Session Recaps

N. began our session by asking if we can concentrate on math today, which I agree makes sense. Picking up where we left off last week with solving word problems with systems of equations, we opened with a particularly challenging word problem. N. needed a little hint in order to create the second equation, but was able to do the problem mostly independently – this looked really good!

Feeling confident in N.’s command of problems with linear expressions, we then moved on to polynomials. We started with a quick review of different kinds of operations with exponents and square roots; Noord demonstrated a strong knowledge of these fundamentals. We then worked through a series of problems from a real past SAT, covering addition and subtraction of long polynomial expressions, rational expressions, and exponential expressions. We talked about how to avoid doing extra work by lumping parts of an equation into a single variable. We saw how we have a variety of tools that we can use to approach questions that require manipulating an expression with exponents: we can knock everything down to the same base, factor, do arithmetic when the exponents get small enough, or combine the bases if the exponents are the same. N. made small mistake initially on the first problem, but completed the second problem flawlessly and confidently; on the third (different) problem she needed a few hints along the way, but did well overall.

We continued to work through a variety of concepts in algebra, covering imaginary numbers and factoring quadratics. We did several word problems with quadratic equations, finding the x-intercept or finding the vertex. We reviewed the equation for finding the vertex, which N. had forgotten. We worked through a challenging problem that required considering a quadratic model for the maximum area for a set perimeter. N. did a great job tackling this one!

We then took a moment to try some problems featuring inequalities with absolute value. It was good that we check this, as we were able to fill in a small gap in N.’s knowledge; after introducing these concepts, a second question with two variables went smoothly. We tried a word problem with an inequality – N. did well with this problem, except she made the same mistake as on the first problem of the day. We discussed how this is a very common mistake, so we should remember to check our work for the relationship equation by trying two easy integers. Whereas usually checking our work is a luxury for which we don’t always have enough time, we should always check this type of equation.

Continuing through our survey of concepts in algebra, we moved on to exponential growth. N. had a great command of this material, and she solved the first two word problems perfectly. On a third problem about bacterial growth it took her a moment to fix her mistake, but she was able to amend it independently. We also tried an exponential decay word problem, which was similarly strong.

At the end of our session we began to work on circle equations. We confirmed that N. remembers the equation for a circle, and then tried a word problem with circle. I encouraged N. to start by drawing a sketch, and she solved the problem well!

N. did a fantastic job today working through a wide swathe of the algebra material for the SAT. We’ll pick up next session with more questions about circle equations, and then we’ll move on to tackle geometry, probability, and other concepts.

It was great to meet A. today and begin working together on SAT and ACT prep. We began by introducing ourselves and briefly discussing A.’s goals and experience thus far with the tests. She took the ACT once without any prep last month and scored a 14 Composite, and she plans to take the SAT through her school next month. (She’ll send me the date of this upcoming SAT.) A. mentioned that while she’s a strong student who normally gets As in school, she feels like she’s “not a good test-taker.” We’ll talk about strategies for test-taking and ways of increasing focus and decreasing test anxiety.

For now, we’re preparing for both the SAT and the ACT, until A. gets a clearer sense which tests plays better to her strengths. This sounds like a great strategic plan to me! Fortunately there’s a lot of overlap between the two tests, so it’s absolutely possible to prepare both at the same time.

Today we focused on math, which A. mentioned was her weakest area. We warmed up by practicing single-variable algebra. I noted some initial confusion about what step to do first, so these problems gave us a chance to review and strengthen A.’s understanding of the order of operations.

We then turned to linear functions. This is an key concept for both tests and the most important place to start, since not only will these concepts lay the foundations for more advanced math topics, but linear functions themselves account for approximately a third of all math questions on the SAT.

We talked about determining the slopes of lines, and then reviewed how to write the equation of a line (y = mx + b). This was initially rusty for A., especially finding the y-intercept, but improved rapidly with a bit of review. We practiced problems where we had to find the slope or equation of a line based on an equation or on a graph, and we saw more complex examples where the line was presented as the line of best fit on a scatterplot (very common on the SAT).

We then introduced how to find the equation of a line based on two points. We worked through one example together, and then A. was able to tackle further examples more independently. We wrapped up with a tougher problem where we had to find the equation of a line given one point and then two points for a line running parallel. A. did a great job using logic to work her way through this problem (with some help).

We noted throughout our session that we need to be extra careful with negative signs, which are a very common source of mistakes for any kind of math!

I also noticed that A. struggled a bit with arithmetic, especially involving fractions. (This is quite common for many students these days, especially as we’re all accustomed to having a calculator in our phones on hand at all times!) While on the ACT it’s possible to use a calculator for the entire Math section, the SAT does have a no-calculator section, so we’ll want to make sure that small arithmetic mistakes don’t cause A. to miss a problem that she otherwise understood.

A. did demonstrate an excellent speed when solving algebra problems, especially after our warm-up. This is a great advantage, since pacing is such a key component of both the SAT and the ACT.

Overall we had a very productive session! We covered a significant amount of core math material from the SAT and ACT, and the review and practice had a clear impact on A.’s ability to answer sample test questions.

G. has some finals (history and math) last week, and she has more finals (English and science) coming up this week. In light of that, she hasn’t been practicing SAT as much this week, but she did do about an hour of practice on Khan Academy. On this practice, she missed some questions about scatterplots, and also word problems with linear inequalities. Khan Academy also advised her to work on her narrative and history passages for the Reading.

We haven’t practiced a Reading passage in a few weeks, so today we began our session with a challenging paired passage from the Reading section, featuring texts from the late eighteenth century by Talleyrand and Wollstonecraft.

At a high level, we talked about how it can be a good strategy to leave the paired passages, or any other passages that look particularly difficult, to do last. We then reviewed our basic strategy for paired passages, in which we answer the questions that are only about the first passage before reading the second passage. G. did a great job with this difficult passage overall! She did miss a vocabulary question, where I think she was too focused on the multiple-choice options and didn’t focus enough on the text itself. She also missed a question about author’s main point, which we worked through together. Finally, she was thrown off by the old-fashioned wording of one multiple-choice option, where “to enjoy” is used for “to have.”

In general, G. needs to remember to ignore the multiple-choice options (which can lead readers astray) and focus instead on the passage and on the questions. She’s an excellent reader, and this will play to her strengths!

With the second half of the session, we moved on to math, continuing our work the past few weeks introducing the main concepts of geometry. We reviewed the properties of isosceles triangles. We then worked through a series of more challenging problems combining angles with algebra. On a right triangle angle problem featuring a system of equations, G. needed some help spotting how she could create her equations; once we had everything set up, she did a great job solving the rest of the problem. (I noted that she prefers to solve systems of equations with elimination, not substitution.) The next question (involving only one equation) went very smoothly.

During the course of our practice, we saw that it’s helpful to assign variable names to things at the beginning; at times G. wanted to assign variables to complicated expressions (e.g. “twice the complement of angle ABC minus 75”), not to the actual angles themselves, and we saw that it’s much easier to assign the variable names to the angles. We also tried a question with no diagram and saw how helpful it is to draw one. We introduced the terms supplementary and complementary angles, and the term “bisects.” We introduced angles within circles that add up to 360 degrees and the concept of arc measures. We saw that we can’t assume something is 90 degrees unless it’s explicitly marked as such. In general G. did a fantastic job of noting what the questions were actually asking her to solve for – she managed not to fall into the traps that the questions set.

Overall G. is doing a great job and making fantastic progress, especially with all of this new geometry material that we’re covering!

This week we picked up where we left off last week in ACT Math, solving systems of equations with two variables.

We started with a more straightforward word problem. We recalled that it’s a good idea to clearly and fully define what our variables mean (e.g.”s = number of sodas sold”). E. was a bit hesitant with all of the steps of the first problem we tried, so we worked together to name and define our variables, create our equations, isolate one of the variables in one of the equations, and then use substitution to solve the system of equations. We noted that often the word problem gives a hint about how to define the variables in the last sentence when they ask the question.

The second straightforward word problem that we tried went really well. In particular, E. did a great job spotting which variable in which equation would be the easiest to isolate — this kind of insight saves time on solving problems, which is key for time management on the ACT. E. also did a great job of remembering to use parentheses when doing substitution.

Feeling more confident after the success of our first two word problems, we then tackled a more complex word problem. We learned how words like “together” or “in total” usually indicate an equation with addition. We talked about how since it’s very easy to accidentally reverse a “relational” equation, we should test our equation with two demo numbers. The few seconds that this extra step takes are well worth the reduced risk of setting up the equation incorrectly!

In the second half of our hour together we moved on from linear equations and began working with exponents. We began by reviewing core concepts about exponential operations. It was good that we started here because we discovered that E. was a little rusty with combining “like terms,” so we did some targeted review to clear up any misconceptions.

Similarly, E. had a hard time remembering how to multiply exponents, so we did some further practice of this to help her recall how to do this operation. After a little bit of practice, she improved significantly with this.

We then wrapped up by reviewing how to divide exponents, work with negative exponents, or layer exponents on top of each other. This sets us up for working with more complex expressions with exponents in our next session.

Great progress with these important algebra concepts today!

Continuing our plan to alternate between the Reading and English sections of the SAT each session, D. and I began today with a challenging Reading paired passage featuring political speeches from the late eighteenth century. We discussed how on the actual SAT D. should do the passages out of order, and he should definitely save a passage like this one—quite old, as well as a paired passage with two different authors—for last, after he’s answered the easier passages on the Reading section. We reviewed our plan of attack for paired passages, where we address the passages one by one as much as possible, answering the questions for passage one before we read the second passage.

We talked about how, after we finish reading a difficult passage, it can be worthwhile to take a moment to figure out the main ideas of the passage before attempting to answer the questions. This was a bit tricky for D., so we talked through the passage together. D. did a great job with the first question, spotting the place in the passage that supported his answer. He also did a great job interpreting a metaphor on the second question! We briefly reviewed the most important figures of speech. We practiced using the four different lines provided by the paired evidence questions to work backwards and find the matching answer. D. noted that this type of passage was particularly challenging for him, and mentioned that he would like to practice reading more passages of this type. I’m emailing him a set of similar passages that he can practice reading and summarizing on his own.

Switching gears to math for the rest of our session, we continued our work from Monday on operations with exponents and square roots. D. was initially a bit puzzled by some of these operations, but with some targeted practice it really “clicked” for him! We’re making great progress with these math fundamentals.

D. initially had forgotten how to use FOIL to multiply out a polynomial expression, so we reviewed this with some targeted practice. Fortunately this material came back to him quickly with some review!

I can tell that D. is becoming more personally invested in his learning, which is great to see! You really can see his sense of accomplishment when he understands and correctly answers a hard question.

C. had asked if we could incorporate review of what we’ve been learning and practicing thus far. It’s great that C. is taking such an active role in his learning and stating what he thinks would be most helpful to him. I absolutely agree that periodic review and repetition is a great way to to solidify what we’ve learned!

With that in mind, we warmed up today with another system of linear equations word problem, which we had introduced and practiced in our previous two sessions. C. did a great job solving this problem, and he seemed confident working through each step of the process.

We then did a review problem featuring operations with exponents. Again, this looked solid for C.!

This review made us feel confident to keep pushing ahead with new material. Today we practiced how to multiply out a polynomial (FOIL) and factor a polynomial. Applying FOIL seemed easy for C.; factoring was a little bit rustier but seemed solid after practice. We worked up to more complex factoring problems and practiced approaching them in an organized way.

We then used our factoring skills to tackle a real SAT problem that’s quite typical for the test, combining function notation, factoring, and rational expressions. C. was initially stumped by this problem, but we worked together to see that the skills we had just honed applied directly. We talked about how whenever we see a quadratic function or polynomial expression on the test, we often have to factor it as part of the solution.

Next, we introduced how to graph a quadratic function in the coordinate plane. Like many students, C. had forgotten how to graph this, although he did remember that a quadratic function will make a parabola. We learned that the phrases “find the x-intercepts,” “find the zeros,” and “find the solutions” all mean the same thing. We recalled that the value of the “a” term dictates whether the parabola will open upwards or downwards. We learned how to find the two x-intercepts by factoring. We then learned two ways of finding the x-value of the vertex of the parabola: either finding the middle between the two x-intercepts or using the formula -b/2a. Once we had the x-value of the vertex, we were able to plug it in to the original equation to find the y-value of the vertex.

After working through all of the steps of the first example together very slowly, C. was then able to complete a second example with confidence on his own. He did a great job absorbing this material!

Now comfortable with factoring quadratics and graphing parabolas, we finally turned to word problems with quadratic functions. Often these word problems involve projectile motion, which is fun! We talked about how often we’re graphing height as a function of time — time is almost always in the x-axis in general in math.

C. did a great job tackling three word problems with quadratic functions. These problems seem intimidating, with lots of physics terminology, but we can approach them the same way that we approach finding the zeros or the vertices of quadratics in general. This seemed to boost C.’s confidence significantly!

We saw how sometimes we don’t need to factor a quadratic when a word problem asks us only for the vertex! This is a great way to avoid doing extra work (and wasting valuable time) on the SAT.

Fantastic job today! We covered a huge amount of ground with these core concepts from Algebra II.

R. and I agreed that it would be good to continue working on grammar concepts together today.

We picked up where we left off last week with an English passage from a real past ACT. R. did a great job answering the first question about verb forms. She did struggle a bit with the next question, one about comma placement — but when I had her read the sentence out loud she was able to put pauses in the right place, which enabled her to realize how she could correct her answer! We talked about how we can answer these grammar questions either by “hearing” the one that sounds right (more instinctively) or by analyzing the grammatical structure. On another question, we talked about how we can try using a word or phrase in a different sentence.

We learned how to use semicolons, which R. said she had never learned in school. (These are commonly featured in ACT questions!) We also learned how to use apostrophes to make possessive forms of words, and we cleared up some questions that R. had about plurals versus contractions versus posessives.

We tackled some questions about subject-verb agreement that were challenging for R.. R. seemed relieved when we saw that taking out the words in between the subject and the verb made it much easier to “hear” the correct answer.

We made great progress today and learned more key grammatical concepts! There’s still a lot of material, both grammar and math, to cover, but it’s great to see R. learning and growing.

Today is our last session before G. takes the SAT for the first time on March 11. (We’ll continue working together after this first test attempt to help G. raise her score even further.)

We began by picking up where we left off yesterday, tackling a more abstract problem involving the areas of a square and circle combined with probability concepts. We saw how it was helpful to draw a picture and assign a variable to a particular segment length within that picture. That is fact is a very common tactic to solving questions — we might have to assign up to three variables to solve a problem. We also experienced how it can be helpful to write out a relevant formula somewhere on the page to jog our thinking and help us organize our thoughts.

We clarified a question that G. had about simplifying rational expressions with exponents and practiced another example to solidify her knowledge.

We then introduced some new geometry concepts that can appear on the SAT involving arc measures, central angles, inscribed angles. We practiced some example questions using these concepts, including problems that combined geometry and algebra. G. did a great job of working with this new material! We also tried a classic challenging problem (that has appeared on the SAT multiple times in recent years and tends to stump most students) featuring a central angle and several inscribed angles. G. did a fantastic job and solved this problem correctly! We also worked through the same problem a second time to see an alternate way of solving it.

We then introduced a new-ish concept for G.: circle equations in the coordinate plane. We learned the formula for a circle function, then worked through three example problems together. On the last problem, we also learned how to complete the square, a more advanced technique in algebra.

We reviewed the Pythagorean theorem. We learned about special triangles: not just the 30-60-90 and 45-45-90 but also the 3-4-5 and the 5-12-13 triangles.

We reviewed the kind of trigonometry questions that can appear on the SAT.

We wrapped up by discussing how G. might continue her SAT prep this week — the best would be to take full timed practice sections from the College Board official practice SAT tests 9 and/or 10. We discussed at a high level the most helpful strategies for the test as a whole.

N. starts school again next week. Over the weekend she did a practice test (all on one day, but not timed), one of the 10 Princeton Review tests (#3). Reading and Math felt the hardest for her.

On her previous SAT, she had 11 Reading questions missed, while on this test she missed 10 questions (so she didn’t feel like she had an improvement).

N. and I chatted about how best to be doing practice tests. I explained the difference between the official practice tests published by the College Board (which were originally real SATs administered in the last few years) and the tests created by test prep services like Princeton Review (which try to mimic the test but aren’t real tests). We talked about how the best tests for her to practice would be either the April 2022 SAT (as she took the March SAT and the May SAT) or the published test #10.

N. asked a a great question about what to do if you have literally no clue how to approach a math problem. We talked about a few of the techniques we can use to jumpstart our brains — on geometry problems we can label the diagram with what we know and extend straight lines outwards, and on algebra problems we can draw a thumbnail sketch of the graph. Drawing a picture to illustrate a word problem can often help. But if we have absolutely no idea what to do, it’s best to put down a guess and circle the problem, and hope that we can return to it at the end.

N. also asked how to approach the hard Reading passages. We talked about how in general we should save the hardest SAT passages for last. We then discussed how we can take a hard passage piece by piece, even annotating in the margins the main idea of each paragraph or chunk of text.

With this in mind, we spent the first portion of our session working through a tough Reading passage from a past official SAT featuring a political speech about women’s involvement in the nineteenth-century abolition movement. We discussed the meanings of the key words “tyrant” and “abroad.” We talked about some common thematic traps to avoid on these passages about abolition and women’s rights. N. missed one question because she misunderstood the meaning of the word “abroad.” She missed another question because she fell for the trap of an answer that was in the right general direction but too extreme; we talked about how to avoid these traps. N. did a great job focusing on keywords on another tough question!

We used a particularly tough paragraph to practice taking a passage line by line and re-phrasing it in our own words to better understand it. On this question N. had misunderstood the paragraph but actually answered the question correctly due to her good application of test-taking strategies — which is actually great news!

N. did a great job on this tough passage, missing only two questions out of eleven. Overall she did a great job focusing on the textual evidence to choose her answers. On Wednesday we’ll review some of the math questions that she missed on her practice tests!

R. has been practicing some of the English and the Reading practice test sections at home, which is great! She’s been focusing on pacing for the Reading section, figuring out how quickly she needs to read to get through all four passages and their questions within the time limit.

We began our session with a quick review of what we worked on last week: graphing quadratic functions. R. was a tiny bit rusty with factoring the quadratic. We then learned about the number of times that the graph of a polynomial turns around, which will always be one fewer than the value of the highest power of x. We worked through a series of more creative questions about the graphs of polynomial functions, including problems using function notation. These questions went well overall!

Picking up where we left off at the end of our session last week, we then turned to exponential growth problems. R. had forgotten the general formula for exponential growth, so we reviewed the formula first. We then tackled a series of word problems with exponential growth applied to different situations like population growth, compound interest, and bacterial growth. After a bit of practice this went really well! We then tackled exponential decay; R. had great intuition that the formula would be nearly identical, but with 1 minus the rate of decay (and not 1 plus the rate of growth). We tried a more adventurous problem about graphing an exponential function; along the way, we reviewed the fact that any number raised to the 0th power will be equal to 1.

I had noted that R. was a little weaker working with function notation, so we did some targeted practice with variety of problems involving function notation. We saw that it’s better to err on the side of caution when adding parentheses around the value that we’re subbing in for x, especially if it’s a negative number. We introduced the concept of domains and ranges and learned about the ways to notate them. We reviewed to be on the lookout for situations where we would end up with zero in the denominator of a fraction, giving us a value of “undefined” (i.e. an “illegal” situation in math). We also talked about domains of functions with square roots, where taking the square root of a negative number is also “illegal” in math.

We spent the last twenty minutes of our session practicing a passage from the Science section. R. did a great job with this! On one question she initially chose the wrong answer but was able to fix it herself when she started to explain her answer. Ruby also missed a question because she didn’t realize that the axes were different across three graphs – we talked about how we always need to check the axes for the ACT Science, because they can try to trick us!