← SHSAT

Math: The Number System

The arithmetic half of the ceiling — signed numbers, fractions, decimals, factors — all of it from NYS Grade 6 and 7, and all of it without a calculator.

15 topics · 64 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Grade-Level Ceiling: What the Math Section Can Ask

  1. How it worksHow do the New York Grade 7 standards draw a hard ceiling over every math question on a Grade 8 SHSAT form?
  2. How it worksHow does the Grade 9 form lift that ceiling one year to let in exponents, slope and systems?
  3. Common mistakeWhy isn't the SHSAT harder just because a prep book spends forty pages on quadratics the test never asks?
  4. Going furtherWhat happens to a study list once you cross off every topic sitting above the Grade 7 ceiling?
Your turneasy1 of 3
A prep book spends 48 pages on solving quadratic equations and promises this will raise a Grade 8 SHSAT score. What is wrong with that promise?

Integers, Opposites, and the Number Line

  1. How it worksHow does the number line turn adding a negative into a step left instead of a step right?
  2. How it worksHow does taking the opposite of the opposite of -6 land you right back on -6?
  3. Worked exampleFind which is greater, -3 or -7, and show why the bigger digit loses once both are plotted.
  4. Common mistakeWhy isn't -12 larger than -5 when 12 is so obviously larger than 5?
Your turneasy1 of 3
A diver is 30 meters below sea level and a gull is 17 meters above it. How many meters apart are they?

Adding and Subtracting Signed Numbers

  1. How it worksHow does p minus q become p plus the opposite of q as a pair of arrows on a number line?
  2. How it worksHow do two arrows pointing opposite ways cancel each other into a sum of exactly zero?
  3. Worked exampleFind the value of -14 + 9 - (-6) + (-3) in one pass down the number line.
  4. Common mistakeWhy doesn't adding two negatives give a positive the way multiplying two negatives does?
  5. Going furtherWhat happens to the gap between two numbers when you subtract them in the other order?
Your turneasy1 of 3
Find the value of negative 18 plus 8 minus negative 4 plus negative 2.

Multiplying and Dividing Signed Numbers

  1. How it worksHow does the distributive property force -1 times -1 to equal 1 and nothing else?
  2. Worked exampleFind the value of (-4)(-3)(-2) and watch what each sign flip does to the running product.
  3. Common mistakeWhy isn't -32 the same number as (-3)2 when both get read out loud as negative three squared?
  4. Going furtherWhat happens to the sign of a product when you multiply five negative numbers together?
Your turneasy1 of 3
Find the value of negative 5 times negative 6 times negative 5.

Order of Operations

  1. How it worksHow does the order of operations decide which chunk of an expression is already a single number?
  2. How it worksHow does a fraction bar act like a pair of invisible parentheses wrapped around its top and bottom?
  3. Worked exampleFind the value of 18 - 2(5 - 8)2 / 6, one operation at a time.
  4. Worked exampleFind 3 + 4 x 2 - (10 - 6) / 2 and name the step most students skip on the way.
  5. Common mistakeWhy isn't multiplication always done before division when the M sits before the D in PEMDAS?
Your turneasy1 of 3
Write the expression, in terms of x, that means this: start with 31, then subtract the result of 2 times the square of (x - 8), divided by 6.

The Four Operations with Fractions

  1. How it worksHow does flipping the second fraction turn a division problem into a multiplication you can do?
  2. Worked exampleFind 3/4 + 5/6 - 2/3 and reduce the result to lowest terms.
  3. Worked exampleFind 2 and 1/2 divided by 3/4 without turning either number into a decimal first.
  4. Common mistakeWhy doesn't dividing by a fraction shrink the answer the way dividing by 5 does?
Your turneasy1 of 3
Find the value of 8/4 + 5/6 - 2/3. Give your answer as a decimal or a fraction.

Complex Fractions

  1. How it worksHow does a stacked fraction turn into a plain division problem between its top and its bottom?
  2. Worked exampleFind the unit rate when someone walks 1/2 mile in 1/4 hour by evaluating the complex fraction.
  3. Worked exampleFind the value of (2/3 + 1/6) divided by (3/4) as a single fraction in lowest terms.
  4. Going furtherWhat happens to a stacked fraction when you multiply its top and its bottom by the same number?
Your turneasy1 of 3
Someone walks x miles in 1/6 of an hour. Write an expression for the speed, in miles per hour, in terms of x.

Simplifying Fractions and Mixed Numbers

  1. How it worksHow does dividing top and bottom by their greatest common factor put a fraction in lowest terms?
  2. Worked exampleFind the simplest form of 84/126 without guessing at common factors one at a time.
  3. How it worksHow does the improper fraction 17/5 become the mixed number 3 and 2/5?
  4. Common mistakeWhy isn't 5/10 a smaller share than 1/2 when the numbers inside it are bigger?
Your turneasy1 of 3
Write an expression, in terms of n, for the numerator of the improper fraction equal to the mixed number n and 4/8.

Decimal Arithmetic

  1. How it worksHow does counting decimal places tell you where the point lands in a product?
  2. How it worksHow does sliding both decimal points the same number of places make 0.84 divided by 0.4 easy?
  3. Worked exampleFind 12.5 x 0.08 and then 4.2 divided by 0.35, both by hand.
  4. Common mistakeWhy isn't 0.45 bigger than 0.7 when it carries more digits after the point?
Your turneasy1 of 3
Find the product of 71 and 0.08.

Converting Among Fractions, Decimals, and Percents

  1. How it worksHow does long division turn any fraction into a decimal that either ends or repeats forever?
  2. How it worksHow does a repeating decimal such as 0.363636 get folded back into a fraction?
  3. Worked exampleFind 5/8 as a decimal and as a percent, then push 0.375 back into a fraction.
  4. Worked exampleFind the largest of 7/16, 0.43 and 44 percent by rewriting all three in one form.
  5. Common mistakeWhy isn't 0.5 percent the same size as 0.5 when the digits written down are identical?
Your turneasy1 of 3
Write an expression, in terms of n, for n out of 30 expressed as a percent.

Absolute Value and Distance on the Number Line

  1. How it worksHow does absolute value measure distance from zero while throwing away direction?
  2. How it worksHow does the absolute value of a difference hand you the gap between two points on a line?
  3. Worked exampleFind the value of |-7 + 3| - |4 - 9| without losing a sign along the way.
  4. Common mistakeWhy doesn't an absolute value bar simply erase every minus sign written inside it?
Your turnmedium1 of 3
Find the value of the absolute value of (-18 + 3), minus the absolute value of (4 - 11).

Factors and Multiples

  1. How it worksHow does pairing divisors let you list every factor of a number without missing one?
  2. Worked exampleFind every factor of 60 by walking up from 1 and pairing each divisor as you go.
  3. Common mistakeWhy isn't every multiple of 6 also a multiple of 4 when both numbers contain a 2?
  4. Going furtherWhat happens to the list of factors of a number the moment you double that number?
Your turnmedium1 of 3
How many whole-number factors does 2 raised to the power 15 have?

Greatest Common Factor and Least Common Multiple

  1. How it worksHow do the prime factorizations of two numbers hand you the GCF and the LCM at once?
  2. Worked exampleFind the GCF and the LCM of 36 and 48 from a factor tree for each.
  3. Worked exampleFind how many minutes pass before buses running every 15 and every 18 minutes depart together again.
  4. Common mistakeWhy do students grab the GCF when a problem about repeating events actually wants the LCM?
  5. Going furtherWhat happens when you multiply the GCF of two numbers by their LCM?
Your turneasy1 of 3
One bus leaves the depot every 3 times 9 minutes and another leaves every 4 times 9 minutes. They have just left together. After how many minutes will they next leave together?

Prime Factorization

  1. How it worksHow does a factor tree grind 504 down into a product of primes?
  2. Worked exampleFind the prime factorization of 1080 and use it to count how many factors 1080 has.
  3. Common mistakeWhy isn't 1 counted as a prime when nothing divides it except 1 and itself?
  4. Going furtherWhat happens to a prime factorization when the number happens to be a perfect square?
Your turnmedium1 of 3
A number's prime factorization is 2 raised to the power 5, times 3 raised to the power 4. How many whole-number factors does the number have?

Estimation and Checking Reasonableness

  1. How it worksHow does rounding each number to one digit reveal the size of an answer before you compute it?
  2. Worked exampleFind a fast estimate for 38.7 x 21.4 and then measure how far the estimate missed.
  3. Worked exampleFind whether 19 percent of 412 lands nearer 40, 80 or 400 without multiplying exactly.
  4. Common mistakeWhy doesn't an answer that looks about right prove the arithmetic underneath it was correct?
Your turneasy1 of 3
Estimate 19 percent of 687 by using 20 percent instead. What does that estimate give?