Which fractions make decimals that stop and which repeat, numbers whose digits never stop or repeat (square roots and π), and the laws for multiplying and dividing powers.
Each lesson works two ways. Students can work through the online tutorial at their own pace (it checks their answers and saves their progress in their browser), or you can teach it to the whole class from the slides, then send students into the tutorial for the Modify and Make steps.
Every lesson guide has plain-language notes, the answers to every check, a solution for every task, and slides and a worksheet for Waypoints Prepare. These guide pages are not linked from student pages, but they are not secret either: keep printed Teacher editions for answers you do not want students to find.
Whole-course cheat sheet: Open to print (every command, grouped by unit: a handout for the final challenge)
Long division shows why a fraction's decimal stops or repeats, and the denominator's prime factors predict which.
A fraction is a division, and dividing it out gives a decimal that either terminates (3/8 = 0.375) or recurs (1/7 = 0.142857142857…). In long division every remainder is less than the denominator, so either a remainder is 0 (it stops) or one comes back (the digits repeat from there). The repeating block gets a bar or dots.
You can predict which without dividing: write the fraction in simplest form and break the denominator into primes. Only 2s and 5s: it terminates, because terminating decimals are fractions over 10, 100, 1000… and 10 = 2 × 5.
It is the first step to the real number system: every fraction is a decimal that stops or repeats, which is what makes irrational numbers (which do neither) different.
terminating, recurring, repeating block, remainder, prime factor, simplest form.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Numbers like √2 and π that are not fractions: their digits never stop or repeat, and where they sit on the number line.
A rational number is a fraction of integers (whole numbers or their negatives); its decimal stops or repeats. An irrational number cannot be written as a fraction; its decimal never stops and never repeats. √2, √3, √5 and π are irrational. The square root of a whole number is irrational unless the number is a perfect square (√16 = 4).
Together the rational and irrational numbers make the real numbers: every point on the number line. To place a root, find the perfect squares either side (25 < 30 < 36, so 5 < √30 < 6), then square decimals to get closer. A rounded value (1.414) is an approximation; √2 is exact.
Exact answers from Pythagoras and circles are often irrational. Knowing when a decimal is only an approximation matters for measuring, building and calculating.
rational, irrational, real number, perfect square, approximation, exact value.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Multiplying, dividing and raising powers of the same base, and why a⁰ = 1.
A power is repeated multiplication: 2³ = 2 × 2 × 2 (base 2, index 3). Writing powers out as factors gives the laws for powers of the same base: multiplying adds the indices (aᵐ × aⁿ = aᵐ⁺ⁿ), dividing subtracts them (pairs cancel), and a power of a power multiplies them ((aᵐ)ⁿ = aᵐⁿ). Dividing a power by itself gives 1, and the quotient law calls it a⁰, so a⁰ = 1.
Simplify one law at a time: powers of powers first, then multiply and divide from left to right.
The laws make very large and very small numbers manageable (scientific notation in science), and they are the base for algebra with powers in Years 9 and 10.
base, index (exponent, power), product law, quotient law, power of a power, zero index, scientific notation.
Cheat sheet: Open to print (A4, with board pictures) · Markdown