©2021 Compass Learning Technologies → Live Mathematics on the Web → GXWeb Showcase → GXWeb Jigsaws and Quizzes

## GXWeb Jigsaws and Quizzes

Symbolic computations on these pages useNerdamer Symbolic JavaScriptto complement the in-built CAS of GXWeb

Saltire Software, home of Geometry Expressions and GXWeb

Algebra Tiles Jigsaw YouTube demonstration

Fractured Fractions Jigsaw YouTube demonstration

Kissing Circles Jigsaw YouTube demonstration

Fancy some mathematical fun?

UsejigsawsandGXWebto 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 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 tilescan 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 morewith Algebra TilesExplore 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 morewith continued fractionsExplore 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, 6and15.

Each number refers to the

curvature- orbend- of the circle it describes - which is the reciprocal of theradius. Anegative benddescribes 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

\[(a+b+c+x)^2 \]\[= 2\cdot (a^2 + b^2 + c^2 + x^2)\]a, b, candx- and hence radii \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) and \(\frac{1}{x}\) - thenSolving for

\[x = a + b + c \]\[\pm\]\[2\cdot\sqrt{(a\cdot b + a\cdot c + b\cdot c)}\]xgives two solutions for the curvature of the fourth circle:

Welcome to the Kissing Circles Jigsaw!

Try the JIGSAW and REVERSE JIGSAW(?) buttons

Play morewith Kissing Circles and mathematical bubbles!Explore the

Kissing Circles Collection

Construct your own Model with GXWeb

Behind the Scenes

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