Percentages as Fractions

Activities to try

1. 1

   Shade a 100-grid to define 'per hundred'

   Why it works: A percentage literally means parts per hundred, so it is the same quantity expressed as a fraction (out of 100) and as a decimal; 100% is 'one whole', not a ceiling, and percentages can go beyond it.

   1. On a 10×10 grid (100 cells, each cell = 1%), shade 45 cells and write the result three ways: 45% = 45/100 = 0.45.
   2. State in your own words what '45 out of 100' means by pointing at the shaded and unshaded parts of the grid.
   3. Shade a second grid partly to show more than one whole: fill one full grid and 50 cells of a second, and write 150% = 1.5 = one and a half wholes.
   4. Convert a small percentage carefully: shade 7 cells and record 7% = 7/100 = 0.07 (watch that it is 0.07, not 0.7).
2. 2

   Same percentage, different wholes — 50% of two bags

   Why it works: A percentage is a proportion of its own whole (50% = half of whatever the whole is). Because the wholes differ, equal percentages give different amounts, so a percentage is not a fixed absolute quantity; a larger whole's 50% can even exceed a smaller whole's 100%.

   1. Predict first: Bag A has 20 sweets and Bag B has 40 sweets. If you take 50% of each, will you get the same amount from both, or more from one — and why? Write your reason before checking.
   2. Compute 50% of each: 50% of 20 = 10, 50% of 40 = 20.
   3. Shade half of two bars (or grids) sized to represent 20 and 40, and compare the shaded amounts.
   4. Explain in one sentence why the same percentage gave different amounts.
   5. Check a trap comparison: is 100% of a small 10-sweet bag more or fewer sweets than 50% of Bag B (40 sweets)? Compute both (10 and 20): 50% of the larger bag gives MORE than 100% of the smaller bag — so '100% always beats 50%' is false, because each percentage acts on its own whole.
3. 3

   Convert fluently between fraction, decimal and percentage

   Why it works: The three forms are interchangeable notations for the same proportion; converting reliably in both directions depends on tracking the 'out of 100' divisor and decimal place.

   1. Match each percentage to its decimal and its fraction-out-of-100 (e.g. 45% ↔ 0.45 ↔ 45/100; 7% ↔ 0.07 ↔ 7/100; 150% ↔ 1.5 ↔ 150/100).
   2. Go in both directions: given 0.08, write the percentage (8%); given 3/100, write the percentage (3%) and decimal (0.03).
   3. Place a given value on a number line that passes 100%, locating 150% at one and a half wholes.
   4. Explain the decimal-placement rule: converting a percent to a decimal divides by 100, moving two places, so a small percent becomes a small decimal (7% → 0.07).