Note: GERMAN versions of the Week 13 tasks are available HERE.
In this week's tasks we take a sideways look at fraction multiplication, using area of a rectangle as a model. A common way of introducing the model is to partition a rectangle into equal parts, and then to partition the partitions, and then finally to ask students to quantify the result. In these tasks we work the other way round, by presenting students with a carefully chosen fractional part of a rectangle in the hope that students will construct suitable partitionings for themselves.
Monday: Here we can arrive at the tinted region by halving the rectangle with a vertical cut, and then cutting one of the pieces horizontally into three equal parts. Or we could proceed the other way round: cut the rectangle into three equal horizontal strips and then halve one of the strips with a vertical cut. Either way, it is fairly obvious that six versions of the tinted piece will exactly cover the rectangle, so the tinted region covers ⅙ of the large rectangle.
It is possible that some students will find the desired fraction by calculating the actual areas of the two rectangles and expressing one area as a fraction of the other (ie 12×42 = 504 / 36×84 = 3024). However, a brief class discussion should soon make it clear that the task is such that there is a quicker way!
Tuesday: Here we make explicit that Monday’s tinted region can be thought of as one third of one half of the large rectangle.
Notice that in our diagram we have only cut one of the halves into thirds rather than the whole rectangle. This is a more accurate representation of ‘⅓ of ½’ than halving the rectangle and then cutting the whole rectangle into thirds, as is commonly done. However, it does not explicitly show that the resulting piece is ⅙ of the large rectangle. This might be seen as a drawback, but it has the beneficial effect of prompting students to think about the relative size of the pieces.
The two tinted regions are very differently shaped rectangles, though students may well predict that they cover the same area, as indeed they do. As with the first tinted region, the second can be thought of as ⅓ of ½ of the large rectangle or as ½ of ⅓.
Wednesday: Given the seemingly random position of the tinted region, students might be tempted to find the actual areas of the blue region and the whole rectangle. However, they might notice that 13 is ¼ of 52 and that 36 is ⅗ of 60, especially if they imagine the tinted region pushed to fit into one of the corners of the rectangle. So the tinted region is ⅗ of ¼ = 3/20 of the whole rectangle.
Some students might decide to draw a diagram, for example like the one below, to help solve the task. It can also be useful to do so after the task have been solved, especially for students who adopted a numerical approach, as this can throw light on the arithmetic.
Thursday: Here we have made things slightly more obscure - they should become clearer if one imagines the tinted region turned through 90˚.
It is fairly easy to spot that 12 is ¼ of 48, but it is probably less obvious that 45 is ³⁄₁₀ of 150. Once these fractions have been found, we can express the tinted region as ³⁄₁₀ of ¼ of the whole rectangle (or ¼ of ³⁄₁₀). How readily do students see that each of the resulting 3 pieces are 40ths of the rectangle? Again, a diagram might help, for example like the one below.
Do any students resort to decimals: 0.25 × 0.3 = 0.075? If so, can they express 0.075 as a vulgar fraction?
Friday: It is sometimes easy, and useful, to make only parallel rather than orthogonal cuts when partitioning a rectangle into a fraction of a fraction. That is the case here, where we investigate fractions of a fraction that result in one half.
In this special set of fractions of a fraction, parallel cuts work very nicely - one can see fairly clearly that the result is always one half. Interestingly, the parallel cuts also make it easier to interpret the situation as scaling, especially if we make the rectangle look more strip-like.
Parallel cuts can also be made to work in less simple cases, such as ⅕ of ⅔. Here it would not be easy, or particularly helpful, to cut the ⅔ region into just 5 equal parts using cuts parallel to the previous cuts (below, left); however, the result is perfectly easy to achieve and to read if two ⅓ pieces are each cut into five (below right): we would then want 2 of the resulting small pieces, of which there would be 15 if the cuts were applied to the whole rectangle.
