Grade 3: Geometry
Three ideas to kick-start enquiries in Grade 3 concerning geometry.
#1  Dotted paper polygons.  Students explore how many unique polygons can be made with 1 dot in the middle.  Further constraints could be added, e.g. how many quadrilaterals can be constructed?

What generalizations could exploration reveal?  Area formulae?

#2 Perimeter 12 triangle problem

When solving this problem many students simply partition 12 into three side lengths, e.g. 3 + 4 + 5.  However, 2 + 3 + 7, whilst summing to 12, is an impossible triangle.  Why?  How would you explain this mathematically?  What can be generalized about the side lengths of any triangle?  If we label the sides of a triangle ABC, then A+B must equal more than the length of C.  Using compasses to construct the triangles helps students understand this relationship, particularly because their drawings will be accurate and arcs drawn with the compasses will only intersect with triangles that are possible.  G3 students will generally use everyday language to discuss these side length relationships, however, as a teacher I like the idea of translating this spoken English into mathematical language (see above).

#3 12 cubes - an inquiry into cuboids (rectangular prisms)

Students are given 12 multilink cubes and invited to make all the different cuboids they can (students from North America are more used to calling these shapes rectangular prisms).  For each cuboid the students make an isometric drawing (common in architecture and design engineering) using dotted isometric paper.  The ITP above, isometric grid useful for modelling this. http://www.taw.org.uk/lic/itp/itps/isoGrid_16.swf

There are 4 uniques cuboids that can be constructed using 12 cubes.  So, whilst all have the same volume, what about their nets?  Encourage students to explore what the nets for these cuboids will look like.  If we were to make these as boxes, which would be the most expensive to make/the cheapest?  And how should we record our findings mathematically so that any generalizations will be revealed?

Grade 4: Happy Numbers Investigation
Think of a number. Any number.  Square each digit and sum the squares.  Follow the same process with each answer and if the resulting number chain resolves to 1, then the number you started with is Happy!  E.g: 31 ..... (3 x 3) + (1 x 1) = 9 + 1 = 10....... (1 x 1) + (0 x 0) = 1.  31 is a happy number!
So, we will need to be able to square all digits up to 9.  Here is a representation of the first seven square numbers:

1,4,9,16,25,36,49,......... so, what are the 8th, 9th and 10th square numbers?
Task #1:  When will your next 5 Happy Number birthdays be?
Task #2:  What are the first two consecutive Happy Numbers?

Recording is important here because any chain leading to 1 will contain other happy numbers.  Chains that do not resolve to 1 will contain other unhappy numbers!

Grade 4 inquiry into number patterns/sequences: The Collatz Conjecture

Here is a representation of numbers 1 to 10 and their iterations following the rule (even n)/2, 

(odd n)x 3 +1

Solving word problems in Grade 2.
The following example is from a research project, 'How old is the captain?', by Christoph Selter, Vol.5 No 1, October 1984

What will the Grade 2 students make of this first question?  This totally redundant question illustrates a common problem with word problems: children often just look at the numbers and calculate.  A further issue is that publishers of maths text books presume the number ranges that are appropriate for a particular grade which can lead to assumptions being made by students as to what operation(s) should be used.  A grade 5 once said to me: 'word problems are easy! If there are single digits you always multiply; if there's a big number and a little number you always do division; if there are two big numbers it's nearly always an adding; If there are two big numbers that are close together in size it's always a subtraction; if there are two 2-digit numbers it's always multiplying.....!
Check out the textbooks!  Is this student correct?

Parents' session, December 6th

Grade 3 example:

The cube model shows the dimensions of the box, aids visualizing and points the way to the generalization/formula.  Earlier learning about arrays and area helped students make connections and have their own eureka moments when they were able to construct a rule and explain it.  

Number Board Games in EY2:
We have been experimenting with board games in EY2 as a way of including number more intentionally through the children's play.

The Space 10 game initially only interested one student, however, many more are now joining in.  Number cards from 0 to 10 are shuffled and students take turns to choose a number card.  Whatever card is selected the student has to choose its complement within 10.  So, if their first card chosen was a 1, they would move their counter to number 9 on the rocket's journey.  Whilst some students had knowledge of addition complements within 10 (bonds), others needed scaffolding (using a visual 20 frame):

Another game, Stars, requires children to choose a number card (1 - 6) and identify a square containing the same number of stars.  The winner of the game is the player who has three of their counters in a line.  

This game helps develop:

• subitizing (knowing a quantity without counting)
• matching numerals to quantities
• Cardinal number principle (last number counted is the quantity).

Maths games are great for practising number skills because of the child's motivation to play the game. In addition, players need to be strategic in order to win the game.  If there is no strategy involved then, technically, it's not a game.  The Stars game was the stimulus for the next 3-in-a-row game:

Here, two 1-6 dice are thrown and their numbers summed.  Players need to score three in a row, horizontally, vertically or diagonally.  With this game interpreting the dice requires subitizing and dice totals will either be known facts or mental addition.  This is a good vehicle for teaching counting-on as an addition strategy (counting on from one of the known dice numbers).  Again, as with the star game, children are learning to match appropriate numerals to the dice sums.  In reality, children will use a range of strategies including counting one-to-one, for some totals.  

With home-made games such as this, it is important to consider the theoretical probabilities of the various dice throws.  A simple sample set will reveal these probabilities:

The game board above was made with reference to these theoretical probabilities.

Shape and Space in Grade 1: Pentominoes Inquiry
Pentominoes are rectilinear shapes made from 5 connecting squares.  The squares must connect by matching side-to-side, and cannot connect just by vertexes.  The first step in the inquiry is to find all possible pentominoes (there are 12 altogether).  The first 1 or 2 could be modeled using the Area ITP: http://www.taw.org.uk/lic/itp/itps/area_2_2.swf

The 12 unique pentominoes (G1 student's recording)

Finding all possibilities type problems help learners develop persistence.  How do you know when you've found all possible pentominoes?  It is common for children to repeat the same shape in a different orientation.  This is an good teaching opportunity - a time to discuss the concept of congruence.  Sorting and classifying according to the shapes the pentominoes make is another good next step.  The set of pentominoes includes rectangle, hexagon, octagon, decagon, dodecagon.  The prefixes of these polygon words all originate in Greek....the suffix 'agon' denotes 2D ('hedron' denoting 3D).  Other inquiry routes include investigating:

• symmetry - which are symmetric which are asymmetric?
• Nets - some pentominoes are nets for open-topped cubes, some are not

Which of these pentominoes is a net for an open-topped cube?  Can you visualize which it is?

• Tessellation - Do all the pentominoes tessellate? - why/why not?

The pentominoes all link together to make rectangles of various sizes, 12 x 5, 10 x 6, 3 x 20 etc:

Another puzzle that requires trial & improvement and persistence!