Sunday, 7 June 2015

Visually Balancing Equations

When I am teaching students a new concept it quickly becomes clear when students have been taught what to do rather than why or at least how it works. Recently this situation become clear when performing Inverse Operations;
The students ask-
Why do we switch the sign when we switch the side of the equal sign? 
Why do we need to perform our Inverse Operations first on the addition, before multiplication? (We have just learnt the Order of Operations (BIMDAS/BODMAS)

I find that when students understand that they need to keep the equation balanced, that they develop a robust understanding of how inverse operations work, and why we can perform the operations we do.

Balancing equations is a pretty simple, and quite visual concept. I have seen a number of sites and games (mostly aimed at primary school level) using scales to consider how changing one side, needs to have an equal change on the other side.

But then I came across a different site. It had a simple, interactive and attractive site that invites students to be play and which flips the scales idea on its head.

Solve Me Mobiles ( uses mobiles (like those hanging above a cot) to balance the equations.

The site allows users to access a hundreds of balancing problems. Running from the simple to quite complex arrangements, with multiple variables and simultaneous equations. It is not intimidating, in fact some students thought it looked too simple (until they started trying to solve the more difficult ones).

On the face of it this is a very simple exercise. Guess the weigh of each shape, while animations of cloud and seagulls gently float by, and at first it is. The first level the students need to find 2 numbers that are equal and together equal 10...

But as they start to become more complicated the tools the site provides becomes valuable.

Students can click and drag the mobile, to create equations.
Once they have the equations, they can subtract(or eliminate) items which naturally happen on both sides.
The site also allows for prime factorisation, with factors also able to be 'eliminated' from both sides (effectively dividing both sides)
Through these operations the maths comes naturally and visually allowing students to solve the equations.

The balancing of equations can then be expanded through moving into simple simultaneous equations, when there are multiple bars, and increasing number of objects/variables.

This structure then allows for substitution, again through natural drag and drop approach. This allows the user to isolate variables to solve.

The students quickly took to the problems, with flexibility for some students to pick a range of difficulties to match their ability.

The lesson flew by, with my time as the teacher, simple offering leading questions to students as they find themselves on a bump, before jumping up to the next level.
"Are you sure you have all of the possible equations from that set up?"
"If the left arm is balanced with the right, the left arm must be half of the total?"
"What could you substitute to isolate an object?"

I found that I needed to direct students towards the correct vocabulary, but whether they were eliminating variables, or "flicking bits", they understood the mechanics and the limitations.

As we summarised the lesson, we found that different students were applying a range of strategies to solve the puzzles, with some focused on equations, others applying a halving logic, and others aiming to isolate each object.
This really highlighted how important communication and willingness to try new approaches was.

I created 2 mobiles (one challenging {} and one impossible{}) and shared them with the students, and as an extension opportunity I suggested they create one challenging and one impossible mobile, and provided a discussion board so they could share them. The discussion and the learning continues as they explore why some may be impossible to solve.

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