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# GXWeb Jigsaws and Quizzes

Symbolic computations on these pages use Nerdamer Symbolic JavaScript to complement the in-built CAS of GXWeb

Saltire Software, home of Geometry Expressions and GXWeb

YouTube Introduction to GXWeb Jigsaws (6:46)

AAMT eConference Presentation 2021: GXWeb and Jigsaws: Free Online Tools for Exploring Beautiful Mathematics (37:30)

Fancy some mathematical fun?

Use jigsaws and GXWeb to explore three mathematical gems drawn from geometry, number theory and elementary algebra - and then follow the links provided to see some of their many and varied applications!

Algebra Tiles

Fractured Fractions

Kissing Circles

Behind the Scenes

## Algebra Tiles Jigsaw

What does $$2x + 1$$ mean to you? What about $$4-3x$$ or even $$x^2=x+1$$?

What dominant image springs to mind?

Do you see an object or a process?

Do you think of a graph? A table of values?

 Students who are successful in algebra tend to have a richer repertoire of images compared to those less proficient. As teachers, we need to build these images deliberately and with care. Algebra tiles can be a powerful tool for building deep understanding, and the virtual variety actually offer some major advantages: they explicitly link the shapes to the symbolic form, and they establish that variables are dynamic rather than static things. Play more with Algebra Tiles Explore the Meaningful Algebra Collection

## Fractured Fractions Jigsaw

(Never heard of continued fractions? Tap here.)

 Every real number, rational and irrational, can be represented by a continued fraction - the rational ones are finite, of course, but both finite and infinite offer some wonderful patterns and opportunities to explore!
 Unlike irrational decimals, however, even irrational continued fractions can be predictable and are an ideal way to calculate approximate values - as accurately as you like! See from the model that $$\frac{10}{7}$$ becomes $$[1,2,3] = 1 + \cfrac{1}{2 + \cfrac{1}{3}}$$. So what number gives the continued fraction $$[1,2,3,4]$$? Can you find some values whose continued fractions are composed entirely of the number 1? Play more with continued fractions Explore the Fractured Fractions Collection Rearrange the squares to fill the given rectangle, and then press INPUT (or the JIGSAW button again) to enter your answer.The first one is a little easier - the target positions and the resulting fraction are shown.

## The Kiss Precise: Kissing Circles Jigsaw

Given the sequence $$[-1,2,2]$$, what do you predict to be the next few numbers?

And what might follow $$[-1,2,3]$$ - or $$[2,2,3]$$?

What about $$[-2,3,6]$$? Or $$[-6,7,42]$$?

If we simplify the model shown here, you might see that the next few terms after $$[-1,2,2]$$ are 3, 6 and 15.

 Each number refers to the curvature - or bend - of the circle it describes - which is the reciprocal of the radius. A negative bend describes the circle surrounding the others! Here we imagine a shape (circle, rectangle, triangle(-ish)... even a Golden (Pappus) chain!) completely filled with bubbles - kissing circles! Rene Descartes found the rule for such patterns, and it lay in a quadratic equation! If four circles have curvature a, b, c and x - and hence radii $$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$$ and $$\frac{1}{x}$$ - then $(a+b+c+x)^2$$= 2\cdot (a^2 + b^2 + c^2 + x^2)$ Solving for x gives two solutions for the curvature of the fourth circle: $x = a + b + c$$\pm$$2\cdot\sqrt{(a\cdot b + a\cdot c + b\cdot c)}$ Welcome to the Kissing Circles Jigsaw!Try the JIGSAW and REVERSE JIGSAW(?) buttons