the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari
kia mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Lesson #3 Revised 3/1/19
Beginning a Year
and Beginning as you Mean to Finish Knowing
to describe how to begin your journey with a
to begin to see beyond mathematics
to the real learning
1. Look for a Pattern
2. Find a Rule
3. Test the Rule.
I began all my Y9 classes with this diagram stating what
mathematicians do. This was the first entry in our Math Workbooks.
I said "We are going to be doing this all year in everything we
do" and I would revisit this statement whenever I saw the
connection. I wanted students to notice patterns and to frame a
rule around the pattern using words and algebra. My
target...nth term, a key indicator of being a
multiplicative thinker. More on this later.
Math Journals are to record important ideas and solutions. They
are colourful and something to treasure from school days. I still
have a few choice selections of my own and by students. For me
they are a place to record formative comments. There are also
write on books which are great for practice. See Sigma for example.
Then after that first entry we went into the the world of
mathematics and statistics...and for Year 9 students into the
world of n-ominoes.
Lesson One in Hogan's class
Task 1 --- My students are asked to rule up a 2cm grid and cut
out five perfect 2cm side squares.
This is an excellent first task to assess use of ruler, use of
scissors, following instructions, CCCC and self management!
I would discover many things while working my way around the room
and linking names to students. How astonished they were, each
year, when at the end of this first period I named each student as
they left the class. I also learned a lot about each student and
if each could measure or had "5 thumbs on each hand"and if their
ears were "painted on"!
Task 2 --- Once each had acquired a tidy set of 5 squares
they create shapes by joining the squares.
Joining squares needed agreement and was our first discussion. A
join is a full side to side connection. No halves or corner joins.
No overlaps. The rules were listed in their words. [in 2017 I did
this using multilink
blocks" with Year 7 students.]
Students worked in small groups at tables or desks and soon
started to record shapes by drawing them. I used a space on the
whiteboard for students to record new shapes as they found them.
Reflection and rotation needed some discussion as well. It is
a;ways surprising what students know. Some shapes are flips or
rotations of others and we just need the one shape that represents
Task 3 --- A Problem, "Have we got all the shapes?" How can we
Pretty soon most shapes have been discovered and I challenge them
to be absolutely sure they have found all the shapes. We remove
the reflections and rotations and name the ones that are left with
How can we know that we have all the shapes? This is critical,
creative, logical, collaborative, communicative thinking all
wrapped in one. This is mathematical proof and students do do not
even know what they are really learning. I know, they are learning
about proof. Knowing. This is the world of mathematics and "being
a mathematician". That is a good start to the year! On Day 1 we
are doing some serious maths.
Does anyone have an idea on how we can prove there are only 12
shapes? It appears we can not make more but that is not a
The complete and named set of Pentominoe Shapes.
I suggest we "Take the Dog for a Walk". It is time to
tell. Telling is about guiding in the right direction and not
about doing the walking. Teacher "lust" often takes control here
and I see teachers taking the pencil, or the scissors, or the
squares, or the problem and suddenly the student becomes a passive
observer. Let the student do the walking and... be patient! It
is not your learning teacher!; that belongs to the student.
Task 4 --- Taking the Dog for a Walk.
Let us start with just the one square, which is the trivial or
obvious case, and walk another square around each side and record
all the different shapes that we make. As a class we agree there
is only one shape that can be made and we shall call that shape a
2-ominoe or a "Dominoe". We know about dominoes, and we will
revisit them when we start algebra. I stick with the Greek names
Around the Dominoe we "walk the dog" and discover three new shapes
called the set of Trominoes.
And so a new list on the whiteboard begins.
And by building logically from one square we soon discover the 12
pentominoe shapes and know there are no others. A proof by
"all possibilities" or "exhaustion".
square 1shape Boring but
very useful, the Muppet Song - Mah Ną Mah Ną...fun
2 squares 1
shape A game called Dominoes and worthy of
a lot of maths investigation.
squares 3 shapes Curious
Quadominoe 4 squares ?
(left as ? for you the reader to rediscover)
Pentominoe 5 squares 12
Hogan's best discovery!
Hexominoe 6 squares A
Homework problem for students.
Heptominoe 7 squares ?
Octominoe 8 squares ?
Nonominoe 9 squares ?
Dekominoe 10 squares ?
In our first lesson we have done algebra and introduced the notion
of infinity, Greek names, and a problem solving method. We
have been creative, collaborative, critical and communicative. We
have been doing mathematics. The first lesson not surprisingly
runs out of time but students are surprised how fast the time
went. We enjoyed concentrating, persevering, contributing. We
struggled. We have fun. I talked with each student. Each student
talked to me. The relationship has begun developing.
At no time did we add or multiply, subtract or divide. But we did
maths! And we had fun. My main vine is secure.
Homework Challenge - Find all the hexominoes! I sometimes
found out what parents said when my students explored and
explained what they were doing. Amazingly many students willingly
did this task and returned the next day with the drawn list of all
Hence Lesson #3
After my first lesson and the main vine is secure, they are all
thinking mathematically, we have looked for a pattern, found a
rule and we are testing it. We are developing the CCCC
competencies. The students are being mathematicians and learning.
In the first lesson above list the different forms of
thinking, communication, participation, contribution and self
management skills that have been build on.
Compare with your own first lesson. How was your Main Vine
Record and share what happens when you start a class for the