One quantity can be written in many forms: a fraction, a decimal that ends, a decimal that repeats, a power of ten. This unit asks which form suits which job. It starts with the numbers themselves: which are fractions, which are not, and how to turn a repeating decimal back into a fraction. It then builds the laws of indices from repeated multiplication, pushes them to zero and negative powers, and uses them to write and calculate with the very large and the very small.
Every idea is something you can move: a fraction long-divided digit by digit, a repeating decimal lined up against a copy of itself, factors counted as they join and cancel, a ladder of powers walked downwards past zero, a decimal point slid along a number. Lesson 7 uses a scientific calculator, once, to check an estimate.
The lessons, in order
- Lesson 1 · Rational and irrational numbers A rational number as p/q, which fractions give ending and which give repeating decimals, irrational numbers, and the nested sets of natural, integer, rational and real numbers. Long-divide any fraction with a numerator up to 20 and a denominator up to 16, and see its decimal end or its block repeat. No calculator
- Lesson 2 · Repeating decimals as fractions Writing a repeating decimal with a one- or two-digit block as a fraction in simplest form by the multiply-and-subtract method, including 0.999... = 1 and decimals where only some digits repeat. Set the block of a repeating decimal, line it against ten or a hundred times itself, and subtract the tails away. No calculator
- Lesson 3 · Multiplying and dividing powers Base, exponent and power; the product law (add the exponents), the quotient law (subtract the exponents), coefficients, and why the base must be the same. Join two runs of a factor into one, or cancel one run against another, and watch the exponents add or subtract. No calculator
- Lesson 4 · Powers of powers and products The power of a power law, the power of a product and of a quotient, powers of negative bases with and without brackets, and why a sum to a power does not split. Draw a power raised to a power as rows of factors, then raise a product to a power and compare it with a sum. No calculator
- Lesson 5 · Zero and negative indices A number to the power 0 is 1, a negative exponent as a reciprocal, evaluating zero and negative powers of whole numbers and fractions, and the laws of indices with negative exponents. Walk a ladder of powers of any base from exponent 4 down to minus 4 and read what each rung is worth. No calculator
- Lesson 6 · Writing numbers in scientific notation Scientific notation a × 10ⁿ with 1 ≤ a < 10, converting between ordinary and scientific notation for large and small numbers, and ordering numbers by exponent first. Pick a number from 120 up to eight billion and down to forty-one billionths, and slide its point to the first digit. No calculator
- Lesson 7 · Calculating with scientific notation Multiplying and dividing numbers in scientific notation, carrying when the front number reaches 10, adding with matched powers of ten, a scientific calculator's EXP key, and sensible accuracy. Multiply two numbers in scientific notation and see the carry, then add two and watch the smaller one shrink to a decimal. Scientific calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.