Essential Math
Two Numbers Have the Same Sum. Why Can Their Products Be So Different?
The pairs 1 and 11, 3 and 9, and 6 and 6 all add to 12. Their products are 11, 27, and 36. A fixed sum does not…
Alex Ion
Articles
Essential Math
What a Quadratic Graph Can Tell You Before You Solve It
The graphs of \(y=x^{2}-6x+5\), \(y=x^{2}-6x+9\), and \(y=x^{2}-6x+13\) have the same shape. Moving the constant term shifts the parabola vertically. Those shifts change whether the curve crosses, touches, or…
Essential Math
Why Successive Percentage Changes Do Not Simply Add
A price falls from $100 by 20%, then the reduced price falls by another 10%. The final price is $72, giving a total decrease of 28%. Adding the…
Essential Math
Two Timers Ring Every 12 and 18 Minutes. When Do They Ring Together?
Two timers ring together at noon. One rings every 12 minutes and the other every 18 minutes. Their next simultaneous ring is at 12:36 p.m. The answer is…
Applied Math
A 10% Road Grade Is Not a 10-Degree Angle
A road that rises 10 feet for every 100 feet of horizontal distance has a 10% grade. Its angle above the horizontal is about 5.71°, not 10°. Percent…
Essential Math
The Same Line Can Look Steep or Flat: Read the Axes First
A line rising from \((0,0)\) to \((4,2)\) has slope \(\frac{1}{2}\). Draw it on two charts with different vertical scales and it can look almost flat on one and…
Essential Math
Why Doubling Every Length Makes the Area Four Times Larger
A rectangular photo measures 4 inches by 6 inches. Enlarge both dimensions by a factor of two and it becomes 8 inches by 12 inches. The length and…
Essential Math
Dividing by a Fraction Can Make a Number Bigger
You have three quarters of a yard of ribbon. Each decoration needs one eighth of a yard. Dividing tells you how many decorations you can make: \(\frac{3}{4}\div\frac{1}{8}=6\). The…
Probability and Statistics
Why Four Times as Much Data Does Not Mean Four Times the Precision
If collecting more observations improves an estimate, four times as many observations should make it four times as precise. For a sample mean under a standard independence model,…
Probability and Statistics
Those Error Bars Might Show Three Different Things
Two charts show a mean of 50, but their error bars have very different lengths. Before deciding which result is more consistent or more precise, read what the…
Probability and Statistics
Should You Delete That Outlier? Three Questions to Ask First
A single observation sits far from the rest of your data. Removing it makes the chart cleaner and changes the average. Neither fact, by itself, tells you whether…
Probability and Statistics
Why Two Correct Calculators Can Give Different Quartiles
You enter the same six numbers into two calculators. Both return the same median, but their lower and upper quartiles differ. Before assuming one is broken, check which…