This unit is about being exact, and about knowing when you cannot be. It starts by sorting numbers into the sets mathematicians use, natural, integer, rational and real, and writing those sets in proper notation. Then it asks how many digits of a measurement mean anything (significant figures) and how a single unit stretches from nanometres to terametres.
The middle of the unit keeps square roots exact. A surd such as √8 is simplified to 2√2, and a fraction such as 6/√2 is rewritten with a whole-number denominator. The last lesson measures distance with absolute value and turns it into absolute, relative and percentage error.
A graphic display calculator (GDC) is optional all the way through. Lessons 2, 3, 4 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 · Sets and the number sets Set notation (∈, ∉, ⊂, n(A), ∅), set-builder notation, and the nested number sets ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ with the irrational numbers. Drop a number into the nested number sets, then change a set-builder condition and watch the set it describes. No calculator
- Lesson 2 · Significant figures and estimating Significant figures and which zeros count, rounding to a given number of significant figures, estimating by rounding to 1 significant figure, and giving an answer to a sensible accuracy. Zoom in on a number, choose how many significant figures to keep, and see which end of the interval it rounds to. GDC optional
- Lesson 3 · Prefixes and converting units Metric prefixes from nano to tera as powers of ten, converting lengths, areas and volumes (factors squared and cubed), litres, and compound units: speed and density. See 1 m² and 1 m³ split into smaller squares and cubes, then climb a ladder of prefixes a factor of 1000 apart. GDC optional
- Lesson 4 · Simplifying surds Surds, √a × √b = √(ab) and √a ÷ √b = √(a ÷ b), simplifying with the largest square factor, adding and subtracting like surds, and why √a + √b ≠ √(a + b). Draw a tilted square on a dot grid and see its side split into equal surds, then compare two legs of a triangle with its hypotenuse. GDC optional
- Lesson 5 · Rationalizing denominators Rationalizing denominators of the form a/√b and a/(k√b), expanding single and double brackets with surds, and (a + √b)(a − √b) as a whole number. Find a square's side from its diagonal two ways, then watch a/√b and its rationalized form land on the same point. No calculator
- Lesson 6 · Absolute value and error Absolute value as distance, |a − b| as the distance between a and b, the properties |ab| = |a||b| and |a + b| ≤ |a| + |b|, solving |x − c| = d, and absolute, relative and percentage error. Walk a then b along a number line and compare |a + b| with |a| + |b|, then move a measurement away from the true value. 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.