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Welcome to the SHSATAcademy math course. These free lessons are designed to teach students various basic skills from the complete math course. If you enjoy what you learn, please consider subscribing to the complete course. If you want additional help, our SHSAT classes are affordable and have a high proven rate of success.

Introduction to Math

This course is designed to teach students the core SHSAT math subject material and how to execute related concepts to solve problems correctly. It is essential to gain a mastery of the broad array of subject material covered for the SHSAT math exam. In particular, it is important to develop a comfort level with geometry and algebra, which are big parts of the test. Unlike many guides, this course will provide background material and video tutorials for each math topic, provide skill exercises to test student knowledge of the lesson material, and then provide SHSAT level problems that include the lesson skills. Full-length SHSAT exams (and mini-exams for those with a busy schedule) are available to measure your progress. The course is organized to:

  1. Teach the Exam – Familiarize students with the exam design and related strategies.
  2. Teach the Content – Make certain students have a summary overview of required concepts and theorems as well as complete, focused lessons covering all required topics on the exam.
  3. Execute What You Learn – Students will have to learn to apply the lessons to real problems including strategies to avoid mistakes.

Although essential, mastering the math does not guarantee a high score. The SHSAT is more than a typical math test. It is a series of problem-solving exercises. The SHSAT asks basic math questions in weird ways, chock full of details and these patterns repeat throughout the exam. In most cases, the cause for difficulty on the SHSAT exam boils down to time management and paying attention to details. Beyond learning the math subject material, this course will help teach students aspects of the test design including a process to follow that will help students avoid the common mistakes. Our answer walkthroughs are unique because most answer explanations do not do justice teaching students who do not already know the correct answer. Exam walkthroughs in this course will review the relevant math concepts covered in each problem, identify different methods to solve the problem, and highlight potential patterns and traps when applicable. Many of these rules and strategies apply equally to the SAT and PSAT, which is developed similarly. In other words, what you learn herein will be helpful down the road as you also look to succeed on college entrance exams. In fact, we like to think of the SHSATAcademy as the ultimate guide for 6th to 9th graders to not only prepare for the SHSAT but to get a head start and learn the necessary habits to succeed on the PSAT and SAT.

What is the new SHSAT exam and why should I prepare for it?

Any student hoping to gain admission to a specialized high school should prepare for the SHSAT. Perhaps more importantly, 6th to 9th graders everywhere can get a head start on the new SAT from this SHSAT course. The new SAT and SHSAT are more alike than ever. This course might be the best way for middle school students to learn SAT design and strategy with age-appropriate material.


SHSAT stands for the Specialized High School Admission Test given each fall toward the end of October* to approximately 25,000 8th grade students and nearly 30,000 students total including 9th grade students in New York City. The specialized high schools include 9 free public schools (listed below) that offer a world class education at no cost to NYC’s gifted & talented students. Course offerings, academic standards, and matriculation to colleges & universities rival the top private or preparatory schools in the country that can charge up to $200,000 for 4 years tuition. 8 of the 9 schools sole admission requirement is performance on the SHSAT (Laguardia HS includes a performance audition).

The format of the SHSAT exam changed in 2017. In previous years, the exam consisted of a 50 question multiple choice math exam for 75 minutes preceded by a 45 question verbal exam also for 75 minutes which included scrambled paragraphs, logical reasoning, and reading skills. Scores were based on a 400 point scale for each exam section or 800 points in total (just over 700 in practice). In 2017, the verbal section was substantially modified and called the ELA (English Language Arts) portion of the SHSAT. Scrambled paragraphs and logical reasoning were removed from the exam and replaced with writing/editing skill questions which reflect the curriculum taught in school. The reading portion of the ELA will remain. The math exam includes 5 grid-in questions which are not multiple choice. Each exam section includes 57 questions and the total exam will take 3 hours. Only 47 questions on each section will be scored. 10 questions are random “field questions”, and students will not know ahead of time which questions are field questions, so they must treat every question the same. Scores will continue to be scaled and the difficulty level is expected to remain the same. More details about the SHSAT exam changes and samples are available in PDF format directly from the DOE press release. has made practice exams available in the new format for the all new SHSATAcademy Course. 



1. The SHSAT has been the focus of discussion in regards to lack of representation from certain demographic groups. The following is an excellent study on the matter. “Pathways to an Elite Education.”

2. PASSNYC is a 501(c)(3) non-profit organization that has partnered with many community organizations & service providers to broaden the opportunities for NYC’s talented, underserved students. They are our partner in providing the dramatically expanded SHSATAcademy free course offering and excellent resource for additional information & training opportunities regarding the SHSAT.

3. The previous student handbook can provide some excellent background SHSAT information. Make certain to note the sample exams and sections will not be the same for 2017.

4. The DOE Frequently Asked Questions press release regarding the new exam format.

  • For additional test dates and more details on the SHSAT consult the NYC DOE website and the SHSAT Handbook release for the coming year.

Why the SHSAT today?


Imagine good athletes from every community in the city had the chance to play, all expenses paid, for the farm team of the New York Yankees which has developed numerous professional ballplayers over time. Would you try out?

The specialized high schools offer entry into world class education opportunities and when it comes to education, your odds of future professional success are significantly better. There are 500,000 masters’ graduates each year who go on to develop lifelong careers as professional doctors, lawyers, engineers, etc. versus about only 2,000 NCAA athletes that get drafted to the major professional sports every year. In the medical field alone there about 800,000 doctors in the U.S., but only about 18,000 pro athletes in all major sports. Why wouldn’t you invest in your education? 

A Comment on Diversity

Black and Latino students represent only about 10% of students entering the big “3” specialized high schools despite representing almost 70% of all NYC students. Historically, high costs to prepare for the SHSAT and lack of access to gifted & talented programs in many communities have created disparities along the lines of demographics, income, and location. Our goal is to level that playing field and we would also point parents and students to many new city sponsored programs in 2017 to help close these gaps. We believe the gaps that develop in childhood too often extend into adult life. For example, the median household wealth for a caucasian household is $111, 145 according to Forbes versus $7,113 for a black family and $8,348 for a Latino family household. The median incomes are $59,270 versus $34,598 and $40,963 respectively for each household as of 2013. It takes time and effort to make change, but change begins at an early age and education is the best path to a brighter future.

The main reason for mistakes on the SHSAT, PSAT, and SAT

SHSAT mistakes very often result from reasons other than lack of prior knowledge in the subject matter, and that is perhaps more prevalent for reading where prior subject knowledge is not critical in the same way it is on the math exam. Most students think good grades in algebra, geometry, or English class will necessarily translate to a high SHSAT math or ELA score. That is not necessarily true. While subject knowledge is an essential requirement, it is not sufficient. Prior exposure to courses like algebra and geometry are a huge plus in math, for example. Both topics comprise a large part of the math test, and there is a chasm between students who have seen these courses early enough to gain familiarity and those who have not seen the material or are currently taking such courses when the SHSAT exam is given in late October in the 8th grade. If this is an issue for you, consider planning and preparing early for the SHSAT. That said, even among students with excellent grades and top-level state math and ELA exams scores, only about half the test takers will gain admission.


The SHSAT — and SAT and PSAT for that matter — ask questions in strange ways. Questions often repeat topics and phrase their wording in ways students are not used to seeing in their classroom setting. Accurately diagnosing what the question prompt is asking is a large part of the battle. Moreover, questions are written in ways that require test takers pay extra careful attention to detail. Every word is important, and a loss of focus will result in mistakes that often separate two students who both have a strong grasp of the subject matter. The following typical reasons for mistakes on the SHSAT explain why student results do not always correlate with school grades. One or two reasons may come as a surprise. In either case, this course will teach each student how to evaluate their results and weaknesses to improve their score on the SHSAT.




  1. Subject-Matter Knowledge – does not know the subject matter or lacks confidence in the procedures to solve
  2. Time pressure – student runs out of time before finishing
  3. Confused by Question Prompt – cannot identify what the question is about
  4. Misses Details – did not pay attention to a particular word or detail in the prompt or answer choice
  5. Calculation Errors – student made an execution error while solving

What tests are best for practice?

In the case of the SHSAT and SAT alike, questions are written according to specific rules and patterns, and the ability to excel on the exam depends on learning to take advantage of these rules and procedures. Many of the difficulties students have on the SHSAT result from poor test-taking skills, not poor subject-matter knowledge. Any material that breaks the rules of the real test will encourage bad habits or prevent good habits from developing. Even “harder” questions will not help unless they are guaranteed to follow the same rules, so don’t assume harder questions will better prepare you for the actual test. Official sources are the best, and anyone who states otherwise is probably trying to sell you something you probably should not buy.

How to Begin

You have already taken your most effective first step: you have begun our SHSATAcademy course for free. We hope you enjoy it. Whether you are new or experienced, chances are the free version will teach you some basic skills or strategies you had not previously considered. Start with a real SHSAT practice exam. Follow the lessons in order or focus only on certain topics. We want you to learn. Follow up with another practice exam. If you like what you see, we welcome you to subscribe to complete the course for only a few dollars.

President’s Letter

Photo: Richard Baxter Foster

This course is dedicated to that modern, handsome, well shaven gentleman in the picture. No, that is not our President – not really. Richard Baxter Foster is a late relative and, much to my chagrin, I could never sport a beard and mustache with quite the same demeanor nor muster the same energy as he did. Richard Baxter Foster was abolitionist, soldier, teacher, Ivy League graduate, minister, frontiersman and the inspiration for this course. As a volunteer in the 62nd U.S. Colored Infantry Regiment since its formation in 1863, he spent many days and evenings teaching soldiers in his unit who typically did not have access to education before the war nor likely would afterward. The troops believed in Richard, and they valued education highly enough to pitch in $5,000 of their hard earned savings (a fortune to the men at the time) to help Richard Baxter Foster start Lincoln Institute – the first African American college in the United States today known as Lincoln University. We too started with $5,000 provided from students or their parents in this case – our early adopters. Richard Baxter Foster made it his mission to provide equal education opportunities to all and, in a small way, we hope to continue his mission today in NYC. 

The Story of Eratosthenes – Critical Thinking Adventure Part 1

In this critical thinking adventure, we intend to place you in the shoes of inventors and historical figures and ask you to try and solve the problems they faced with only the same observations and information available to them. In short, we will pit your creative mind against that of scientists who lived hundreds, or in the case of Eratosthenes, over 2,000 years ago. Is your modern brain power up to the challenge? You could Google their discoveries, but that is not in the spirit of the effort, which is to give you a break from SHSAT preparation and get your creative juices flowing to solve a puzzle while hopefully having a little fun. 

Eratosthenes was an accomplished inventor and intellectual who at the young age of 30 in 245 B.C. was hired by the Pharaohs to be the librarian for the Alexandria (Egypt) Library, one of the prestigious centers of intellectual history and pursuits at the time. While rubbing elbows with the likes of Archimedes, Eratosthenes most likely read library documents indicating that during the summer solstice, June 21, there could be observed no shadow from columns or a vertical stick in any direction at high noon in the city of Syene approximately 500 miles from Alexandria also in Egypt. If a person looked down a water well, they would observe no reflection from the water below. This was notably not the case in Alexandria at the same time. Vertical columns did cast a shadow. Eratosthenes used these observations to calculate the first accurate measurement of the distance around the globe to 252,000  stadia (a Greek unit of distance). The question to students is how did he do it based on the observations made? Could you make the same discovery and measurement with only the information and tools available to Eratosthenes in 245 B.C.?

Some background information is required as Eratosthenes was a leading scientist of the age and not flying completely blind. For example, and this may surprise you, it was already known to Eratosthenes and others for over two centuries prior to 245 B.C. that the earth was round. Pythagoras asserted the earth was round in 500 B.C. based on the round shape of the moon and others came to the same conclusion based on different means. Aristotle determined from the movement of constellations in 350 B.C. that the earth was round, and he even estimated its size as 400,000 stadia around. Eratosthenes knew rays from the sun approached earth in parallel, and he knew the earth rotated on a tilted axis where the northern hemisphere was tilted closest to the sun on June 21, the summer solstice. In fact, Eratosthenes is also credited with the first accurate measurement of the earth’s axial tilt. He knew the distance from Alexandria to Syene fairly accurately. He understood the mathematics of the day which included analysis of right triangles and the work of Pythagoras and others who lived before him. 

Given the similar information, what conclusions can you draw from the observations of shadows and how would you design a measurement of the earth’s circumference?

Feel free to share thoughts and ideas below but not answers sourced from Google or elsewhere. The objective is to put on your own thinking cap and proceed. Keep an eye out for Eratosthenes Part 2 on an upcoming Lectures page at the start of each lesson group for answers to these questions and more.


Part 2 continues on Math Fundamentals.

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