Friday, 21 November 2014

Magic Squares

Magic Squares are so named because each of the rows, columns and diagonals add to the same sum. BUT does it stop there.

The earliest known magic square is Chinese, recorded around 2800 B.C. Fuh-Hi described the "Loh-Shu", or "scroll of the river Loh". It is a typical 3x3 magic square except that the numbers were represented by patterns not numerals

Magic Squares seem to crop up in some unfamiliar places. 
     Perhaps the most famous of them all is Durer’s magic square in the back of a picture he drew in 1514. 

More recently there is a Magic Square with a difference on the doors of Familia Sagrada, Gaudi’s Cathedral in Barcelona, which is still under construction.  (photo by Len)

Can you see how this Magic Square is different to most others?

Primary Students are often asked to use the digits 1-9 and place them on a 3x3 grid so that each row, column and diagonal add to the same total.

Beside trial and error, or thinking laterally-“put the middle number in the middle square and then use pairs of numbers like 1 and 9 2 and 8, there is a ‘cheat’s way’ especially for teachers!
1. Draw a 3x3 grid

 2. Now put a square next to the middle square on each side
3. Take any consecutive set of nine numbers and put numbers in each square working diagonally upwards from the middle square on the left.
4. Now take the numbers in the outside squares and move them across to the empty square opposite
5. Each row, column and diagonal adds to 30!
6. Can you find a way of easily creating a 4x4 Magic Square like Durer's?

Durer’s Magic Square has the 'magic' sum of 34.  This total is easily found in the two diagonals.  Set you students the task of seeing how many sets of 4 numbers add to 34 throughout the whole Magic Square.  (I found in excess of 25!)
And here is one to get you started
Finally, senior students may wish to explore this page about Magic Squares:

Wednesday, 29 October 2014

The Mathematics of Spirals

Much our mathematics teaching does not have a direct connection with the world that the mathematics came from. This activity tries to give a context for students to see spirals and 
hopefully a desire to some of the mathematics that is naturally occurring around them 

Chinese and Lattice Multiplication

I was sent this You Tube clip recently explaining how the Chinese work out their multiplication problems, all through drawing and counting the number of line intersections. (There are a number of other similar clips)  For some students they will enjoy exploring alternate ways of working out multiplication.
I regularly used and encouraged the "Lattice Method" method. This shows partial products but does need students to know their Basic Facts!

In response to the Chinese Drawing method of Multiplication.   I have regularly encouraged a Lattice Method of multiplication that also shows the Place Value and can also be modelled by Place Value Blocks.
The difference here is that the students needs to know their ‘Basic Multiplication Facts’ but there are no rules such as “multiply, put down the ones and carry the tens  OR just add a ‘Nought’ or Zero”
It is a schematic way of showing ‘Partial Products’

24 x 56 =

put 24 across the top two columns
Put the 56 on the right hand side of the two rows
Now multiply each pair of numbers and place the answer in the box, tens above the diagonal ones below
Add down the diagonals, carrying the an tens into the next diagonal to the left and above
The answer is 1344
The Lattice can be increased for all multiplication like 24 x 569, or 274 x 4590

Saturday, 25 October 2014

Do We Celebrate Math(s) Achievement?

Here in New Zealand many schools have a large notice board(often electronic) at the school entrance or a prominent place for the passing public to see.  The notice board regularly has postings of student achievement, which is great as in this negative focusing world it is great to see positive achievement.

I do have a concern that the majority of Posts of student achievement are sports and cultural orientated, this in itself is great but in my experience most parents see schools as places for the learning of subjects such as Maths, Reading, Language, Science etc.

Why then do we not see postings of student achievement in these subjects?

Some years back I was discussing this lack celebration of Maths with Principals and they agreed that they did not mention student achievement in the "Core Subjects" as much as they did with Sporting and Cultural Achievement.

Is it any wonder that a subject like Maths, has such a low interest level with many students when there is little or no celebration of their achievement?

  • What Mathematics Achievement can we celebrate on a regular basis?   
  • A student's improvement from one test to another.
  • Students' creative ability in using geometry to create patterns
  • The creating of student problems for others to solve
  • The students who solve "Problem of The Week " with a great explanation
  • A student who helps others with maths learning

I look forward to seeing (and hearing about) regular posting of Maths Achievements and successes, so that all of the present school generation see maths as a fun and exciting subject.

The Perfume: OR Are you teaching a curriculum or children?

This article, not sure if it is true or not (SNOPES suggest not), came my way some years ago as an email to pass on. I hope that it you are a teacher then it asks you to reflect on how you are teaching? 
  • Is the curriculum more important than the children you are teaching?
  • Do ypu take time to find out about the kids in your class?
As she stood in front of her 5th grade class on the very first day of school, she told the children an untruth . Like most teachers, she looked at her students and said that she loved them all the same.

However, that was impossible, because there in the front row, slumped in his seat, was a little boy named Teddy Stoddard.

Mrs Thompson had watched Teddy the year before and noticed that he did not play well with the other children, that his clothes were messy and that he constantly needed a bath . In addition, Teddy could be unpleasant.

It got to the point where Mrs. Thompson would actually take delight in marking his papers with a broad red pen, making bold X's and then putting a big "F" at the top of his papers.

At the school where Mrs. Thompson taught, she was required to review each child's past records and she put Teddy's off until last. However, when she reviewed his file, she was in for a surprise.

Teddy's first grade teacher wrote, "Teddy is a bright child with a ready laugh. He does his work neatly and has good manners... he is a joy to be around."

His second grade teacher wrote, "Teddy is an excellent student, well liked by his classmates, but he is troubled because his mother has a terminal illness and life at home must be a struggle."

His third grade teacher wrote, " His mother's death has been hard on him He tries to do his best, but his father doesn't show much interest and his home life will soon affect him if some steps aren't taken."

Teddy's fourth grade teacher wrote, "Teddy is withdrawn and doesn't show much interest in school. He doesn't have many friends and he sometimes sleeps in class."

By now, Mrs. Thompson realized the problem and she was ashamed of herself. She felt even worse when her students brought her Christmas presents, wrapped in
beautiful ribbons and bright paper, except for Teddy's .

His present was clumsily wrapped in the heavy, brown paper that he got from a grocery bag.

Mrs. Thompson took pains to open it in the middle of the other presents. Some of the children started to laugh when she found a rhinestone bracelet with some of the stones missing , and a bottle that was one-quarter full of perfume . But she stifled the children's laughter when she exclaimed how pretty the bracelet was, putting it on, and dabbing some of the perfume on her wrist . Teddy Stoddard stayed after school that day just long enough to say, “Mrs Thompson, today you smelled just like my Mom used to."

After the children left, she cried for at least an hour. On that very day, she quit teaching reading, writing and arithmetic. Instead, she began to teach children.

Mrs. Thompson paid particular attention to Teddy. As she worked with him, his mind seemed to come alive. The more she encouraged him, the faster he responded.
By the end of the year, Teddy had become one of the smartest children in the class and, despite her lie that she would love all the children the same, Teddy became one of her "teacher's pets."

A year later, she found a note under her door, from Teddy, telling her that she was still the best teacher he ever had in his whole life.

Six years went by before she got another note from Teddy. He then wrote that he had finished high school, third in his class, and she was still the best teacher he ever had in his whole life.

Four years after that, she got another letter, saying that while things had been tough at times, he'd stayed in school, had stuck with it, and would soon graduate
from college with the highest of honors . He assured Mrs. Thompson that she was still the best and favorite teacher he had ever had in his whole life.

Then four more years passed and yet another letter came. This time he explained that after he got his bachelor's degree, he decided to go a little further. The letter explained that she was still the best and favorite teacher he ever had But now his name was a little longer....The letter was signed, Theodore F. Stoddard , MD.

The story does not end there. You see, there was yet another letter that spring. Teddy said he had met this girl and was going to be married. He explained that his father had died a couple of years ago and he was wondering if Mrs. Thompson might agree to sit at the wedding in the place that was usually reserved for the mother of the groom.

Of course, Mrs. Thompson did. And guess what? She wore that bracelet, the one with several rhinestones missing. Moreover, she made sure she was wearing the perfume that Teddy remembered his mother wearing on their last Christmas together.

They hugged each other, and Dr. Stoddard whispered in Mrs. Thompson's ear, "Thank you Mrs. Thompson for believing in me Thank you so much for making me feel important and showing me that I could make a difference."

Mrs. Thompson, with tears in her eyes, whispered back. She said, "Teddy, you have it all wrong. You were the one who taught me that I could make a difference. I didn't know how to teach until I met you."

Warm someone's heart today . . .

I love this story so very much, I cry every time I read it. Just try to make a difference in someone's life today? tomorrow? just "do it".

Random acts of kindness, I think they call it. "Believe in Angels, then return the favour"

Tuesday, 14 October 2014

Curves From Lines

This was prepared for the Family Maths Newsletter 2007. Copies of past issues are available for download from  The Family Maths Trust is an Offspring from Family Math, Lawrence Hall of Science Berkeley California.  Organisations that encourage families to participate in fun math activities at home.


I was sent the following YouTube link which shows a series of 'balls' moving around a circle.  The caption with this link was: "Eight Balls Moving in Straight Lines's all in the timing.  Eight Balls Moving in Straight Lines Who would think of this?"

A Cycloid created by tracing a point on a circle as it travels along a line, but they become more interesting when the point is traced as the wheel of circle is inside or outside of a circle.

The mathematics of cycloids is too difficult for most primary students, involves Trigonometry ratios but these students could have fun creating their own shapes, using a variation of the cycloid with the use of a drawing compass.  
Many moons ago I created a number of these off shoots and had them prepared on Overhead Projector Slides.
The ones I created consisted of drawing circles from equidistant points on a circle to a point inside or outside the circle and similarly with a point on a line with the points on the circle.
A great practice for measuring and using the compass and the students end up with interesting designs.
This from:

Monday, 13 October 2014

Halloween Problems

Some more problems from Charlotte Wilkinson
Use these as starters and frameworks and then get your students to create and share their own Halloween Problems.  Remind them that they have to have worked out their own solutions before sharing!

Embrace Educational Change

This article from Charlotte Wilkinson arrived by email today and I thought it was well worth sharing.  Her quotes remind me of a similar one, (I have lost) but referred to the likelihood of handwriting skills deteriorating with the advent of the Ball Point Pen as it replaced the dip or fountain pen.

Thursday, 9 October 2014

CSI – Can you solve the case of the discovered bone?

Detectives have found a bone buried in a hole on the banks of the Waikato River. Forensic anthropologists have established that the bone is a radius bone, and that it is almost certainly human. The bone measures 28cm long. From some scraps of material found with the bone, detectives can conclude that the mystery person is almost certainly female. Another bone, a 44cm tibia, possibly belonging to the same person, has been found
  • The detectives now need your help. 
  • They need to find out the height of the mystery person along with some other measurements 
  • Circumference – head (around forehead); 
  • shoulders; 
  • hips; & 
  • knees (both knees together). 
  • From these, the detectives need you to create a ‘skeleton’ model.
Can you solve it?

Teachers the students will need to collect data and find averages etc so that they can use the given information to work out the 'size' of the mystery person.   One way to may the 'skeleton' could be to have a length of adding machine tape(or card) for the backbone and then attached strips of card for the other details.  

I would love to hear the results of the investigations

Adapted from ‘The Case of the Mystery Bone’ by Doug Clarke (possibly out of print)

Tuesday, 7 October 2014

Another Quick and Easy Warm Up

As a visiting consultant one of the the things I always had in my pocket was a ten sided die (0-9), because invariably a teacher or class would want me to 'play' a maths game.  This one is so easy to set up but has the element of chance(probability) and place value.

Have the students to draw four boxes on a piece of paper.

Explain to them that you are going to roll the die 4 times and after each roll call out the number.  Their task is to (make the Largest 4 digit number, Smallest 4 digit number, Number closest to 5000 etc) The students are to decide which box to write the number in.  Once written it can not be moved.


  • Have 4 boxes with a discard box and roll the die 5 times, students are allowed to discard at any time one of the numbers
  • Introduce a decimal point between one of the boxes, this will show you the teacher, that many students just do not understand very well our place value system
  • For younger children just have two or three boxes
  • For younger children have them record their numbers on cards and then use their cards as a lesson for 'Ordering" the numbers and statistics like Range, Mode.  (I had students display the cards in order on the classroom carpet with them sitting around so that they could all be involved)  Children love 'owning' their number

Thursday, 2 October 2014

Len's Warm Up/Warm Down Maths Activity

This activity has Addition, Place Value and Probability all rolled in to one Great Warm Up.

Prepare a grid as below(this size is best for Upper Primary and Junior Secondary)  Younger children could just have 4 rows and then a total row. Distribute  OR have students prepare their own

Object: to be the student who has the highest total after all the dice rolls.

  1. Teacher or student rolls a die (preferably a 0-9) the students use this number to fill in the square in ROW 1
  2. The Die is rolled a second time and the students can now put the rolled number in either of the Ten's or the One's Place.
  3. On the 3rd roll the number is placed in the vacant square in row 2.
  4. Now it is Row 3 to be filled. After each roll of the die, the students decided whether to put the number in the Hundreds, or Tens or Ones.  (No storing of numbers OR changing once the number has been recorded.)
  5. Continue in this manner for row 4 Th, H, T, O and then Rows 5-7
  6. Hardest part is now to get the students to add each column to find their Grand Total!
  7. The student who has the Largest Total wins, or scores points for their ‘school house’ etc.

Teachers: Worried about how to check the accuracy of the students answers?
  • Have all students find the digital root of the TOTAL Row; This is when all the digits are added to get a single digit.  e.g. 14 666 789  becomes 1+4+6+6+6+7+8+9 OR 47 and then 4+7 is 11 and 1+1 is 2.
  • Students with correct total should have the same digital root.
  • Alternatively, ask one child to call out their answer and see how many have the same (if half the class or more)  ‘assume’ this is the correct one!