How many spots are there in a complete set of dominoes?

It’s a lovely problem that offers the opportunity to discuss methods of calculation, and methods for avoiding mistakes. It links to combinations and permutations, as well as triangle numbers, and is a really simple question to pose. An intermediate question is to ask how many dominoes there are in a set, and establish that solution before continuing to the original question.

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This activity is for a first introduction to expansion of brackets or for a review activity.

Imagine telling a class that 4(x+3) = 4x+12 and them asking you “why?”. What if instead you presented the information to them in a way that they generated the idea that they were equal and the explanations for their ideas themselves?

This sheet generates pairs of equations (for just expressions, read on)*, one with the bracket in place, one with it expanded. It is a very simple spreadsheet. It contains simple macros that allow either equation to be hidden. Though you can use it however you like, my suggestion follows…

I would use the sheet with both equations or expressions* visible and change the values a few times. I would ask students to try to explain the relationship between the two expressions.

Most likely the first thing that they notice is the methodological relationship the product of the two numbers in the first expression is the last number in the second expression. A brief glance at some other variations will demonstrate that this is the case.

However, they are unlikely to have grasped the numerical relationship; that they each have the same value, irrespective of the value of x. This will probably require some prompting. You could ask each student to choose a value of x and substitute it into the two values. What do they notice. Choose random numbers a few more times. Repeat the activity. Once they have established that they are always getting the same numbers then you can move on to the big one: the conceptual relationship.

The concept we’re aiming at is surely for the students to grasp that the link between the two expressions they first noticed is necessary for them to have the same value. It will be difficult for this activity alone to convince them of that, but I would at this point urge them strongly to consider whether the two things they’ve noticed are related. Since they’re expecting a relationship, they may trust that linkage. I haven’t linked it for them however, at least not in their minds.

I would then remove the expanded expression and ask them to construct it themselves, making sure that for whatever number they make x the two expressions match. This is a crucial check that should be encouraged early.

* The ‘y=’ on both the equations can be easily removed to just give the expressions. To be honest, that’s my preferred usage, because of the problems students have understanding when we’re using equality as a question signifier and when we’re using it as a binary relation between the two expressions on either side. To get rid of the ‘y=’ click on the ‘control’ tab and delete G4 and G5, where the ‘y=’ text is seen. Go back to the ‘display’ tab and they will have gone.

Also note that you must have macro security set to medium and allow macros for the buttons to function. Please disable macros and view the macro code if you are unsure whether this is safe, or if you do not understand code, ask someone who does.