Mixed numbers and improper fractions

Shortcut for getting improper numbers from mixed numbers


Forty years ago, when I was my youngest daughter’s age, we were taught to multiply the whole number in a mixed number with its denominator and to add the numerator when we want to change it into a mix number. I don’t recall being taught why, but maybe I forgot or was not paying attention in class.
As part of our online class, I decided to introduce this shortcut to my students and ask them to figure out why this formula works. Prior to that students were asked to visually represent a mixed number and an improper fraction. They were not told both these numbers represent the same value or are equal and the very small class of two did not themselves see this relation.

When asked to show how we can use the visuals to show why the formula works, the 2 students were at a loss to show how to.
So, I prompted the students to put together all the halves- they will get the same model as the first one and when they divide the 3 wholes in the visual into two for the mixed number they get the same as the visual for 7/2. We have thus established that they are the same; but still why the sequence in the formula?
To do so we need to look at what 3 1/2 represents. The denominator ‘2’ represents how many parts there are in a whole, so if we have 3 wholes to know how many parts there are we multiply ‘3’ to ‘2’ to get 6, but this only gives us how many parts there are in the ‘wholes’. To get the complete answer we need to add the number of parts that do not make a whole to the product. Hence this process gives us the total number of parts out of the number of parts in one whole.
I’m not quite sure if my explanation made sense to the girls (no worries, they take time) but I think pondering about such simple short cuts such as why you add a zero to a number if you multiply by ten and so forth, firstly gives you that ‘aha!’ moment and on a deeper level, it makes you wonder about what is this order in Mathematics that makes such formulas possible. Hopefully, this can spark their interest in Math. Asking ‘why?’
You also hope that this allows them to access Math at a more conceptual level especially as they move on to more advanced math. Finally, this of course benefits me who wonders why these formulas I learned in school decades ago are the way they are. How would you teach this formula to nine-year-olds?

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