← SSAT® Middle Level

Algebra, Geometry and Other Math: Algebra, Patterns and Logic

Variables, equations, inequalities, exponents cancelling factor by factor, patterns, sequences, and the logic and flowcharts E3n names in its own subscore.

11 topics · 50 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.

What a Variable Stands For

  1. How it worksHow does a box with a question mark in it turn into the letter x with nothing else changing?
  2. How it worksHow do 3x, x + 3 and x divided by 3 do three completely different things to the same number?
  3. Worked exampleFind the value of 4n - 7 when n is 5, and again when n is negative 2.
Your turneasy1 of 3
Find the value of 4n - 7 when n is 7.

Writing Expressions from Words

  1. How it worksHow does each phrase in a sentence become one piece of an algebraic expression?
  2. Worked exampleFind the expression for the cost of b books at 7 dollars each plus a 4 dollar shipping fee.
  3. Worked exampleFind the expression for a number doubled, then increased by 5, then cut in half.
  4. Common mistakeWhy isn't twice the sum of x and 4 written as 2x + 4?
  5. Going furtherWhat happens to an expression when a second unknown joins in, like c cats and d dogs?
Your turnmedium1 of 3
Books cost 10 dollars each and there is a flat shipping fee of 7 dollars on the order. Write an expression for the total cost of b books.

Solving Equations Step by Step

  1. How it worksHow does doing the same thing to both sides keep the scale balanced while x gets alone?
  2. How it worksHow do you undo operations in reverse order to peel a variable free?
  3. Worked exampleWalk me through solving 5x + 8 = 33 for x.
  4. Worked exampleWalk me through solving 3(x - 2) = x + 10 for x.
  5. Going furtherWhat happens when you solve an equation and end up with 5 = 5, or with 3 = 7?
Your turneasy1 of 3
Solve for x: 5x + 15 = 72. Round to two decimal places.

Inequalities and the Flip

  1. How it worksHow does an inequality answer become a whole ray of the number line instead of one point?
  2. Worked exampleFind every x that makes 2x + 3 less than 11, and draw the answer on a number line.
  3. Worked exampleFind the solution to negative 3x greater than 12 and show exactly where the sign flips.
  4. Common mistakeWhy isn't an inequality solved the same way as an equation all the way through?
Your turnmedium1 of 3
Every x that satisfies 2x + 9 < 37 also satisfies x < c for one particular number c. Find c. Round to two decimal places.

Properties of Exponents

  1. How it worksHow does writing out the repeated factors show why exponents add when powers multiply?
  2. How it worksHow do repeated factors cancel top against bottom when you divide two powers of the same base?
  3. Common mistakeWhy isn't anything raised to the zero power equal to zero?
  4. Worked exampleFind 34 divided by 32 by cancelling the repeated factors one at a time.
  5. Worked exampleFind the value of (23)2 and show why the two exponents multiply.
Your turneasy1 of 3
Find 3 raised to the power 7, divided by 3 raised to the power 2.

Algebraic Word Problems

  1. How it worksHow does one English sentence turn into one equation, phrase by phrase?
  2. How it worksHow do you set up a problem where two quantities are each described using the other?
  3. Worked exampleFind the two numbers that add to 40 when one of them is three times the other.
  4. Worked exampleFind Maya's age when she is 4 years older than her brother and their ages total 26.
  5. Common mistakeWhy isn't the answer to an algebra word problem always the value of x you just solved for?
Your turnhard1 of 3
Two numbers add to s, and the larger one is 5 times the smaller. Write an expression for the smaller number in terms of s.

Equations of Lines as Input-Output Rules

  1. How it worksHow does an equation of a line turn every input you feed it into exactly one output?
  2. Worked exampleFind y when x is 7 in the rule y = 4x - 5, then find x when y is 15.
  3. Worked exampleFind the rule that turns the inputs 1, 2 and 3 into the outputs 5, 8 and 11.
  4. Going furtherWhat tells you a rule draws a straight line rather than something that bends?
  5. Common mistakeWhy doesn't a table of x and y pairs have to come from a straight-line rule at all?
Your turneasy1 of 3
In the rule y = 4x - 10, find y when x is 11.

Patterns and the Rule Behind Them

  1. How it worksHow do you test a guessed rule against every term instead of only the first two?
  2. Worked exampleFind the next two terms of 1, 4, 9, 16, 25 and name the rule that makes them.
  3. Common mistakeWhy doesn't a pattern that opens 2, 4, 6 have to keep adding 2 forever?
Your turnmedium1 of 3
In a pattern, each term is 6 times the square of its position number. Write an expression for the term at position n.

Arithmetic and Geometric Sequences

  1. How it worksHow does each term of a sequence get built out of the term standing right before it?
  2. Worked exampleFind the 8th term of the sequence 4, 7, 10, 13 and show the shortcut that skips the rest.
  3. Worked exampleFind the 6th term of the sequence 2, 6, 18, 54.
  4. Common mistakeWhy isn't every sequence either an adding one or a multiplying one?
  5. Going furtherWhat changes about a sequence when the amount being added each time is negative?
Your turneasy1 of 3
A sequence starts at 11 and adds 8 each time. Find the term at position 8.

Series and Repeating Cycles

  1. How it worksHow does pairing the first and last terms of an evenly spaced list make every pair total the same?
  2. How it worksHow does a repeating cycle let you predict the 100th item using only a remainder?
  3. Worked exampleFind the sum of 1 + 2 + 3 all the way up to 50 by pairing the ends.
  4. Worked exampleFind the 47th letter in the endlessly repeating pattern A, B, C, D.
  5. Going furtherWhat changes in the pairing trick when the list holds an odd number of terms?
Your turnmedium1 of 3
Find the sum of 1 + 2 + 3 + ... all the way up to 69, by pairing the ends.

Logic and Flowcharts

  1. How it worksHow does a flowchart send a number down a different branch depending on a yes or no test?
  2. How it worksHow do overlapping circles show which statements must be true and which only might be?
  3. Common mistakeWhy doesn't if it rains the game is cancelled mean a cancelled game proves it rained?
  4. Worked exampleWalk me through a machine that doubles 14 and then subtracts 3 whenever the result passes 20.
  5. Worked exampleFind the output when 7 enters a chart that triples it and then adds 1 if the result is odd.
Your turnmedium1 of 3
A machine doubles the number it is given, and then subtracts 3 if that result is greater than 20. What comes out when 35 goes in?

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