Mathematics from zero
Fractions: a three-way budget chart
Reading that a fraction, a decimal, and a percent name the same amount is not the same as building a chart where all three columns line up and the parts sum to one whole. Take a real amount of money, split it into categories, and write every share three ways — then prove the whole thing balances.
Turn the unit’s mental model into one artefact: a budget table where each category is written as a fraction in simplest form, as a decimal, and as a percent, the percents add to 100, and the actual money amounts add back to the total — every conversion shown, not guessed.
Take a whole amount of money (e.g. a 100-unit allowance, or any total you choose) and split it across 4–6 spending categories, then express each category's share three ways — fraction in simplest form, decimal, and percent — proving the parts combine back into the whole.
- A table with one row per category and four columns — money amount, fraction (simplest form), decimal, percent — every cell filled.
- Each fraction is shown simplified, with the dividing step visible, and each fraction, decimal, and percent in a row names the same share.
- At least one worked sum of two categories using a common denominator, with the fraction result matching the percent result.
- A final check line: the percents add to exactly 100% and the money amounts add to the starting total.
- Redo the entire chart with a total that is not 100 (e.g. 60), so the fractions no longer convert to percent by inspection and you must rewrite over 100 — confirm the percents still sum to 100.
- Add a 'spent so far' row: find a percent of the total (e.g. 30% of the amount) and show it as a fraction, decimal, and money figure that all agree.
- Introduce one category whose share is greater than the others combined, and use a bar split into equal parts to show its fraction visually next to its percent.
- Write a short note explaining, for one row, why the fraction, decimal, and percent are the same amount — referencing equal parts and place value.
This is the loop behind every real ‘what share is that?’ question: assign the parts, write each as a fraction and simplify it, convert through decimal to percent by place value and multiplying by 100, and combine shares with a common denominator. The whole-balances check — percents summing to 100, money summing to the total — is your proof that the three names really describe one amount. Doing it once on a budget makes the conversions automatic everywhere else.