This unit asks how sure you can be of a number. It starts with proof: a recurring decimal is turned into a fraction by lining up its tails, and a short argument by contradiction shows that a square root is not a fraction at all. Then it moves to the arithmetic of very large and very small quantities, in standard form and in compound units such as density.
The middle of the unit keeps roots exact. Brackets with surds are expanded, and a denominator with two terms is cleared by multiplying by its conjugate. Absolute value becomes a distance, which solves equations and measures how an error grows through a product.
Lessons 5 and 6 are entirely new to Year 10. A fractional index is a root and a power in one, and a rounded value is not a number at all but an interval, whose upper and lower bounds decide what can be claimed with certainty. A graphic display calculator (GDC) is optional all the way through. Lessons 2, 4, 5 and 6 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Recurring decimals, closure and proof Writing terminating and recurring decimals as fractions, closure of the number sets under the four operations, and proving that a square root is irrational by contradiction. Shift a recurring decimal and watch the tails cancel; then combine two numbers and see which set the result lands in. No calculator
- Lesson 2 · Standard form and compound units Multiplying, dividing and adding numbers in standard form, converting compound units by moving each unit separately, and how length factors become area and volume factors. Slide a quantity along an exponent line one unit at a time, then multiply two standard form numbers and watch the carry. GDC optional
- Lesson 3 · Surds, brackets and conjugates Expanding brackets that contain surds, conjugate pairs and the difference of two squares, squaring a sum of roots, and rationalizing a denominator with two terms. Cut a corner from a square and re-form it as a rectangle, then split the square of a sum into four areas. No calculator
- Lesson 4 · Absolute value, equations and error Absolute value as distance, solving |ax + b| = c, |A| = |B| and |x − a| < r, the triangle inequalities, and absolute and percentage error through products. Move a test point on a number line to see whether it is within r of a, then see how a length error grows in an area. GDC optional
- Lesson 5 · Fractional indices Defining x to the power 1/n as a root, x to the power m/n as a root and a power, negative fractional indices, and the laws of indices applied to roots. Cut the step from exponent 0 to 1 into equal multiplications, then compare the root-first and power-first routes. GDC optional
- Lesson 6 · Upper and lower bounds Bounds of a number given to a stated accuracy, and the upper and lower bounds of a sum, difference, product and quotient, with the rounding that a context demands. Zoom in on a rounded number to see the interval it stands for, then choose which ends of two numbers give the largest and smallest result. GDC optional
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.