D03: Quarters and things as a decimal

Here we use diagrams similar to Tasks 01B and C where we were expressing 1/8 as a decimal. This time the diagrams include a part of which a quarter is shaded. A quarter is a fairly familiar fraction as a decimal.

TASK 03A: Here one quarter of one tenth of a rectangle is tinted yellow. A strength of the task is that we can find the desired fraction (common and decimal) in various ways, where each method can cast light on the other methods.

Some pupils might use the fact that 1/4 of 1/10 is the same as 1/4 × 1/10 which equals (1×1)/(4×10) = 1/40.
Some might use a much more grounded approach: in the diagram, if each of the tenths is split into 4 equal parts, then the blue rectangle can be split into 4×10 = 40 such parts, one of which is yellow; so 1/40 of the rectangle is tinted yellow.

When it comes to representing the fraction as a decimal, some pupils might simply use the fact that ¼ = 0.25, and a tenth of that is 0.025.
Others might use a much more grounded approach by dividing the tenth containing the yellow part into ten equal parts (ie into 100ths) and seeing how many of these are covered by the yellow region. To do this in a productive way, and with such a thin rectangle, is quite challenging! One possibility is shown below, in Task 03B, where the thicker (taller) given rectangle makes it easier to see what is going on. Here the yellow region covers 2½ of the small rectangles, each of which is one hundredth. How do we represent 2½ hundredths as a decimal?!
[Some pupils might express 2½ hundredths as 0.02½, which could lead to an interesting discussion. What might the ½ mean? Is it in the 100ths column, along with the 2, or in the 1000ths column?]
Some pupils might find formal ways of transforming 1/40 into hundredths or, as we don't get a whole number of 100ths, into 1000ths. We can do this by multiplying the numerator and denominator by 25, giving 25/1000 which is 0.025 as a decimal.

Task 03B: a simpler variant of Task 03A, where the diagram is larger and some useful extra partitions have been drawn.

 

Task 03C: This might, at first sight, look more challenging than the original task. However, it falls out very neatly: a quarter of one 25th is one 100th.

Of course, we can get to this result in other ways. For example, we could create a horizontal one-tenth strip and then show that the yellow region is one tenth of this strip.

Task 03C: A slight variant on the previous tasks. This time the the yellow region is one quarter of one twentieth, which is 1/80. This might make the task more challenging than the first task, where the fraction was 1/40.

Task 03D: Two tasks for the price of one. The first part is essentially the same as Task 03A. In the second part, the fraction is one tenth of the first, ie one tenth of one quarter of one tenth! How do pupils tackle this? There are lots of ways.