# Pears Make Squares

## A Maths Starter Of The Day

Let a, b and c be the pear numbers.

a + b is a square number

b + c is a square number

c + a is a square number

Topics: Starter | Arithmetic | Number

• COLA, London Y9 Set 1
•
• 0.5, 0.5, 0.5.
• Matthew, 3D, Craigslea State School
•
• I figured out another solution. All the pears' numbers are 0!
• R G, City Of Leeds
•
• I believe it works for all values to equal a square number divided by 2. eg 2, 2, 2 or 8, 8, 8.
• Transum,
•
• As you can see there are many answers below. It may be interesting to group the answers to help understand the patterns. There are some answers where all three numbers are the same. How would you describe these solutions? The comment above suggests the rule for even square numbers. What about odd square numbers? Must the pears be thinking of integers or could they have mixed numbers in their fruity minds (as proposed in the first comment above)?
• Y4GB, Leigh, England
•
• My Year 4 primary top set found 1 , 120 and 24 to work. Mr B.
• Wendover C.E Junior 6J,
•
• Freddy found 2,2,14 and 0, 16, 9! Connor found 2,34,2!
• 4W, Wendover C E Junior School
•
• Dylan found 20, 5 and 44.
• Sam, Carnoustie
•
• - half a square number, write it twice subtract that from any other square to find the third number eg. 2,2,7 2,2,14 2,2,23, 2,2,34... 8,8,1 8,8,17 8,8,28 8,8,41... 12.5,12.5,3.5
if you want three different numbers
- find two numbers with an odd difference which add up to an odd square: write them down
- find the two squares with that same odd difference
- subtract the numbers you wrote down from the two squares you just found: you should get the same answer twice. write it down.
eg. 1, 24, (with a difference of 23: just like 121 and 144) 120
2, 23 (with a difference of 21: just like 100 and 121) 98
3, 22 (with a difference of 19: just like 81 and 100) 78
4, 21 (with a difference of 17: just like 64 and 81) 60
5,20,44 6,19,30 7,18,18 8,17,8 9,16,0 10,15,-6 11,14,-10 12,13,-12
notice the pattern: first number increases by 1 second number decreases by 1 third number decreases by previous even number
1,48,528 2,47,482 3,46,438 4,45,396 5,44,356 6,43,318 7,42,282 8,41,248 9,40,216 10,39,186 11,38,158 12,37,132 13,36,108 14,35,86 15,34,66 16,33,48 17,32,32 18,31,18 19,30,6 20,29,-4 21,28,-12 22,27,-18 23,36,-22 24,25,-24
the solutions are infinite!
•

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.

Previous Day | This starter is for 8 January | Next Day

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... #ad

## 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 GĂ¶del'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... #ad

## 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... #ad

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 gift for anyone.

The redesigned Retina display is as stunning to look at as it is to touch. It all comes with iOS, the world's most advanced mobile operating system. iPad Pro. Everything you want modern computing to be. more... #ad

## Craig Barton's Tips for Teachers

Teaching is complex. But there are simple ideas we can enact to help our teaching be more effective. This book contains over 400 such ideas." more... #ad

"The ideas come from two sources. First, from the wonderful guests on his Tips for Teachers podcast - education heavyweights such as Dylan Wiliam, Daisy Christodoulou and Tom Sherrington, as well as talented teachers who are not household names but have so much wisdom to share. Then there's what he has learned from working with amazing teachers and students in hundreds of schools around the world.

## 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... #ad

## Christmas Maths

This book provides a wealth of fun activities with a Christmas theme. Each photocopiable worksheet is matched to the Numeracy Strategy and compatible with the Scottish 5-14 Guidelines. This series is designed for busy teachers in the late Autumn term who are desperate for materials that are relevant and interesting and that can be completed with minimun supervision.

All the activities are suitable for use by class teachers, supply teachers, SEN teachers and classroom assistants and cover topics such as 'How many partridges did the true love give all together?' and 'Filling a sleigh with presents by rolling a dice!'. Children will have lots of fun working through the Christmas Maths themes but also gain valuable skills along the way.

A great source of ideas and another reasonably priced stocking filler. more... #ad

## A Compendium Of Mathematical Methods

How many different methods do you know to solve simultaneous equations? To multiply decimals? To find the nth term of a sequence?

A Compendium of Mathematical Methods brings together over one hundred different approaches from classrooms all over the world, giving curious mathematicians the opportunity to explore fascinating methods that they've never before encountered.

If you teach mathematics to any age group in any country, you are guaranteed to learn lots of new things from this delightful book. It will deepen your subject knowledge and enhance your teaching, whatever your existing level of expertise. It will inspire you to explore new approaches with your pupils and provide valuable guidance on explanations and misconceptions. more... #ad

I had been tutoring the wonderful Betsy for five years. When the day came for our last ever session together before the end of her Year 13, I received this beautiful book as a gift of appreciation.

This a very readable book by Ben Orlin. I'm really enjoying the humour in the writing and the drawings are great.

Ben Orlin answers maths' three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that mathematics should belong to everyone. more... #ad

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

## Maths T-Shirts

Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.

Educational Technology on Amazon

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

Transum.org/go/?Start=March20

Here is the URL which will take them to another activity involving square numbers.

Transum.org/go/?to=Square

## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

For Students:

For All: