Using structure in shapes and expressions
Look for and use mathematical structure: exploit the hierarchy of 2-D shapes to deduce properties; use order of operations and algebraic structure to simplify expressions; connect fraction–decimal–percentage equivalences; use ratio structure to solve proportion problems efficiently
What a learner can do afterwards
- Use the property that all rectangles are parallelograms to deduce missing angle facts
- Recognise that 3 × (n + 5) = 3n + 15 by applying distributive structure
- Use the equivalence 1/8 = 0.125 = 12.5% flexibly to solve a comparison problem
1 · Read
Look at 2 + 3 × 4. If you work left to right, you get 2 + 3 = 5, then 5 × 4 = 20. But if you multiply first, you get 3 × 4 = 12, then 2 + 12 = 14. Two different answers is a problem, so mathematicians agreed on one order to always use.
The order is: brackets first, then multiply and divide (left to right), then add and subtract (left to right). Follow this order every time and everyone gets the same answer.
Brackets have structure you can use. In 3 × (n + 5) the 3 multiplies everything inside, not just the n. So 3 × n = 3n and 3 × 5 = 15, and the whole thing becomes 3n + 15. That works for any number outside the brackets.
Shapes fit inside each other. A parallelogram is any four-sided shape with two pairs of parallel sides, so every rectangle is also a parallelogram. That means every parallelogram fact works on a rectangle too, including the one that opposite angles are always equal.
Priya compares 1/8 of a pizza with 12% of a cake. She remembers that 1/8 = 0.125 = 12.5%. Since 12.5% is more than 12%, the pizza slice is the bigger piece.
Follow the agreed order in a calculation, expand brackets by multiplying every part, and use the links between shapes and between fractions, decimals and percentages.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.