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!

Outdoor Learning with KG.  Introduced a skittle bowling game with 12 skittles distributed into 3 hoops (4 in each).  The skittles the furthest away (blue) score the highest (5).  All other skittles score 1.  Children take turns to bowl and scores are recorded using tally marks on a chalkboard.

It was a great opportunity to assess the children's counting skills and known addition facts.  How could the scoring be changed to develop their counting skills further?  Skittles could be worth 2, 3 and 5 to create a wider variety of scoring options.

One player, was able to remember his previous score and mentally add the new score to it, whilst others needed to count from the beginning each time.  The game provided motivation to engage with the numbers of the scoring system.  Once familiar with the game, including taking turns recording the scores and totaling them, children could be engaged with problems in the classroom that relate to the real game outdoors: (from NNS Mathematical Challenges booklet. Crown Copyright 2000)

Fractions and Music.
All students from Grade 2 onwards learn a musical instrument.  By looking at musical note values from a mathematical perspective children are gaining greater depth in their understanding, both of the notes in a music context, and the nature of fractions themselves:

Of course, the order of the notes is also of huge importance in music, so how many more possibilities will there be if we change the order of the examples above?

...And what if we include 1/16 notes?:

As a maths problem/inquiry, students are required to find all possibilities of the note combinations.  Consequently, systematic recording is important and this problem is an excellent vehicle for modelling this with students.  Investigating the possibilities is important from a musical perspestive and students will be able to build more variety into their compositions, especially when the order of notes is a consideration:

The following ITP, Fraction, is a good way of exploring the relationships between whole notes, half notes, quarter and eighth notes:

Dominoes as fractions

Starting to think about FDPRP (much easier to say than fractions,decimals, percentages, ratio and proportion!) in grade 5 today... The following represent some tuning-in activities:

I was wondering about a normal set of dominoes.  This game is familiar to children, however, what about the actual dominoes themselves and what they can represent in terms of fractions?:

Using a range of visuals, models, e.g: long strip of till roll/border paper; counting stick, fraction ITP , large display dominoes, discuss how a domino can be seen as a common fraction with a numerator and a denominator.

Draw a number line 360mm long and order the dominoes, recording what fraction they represent, and place them accurately on the number line.  What decisions do we need to make to show where these fractions should go?    In addition, the fraction can be shown in its decimal form, as a percentage and as an equivalent fraction.  It is important to include in the range of representations, the actual measurements needed, e.g. 3/4 = 270/360.  The opportunity here is to make connections within FDPRP: 3/4...0.75....75%.....(provide calculators for fraction/decimal equivalents, however, some may be known facts or easily derived without the calculator).

This idea was provoked by EY2's visit to the Zytturm, Zug.  EY2 are investigating structures and the visit provided an excellent stimulus for the students' own structure building.

For EY2 students, 5m (five 10 x 10 squares) can be measured using cubes.  This new 5m scale measuring tool can be used to build a scale model, i.e., 10 times higher than the "5m" stick of cubes.  The resulting model will of course be to scale and look more realistic.  For older students, the floor plan can be interpreted with accuracy to determine the height needed to represent the dimensions of the tower using art straws as on-site profiles.  On-site profiles are used in Switzerland by architectural companies and builders to show to the public the space that will be occupied by the intended structure.  This must be done to comply with Swiss building law.  The profiles indicate the key dimensions to help observers visualize the intended structure.  Do other countries use these profiles?

UPPER PRIMARY

The floor plan also gives us another example of arrays in context.  Given that the dimensions of this rectangular floor are: 6.3 x 5.8 metres, calculate the area of the floor (an essential calculation for all fitters of carpet/flooring).  The floor plan is an interesting model to help students understand the decimal components of the calculation, i.e., 0.3 x 0.8 = 0.24.  This is counter-intuitive for many students who expect multiplication to be making quantities bigger!  To understand this, it is useful to think of the expression as meaning 0.8 of 0.3.  0.24m is of course 24 cm..or 24 out of 100 (each square represents 100cm squared.

The analogy of area when helping students understand multiplication begins with the introduction of simpler arrays:

A Grade 3 inquiry into Multiplication and Division

                                 

Arrays are all around us.  

We organize objects into arrays, to sort them, to store them, to count them.  We may see them differently from each other, for example, if I write an expression to describe the solitaire board above I see it thus:  4(3 x 2) + (3 x 3 - 1) or  (7 x 3 - 1) + 2(3 x 2)  What other expressions describe this array?

Investigating Arrays

Organizing objects into arrays can be a good way of quickly seeing how many there are.  Arrays can be of many different shapes, however, rectangles are the most common shapes for arrays.

If I have 12 objects I can organize them into rectangular arrays: 

This array of 12 is a rectangle.  Rectangles have 2 dimensions: length and width.  4 x 3 = length x width = 12

Enquiry Task:

1) Using counters or cubes and squared paper, find all possible rectangular arrays for the numbers:

·      24

·      36

·      48

·      90

2) One of the numbers above is special because of one of its arrays.  Investigate into why one of these numbers is a special number!

·      24

·      36

·      48

·      90

 By finding all the possible rectangular arrays for these numbers we have also discovered their factors.

Factors of 24: 1,2,3,4,6,8,12,24

Factors of 36: 1,2,3,4,6,9,12,18,36  (odd number of factors! Why?)

Factors of 48: 1,2,3,4,6,8,12,16,24,48

Factors of 90: 1,2,3,5,6,9,10,15,18,30,45,90

6 x 6 = 36  which describes a square.  This is why 36 is a square number!

Using this discovery, find the first 10 square numbers.

Factors and Multiples

The factors of 24 are 1,2,3,4,6,8,12,24.  Another way of saying this is that 24 is a multiple of 1,2,3,4,6,8,12 and 24. 

Investigate the number 64 and find what it is a multiple of.   Can we do this without having to build all the possible array?  Could simple recording of numbers help us?  Start with the two easiest factors you can think of:

The logic of the recording gives us the answer!

64 x 1

32 x 2  (What’s the relationship between this and 64 x 1?)

16 x 4 

8 x 8

Bigger arrays and partitioning 

Here is a 56 array.  Its difficult at a glance to count all the dots so we can make it easier by partitioning:

                                                                         10                                            4

This rectangle is 14 cm long and 4 cm wide and it covers an area that we can measure in squares:

To find out how many squares the rectangle covers we simply multiply the length (14) by the width (4).  Partitioning the rectangle makes the calculation easier to do:

Writing this as an equation looks like this: 14 x 4 = (10 x 4) + (4 x 4)

This is a very effective strategy to use when multiplying 2-digit numbers.

23 x 5 = (20 x 5) + (3 x 5)

           = 100 + 15

           = 115

There are 3 laws of arithmetic that pertain to multiplication: The commutative, associative and distributive laws.  The mental strategy of partitioning exploits these laws.

Tweaks

Who redistributes the skittles, teacher first then students.

Children only start to be 'out' after playing for a few times, or if they deliberately kick over the skittles.

Extension

Can extend by making the hoop sizes smaller (physical dexterity) or by adding more skittles up to around 20 with 5 hoops.
Can also say a number of skittles that the children have to stand in the hoop which contains that amount.

Early Years 1 and 2: PE and Mathematics

EY and Lower Primary PE teacher Laurence Trengove designed this exciting and engaging PE/maths activity.  Here I will briefly outline the mathematical concepts being developed:

A game is something you can get better at because it may require specific skills or strategies.  
Five hoops are placed on the ground and 10 skittles are distributed evenly between them.  The children run, skip or dance around the hoops, moving to the rhythm of a drum.  When the drum stops, the children must move into a hoop and stand still, without knocking over any of the skittles.  This requires poise, balance and awareness of others.

Children who knock over skittles are 'out'.  One hoop is now removed and the 10 skittles re-distributed within the remaining hoops.  After each round a hoop is removed, until, at round 5, only one hoop remains with 10 skittles.  The winner (or winners) is the student(s) standing inside the final hoop who hasn't knocked over any skittles.

Mathematical concepts:

• The Counting Principles: Cardinality and Conservation
• Subitizing, i.e. knowing a quantity without counting
• Partitioning 10 and addition complements
• Division - how can i divide 10 into 5,4,3,2 groups?