## Starter of the Day

By how much would the area of this triangle increase if its base was enlarged to 8cm?

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• Transum,
•
• At the time of writing this is a new starter and not many people have had a chance to score it. Though it is going to be perhaps out of reach of the younger pupils, those with a knowledge of Pythagoras' Theorem should find it quite easy. The height of the triangle works out to be a whole number in both the before and after situations and from that, the surprising result can be found. A complete explanation can be found in the answers below. Please feedback any thoughts on this.
• Dave, Australia
•
• It is unclear whether the other lengths of the triangle adjust along with the base or if they remain constant. If they (the 5cm sides) remain constant the area will be unchanged - however if they do not the area will be increasing...
• Joe Young, Chirstchurch
• Monday, November 6, 2017
• This doesn't help, it doesn't really tell us what to do. Also it gives no place to answer and find out if we have even got it right. My students are unsure what to do.

[Transum: Thanks for your feedback Joe. This activity is designed as a lesson Starter to be projected in front of a class at the beginning of a lesson. The teacher can then lead a discussion with pupils about what the solution might be. The answer appears lower down this page for teachers who are signed in.]
• Elliot Oliver, Temuka
• Monday, November 6, 2017
• Rephrasing what Dave and Joe said, this starter isn't the best. It is not very clear about if the hypotenuses of the two triangles stay the same, as well as if the height changes. To clarify that would be great.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
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This starter has scored a mean of 3.2 out of 5 based on 107 votes.

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## Extension

Can you find another pair of triangles that, like those above, surprisingly have the same areas?

What is the length of the base which gives the triangle mentioned above the maximum area?

Christmas Present Ideas

It is often very difficult choosing Christmas presents for family and friends but so here are some seasonal, mathematics-related gifts chosen and recommended by Transum Mathematics.

## Equate board game

Here's a great board game that will give any family with school-aged kids hours of worthwhile fun. Christmas is a time for board games but this one will still be useful at any time of year. Games can be adapted to suit many levels of Mathematical ability.

For Maths tutors working with just one or small groups of pupils this game has proved to be an excellent activity for a tutorial. Deciding on the best moves can spark pertinent discussions about mathematical concepts.

Equate looks a bit like Scrabble--for aspiring mathematicians, that is. Designed by a real mathematician, it works like this: You put down tiles on a board and make points by correctly completing simple equations. Your nine tiles include both numbers and mathematical symbols; you can add on to previous plays both vertically and horizontally. more...

## How Not To Be Wrong

The maths we learn in school can seem like an abstract set of rules, laid down by the ancients and not to be questioned. In fact, Jordan Ellenberg shows us, maths touches on everything we do, and a little mathematical knowledge reveals the hidden structures that lie beneath the world's messy and chaotic surface. In How Not to be Wrong, Ellenberg explores the mathematician's method of analyzing life, from the everyday to the cosmic, showing us which numbers to defend, which ones to ignore, and when to change the equation entirely. Along the way, he explains calculus in a single page, describes Gödel's theorem using only one-syllable words, and reveals how early you actually need to get to the airport.

What more could the inquisitive adult want for Christmas? This book makes a cosy, interesting read in front of the fire on those cold winter evenings. more...

## Graphic Display Calculator

This handheld device and companion software are designed to generate opportunities for classroom exploration and to promote greater understanding of core concepts in the mathematics and science classroom. TI-Nspire technology has been developed through sound classroom research which shows that "linked multiple representation are crucial in development of conceptual understanding and it is feasible only through use of a technology such as TI-Nspire, which provides simultaneous, dynamically linked representations of graphs, equations, data, and verbal explanations, such that a change in one representation is immediately reflected in the others.

For the young people in your life it is a great investment. Bought as a Christmas present but useful for many years to come as the young person turns into an A-level candidate then works their way through university. more...

## iPad Air

The analytics show that more and more people are accessing Transum Mathematics via an iPad as it is so portable and responsive. The iPad has so many other uses in addition to solving Transum's puzzles and challenges and it would make an excellent Christmas gift for anyone.

You have to hold iPad Air to believe it. It’s just 7.5 millimeters thin and weighs just one pound. The stunning Retina display sits inside thinner bezels, so all you see is your content. And an incredible amount of power lies inside the sleek enclosure. So you can do so much more. With so much less. more...

Before giving an iPad as a Christmas gift you could add a link to iPad Maths to the home screen.

## Aristotle's Number Puzzle

It’s a bit of a tradition to give puzzles as Christmas Gifts to nieces and nephews. This puzzle is ideal for the keen puzzle solver who would like a challenge that will continue over the festive period (at least!).

This number puzzle involves nineteen numbers arranged into a hexagon. The goal of the puzzle is to rearrange the numbers so each of the fifteen rows add up to 38. It comes in a wooden style with an antique, aged look.

Keep the Maths in Christmaths with this reasonably priced stocking filler. more...

## The Story Of Maths [DVD]

The films in this ambitious series offer clear, accessible explanations of important mathematical ideas but are also packed with engaging anecdotes, fascinating biographical details, and pivotal episodes in the lives of the great mathematicians. Engaging, enlightening and entertaining, the series gives viewers new and often surprising insights into the central importance of mathematics, establishing this discipline to be one of humanity s greatest cultural achievements. This DVD contains all four programmes from the BBC series.

Marcus du Sautoy's wonderful programmes make a perfect Christmas gift more...

Click the images above to see all the details of these gift ideas and to buy them online.

 Teacher, do your students have access to computers?Do they have iPads or Laptops in Lessons? Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Here a concise URL for a version of this page without the comments.

Transum.org/go/?Start=August5

Here is the URL which will take them to a related student activity.

Transum.org/go/?to=pythag

## Extension Answers

A first thought might be that the graph of area against length of base is symetrical but it is not. A graphic calculator will help you identify equal-area bases and the length of base giving the maximum area:

For Students:

For All:

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