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.