Before exponents, equations, or graphs, there's just numbers. This section is the toolbox everything else is built from: what kinds of numbers there are, what order to do things in, how numbers break into primes, how fractions really work, and how to spot a pattern. Most of it you've seen before. The ISEE's trick is to test it in ways that punish anyone running on autopilot, so let's make it automatic. The practice is mixed into the lesson: each question sits right after the idea it needs, and the mixed set at the bottom takes the hint away.
Numbers come in families, and each family lives inside a bigger one. Start with the ones you count with.
The natural numbers (counting numbers) are $1, 2, 3, 4, \ldots$
Add zero and you get the next family up:
The whole numbers are $0, 1, 2, 3, \ldots$
Now let the numbers go below zero:
The integers are the whole numbers and their negatives: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
Then fill in everything you can make by dividing one integer by another.
A rational number is a number that can be written as a fraction $\frac{a}{b}$ of integers, with $b \neq 0$.
Here's a fact that does a lot of work: every decimal that ends or repeats is rational. Test it on something that doesn't look like a fraction.
Is $0.75$ a rational number?
Yes. It stops after two decimal places, so it's $\frac{75}{100}$, which is $\frac{3}{4}$: a fraction of integers. Any decimal that stops is a fraction with a power of $10$ underneath.
So far every family sat inside the next. Now zoom all the way out to every number that has a spot on the number line.
The real numbers are all the numbers on the number line: every rational number, and everything in between them.
Most of the real numbers you've just met. The rest are the rebels.
An irrational number is a real number that cannot be written as a fraction of integers.
You can spot one by its decimal: it never ends and never repeats. The two most famous ones:
$\sqrt{2}$'s decimals never repeat and never end. Main-character energy.
Are $\sqrt{2}$ and $\pi$ rational or irrational?
$\sqrt{2} = 1.41421356\ldots$ and $\pi = 3.14159265\ldots$ Neither decimal ever ends, and neither one settles into a repeating block, so neither can be written as a fraction of integers. Both are irrational.
Here they all are, dropping onto one line: the rationals from above, each with the fraction it equals, and the irrationals from below.
Every dot has a spot on the line, so every one of them is real. The split is only about whether a fraction can name it. Two things trip people up here: roots of perfect squares, and decimals that go on forever.
A square root is only irrational if the number under it is not a perfect square. $\sqrt{16} = 4$ is a plain counting number in disguise. And a repeating decimal like $0.\overline{6}$ is rational: it's exactly $\frac{2}{3}$. The bar means "repeats forever."
The ISEE likes to hide that second trap inside a lineup of equal-looking forms.
Which value is NOT equal to $\dfrac{2}{3}$? $0.6666667$, $0.\overline{6}$, $\dfrac{1}{1.5}$, $\dfrac{2.4}{3.6}$
$0.\overline{6}$ repeats forever, so it's exactly $\frac{2}{3}$. $\frac{1}{1.5}$: multiply top and bottom by $2$ to get $\frac{2}{3}$. $\frac{2.4}{3.6}$: multiply by $10$ to get $\frac{24}{36} = \frac{2}{3}$.
$0.6666667$ stops, so it's only very close. That's the one that's not equal.
Why does every fraction's decimal either stop or repeat? Watch a long division run. There are only so many remainders it can leave, so sooner or later one comes back, and from then on the digits loop.
That's one ending. Here's the other: the remainder hits $0$ and the division just stops.
Stop or repeat, those are the only two endings, and both come from a fraction. That's the whole reason "ends or repeats" means rational. Try it: the question below hands you a fraction and asks for its decimal, or hides a rounded decimal among equal forms.
Loading practice engine...
That loop works backward too: any repeating decimal can be turned back into a fraction.
Write $0.\overline{27}$ as a fraction in lowest terms.
The block $27$ repeats right from the decimal point, so put it over one $9$ for each repeating digit: $0.\overline{27} = \dfrac{27}{99} = \dfrac{3}{11}$. (Why 9s? $\dfrac{1}{99} = 0.\overline{01}$, so $\dfrac{27}{99}$ is $27$ copies of that: $0.\overline{27}$.)
If a digit comes before the repeat, like $0.1\overline{6}$, write the digits through the first repeat over a $9$ for the repeating digit and a $0$ for the one before it, then take away the part that doesn't repeat: $\dfrac{16 - 1}{90} = \dfrac{15}{90} = \dfrac{1}{6}$.
The rule has three moving parts, so here it is as a machine. Pick the digits and watch the 9s, the 0s, and the subtraction fall into place.
Now you run the machine by hand.
Loading practice engine...
Back to the families. Sort them yourself: every number goes in the smallest family that fits.
The test won't draw the boxes for you. It flashes a number and expects you to know. Call each card before it flips.
Same skill, test wording.
Loading practice engine...
One more kind of question about the families: what do you get when you combine numbers? Start with a surprising one.
Can the difference of two irrational numbers be rational?
Yes: $\sqrt{2} - \sqrt{2} = 0$, and $0$ is rational. But $\sqrt{3} - \sqrt{2}$ is irrational. So the difference of two irrationals can go either way, while a rational plus an irrational is always irrational.
Here's the whole cheat sheet for what you get when you combine numbers:
The "sometimes" lines are the ones people get wrong, and there's a trick to proving each kind of line.
To show something only sometimes happens, one example each way is enough. To show it always happens, an example isn't enough: you need a reason (like "a fraction times a fraction is still a fraction").
The ISEE asks this as a fill-in-the-blank sentence.
Loading practice engine...
$5 + 2 \times 3$ is either $21$ or $11$, depending on what you do first. Math settled the argument long ago:
So $5 + 2 \times 3 = 11$: the multiplication goes first. Here's the order on a problem the ISEE actually asks.
Which is greater: $5 + 2 \times (4 + 3)$ or $19$?
Parentheses: $4 + 3 = 7$. Multiply: $2 \times 7 = 14$. Add: $5 + 14 = 19$. They're equal. (Going left to right, $(5 + 2) \times 7 = 49$, is the trap.)
The letters in PEMDAS hide one more trap, in the middle.
PEMDAS lists M before D, but they're a tie: do whichever comes first, left to right. $24 \div 4 \times 2 = 6 \times 2 = 12$, not $24 \div 8 = 3$. Same goes for A and S: $10 - 3 + 2 = 9$, not $5$.
A cheap calculator doesn't know any of this: it just reads left to right. Build an expression and watch when the two calculators disagree.
A fraction bar is a sneaky grouping symbol: it puts invisible parentheses around the whole top and the whole bottom.
What is the value of $\dfrac{3(3^2 + 3^3)}{9(3 + 9)}$?
Top: $3^2 + 3^3 = 9 + 27 = 36$, so the top is $3 \times 36 = 108$. Bottom: $3 + 9 = 12$, so the bottom is $9 \times 12 = 108$. The value is $\frac{108}{108} = 1$.
Now you pick the order. Each wrong click explains which rule you skipped.
Flip it around: the same four numbers can be worth four different things, depending on where the parentheses go.
Time to compute some on your own. The harder versions nest brackets inside brackets.
Loading practice engine...
Multiplication runs in two directions. Going down, you break a number apart:
A factor of a whole number divides it evenly, with no remainder.
Going up, you build bigger numbers out of it:
A multiple of a number is that number times a counting number.
Both directions at once, starting from $12$:
Factors go down and get smaller; multiples go up and get bigger. Mixing the two words up is the most common slip in this section, so write out both lists once.
List every factor of $12$, and the first five multiples of $12$.
Factors come in pairs that multiply to $12$: $1 \times 12$, $2 \times 6$, $3 \times 4$. That's six factors: $1, 2, 3, 4, 6, 12$. The multiples are $12 \times 1, 12 \times 2, \ldots$: $12, 24, 36, 48, 60$. (Factors stop at $12$; multiples go on forever.)
Pairing up the factors is also how you count them without missing one.
Loading practice engine...
Some numbers refuse to break apart at all.
A prime number is a whole number greater than $1$ whose only factors are $1$ and itself.
Everything else (above $1$) can be split:
A composite number is a whole number greater than $1$ that has more than two factors.
Primes are basically the Avengers: they can't be broken down, and every other number is a team-up of them.
Where do the primes hide among the first hundred numbers? Over two thousand years ago, Eratosthenes found them by crossing out everything that isn't one.
Twenty-five survivors, and $2$ is the only even one. Every number that got crossed out is a product of the primes that crossed it out, which brings us to the most important fact in this section:
Prime factors are how you simplify roots in Section 2: $\sqrt{72} = \sqrt{2^3 \cdot 3^2}$ $= \sqrt{36 \cdot 2}$ $= 6\sqrt{2}$.
Every whole number greater than $1$ can be written as a product of primes in exactly one way (ignoring order). For example, $360 = 2^3 \cdot 3^2 \cdot 5$.
"Exactly one way" is the useful part: split $360$ however you like, and the same primes land at the bottom. Try a few trees and see.
The test asks for the whole factorization, or just its biggest prime, or the sum of the different primes. All three start with a tree.
Loading practice engine...
Growing a tree means spotting small factors fast, and there are tests for that which skip the dividing.
The digit-sum tests feel like magic the first time. Turn the dials and watch which lamps light, and why.
The ISEE's favorite version hides one digit and asks which digit makes the number divisible by $9$.
Loading practice engine...
Two numbers' prime factors also tell you what they have in common.
The greatest common factor (GCF) of two numbers is the largest number that is a factor of both.
And the smallest number they both fit into:
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
Both come straight off the prime factorizations.
Find the GCF and the LCM of $12$ and $18$.
Prime factors: $12 = 2 \cdot 2 \cdot 3$ and $18 = 2 \cdot 3 \cdot 3$.
GCF: the primes they share, $2 \cdot 3 = 6$. LCM: every prime either one needs, $2 \cdot 2 \cdot 3 \cdot 3 = 36$.
The Prime Factors widget above has a second mode for exactly this: match the shared primes into the middle of a Venn diagram, and the GCF is the middle while the LCM is everything.
The ISEE rarely says "GCF" out loud. It hides it in a story about splitting things into equal groups.
A teacher has $20$ apples and $30$ cookies to put into identical snack bags, with nothing left over. What is the greatest number of bags?
The number of bags has to divide both piles, and we want the biggest one: that's the GCF. $20 = 2^2 \cdot 5$ and $30 = 2 \cdot 3 \cdot 5$ share $2 \cdot 5 = 10$. So $10$ bags, each with $2$ apples and $3$ cookies.
Deal them out yourself. Some numbers of bags leave apples on the table.
The LCM has its own disguise: waiting for two things to happen together again.
A red light flashes every $4$ seconds and a blue light every $6$ seconds. They just flashed together. How many seconds until they flash together again?
The red light flashes at $4, 8, 12, \ldots$ and the blue one at $6, 12, \ldots$. The first time they match is a multiple of both, the smallest one: that's the LCM, $12$ seconds.
Telling the two disguises apart is half the battle, and the size of the answer gives it away.
Splitting things into equal groups (bags, rows, tiles) is the GCF, and the answer is never bigger than your numbers. Waiting for things to happen together again (lights, buses, laps) is the LCM, and the answer is never smaller than your numbers. If your answer lands on the wrong side, you used the wrong one.
Start with the GCF stories.
Loading practice engine...
For the LCM, slide the two lights' timings and watch where the flashes first line up. Sometimes it's the product of the two numbers, and sometimes it comes much sooner.
Lights, packs of hot dogs and buns, runners on a track: all the same question.
Loading practice engine...
Factor pairs show up in geometry too: every whole-number rectangle with a given area is a factor pair.
A rectangle has an area of $110 \text{ square inches}$, and its sides are whole numbers of inches. What is its least possible perimeter?
Factor pairs of $110$: $1 \times 110$, $2 \times 55$, $5 \times 22$, $10 \times 11$. The closest to a square is $10 \times 11$, with perimeter $2(10 + 11) = 42$ inches.
Why the squarest pair? Slide the area and watch the perimeters of every rectangle that fits it.
The shortest bar is always the squarest pair, and the longest is always the skinny $1 \times n$ one.
Same area, closer to a square means a smaller perimeter. The skinny $1 \times 110$ rectangle has a perimeter of $222$! And if the answer choices include $21$, that's half the perimeter: somebody added the length and width once.
The harder versions ask for the greatest perimeter instead.
Loading practice engine...
A fraction is a part of a whole. In Section 1, the same idea comes back as decimals, percents, and ratios.
Here's where the LCM pays off. You can only add or subtract pieces that are the same size.
Three rules, three questions coming up. Adding first, since it's the one with the famous mistake in it.
What is $\dfrac{1}{4} + \dfrac{1}{6}$?
The LCM of $4$ and $6$ is $12$. So $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{6} = \frac{2}{12}$, and the sum is $\frac{5}{12}$. (Adding straight across, $\frac{2}{10}$, is the number one fraction mistake on Earth.)
Why $12$? Cut both pizzas into the same number of slices and the dashed cuts only line up with the real ones at certain numbers.
The ISEE also likes to glue a whole number onto the front of a fraction, so two more names before the practice.
A mixed number is a whole number and a fraction written together; it means the whole number plus the fraction.
So $2\frac{1}{3}$ means $2 + \frac{1}{3}$. The same amount can be written as a single fraction with a big top:
An improper fraction is a fraction whose top is at least as big as its bottom.
That makes it worth $1$ or more. Turning a mixed number into one is a two-step trick.
Write $2\frac{1}{3}$ as an improper fraction.
The $2$ wholes are $\frac{6}{3}$, so $2\frac{1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$. The top, $7$, is bigger than the bottom, $3$, so it's improper, just as it should be.
To add mixed numbers: add the wholes, add the fractions, then tidy up.
What is $2\dfrac{1}{3} + 1\dfrac{3}{4}$? Write the answer as an improper fraction.
Wholes: $2 + 1 = 3$. Fractions, over the common denominator $12$: $\frac{4}{12} + \frac{9}{12} = \frac{13}{12}$, which is $1\frac{1}{12}$. Altogether $3 + 1\frac{1}{12} = 4\frac{1}{12}$.
As an improper fraction: $4$ wholes is $\frac{48}{12}$, so $4\frac{1}{12} = \frac{48 + 1}{12} = \frac{49}{12}$.
Adding and subtracting, plain fractions and mixed numbers, with and without a story:
Loading practice engine...
Multiplying is the easy rule, and it needs no common denominator at all. "Of" in a story means multiply.
Leo has $\dfrac{4}{5}$ of a pizza left and eats $\dfrac{1}{2}$ of what's left. What fraction of a whole pizza did Leo eat?
Half of four fifths: $\frac{1}{2} \times \frac{4}{5} = \frac{1 \cdot 4}{2 \cdot 5} = \frac{4}{10} = \frac{2}{5}$. Straight across, top times top and bottom times bottom, then reduce.
The traps here are all borrowed from the other rules: adding the bottoms, or flipping something that shouldn't flip.
Loading practice engine...
That "flip the second" step in the division rule has a name.
The reciprocal of a nonzero number is $1$ divided by that number.
To get it, flip the fraction.
What are the reciprocals of $\frac{2}{3}$ and $5$?
Flip each one: the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, and since $5 = \frac{5}{1}$, its reciprocal is $\frac{1}{5}$. Check: $\frac{2}{3} \times \frac{3}{2} = 1$ and $5 \times \frac{1}{5} = 1$. A number times its reciprocal is always $1$.
So dividing by a fraction is multiplying by its reciprocal.
What is $\dfrac{2}{3} \div \dfrac{1}{8}$?
Keep, change, flip: $\frac{2}{3} \times \frac{8}{1} = \frac{16}{3}$. That makes sense: there are a lot of eighths in two-thirds.
"A lot of eighths in two-thirds" is a real picture, not just a saying. Pick the amount and the piece size, and count what fits.
That picture explains something that surprises everyone the first time: dividing can make a number bigger, and multiplying can make it smaller. Call each one before the bars reveal it.
Now the division questions: how many pieces fit, how many servings, how many days.
Loading practice engine...
A sequence is a list of numbers that follows a rule, and each number in the list is a term. The ISEE's two favorite rules: keep adding, or keep multiplying.
An arithmetic sequence adds the same number (the common difference $d$) to get each new term.
Multiplying instead:
A geometric sequence multiplies by the same number (the common ratio $r$) to get each new term.
Start both rules from the same number and see how differently they grow.
Start at $3$. Write the first four terms of the arithmetic sequence with common difference $4$, and of the geometric sequence with common ratio $2$.
Arithmetic: add $4$ each time, so $3, 7, 11, 15, \ldots$
Geometric: multiply by $2$ each time, so $3, 6, 12, 24, \ldots$
An arithmetic sequence is a line in disguise: plot the terms and they climb by $d$ every step. In Section 5, that $d$ gets a new name: slope.
The difference between the two rules is small at first and enormous by term 7. Watch them race.
To jump straight to a far-off term without listing every step, count the steps: from term $1$ to term $n$ is $n - 1$ steps.
Here $a_1$ is the first term and $a_n$ is the one you want. That $n - 1$ is the whole trick, and the whole trap.
Why $n - 1$? To get from the 1st number in the list to the 20th, you take $19$ steps, not $20$. Using $n$ instead of $n - 1$ lands you one step too far, and that answer is always waiting in the choices.
Count the hops yourself: slide the term number and watch the arcs between the dots.
With the formula in hand, a far-off term is one line of work.
What is the 20th term of $3, 7, 11, 15, \ldots$?
It adds $4$ each time, so $a_1 = 3$ and $d = 4$. From the 1st term to the 20th is $19$ steps: $a_{20} = 3 + 19 \cdot 4 = 3 + 76 = 79$.
The ISEE often draws the sequence instead of listing it: a pattern of squares that grows by the same amount each figure. It's the same formula.
Figure $1$ of a pattern has $3$ squares, figure $2$ has $5$, and figure $3$ has $7$. How many squares are in figure $10$?
Each figure adds $2$ squares, so this is an arithmetic sequence with $a_1 = 3$ and $d = 2$. Figure $10$ is $9$ steps from figure $1$: $3 + 9 \cdot 2 = 21$ squares. (The trap answer, $3 + 10 \cdot 2 = 23$, counts ten steps.)
Predict figure $10$ for a few patterns, then watch them grow to check.
Lists, stories, and growing figures, all arithmetic:
Loading practice engine...
Geometric sequences use the same $n - 1$, but as an exponent.
What is the 6th term of $3, 6, 12, 24, \ldots$?
It doubles each time, so $a_1 = 3$ and $r = 2$. From the 1st term to the 6th is $5$ doublings: $a_6 = 3 \cdot 2^5 = 3 \cdot 32 = 96$. (Check by listing: $3, 6, 12, 24, 48, 96$.)
Doubling lily pads, tripling leaves, and plain lists:
Loading practice engine...
Legend says a young Carl Friedrich Gauss was told to add $1 + 2 + \cdots + 100$ to keep him busy, and he answered almost instantly. His trick: pair the ends.
Gauss summed 1 to 100 in seconds as a kid. Meanwhile I'm still waiting for the calculator app to open.
Where does the $n + 1$ come from, and why divide by $2$? Build the sum out of blocks and the formula draws itself.
Two staircases make a rectangle, so one staircase is half of $n(n + 1)$. Pairing the ends works for any list that goes up by the same amount, not just one that starts at $1$. Here it is on $20$ through $29$.
Count the numbers, pair the ends, multiply. The counting step is where the test catches people.
What is the sum of all the integers from $20$ to $49$, inclusive?
"Inclusive" means both ends count, so there are $49 - 20 + 1 = 30$ numbers. Each pair of ends adds to $20 + 49 = 69$, and $30$ numbers make $15$ pairs: $15 \times 69 = 1{,}035$.
Same trick for odd numbers: $1 + 3 + 5 + \cdots + 19$ has $10$ numbers, and each pair of ends makes $20$, so the sum is $5 \times 20 = 100$. (The first $n$ odd numbers always add up to $n^2$.)
That "$+ 1$" in the count is worth its own warning.
Counting from $20$ to $49$ gives $30$ numbers, not $29$. Subtracting the ends counts the gaps between them; add $1$ to count the numbers themselves.
Multiplying a countdown instead of adding it, $5 \times 4 \times 3 \times 2 \times 1$, gives a factorial, which counts lineups in Section 8.
One last twist: sometimes the test gives you a sum with a letter and asks for a nearby one. No formula needed.
If the sum of all integers from $1$ to $1{,}000$, inclusive, is $x$, what is the sum of all integers from $1$ to $998$?
Don't compute $x$! The second sum is the first one minus its last two numbers: $1{,}000 + 999 = 1{,}999$. So it's $x - 1{,}999$.
Staircases of cans, rows of seats, runs of odd numbers: pair the ends.
Loading practice engine...
Last stop: units. The ISEE asks two kinds of questions: "which unit makes sense?" and "convert this."
Which is the most reasonable unit to use when measuring the length of a leaf: centimeters, grams, kilograms, or meters?
Grams and kilograms measure mass, not length, so they're out. A leaf is a few centimeters long; in meters it would be a tiny decimal. Centimeters.
Two decisions, in order: what kind of thing is being measured, then what size of unit gives a sensible number. Sort a few.
The test's version also asks for a reasonable size: is a door 2 meters tall or 20?
Loading practice engine...
Metric units are friendly because every step is a factor of $10$. Watch a number hop down the ladder and back up.
Down a rung, times $10$; up a rung, divided by $10$. Three rungs from kilometers to meters is three tens, a thousand.
How many meters are in $3.5$ kilometers? How many meters are in $250$ centimeters?
Kilometers to meters is a step down to a smaller unit, so you need more of them: $3.5 \times 1{,}000 = 3{,}500$ meters.
Centimeters to meters is a step up: $250 \div 100 = 2.5$ meters.
Whichever way you go, the number and the unit move in opposite directions.
Sanity check every conversion: going to a smaller unit, the number should get bigger; going to a bigger unit, it should get smaller. If $3.5$ km came out as $0.0035$ m, you divided when you should have multiplied.
Drag an amount around the ladder and watch the number rewrite itself at every rung.
The US customary units don't come in tens, so these few facts just have to be memorized.
The direction rule is the same as on the ladder, and heights come with a twist: feet and inches.
How many inches tall is someone who is $5$ feet $3$ inches tall? How many feet is $66$ inches?
Feet to inches is a step down, so multiply: $5 \times 12 = 60$ inches, plus the extra $3$ makes $63$ inches.
Inches to feet is a step up, so divide: $66 \div 12 = 5.5$ feet. (That's $5$ feet $6$ inches, since half a foot is $6$ inches.)
Metric, customary, two steps at once, feet and inches:
Loading practice engine...
That's the whole toolbox.
One more trick before the mixed set. The test sometimes hands you a number with a letter in it, like $8n$ or $6nm$, and tells you the letters are prime. Nothing new is happening. A prime letter is just a prime you don't know the name of, so it goes into the factor tree like any other prime, and the LCM and GCF rules from earlier work word for word.
If $n$ and $m$ are prime numbers, what is the least common multiple of $6n$, $4nm$, and $9n^2$? And what is their greatest common factor?
Factor each term, letters included: $6n = 2 \cdot 3 \cdot n$, $4nm = 2^2 \cdot n \cdot m$, $9n^2 = 3^2 \cdot n^2$.
The LCM takes every prime at its biggest power: $2^2 \cdot 3^2 \cdot n^2 \cdot m = 36n^2m$.
The GCF takes only the primes in all three terms, at their smallest power. The only one in all three is $n$, so the GCF is $n$.
A letter's power counts exactly like a number's: $n^2$ beats $n$ the way $2^2$ beats $2$.
Multiplying all the terms together ($216n^4m^2$ above) is the "common multiple, but not the least" trap wearing a disguise. And if a term has an $m$ and the others don't, $m$ can't be in the GCF, but it has to be in the LCM.
Loading practice engine...
Every question so far came right after its idea, which is a hint the real test never gives. Down here the hint is gone. First, the ISEE's quantitative comparison: two columns, four fixed choices, and any idea from this section fair game.
Loading practice engine...
Next, the mixed set. Every question type on this page except the quantitative comparison goes into one hat, and each click draws a new one at random. Flip the switch to type your answer (fractions like 5/12 are fine); a few types need their answer choices, and they'll say so.
Loading practice engine...
Alright, if you've made it this far, it's time for something with a little heat. Letters show up in a number theory problem again, this time with more going on. Don't panic: treat them like any other prime.
Loading practice engine...