Cross Products

Using Materials

Provide the students with a copy of the dotty array. If you are using laminated copies, whiteboard pens are very practical.

Write 23 x 37 on the board or modelling book and ask the students to draw a border around the array that represents it. Tell them to partition the array in any way they  want that will make it easy to find the total number of dots.

Students are likely to use a variety of partitions including use of place value:

Discuss how the partitions relate to each other in that 3 x 37 = (3 x 30) + (3 x 7) and 20 x 37 = (20 x 30) + (20 x 7).

A key point is that the multiplication involves the idea of a cross-product. 23 x 37 can be calculated in this way, with each arrow representing a  separate multiplication.

Note that students may suggest valuable strategies for 23 x 37 that are efficient but not transferable to more complex examples, for example, 23 x 37 = (25 x 37) – (2 x 37) = 925 – 74 = 851, or 23 x 37 = (23 x 40) – (23 x 3) = 920 – 69 = 851.

Give the students other problems that can be modelled by partitioning the array.  Good examples are: 42 x 15 = 630,  26 x 24 = 624,  18 x 33 = 594,  45 x 35 = 1 575

Ensure the students have established a clear idea of the cross-product idea before proceeding to imaging.

Using Imaging and Number Properties

Provide further similar examples and ask the students to image the array and record the products that need to be calculated to ease memory load. Draw a border around the required array and hide it.

28 x 17 = 476,  19 x 43 = 817,  34 x 22 = 748,  40 x 34 = 1 360

Generalise the cross-product to include three-digit factors.

173 x 26 = (100 x 20) + (100 x 6) + (70 x 20) + (70 x 6) + (3 x 20) + (3 x 6) =
2 000 + 600 + 1 400 + 420 + 60 + 18
= 4 498