Balancing Equations

'Everyone knows how to make a set of scales balance, so everyone can solve equations!'

The need for solving equations arises from a desire to know what values of an unknown will give a required quantity e.g. the exact quantity of  salt peter required to make 20kg of gunpowder or, as in the video at the bottom of this page, working out when/where the next "Freak Wave" will strike by substituing in wind speeds, current strengths etc. into a series of mathematical equations. When we solve mathematical equations we often solve physical world problems. Many students find the symbols and meanings used in equations difficult to understand. If we think of an equation as a set of scales balanced on the equals sign this provides a good intuitive insight into how we solve equations. This resource provides a structure for using the excellent NVLM "balancing scales" virtual world from Utah State University, US.

Watch the video below which shows you how to use the activity on the website below:

Resources

Add or take off weights form each side, keeping the scales balanced, until you have isolated the the unknown weight "x" on one side and known weights on the other.

"Balancing Scales" solving equations virtual world:  POSITIVE values only [http://nlvm.usu.edu/ [National Library of Virtual Manipulatives, Utah State University]

 Balancing Scales NEGATIVE and positive values.

Applications for Solving Equations

Algebra and equations represent real-life situations - they "model" the real world. Just like an architect builds a scale model of a building either on his computer or with a modelling set, so mathematicians use algebra and equations to  "model" reality e.g. modelling the path mobile phone waves take through the air, describing the bends and folds of a human's brain so as to  create a flattened out 2D model of your 3D brain to better understand which areas are responsible for what activities; analysing currents and wave formations to predict the occurence of "Freak" or "Rogue" waves, previously thought to be a myth, but now confirmed as very real phenomena. Watch the video below (4m30 to 6m30 for linear model predictions and a good insight into how scientifc data collection and mathematical equations work together) and the TED talk  link on "laying the brain flat"  above to get an  inspiring insight into some of the applications algebra is used to model:

Description

  • We can use our intuition for keeping a set of scales, or a friend on a "see-saw", balanced to help us find the weight of the blocks marked "x".
  • When you've solved a few using your intuition about balancing scales, click on the "continue" button to start working with algebra in place of the balloons and weights. When you get the hang of it, it's much quicker than drawing out a set of scales, putting the weights and balloons either side etc.
  • Watch the two videos: the TED talk on laying the brain flat and the "Freak Waves" BBC episode, to get an insight into how equations help us to model the physical world. Once you have worked out the correct equation to describe a physical process, it can then be solved to determine which part of the brain is used for what activity or where, and at what date, the next "Freak" wave is due to occur.
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