My Math Tools: Graphing Systems of Inequalities

coordinate graph system

Graphing systems of equalities is a concept that pops up in algebra 1, algebra 2, and in college and university, beginning and intermediate algebra classes. The method most textbooks and instructors use to graph inequalities is so painful because it takes so much time, effort, and sheer arithmetic. Graphing systems of inequalities is like climbing Mount Everest!

What are systems of inequalities?

Systems of inequalities are similar to systems of equations. However, the main difference is that inequalities have an area or range of values as a solution set, while equations either have 1 solution, many solutions, or no solution. A system of inequalities has an intersecting range of values that, when “plugged in,” make both inequality statements true. Neither can be false, or that value is not a solution.

How to graph a system of inequalities

Inequalities are simple to graph. Remember that when you graph a linear equation, you always have a solid line as the solution. However, a shaded area is the solution to an inequality, whether or not the line is included as part of it. The symbols (greater than or equal to) and (less than or equal to) signal that the line you graph must be a solid line, while the symbols > (greater than) and < (less than) mean a dotted line. That means that the line itself is either included (solid) or excluded (dotted) as part of the solution, which is the shaded area.


graphing system of inequalities

You graphed the line, but what about shading?

In the olden days of algebra, if you graphed an inequality, you just needed to pick two sets of ordered pairs (a.k.a. points) on either side of the line. Then you plugged them in for x and y, respectively, and checked if each pair made the inequality true or false. Finally, you shaded the right area: above or below the line.

However, it was much worse to do the same process for a system of inequalities. You ended up with 4 areas you needed to test, which meant 4 different pairs of points!

Happily, during my years of teaching, I discovered a much shorter way to approach the shading step. I personally didn’t invent the shortcut, but I adopted it into my book of math tricks all the same.

It’s not where you shade but how you shade

First, ask yourself the right question.

Instead of wondering how much time it will take to pick 4 different points, plug them in, simplify all the numbers and mark the answers true or false…

Think outside the box. After all, math is supposed to make sense.

Arrows point up, down, left, or right. Usually, when arrows point to the left, that means that numbers are getting smaller or we’re going in a negative direction. If arrows point to the right, numbers are getting bigger or we’re going in a positive direction. Greater than, >, points to the right. Less than, <, points to the left.

Greater than means shade above. Less than means shade below.

Yes, it is really that simple. And it doesn’t matter if the slope of a line is negative or positive. Above and below still work the same way.

Next, shade for one line at a time.

Don’t worry about the final answer just yet.



Now it’s time to see where the shaded areas meet.

After you’re done shading, you’ll notice that there must be an intersecting area. Where does the shading for each line overlap?


shading systems of inequalities graph

If you use different colors like I did, the answer is obvious. Out of 4 possible areas, only one can work. The overlapping area that follows the greater than/less than rules of both inequalities is the solution area.



In the words of the great Archimedes: Eureka! The intersecting shaded area is your solution set. I hope this easy-as-pie shortcut helps you on your graphing journey, algebra course or whenever math takes you!


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“Natalie is patient and understands her students. She is thorough with her explanations and will repeat the problem until you understand. She has great math tricks to simplify difficult concepts.”

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My Math Tools: Lattice Multiplication

Multiplication is a brilliant shortcut for addition. Long multiplication, on the other hand, can seem like tough math love because the most popular method, “block” form, is tedious and prone to arithmetic mistakes. That is where lattice multiplication comes in to save the day.
lattice multiplication 3 digit example
 

The exact origins of lattice multiplication are unknown, though scholars argue that it was developed in the Middle East. No matter who or where its creator was, the concept seems to predate the widely used “block form.” Their astounding similarities even suggest that lattice multiplication may be its unwitting parent. I first learned about this highly manual, ingenious strategy from a middle school student who absolutely refused to use block form – he had never learned it! Lattice multiplication was the only way he performed long multiplication. Then a college student, taking a math course designed for aspiring elementary school math teachers, showed me how her professor wanted her to practice this method. After some experimentation and soul searching, I was converted.


 

First, construct the boxes.


You need as many boxes as you have digits. For example, if you are multiplying two 2-digit numbers, you need 4 boxes. If you are multiplying a 2-digit number by a 3-digit number, you need 6 boxes, and if you multiply two 3-digit numbers together, you need 9 boxes.


 

first step for lattice multiplication

 


You can place either number on the top or side. Remember that only 1 number goes to each column or row, and you must only number the right side, not the left side.
Then draw a diagonal in each box. Each diagonal must go from the bottom left corner to the top right corner. This is done to prepare for the addition we do near the end of the process, adding from right to left.


 

second step for lattice multiplication

 


Next, start filling in your boxes.


Each row or column acts like a times table. For example, the 5 column multiplies each box in the intersecting rows by 5, while the 8 row multiples each box in the intersecting columns by 8. You can only have 1-digit or 2-digit numbers in the boxes. If the product has 2 digits, both fill the box. If there’s only 1 digit, you place a “0” in the left-hand triangle of the diagonal and the actual number in the right-hand triangle.


third step for lattice multiplcation

 


The hardest step is adding the diagonals.


The bottom right corner of the grid always drops down, and if you don’t carry over to the top left corner, that number also drops down unless it is “0.” If a sum of a diagonal is more than 9, you do have to “carry over” the extra digit to the next diagonal – just like in block form, where you carry over to the next column during long addition.


final answer lattice multiplication

 


The beauty of lattice multiplication is that it makes so much sense! Notice that in block form, the very first row matches the very last row of your grid, and that is an unbreakable rule for lattice form. Each row above the last row in the lattice grid matches the next row underneath the very first row in block form multiplication. In other words, lattice multiplication is the key to block form multiplication and a bona fide shortcut!


 

block form versus lattice multiplication
compare block form lattice multiplication

 

Another function is to multiply decimals easily and quickly. The steps are no different than for whole numbers. Keep in mind how many decimal places you need to apply at the end, after you have reached your final product, and move the invisible but ever present decimal point after the last digit from right to left until all the decimal places are accounted for.


 

decimals lattice multiplication
lattice multiplication decimal answer

 

Not only a quick and great mnemonic, lattice multiplication is a lifesaver on tests where calculators are not allowed. I’ve taught this method to children taking entrance exams and adults taking the ASVAB and CBEST. Doesn’t drawing a grid and filling in boxes take more time than the habitual block form? Actually, it takes less time! The genius of lattice multiplication is how the more you practice using it, the faster it goes. I can beat students who use block form and be finished long before them. Try lattice multiplication today, for school, test prep or everyday life, and leave the hassle of long multiplication behind!



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Stop Using the 5 Paragraph Structure

5 paragraph structure example

The 5 paragraph structure is a timeless teaching and brainstorming tool to show young students how to form an essay. The first paragraph is the introduction with the classic thesis statement, while 3 middle paragraphs are the support and the fifth, final paragraph is the conclusion. Hence, the 5 paragraphs. The name is self-explanatory!

However, this outline is not meant to be used beyond high school.

Many college students and adults I tutored were all under the false impression that there is no other way to write essays. Furthermore, the 5 paragraph structure became an obstacle hindering the development of their writing potential. How did a simple technique turn into such a problem?

Introductions and conclusions do not mix


First of all, introductions and conclusions are tricky to write well. My theory is that most contemporary teachers do not understand or do not know how to correctly build either paragraph type. In this way, when they teach children or even college students, they are leading them down the wrong path. The main consequence is that these same people will never be able to ace a test with a timed writing section or move forward from composing college level essays to submitting undergraduate or graduate level academic papers. The 5 paragraph structure has specific advice for introductions and conclusions, but the results are insipid, monotonous paragraphs that weaken their parent essays.

Except for certain tests, introductory paragraphs must introduce the essay topic, lead into the main argument and present the thesis in a smooth flow of sentences. Similarly, concluding paragraphs must end the discussion, but an ending is never the same as the beginning. Conclusions cannot summarize the essay’s main points or restate the thesis statement! This is a major mistake and flaw in the 5 paragraph structure. A conclusion transitions from the final supporting paragraph and is Point B to the thesis’s Point A. It answers the question “What’s next?” and makes a prediction or offers a solution. You cannot repeat or paraphrase any statements you have already made in previous paragraphs; while a conclusion does “hook” back to the thesis and still has to be on topic, it is meant is wrap up the essay and answer any indirect question(s) in the thesis statement – in a meaningful, original way.

 

Supporting paragraphs do’s and don’ts


Support is the heart of any essay or writing assignment: you must have a reason for the way you think or how you have drawn your conclusions. Supporting paragraphs are meant to be a direct response to and backing for the thesis statement in any essay. Whether you are taking a side in an argument or tackling an analytical prompt, you must show logical reasoning and development in each paragraph.

Being descriptive and non-repetitive counts, too! Think of each paragraph as a boat, with the sentences as cargo items. Every sentence must add to the development of a paragraph and hold its weight. If not, that sentence should be thrown off the boat! Also, the supporting paragraphs must relate to each other, so using transition words correctly is important to keep the paragraph organization smooth and connected.
What about the number of paragraphs, you ask? Should there be 3 paragraphs? The answer is no! If you create 2 strong supporting paragraphs, that can be sufficient. More on that next!

 

Remember, length is not strength


Finally, what is the necessary length for each paragraph type? This is a hard question to answer because there is unfortunately no right answer. Generally, a paragraph must have at least 4 sentences but no more than 8 sentences. Test essay rubrics differ, but I recommend that introductions and conclusions have no more than 4 to 5 sentences, while supporting paragraphs should be about 6 to 8 sentences long. Like I previously mentioned, it is better to aim for 2 supporting paragraphs at first, but they do need to be bulky, around 8 sentences long.

Is it easier to have 3 shorter paragraphs than 2 long paragraphs? That choice depends on the strength of your supporting reasons. When people use the 5 paragraph structure, their third supporting reason is usually the weakest and shows that they have “run out” of support for their thesis. Therefore, the third paragraph is clearly weaker than its predecessors and falls flat. Most of the time, my students tend to repeat themselves in that third paragraph, sharing more evidence for their two other supporting reasons but no new support. We decide to cut up that paragraph and distribute those sentences to the paragraphs they truly belong to. However, if you have 3 distinct supporting reasons, you can indeed have 3 supporting paragraphs.
On timed essays for tests, it is better to aim for 2 paragraphs because you save time on brainstorming and craft a much stronger rough draft. During any leftover time for editing, you can still insert a third paragraph if it fits well and makes sense. Don’t waste time on that 3rd paragraph for an outline’s sake! You will defeat the purpose of the timed essay(s), which is to show that you can write a college-level essay. Length does not determine how well you write. Instead, it can work against you and show that you are running in circles with nothing new to say.

Conclusion time


Do I ever recommend the 5 paragraph structure? Yes, since I also use it as a teaching tool! Again, the 5 paragraph structure is not a bad method, but it has flaws. Teachers use it to guide students through essay writing, but no one can rely on this structure forever. Critical writing is the stellar technique you need to use after you have mastered the 5 paragraph structure. You don’t write more than you need to express yourself, and you evaluate your thoughts to form succinct, clear sentences that build strong, interrelated paragraphs.

Some final pointers:

  • Write the way you think, not how you speak. The latter works great in creative writing, but not in academic writing. You must consider your audience to use the right tone (formal or informal) in your essay. Essays are generally formal and academic in tone. Focus on translating your thoughts into words, not monologue. Would you ever talk the way you write? Likewise, you cannot write like you talk.
  • The cardinal rule of all essays is to never, ever use the word “you.” Essays cannot directly address their audiences. You are not writing an article, blog entry, or advertisement when direct connection with your audience is necessary to be persuasive. Essays only use first person (I, we, my, our, us, me, mine, ours) or third person (he/she/it, his/hers/its, him/her/it, they, theirs, them) POV, i.e. point of view. Even so, first person is only used if a prompt poses a personal question or requires a personal response.
  • Aim for strong sentences that “show” and do not “tell” your audience what happens. This is harder to do in argumentative and analytical essays, which mostly have statements. However, I recommend that you always start a sentence with the actual subject, use a precise verb (avoid “is,” “do,” “have,” and horror of horrors, “get”) that means exactly what you want to describe, colorful adjectives, and a minimum of adverbs (none at all if possible). Complex, compound, and compound-complex sentences show your skills as a writer, but remember to vary your sentence length and use short, simple sentences as well. Do not use weak setups like “there is/are,” “it is,” “this is,” or “these are.” Avoid common phrases like “very,” “fun,” and “a lot [of],” all which weaken your writing.
  • Transition words are so important. Most of us think of fancy words like “however” and “moreover,” but simple conjunctions like “while” and “since” can also work brilliantly to make the reader keep reading your writing. See a full list.

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The Secret to Passing the CBEST

cbest test logo

Passing the CBEST is a challenge for the majority of test-takers. When I started tutoring back in 2010, my first encounter with CBEST test prep was to help a student pass the math section. Over the years, I extended my knowledge and expertise to tutoring the CBEST reading and writing sections as well. In my experience, most of my students struggled to pass one or more sections of the test. Some were aspiring teachers or substitute teachers, while others needed to pass it for their jobs. A college-level exam was turning their dreams into nightmares!

The breakdown

The CBEST has 3 sections: math, reading, and writing. The math section covers basic math, measurement, data and statistics, probability, algebra, and logic – all applied in word problems. Similarly, the reading section focuses on all 4 key areas of reading comprehension – main idea/purpose, details, inferences, and vocabulary-in-context – while including special types of questions, like reading and interpreting indexes and tables of content. Last but not least, students must be able to deliver 2 college level essays, one analytical/argumentative and the other a personal narrative, to pass the writing section. You must score a scaled score of 41 on each section, for a total of 123 points, to pass the CBEST. For math and reading, that translates to a raw score of 35 correct questions out of the total 50 questions to reach the scaled 41 score. If you score higher on any sections, your necessary scores for the other sections can be lower (with the minimum at 37). The total points for all 3 sections must still reach 123.

Why is the CBEST hard?

Arguably, the CBEST has a lenient time limit, since test-takers have 4 unrestricted hours per exam date and can choose how many sections they want to take in one sitting (edit: As of June 1, 2021, anyone who takes the computer-based test or proctored at-home test can only take 1 section per test date; the reading and writing sections each have a new time limit of 1 hour and 30 minutes, while the math section has a time limit of 2 hours. Multiple sections per test sitting are no longer supported, unless you take the paper test. The paper-based test has the previous 4 hour time limit but will be permanently retired in 2022.). However, they still can’t pass the test! As I’ve researched the web for better practice questions and practice tests, I’ve also come across specialized articles that lament how hard the CBEST is. One such writer claims that all available prep books have erroneous practice material, with nothing imitating the actual test experience. Others are sure the concepts in each section are designed to purposely trick them. Who is right? What is the trick to passing the CBEST?

Here’s what I have learned

It’s been a long, dusty journey, but I think I have unraveled the mystery behind CBEST success or failure.
  1. Most people are poor test takers. Test taking is an unpleasant, anxiety-inducing experience for 95 percent of the world’s population. It is a skill you have to practice and perfect, and it depends on your learning needs. Process of elimination, double-checking your answers, finding a strategy to maximize your time for answering questions and to free yourself from stress and self-doubt – you need to be armed with the right tools to conquer any test, let alone the CBEST.
  2. It’s not what you study: it’s how you study. Knowledge plays a great part, but you cannot just cram for the exam and hope that all that studying will stick. Breaking down concepts and trying practice questions can happen in just 30 minutes every day, as long as you fully commit to studying the same amount of time every day. You can even figure out a practical study schedule that meets your needs. Commitment to test prep is a powerful step to reach that passing score of 41.
  3. Prep books can be your best friends or your worst enemies, depending on how you well use them. I’ve read many complaints about this, so here are my two cents. All CBEST prep books published after 2005 are helpful resources that can maximize your score. Based on my own students’ feedback and test experiences, Princeton Review has practice tests with practice questions most similar to the CBEST paper test version, while Cliff Notes imitates the computer test version. The official practice test offered by the official CBEST website is a good start if you are preparing for either the paper or computer version of the test. Are the practice questions in these books close to the actual test questions? Yes and no. A few oddballs exist, even on the official practice test (which hasn’t been updated since 2005). Since the test makers claim to update the test every so often, no practice test is created equal to an actual test. However, you will notice patterns and familiar question types that keep appearing on all the practice tests. Bookmark these and try to master them.
  4. Concept review counts more than you think. For math, you need to be a pro at solving word problems using a simple, common sense approach. Questions are designed to be solved quickly and easily. You will need to invest in a workbook or app that throws hundreds of word problems at you, separated by concept or category (for example, fraction word problems, percent word problems, etc). Figure out which concepts you struggle with and keep practicing those exclusively. On the other hand, reading relies heavily on process of elimination (all wrong answers eliminated first, until you are left with the only possible right answer) and referring back to the text for every question because the right answer must be fully supported by information in the passage. You need to understand what questions are asking you to find and where you need to look in passages.
  5. Few people know how to write a essay. In my opinion, the writing section is the hardest section for most test takers. One reason is that most people can’t write! The truth hurts, but I have seen this happen again and again. They either have never been taught how to write correctly or they are convinced they are writing correctly and being failed by the CBEST test scorers on purpose. Each essay type requires good organization, development (paragraphs and sentences), syntax, sentence structure, transitions, word usage, grammar, and punctuation. Your brainstorming and logical reasoning always have to be on point. As I always say, length is not strength! Writing a long, poorly developed essay is worse than a shorter, well formed essay. “Show, don’t tell” and “Write how you think, not how you speak” are other favorite mantras of mine when it comes to effective writing. Also remember that editing is important – you have to self-edit and proofread your essays! Test scorers expect final draft material, not rough drafts.

My summary

The facts stand: I have helped students who scored a low 20 reach 41. Students who failed the writing section 3 or 4 times finally passed after working with me. One of my most successful students raised her score from a low 30 to a high 50 for math and reading.

 

The CBEST can be hard, but so can any test. However, it is only as hard as you make it. Becoming a better test taker, balancing your study time, and targeting struggle areas are the keys to success. You may not be able to pass all three sections in one take, but you can take each section only once and pass the CBEST the first time!


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