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Powerful Quadratics
This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.
Discriminating
You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.
Factorisable Quadratics
This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.
Age 16 to 18
Challenge Level
Warm-up
Which is bigger, $\frac{2}{5}$ or $\frac{1}{3}$?
How do you know?
Is there more than one way of convincing yourself that one is bigger than the other?
What if you had the fractions $\frac{2x}{5}$ and $\frac{x}{3}$? Would your answer be the same?
Main problem
Here are some more pairs of fractions for you to try.
If you are working with others you may wish to tackle examples separately and then compare approaches and conclusions.
You may like to print and cut out the pairs of fractions on cards, so that they may be attempted in any order.
- $\dfrac{4x}{7}$ and $\dfrac{9}{14}$
- $\dfrac{5}{9}$ and $\dfrac{2x}{12}$
- $\dfrac{3x}{4}+1$ and $\dfrac{x}{4}+3$
- $\dfrac{8(1-x)}{5}$ and $\dfrac{x}{6}$
- $\dfrac{8}{2x}$ and $\dfrac{4x}{16}$
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
NRICH is part of the family of activities in the Millennium Mathematics Project.