. However, some problems are designed so that factoring will take the student down a long, dark rat hole. They do this to see if students are able to take another path. As a result, I recommend checking, as a quick first step, to see if factoring quadratic equations in standard form can be easily avoided. Edited February 20, 2021 by RegGuheert Added "Avoiding Factoring" section. 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 (edited) Writing Rules From @ElizabethB: (I do not know how to put her name into a header of the quote box below since I am editing instead of replying. If someone knows how to do that, please post instructions below so that I can properly credit her and all of the great ideas from the thread. TIA!) Quote PrepScholar has a decent list of Grammar rules. https://blog.prepscholar.com/the-complete-guide-to-sat-grammar-rules I prefer the SAT Black Book for the Grammar rules, I also like the way they are linked to question number, makes it easier to go back through tests and combine similar error types. If you have rules for the writing section which you think should be included, write them in this thread and I can collect them here. Edited February 17, 2021 by RegGuheert Added ideas from ElizabethB 1 Quote Link to comment Share on other sites More sharing options...

daijobu Posted February 16, 2021 Share Posted February 16, 2021 8 hours ago, RegGuheert said: The Line 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. Since you offered, why is the "S" in slope written in parens? 1 Quote Link to comment Share on other sites More sharing options...

daijobu Posted February 16, 2021 Share Posted February 16, 2021 Just now, daijobu said: Since you offered, why is the "S" in slope written in parens? ETA: Ooh, I know! Is it because we typically abbreviate slope with an "m"? 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 16, 2021 Author Share Posted February 16, 2021 2 minutes ago, daijobu said: Since you offered, why is the "S" in slope written in parens? Good catch! 1 minute ago, daijobu said: ETA: Ooh, I know! Is it because we typically abbreviate slope with an "m"? The real answer is that it doesn't really need to be. It is only in parentheses to be somewhat consistent with the parabola sheet, which was the first one I made. There is a reason for it on the parabola sheet, but I cannot think of one for the line sheet. BTW, I noticed the same silliness on the circle sheet when I posted that today. 1 Quote Link to comment Share on other sites More sharing options...

daijobu Posted February 16, 2021 Share Posted February 16, 2021 9 hours ago, RegGuheert said: 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?" LOLOLOL! Quote Link to comment Share on other sites More sharing options...

Garga Posted February 17, 2021 Share Posted February 17, 2021 Is this pinned on the high school board? Or is there a way to do something like that so it’s easy for people to find in the future? 1 Quote Link to comment Share on other sites More sharing options...

Lori D. Posted February 17, 2021 Share Posted February 17, 2021 (edited) 1 hour ago, Garga said: Is this pinned on the high school board? Or is there a way to do something like that so it’s easy for people to find in the future? I have linked this thread in THREE places in TWO pinned threads at the top of the High School Board:PAGE 2 of the pinned thread "High School Motherlode #1", under the "PSAT: Test Prep" section AND under the "SAT: Test Prep" sectionPAGE 5 of the pinned thread "High School Motherlode #2", under the "MATH" section 😄 Edited February 17, 2021 by Lori D. 6 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 Bless you, @Lori D. , for liking the parabola sheet post!! That really was the raison d'etre for this entire thread. Yeah, the others should be useful for those doing PSAT and SAT prep, but that one is the real stand-out to me. (But, I admit it: I seem to be alone in my opinion, here!) 1 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 1 hour ago, RegGuheert said: Bless you, @Lori D. , for liking the parabola sheet post!! That really was the raison d'etre for this entire thread. Yeah, the others should be useful for those doing PSAT and SAT prep, but that one is the real stand-out to me. (But, I admit it: I seem to be alone in my opinion, here!) That was my favorite part!! 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 13 minutes ago, ElizabethB said: That was my favorite part!! Really? You're not just saying that?? 🙃 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 3 hours ago, Garga said: Is this pinned on the high school board? Or is there a way to do something like that so it’s easy for people to find in the future? Unfortunately, the high school board is already too full of pinned things, IMO. I did add some tags. Are there other tags that would be helpful for the future? Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 (edited) 6 minutes ago, RegGuheert said: Really? You're not just saying that?? 🙃 No, I'm not just saying that!! That is the only area of SAT prep my son is struggling with, I printed out the graph to use for SAT study. My son is taking the May SAT, I have a folder of things to use for the last month of prep, that is going in it!! (Also printed the circle picture because he only recently learned circles so it'll be an area he needs review in and that is handy, but he understands circles.) Edited February 17, 2021 by ElizabethB 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 PrepScholar has a decent list of Grammar rules. https://blog.prepscholar.com/the-complete-guide-to-sat-grammar-rules I prefer the SAT Black Book for the Grammar rules, I also like the way they are linked to question number, makes it easier to go back through tests and combine similar error types. 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 1 minute ago, ElizabethB said: That is the only area of SAT prep my son is struggling with, I printed out the graph to use for SAT study. My son is taking the May SAT, I have a folder of things to use for the last month of prep, that is going in it!! (Also printed the circle picture because he only recently learned circles so it'll be an area he needs review in and that is handy, but he understands circles.) I recommend that you also print out my notes here to help guide you and/or your son about how to approach the sheet. I found that some of the students' eyes glazed over because it looks like so much. The notes above break it down so the student can know where to focus their attention. The circle sheet is good for about one question on the test, two if he is luck. It probably should be redrawn with the incorporation of my notes above to make the section on completing the square much more straightforward. I wish your son the best on his SAT in May! If you have any questions for me and I have gone back off the boards (I might soon), I think Garga knows how to get in contact with me or my wife. I'd be happy to try to help. (No charge. I am trying to GET OUT of doing regular SAT prep, but I do not mind helping where I can.) 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 14 minutes ago, ElizabethB said: PrepScholar has a decent list of Grammar rules. https://blog.prepscholar.com/the-complete-guide-to-sat-grammar-rules I prefer the SAT Black Book for the Grammar rules, I also like the way they are linked to question number, makes it easier to go back through tests and combine similar error types. Added above. Thanks! 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 12 minutes ago, RegGuheert said: I wish your son the best on his SAT in May! If you have any questions for me and I have gone back off the boards (I might soon), I think Garga knows how to get in contact with me or my wife. I'd be happy to try to help. (No charge. I am trying to GET OUT of doing regular SAT prep, but I do not mind helping where I can.) If you do contact Garga for my contact information, just tell her you need to get in touch with the "creepy man-troll"! 😀 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 (edited) 16 minutes ago, RegGuheert said: I wish your son the best on his SAT in May! If you have any questions for me and I have gone back off the boards (I might soon), I think Garga knows how to get in contact with me or my wife. I'd be happy to try to help. (No charge. I am trying to GET OUT of doing regular SAT prep, but I do not mind helping where I can.) Thanks! We should be good, though. My husband is going to take over prep for the last month, he is making his own parabolas from some data he has lying around to make an app. He and my son had a discussion about why certain parabola's needed 19 to 21 digits of precision while other needed less... My husband also knows a bunch of math equations off the top of his head, he can spout off the formulas for a sphere still, who knows that!??! Edited February 17, 2021 by ElizabethB 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 2 minutes ago, ElizabethB said: Thanks! We should be good, though. My husband is going to take over prep for the last month, he is making his own parabolas from some data he has lying around to make an app. Great! Please have him post a link to the app when it is ready! 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 (edited) I meant polynomials, not parabolas! Also, I started a thread about it, but for getting good problems at your level and getting faster at doing them, mathchops is excellent for the price. They collected all questions from last 10 SATs and arranged them by how often they occur and difficulty, after 30 min or so to get at your level, all the questions are good. https://www.mathchops.com They have a free version that goes to the 500 level so you can see how the system works without giving them a credit card, the questions are different at each level, you get to higher levels by answering questions at the level challenge section--a section you must always do without help, the other sections you can get help. I was having trouble finding questions at a good level without wasting a bunch of time, it gets harder and harder to find good questions as you get better. Edited February 17, 2021 by ElizabethB 1 Quote Link to comment Share on other sites More sharing options...

Garga Posted February 17, 2021 Share Posted February 17, 2021 44 minutes ago, RegGuheert said: If you do contact Garga for my contact information, just tell her you need to get in touch with the "creepy man-troll"! 😀 It was very disappointing to find out you weren’t a troll. I was all ready to go home and tell my dh, “I met a real live troll today!” Ah well. Unfortunately, I don’t have your contact info. I think I used the private messaging system here on WTM in the past. Or if I had everyone’s contact info, I lost it. 😞 2 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 1 minute ago, Garga said: It was very disappointing to find out you weren’t a troll. I was all ready to go home and tell my dh, “I met a real live troll today!” Ah well. Unfortunately, I don’t have your contact info. I think I used the private messaging system here on WTM in the past. Or if I had everyone’s contact info, I lost it. 😞 My wife was there so I had to do my best to hide my "inner troll"! 😈 If either of us can find your contact information, one of us will send you an email to re-establish contact. I seem to remember that you sent out photographs after one of the meetups. 3 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 40 minutes ago, ElizabethB said: I meant polynomials, not parabolas! O.K. That makes more sense to me now! Now--on to parabolas! My youngest took the PSAT in October. I am happy to report that he got a 1510!! Another student (A BETTER student. Shhh!!) also got a 1510 on that test. Those two would banter with each other every week about how dumb the other one was to the point that I had parents of other students tell me that their child would come home demoralized because those two boys got over 1500 every week! Oops!! I will pass on your recommendations to the parents of the students whom I will not be teaching next year! 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 (edited) On 2/17/2021 at 2:02 PM, RegGuheert said: I will pass on your recommendations to the parents of the students whom I will not be teaching next year! For those who are diligent, the Black book for everything, the Panda book and 1600io orange SAT book (2 volumes) for math, Erika Meltzer books for Reading and Writing/Grammar. I also like Panda for writing instead of Meltzer for the more analytical folks. For the not diligent or unable to figure out on own/from a book, the online Prepscholar program is good overall, walks you through SAT Black Book type of explanations, nags you, easy for parents to track and implement. I like the mathchops for everyone, probably better than the books actually if you just get one thing for math. I haven't tried Kahn Academy, it seems to work for some but not others. It is free, though. Uworld is supposedly good for the price. Edited March 20, 2021 by ElizabethB 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 4 minutes ago, ElizabethB said: For those who are diligent, the Black book for everything, the Panda book and PWN the SAT: Math Guide for math, Erika Meltzer books for Reading and Writing/Grammar. For the not diligent or unable to figure out on own/from a book, the online Prepscholar program is good overall, walks you through SAT Black Book type of explanations, nags you, easy for parents to track and implement. I like the mathchops for everyone, probably better than the books actually if you just get one thing for math. I haven't tried Kahn Academy, it seems to work for some but not others. It is free, though. Thanks! I will also pass these on! I used Kahn Academy with my students as available drill if they wanted it. I tried Erica Meltzer with one of my children but his scores went down a bit. Reading is like that, so I try to help students find what works for them. The biggest issue my students had with reading is if a student is a slow reader and cannot get done on time. 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 One other thing that I did that was quite successful was "the SAT sandwich": I had all of my students take the SAT at the beginning of October, the PSAT in the middle of October, and another SAT at the beginning of November. I considered the first SAT as practice for the PSAT and the second SAT was there to have a chance to improve the superscore. This allowed the students to prep up until the beginning of November and then get on with their lives for the rest of the school year. 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 10 minutes ago, RegGuheert said: I tried Erica Meltzer with one of my children but his scores went down a bit. Reading is like that, so I try to help students find what works for them. The biggest issue my students had with reading is if a student is a slow reader and cannot get done on time. My daughter reads super fast and is naturally good at language, the strategies that work for her for reading are different than for my son. He is more logical and a fast reader but not super fast, depending on the passages he may not be able to finish one. His worst is always the fiction passage, so he skim reads that one and goes back if he has time. He actually scores about the same if he skims it for most passages vs. reading it, LOL. He is better with old passages than current meandering fiction, that style confuses him. For students with a lot of time to build up their reading speed, I've had siblings of poor readers who were at grade level but slow improve their speed by working through my lessons and then doing daily nonsense word drill--it only takes a minute a day to make a difference after working through the lessons, the lessons take 10 hours tops, usually less, 10 hours for a group class though. http://thephonicspage.org/On Reading/syllablesspellsu.html For students who struggle with reading older passages, the McGuffey 1879 Readers gradually build in difficulty, you start from the 3rd or 4th reader and work up, reading a few passages a day. This takes a bit of time, too. They are free online and also cheap on Kindle, you want the blue and orange cover 1879 version. http://www.gutenberg.org/ebooks/author/5671 I also have links to the Parker Readers for students who need extra practice building up reading skills with older readers, linked near the end of my Webster page: http://thephonicspage.org/On Reading/webstersway.html 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 4 minutes ago, ElizabethB said: For students who struggle with reading older passages, the McGuffey 1879 Readers gradually build in difficulty, you start from the 3rd or 4th reader and work up, reading a few passages a day. This takes a bit of time, too. They are free online and also cheap on Kindle, you want the blue and orange cover 1879 version. http://www.gutenberg.org/ebooks/author/5671 Ha! That would be me! I'm impressed that your son does well on those! I was very proud of one of my slow readers this year. He managed to complete the reading section on the PSAT and ended up with a 680 on the R&W section (versus a 560 the previous year). His year-younger brother, who could ace any section of the test, but was all over the map, didn't get through the reading section that day. It amazes me: some of my students are super consistent day-in and day-out, while other ones are all over the map from week to week and from section to section. I even had one young lady improve her SAT score from 1350 to 1490 just between October and November! Crazy! 1 Quote Link to comment Share on other sites More sharing options...

ElizabethB Posted February 17, 2021 Share Posted February 17, 2021 15 minutes ago, RegGuheert said: I was very proud of one of my slow readers this year. He managed to complete the reading section on the PSAT and ended up with a 680 on the R&W section (versus a 560 the previous year). His year-younger brother, who could ace any section of the test, but was all over the map, didn't get through the reading section that day. It amazes me: some of my students are super consistent day-in and day-out, while other ones are all over the map from week to week and from section to section. I even had one young lady improve her SAT score from 1350 to 1490 just between October and November! Crazy! It is really interesting. Also, my daughter scores one standard deviation higher on the ACT than the SAT, my son the reverse. Most people score closer to the same on both. I didn't even figure out the fiction passage was always first until reading through Melzer for my son--my daughter needed no special study for this section, she needed more work on math. He gets 50% right on rambling modern fiction passages, 70 - 80% right on older fiction passages, but can almost always get 100% on any science passage. 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 17, 2021 Author Share Posted February 17, 2021 45 minutes ago, ElizabethB said: I didn't even figure out the fiction passage was always first until reading through Melzer for my son--my daughter needed no special study for this section, she needed more work on math. He gets 50% right on rambling modern fiction passages, 70 - 80% right on older fiction passages, but can almost always get 100% on any science passage. One of my best students from a previous year was very good at reading and typically got all--or nearly all--of the questions correct. Then one day he got everything correct except he missed six questions out of eleven on section four. After I read the section, I knew what to tell him: "You CANNOT let a passage which is contrary to your worldview cause you to completed blow an entire passage." Fortunately, he never did that again! 2 Quote Link to comment Share on other sites More sharing options...

Sarah_hopes Posted February 19, 2021 Share Posted February 19, 2021 These are so helpful, thanks for sharing your resources!! 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 20, 2021 Author Share Posted February 20, 2021 Bump. I added the "Avoiding Factoring" section in the post about factoring. 1 Quote Link to comment Share on other sites More sharing options...

RegGuheert Posted February 25, 2021 Author Share Posted February 25, 2021 Bump. A friend of mine has just shown me the website desmos.com . I have now added links to the posts above for the line, the parabola, and the circle to online interactive graphs that I created to match my equation sheets. There are five graphs in total. These graphs contain sliders which allow you to adjust the coefficients in the equations and see their effects on the graphs. Please try out these links and let me know whether you think these graphs will be useful learning aids for students. Quote Link to comment Share on other sites More sharing options...

daijobu Posted February 26, 2021 Share Posted February 26, 2021 those parabolas are pretty genius. Especially when you are teaching about translating circles and other conics. 1 Quote Link to comment Share on other sites More sharing options...

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