# Lesson 132 of Doodles Do Algebra

Today you will need to work alongside your child. She is going to learn the Binomial Theorem and the best way for her to learn it is to be shown how the relationship is derived in brief form.

So walk her through the process of raising “x+y” and “x-y” to the first, second, third, fourth, and fifth powers, side by side each other. I have outlined these on today’s worksheet. The important part is for her to understand how you get from one power to the next and follow through the process.

Then when you have finished going through it with her, read over the three parts of the pattern summarized at the bottom of the worksheet (these are basically the parts of the Binomial Theorem as discovered by Sir Isaac Newton)

The first point she needs to take away is the pattern of decreasing exponents in the x quantity and increasing exponents in the y quantity for each term.

Then she needs to look at the coefficients of the terms and how they change from power to power. The best way to show her the relationship is to write out the coefficients by reading them off for the first 5 powers as written on the worksheet onto another sheet of paper. You can explain that this is called Pascal’s Triangle.

This is what the triangle should look like:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Now just sit and look at it for a while with your child. Ask her if she sees any patterns. Point out, unless she finds it, that the “outsides” of the triangle are ones. Then you can show her that the inside numbers come by adding the two numbers above them. So the 2 in the second row comes from adding the two 1’s above. and the 3 in the third row comes from adding the 1 and the 2 above it, and the 6 in the middle of the fourth row comes from adding the two threes above that. At this point a little light bulb should come on in your child’s eyes. My boys squealed when he realized the relationship and the implication that knowing this pattern and the first rule about decreasing and increasing exponents, you can raise the sum or difference of two quantities to any power you like without doing any math – you just write down the answer. Now, if your child gets confused by the whole exercise and becomes frustrated, not to worry. If your child is dyslexic, you may find the opposite reaction and end up with a very frustrated and confused child on your hands. If that happens, tell her that she should not worry about it and move on. From experience, I have found that children like this will come back to the concept in time and it will make sense for them. Just don’t push it if they are not ready yet.

The last of the patterns to look at is the difference between raising “x+y” to a power and “x-y” to the same power. For the latter case your child can discover by looking at the worksheet that the sign of each term in the final answer alternates: first term positive, second term negative, third term posititve, fourth term negative, and so on.

Once your child understands all these concepts, and if she has the attention span to continue, you can finish up by writing on a blank piece of paper: $(x+y)^6$

Now, given the worksheet and Pascal’s Triangle and the patterns you discovered together, ask her if she can try writing out the answer. If she stumbles, start her out and, supporting her along the way, write out the answer and talk through the reasoning of choosing coefficients and exponents as we discussed above.

The answer is $x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6$