Twenty Second of February Twenty Two - A lovely Palindrome

 This quote by Ian Stewart, author of many great mathematics Books sums Up what I feel is often missing in our classrooms.

Traditionally we have focused on skills and rules without seeing the patterns and connections that are there if we look, and our students laterally and beyond the numbers.

How many of our students have been encouraged to see the patterns of opposites?  Addition is the OPPOSITE to Subtraction and vice versa.  Multiplication and Division are Opposites (as I was explaining to my granddaughter last week to help her with division)

Using 22 2 22 (the twenty second of February Twenty two) as a starter would nt it be a great investigation to find other Numerical Palindromes? You know words and numbers that read the same forwards and backwards!

I would put 22 2 22 and ask my students what do they notice about it Probably using:

THINK  (on your own no discussion)

PAIR     (discus with a partner (or small group) your findings, observations)

SHARE (share with a larger group or the class) Perhaps list the findings on the board

It would surprise me if the idea of it reading the same forward and backwards was not on the list.

Using this idea, ask the students again(Think, Pair, Share) to come up with other Palindromic Numbers.

(11, 22, 33   121, 232 .....)

Pose the question:  I wonder if a Non- Palindromic Number can be converted into a Palindrome?

Again use Think Pair Share and if no one comes up with a response suggest something along the lines of, What would happen if we reversed the numbers and added (or subtracted)?  e.g. 23  -- 32  thus 23 + 32 = 55 wow a Palindrome!!!

Now we have the basis for a longer investigation: "Can all the numbers from 1-100 be converted into Palindromes?  and secondly How Many Steps for each one? 23 + 32 = 55 is one step 58 + 85 = 143;    143 + 341 = 484  thus 2 steps.    (Be aware there may be a couple which might not convert.)

A wall chart or student chart along the lines of this could be used to record the results of the investigation.

Now what will be the next set of Patterns to be investigated? Fibonacci Numbers? Kaprekars Constant? (these are other Blog Posts.

Patterns on a Calendar? ......

Check out this website NRich palindromes

Time for some Mathematics Inspiration and not just Calculation!!

 For many years as a Mathematics Education, Consultant/Adviser I encouraged teachers to look beyond the calculations that many maths' programs seem to focus on.

"Mathematics is the language in which God has written the universe."

 

In fact Galileo suggested; 

As I look around me I don't tend to see numbers everywhere(unless put there by people!). What I see are flowers and trees and natural things. 

So why is it in maths class we tend to start with the "abstract numbers" rather than the real thing as Galileo would suggest?

In 1987 I had the opportunity to visit California investigating maths approaches. I was fortunate to visit a class in Santa Barbara where for the start of the lesson all the groups of students were given a pineapple!  I did not notice any WALTs or AOs, but I heard the teacher ask the students to explore the pineapple and to see what sort of patterns they saw.  The students were then engaged in debate and investigation for the rest of the lesson.

 Of course we could use Pine Cones and do the same but we cant eat them at the end of the lesson

I have just re watched a TED Talk called : The Magic of Fibonacci Numbers admittedly it goes on into the number patterns that Leonardo of Pisa (Fibonacci) found many years ago but depending on out age group will depend on how far you take the abstract numbers.

A couple of years back I was asked to present at the PMA Maths Seminar Day and I started the session with Water Lillies, Bananas, and pine cones on the desks for the teachers to explore.

I am sure for some teachers they had never started with the "Real Thing"

Why not give it a go-dont start with the Fibonacci Numbers - 1 1 2 3 5 8... but start with various natural objects that the students should be able to come up with similar numbers 

    (the number or clockwise and anticlockwise spirals on a Pine Cone, 

    How many seed heads in one of the spirals? 

    How many petals on a Tulip? Daffodil? 

    How many parts to a Banana?

Enjoy watching the students' curiosity and interest in maths soar.

Do an Internet Search for more ideas and Approaches for the Fibonacci Numbers

Kaprekar's Constant

 This number exploration, uses Largest, Smallest Four Digit numbers and practices subtraction. all an endeavour to find the Constant that many Four Digit Numbers end up at.

 

As the Card suggests all numbers will end up at the Constant within 1 to 7 steps.  I would challenge the students to see if there is one starting number that does not conform.

In the Classroom I would consider having 8 wall charts Labeled 1 through 7 the 8th Others!!  As Students check their four digit numbers and count the steps the record their discoveries on the appropriate chart.

Other students may wish to find out more about Kaprekar, who he was, what mathematical truths he found.

Prerequisites: Some understanding of four digit numbers and four digit substraction(or allow use of calculator) and possibly Flow Diagrams

 

Digital Roots, Patterns and Geometry

 I have long been an advocate of teaching mathematics from a known situation as well as in a context that helps students understand that mathematics is all around us.

We often do not connect Number with Geometry and Vice Versa but they are linked, intertwined, but we need to peel back a layer and see the connections.  I like the adage, "The Maths Behind The Door!" I encourage all teachers to help students find the connections.

Another area we do well, initially(Junior School) is with patterns.  We use shapes, counters and ask children to continue given patterns, create their own etc. This is a great foundation for exploring numbers later, but why do we forget patterns when exploring Number?

Digital Roots of numbers is a good way for students to practise their simple computation skills as well as look for patterns and then use them for Geometry explorations "Patterns in Circles" "Spirolaterals"

I hope these activities help you springboard the connections between Numbers, Patterns and Geometry.(There are more to come)

Indy Anna Jones and the Temple of Gloom

This activity seems logical to post as TV3 (NZ) is replaying the Indiana Jones Movies.

The idea was presented some years back at a conference I was attending, I have tweaked it for using with Teachers and Students I have worked with.

I hope you enjoy using this with your students.

Modulo Arithmetic (Clock Arithmetic) with an Art Component

 Modulo (clock Arithmetic) are Finite Number Systems, and for older students it is worth exploring a Finite System so that they may get a better understanding of an Infinite System like our Base 10(Decimal System)

Putting an Art Component with it, may encourage some reluctant mathematicians to become more interested and involved.

In the attached pics, I mention "Donald in Mathmagicland" (A Disney film 1959) I can remember showing often using 16mm film when I started teaching in '65. It gave me and my students a better understanding that maths is not just numbers but is intertwined in everything we do.

Donald in Mathmagicland

Other websites about Modulo Arithmetic for your interest and background information 

https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/what-is-modular-arithmetic

https://www.mathsisfun.com/numbers/modulo.html 

https://nrich.maths.org/4350   (this great Maths site, suggests students 14-18, I have worked with students 10+ on these activities. Often it is how the activity is approached)

Enjoy Exploring Modulo Arithmetic

An Interesting Question to follow on from a discussion about TIME

 This activity has been adapted from a Puzzle activity prepared by AIMS (Activities in Maths and Science) Fresno USA.

I can assure there are probably 3 solutions but students will come up with one or perhaps two.

Try it yourself first to see what you can come up with?