Pears Make Squares

A Maths Starter Of The Day

Pears Make Squares


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.
Click here to enter your comments.

If you don't have the time to provide feedback we'd really appreciate it if you could give this page a score! We are constantly improving and adding to these starters so it would be really helpful to know which ones are most useful. Simply click on a button below:

Excellent, I would like to see more like this
Good, achieved the results I required
Didn't really capture the interest of the students
Not for me! I wouldn't use this type of activity.

This starter has scored a mean of 3.1 out of 5 based on 409 votes.

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



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 search box and some items chosen and recommended by Transum Mathematics to get you started.

Have you read Craig's book yet?

Craig Barton must surely be the voice of Mathematics teachers in the UK. His wonderful podcasts interviewing the industry experts have culminated in this wonderful book. As Craig says: "I genuinely believe I have never taught mathematics better, and my students have never learned more. I just wish I had known all of this twelve years ago..." more...

"How I wish I'd taught Maths" is an extraordinary and important book. Part guide to research, part memoir, part survival handbook, it’s a wonderfully accessible guide to the latest research on teaching mathematics, presented in a disarmingly honest and readable way. I know of no other book that presents as much usable research evidence on the dos and don’ts of mathematics teaching in such a clear and practical way. No matter how long you have been doing it, if you teach mathematics—from primary school to university—this book is for you." Dylan Wiliam, Emeritus Professor of Educational Assessment, UCL.

The Craig Barton Book

Casio Classwiz Calculator

There is currently a lot of talk about this new calculator being the best in its price range for use in the Maths classroom. The new ClassWiz features a high-resolution display making it easier to view numerical formulas and symbols but it isn't a graphical calculator as such (it has the capacity to draw graphs on your smart phone or tablet, via a scannable QR code and an app).

As well as basic spreadsheet mode and an equation solving feature you also get the ability to solve quadratic, cubic or quartic polynomial inequalities and the answer is given just as it should be written down, using the correct inequality symbols!

This calculator has a high-performance processor and twice the memory of previous models ensuring speedy operation and superior computational power.more...

Online Maths Shop

Laptops In Lessons

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).

Laptops In Lessons

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

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

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

Student Activity



©1997-2019 WWW.TRANSUM.ORG