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May 2013

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How to help students with multiplication tables?

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Using the balance model with negative terms in an equation

It is not so easy to portray negatives in an equation with the balance model because in the natural we don't have any 'negative weight', but you can think of the negatives as holes or empty spots that when filled with positives, they become nothing or zero.

x - 4 = 3

Add 4 balls to both sides to 'fill'
the negative balls.

x - 4 + 4 = 7

Now we have added 4 balls to both sides of the balance/equation.

x = 7

The 4 balls 'fill' the negative balls, or
cancel the negatives. So in the end
there is nothing - or zero - left.

Another example 2x + 3 = -5

2x + 3 = -5	We want the left hand side to contain ONLY blocks therefore we have to take away three balls from both sides.
2x = - 8	But since there are no balls from which to take away on the right side, we have to 'add' negative balls.
x = - 4	In the end we take half of each side to arrive to the solution x = -4.

The same thing can happen with the blocks or x's - they can be negative too! Try the following example:

2x - 3 = -x + 3	To have ONLY x's on the left side we get rid of the -x on the right side. Since it is negative, we add one x to both sides.
2x - 3 + x = -x + 3 + x.	Now -x and x 'cancel' each other.
2x - 3 + x = 3	Then we can combine 2x and x on the left side - together they are 3x.
3x - 3 = 3	The next step is to get rid of the number - 3 on the left hand side. For that end we have to add 3 on both sides.
3x - 3 + 3 = 3 + 3	And again - 3 and + 3 cancel each other (or produce a zero).
3x = 6	As the last step, since there are three x's on the left side and we want to know how much one is worth, we have to take a third part of both sides, or divide both sides by three.
x = 2.

Checking the solution: we substitute x = -4 to the equation 2x + 3 = -5:

2(-4) + 3 = -5

-8 + 3 = -5

-5 = -5

So it was the right solution.

More exercises - you can solve these equations with the help of the balance model or just with pencil and paper

x - 5 = 5
2x - 5 = 5
3x - 4 = 2x + 4
5x - 3 = -x + 3
5x + 3 = - 2x + 4

A Balanced Equation Model from Absorb Mathematics
An interactive animation illustrating solving the equation 4x + 6 = x - 3. Drag the green handles to balance each side. Click the arrow button to reset the animation. On the right side, you'll see links to similar animations of equation solving using a balance.

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New! Times Tales is now on DVD!

The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.

Using the balance model with negative terms in an equation

Another example 2x + 3 = -5

More exercises - you can solve these equations with the help of the balance model or just with pencil and paper

See also