Why multiply everything inside the brackets?
If you multiplied the x but left the other number alone, you have found a useful place to pause. The outside number counts copies of the whole group.
Try this lessonMake sense of math, one step at a time.
From fractions to equations and graphs. Pick a topic, try a step, and ask for a clue when you need one.
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Ask about your own questionExpressions & equations 8
Where did the negative sign go?
A negative sign belongs to the number next to it. Keeping that number together can prevent an almost-correct equation from turning into the wrong answer.
Try this lessonWhy can't I add the x's to the plain numbers?
Lena simplified 5x + 3 + 2x + 4 and wrote 14x. She combined everything into x's, mixing apples with bananas.
Try this lessonWhat is the opposite of adding 7?
Owen solved x + 7 = 12 by computing 12 + 7 = 19. He added again instead of undoing the addition.
Try this lessonWhich move comes first?
Isla solved 3x + 4 = 19 by writing 3x = 23, giving x = 23/3. She never removed the +4 before dividing.
Try this lessonWhy can't I just move the x?
Milo solved 3x + 2 = x + 10 by adding x to both sides: 4x = 12, so x = 3. Substitution showed 11 does not equal 13.
Try this lessonWhich way does the inequality flip?
Sofia divided -2x ≥ 10 by -2 and wrote x ≥ -5, keeping the symbol the way it was. The rule says it flips to ≤.
Try this lessonIs it rise over run or run over rise?
Kai found the slope through (2, 3) and (6, 11) and wrote 1/2. He divided the run by the rise.
Try this lessonFractions & decimals 9
Why can’t I add the denominators?
Before you add pieces, make sure they are the same size. The denominator names the pieces; the numerator counts how many you have.
Try this lessonWhy do I multiply straight across?
Sam computed 1/2 * 2/3 and wrote 3/5. He added the top numbers and the bottom numbers, which is how you combine, not how you multiply.
Try this lessonWhy do I flip the second fraction?
Amara shared 1/2 of a sandwich into 2/3-sized portions and wrote 1/3. She multiplied by 2/3 instead of flipping it first.
Try this lessonWhy can't I subtract straight across?
Leo worked out 3/4 - 1/3 by subtracting the tops and adding the bottoms, writing 2/7. The pieces were different sizes.
Try this lessonWhy does 18/24 turn into 3/4?
The fraction looks different and describes the same amount. The rule that allows this is the same rule that makes adding fractions possible at all.
Try this lessonWhy is 5/8 smaller than 2/3?
Five is a bigger number than two and eight is a bigger number than three, yet five eighths is the smaller amount. Bigger numbers on the page are smaller pieces of the whole.
Try this lessonWhy write 7/4 as 1 and 3/4?
Both forms describe exactly the same amount. One is better for seeing how much it is, and the other is better for calculating with it.
Try this lessonWhy is 0.4 × 0.3 smaller than both?
Multiplying is supposed to make things bigger, and here it made something smaller. That is not a trick, it is what happens when both numbers are less than one.
Try this lessonWhy is 3/5 the same as 60%?
Three names for the same amount, and one short chain that gets you between them. The confusion is rarely the arithmetic — it is which direction you are going.
Try this lessonNumber sense 7
Why can't I just add 7 and 5?
Maya added -7 and 5 and got 12. She counted the sizes of both numbers but forgot that they move in opposite directions.
Try this lessonWhat does minus a negative really mean?
Diego saw 5 - (-3) and wrote 2, as if the problem were 5 - 3. But the two minus signs are doing different jobs.
Try this lessonWhat do two negatives in a row do?
Priya multiplied -3 by -4 and wrote -12. She remembered that negatives matter, but she got the rule backwards.
Try this lessonWhy can't I just go left to right?
Jordan computed 3 + 4 * 5 by moving left to right: 3 + 4 = 7, then 7 * 5 = 35. The convention says multiply first.
Try this lessonWhy do exponents add when I multiply?
Rafi computed 2^3 * 2^2 by multiplying the exponents: 3 * 2 = 6, so he wrote 2^6 = 64. The exponents should add.
Try this lessonWhat happened when the exponent went negative?
Noor computed 2^-3 as -8. She treated the exponent's sign as the answer's sign, but exponents and values are different things.
Try this lessonWhere does the decimal point go?
Evan wrote 4500 as 4.5 * 10^4. He counted the digits instead of the decimal jumps.
Try this lessonRatios & proportions 5
Which number goes first in a ratio?
Nia counted 10 bus riders and 6 walkers and wrote the ratio of walkers to bus riders as 5:3. She answered a different question than the one asked.
Try this lessonHow much does one apple cost?
Ben paid $2.40 for 3 apples and divided 3 by 2.40, getting 1.25. He computed apples per dollar instead of dollars per apple.
Try this lessonWhere do the cross products come from?
Tess solved 6/10 = 30/x by computing 6 * 30 / 10 and got 18. The cross products were paired wrong.
Try this lessonWhat does percent of really mean?
Ivan found 15% of 80 by computing 1.5 * 80 and wrote 120. He moved the decimal the wrong way when converting 15%.
Try this lessonWhich number is the new price?
Zara found that 20% of $50 is $10 and handed me $10 as the new price. The $10 was the increase, not the result.
Try this lessonGeometry 4
Why can't I just add the two legs?
Frida found the hypotenuse of a right triangle with legs 6 and 8 by adding 6 + 8 = 14. The lengths must be squared first.
Try this lessonWhere did the 180 degrees go?
Hana found the third angle of a triangle with angles 65 and 42 by writing 107. She added the two she had and never finished the problem.
Try this lessonHow far apart are A(1, 2) and B(4, 6)?
Two points give you a right triangle in disguise. The legs are the changes in x and y; the distance is the hypotenuse.
Try this lessonIn a 30-60-90 triangle, what is the hypotenuse if the short leg is 5?
A 30-60-90 triangle has fixed ratios between its sides. Know one side and the other two follow without any trigonometry.
Try this lessonData & probability 1
How many numbers do I divide by?
Mateo found the mean of 4, 8, 6, 5, 7 by adding to 30 and dividing by 4, writing 7.5. There are 5 numbers.
Try this lessonAlgebra 11
What is the greatest common factor of 12x^3 and 18x^2?
When every term of a polynomial shares a factor, you can write the whole thing as a product. The tricky part is knowing exactly how much to pull out.
Try this lessonWhich pair of numbers factors x^2 + 7x + 12?
A trinomial like x^2 + 7x + 12 hides a pair of numbers. Find the pair, and the factored form falls into place.
Try this lessonWhy is x^2 - 36 not (x - 6)(x - 6)?
x^2 - 36 looks like an ordinary binomial, but it carries a special pattern. A difference of two squares always factors into a sum times a difference.
Try this lessonWhy does x^2 - 5x + 6 = 0 have two answer boxes to fill?
x^2 - 5x + 6 = 0 has two solutions, not one. Once the quadratic is factored, a product equal to zero tells you exactly where they hide.
Try this lessonWhat is x when x^2 - 2x - 4 = 0?
Not every quadratic factors over the integers. The quadratic formula works for all of them, and it only asks you for three careful numbers.
Try this lessonWhat do you add to make x^2 + 6x a perfect square?
A perfect square like (x + 3)^2 is easy to solve. Completing the square builds one on purpose, right in the middle of an equation.
Try this lessonHow many solutions does x^2 - 5x + 3 = 0 have?
Before solving a quadratic, the discriminant can tell you how many real answers it will have. One number, three cases.
Try this lessonWhat is the 20th term of 5, 9, 13, ...?
An arithmetic sequence changes by the same amount every time. With the formula, the 20th term is one calculation away.
Try this lessonWhat is the 5th term of 2, 6, 18, ...?
In a geometric sequence, each term multiplies the last by the same ratio. That ratio is the engine of the whole sequence.
Try this lessonWhat is (x + 3)(x + 5) expanded?
Every part of one binomial multiplies every part of the other. Four little products, then one tidy sum.
Try this lessonWhat is (x^2 - 9)/(x^2 + 5x + 6) in lowest terms?
A fraction like this simplifies only after you factor the top and the bottom. Cancellation happens between factors, never between single terms.
Try this lessonEquations & inequalities 2
How does y = 2x + 1 help you solve x + y = 10?
Two equations with two unknowns feel harder than one. Substitution turns the pair into a single equation with one letter.
Try this lessonHow does subtracting the equations remove a variable?
When both equations share a term like 2y, that term becomes your exit: subtract the equations and it disappears.
Try this lessonFunctions 6
What slope makes a line perpendicular to y = 2x + 7?
Two lines can look very different and still be perfect partners. The slope alone tells you whether they are parallel or perpendicular.
Try this lessonWhat is f(-2) when f(x) = x^2 - 4x?
f(x) is a rule with a name, not multiplication. f(-2) means: feed -2 into the rule and see what comes out.
Try this lessonWhich x values are allowed in f(x) = 1/(x - 2)?
A function's domain is the list of numbers you may feed it. A zero denominator is one number that is never allowed.
Try this lessonWhat equation undoes f(x) = 2x + 3?
An inverse function undoes whatever the original did. Here f multiplies by 2 and adds 3, so the inverse must unwind both in reverse.
Try this lessonWhy does 10% growth beat adding 10 each year?
Adding the same amount each year is linear. Multiplying by the same number each year is exponential, and the difference grows fast.
Try this lessonWhat power solves 2^x = 32?
A logarithm asks one question: what exponent do I need? log3(81) says: 3 to what power gives 81?
Try this lessonTrigonometry 2
What is sin A when the opposite side is 3 and the hypotenuse is 5?
Trigonometry begins with one idea: a ratio of two sides. Label the sides for your chosen angle, and the ratio is mechanical.
Try this lessonWhat is sin of a 30-degree angle on the unit circle?
On the unit circle, every point's coordinates are (cos, sin). Learn a few landmarks and the special angles are solved.
Try this lessonStatistics & probability 1
What is the z-score of a test score of 85 when the mean is 80?
A z-score measures a data point in standard deviation units: how far it sits from the mean, and in which direction.
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