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.
Download Lesson 132 of Doodles Do Algebra HERE
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 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:
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
Today your child learns to raise fractions to a power. How do you square a fraction? You square the top of the fraction (numerator) and then you also square the bottom of the fraction (denominator). This is really very easy and in a way is additional practice of the concepts learned over the last few lessons about powers.
Download Lesson 131 HERE
Yesterday your child learned to raise a polynomial to a power, and today he learns to raise a polynomial to a power. It is essentially the same, only he will need to use the distributive property (what we call hippo hopping) to do the multiplication of the terms.
DoodlePoodle explains this very well on the worksheets today, so I don’t think I need to add anything. If you have issues explaining it to your child, just shoot me a question. I will reply quickly and give you some help.
Download Lesson 130 of Doodles Do Algebra HERE
Today your child learns to raise a monomial to a given power. The best way to explain to her is to read the definition that DoodleTwo gives on the worksheet out loud to her and then walk her through the example. This is actually a pretty easy concept to learn, just examples tend to be the best way to show her how the process works.
Download Lesson 129 of Doodles Do Algebra HERE
Today your child begins learning about algebra with powers and radicals.
We are starting by learning (or reviewing) the concept behind raising a variable to a power. What does “power” mean in mathmatics? A power is a product that comes from multiplying a quantity by itself, a certain number of times. The number of times you multiply the quantity by itself is the index or exponent of the power.
You can explain to your child that all powers are is a simpler way of keeping track of the math of certain big numbers. The first step is for him to learn the new vocabulary and then the math will be easy to do.
Today’s worksheet is a simple matching between the mathmatical expression of six different powers and their corresponding description, written out in english. I also add an extension to get your child thinking about expanding the idea of powers with one problem that involves raising an expression to a power, instead of just raising a quantity to a power.
All in all, this sheet is a fun break from the intensity of the last few lessons. So encourage your child to have fun and be creative (maybe he wants to draw lines of cat footprints to match the terms instead of lines).
Download Lesson 128 of Doodles Do Algebra HERE
Today DoodlePig takes your child through the problem of the two messengers running at each other and answers the age-old question of, “When will they meet?”
The work your child does today is first following through the explanation and logic of the problem, and next finishing the problem by solving for x, or the distance the first messenger travels.
If you want to take the problem further with your child, you can work out the answer when both messengers are traveling at the same speed towards each other. In that case the m and the n terms are the same and cancel out of the equation.
This is the problem that has plagued crowds and crowds of high school and first year college students in physics class. And if you explain it clearly and make certain your child understands it, then he will be set for life. That is an effort well worth the time to expend!
Download Lesson 127 of Doodles Do Algebra HERE
First step is multiplying through by nm to “clear the fraction” and you get
Second step is moving the terms with the unknown, x, to one side of the equation and you get
And finally you divide both sides by the coefficient of x (in this case ‘n+m’) and you get to th point that they meet. That means that the second messenger has traveled 2 miles, since the total is 3 miles between them at the start.
It is really not hard, unless confusion and dread was sewn into your head by a high school teacher who was amazed that women and girls could form a complete thought since they were, after all, women and was thus completely convinced that you could never do algebra like this. Then you will be left with a life-long sense that this kind of problem is just to complicated. From experience, you can avoid passing this badness onto your children if you walk them through the steps carefully and slowly. Your kids will grow up thinking it is all quite easy – and that is a great thing!
Today we teach your child about the concept of a negative solution to a problem. What does it mean for a number to be negative in real-life situations?
As DoodleCat explains on the worksheet, if you find the solution to an unknown is a negative number, say -5, that means you have a deficit of 5 or you owe 5, or you need 5 of something to get back to the start.
If your child has a hard time internalizing the concept of negative, you can pull out an ice cube tray. This works best if there are ice cubes in the tray to start with. Point out to her that there are 12 (that is the size of our ice cube trays) cubes of ice in a full tray. So one tray equals “plus, or positive, twelve” ice cubes. Now dump the tray out onto the kitchen table. Depending on her age, let her play with the ice a bit (my daughter always wants to eat the ice…but that is a totally different issue). After a bit, ask her how many ice cubes are in the tray. She will probably answer, “none.” Now point out that actually there are spots for 12 ice cubes which are empty in the tray so you could say that there are negative twelve ice cubes in the tray. This is especially true since you just took the ice cubes out of the tray while you were sitting there. You can explain to her that if you were both aliens from another planet who had never seen an ice cube tray before and if you found the tray empty, you would have no context for understanding that there were negative 12 ice cubes in the tray. In that way, many word problems and much of applied math and physics and chemistry is actually dependent on a shared culture and language. It is one of those questions that leads to much discussion if you have teenagers about the invalidity of relativistic ideals and morals, but if you have younger children, it is probably best to just stick with the ice cubes.
Once your child gets the idea, you can move her through the problems on the worksheet, or let her do them by herself.
Download Lesson 126 of Doodles Do Algebra HERE
1. The equation that comes out of the statement of the problem is and the answer is
2. The equations that come out of the problem statement are
the first equation is so dad was 20 when son was born, or you can rewrite it as
the second equation is that at some point in time, the son will be 1/4 of dad’s age, or
Now substitute the second equation into the first, or vice versa if you want to. If you do it the first way, then
or or So if Dad was 26 and 2/3 years old when his son was 1/4 as old as he was, then that occurred 6 and 2/3 years after his son was born, since dad was 20 when son was born. That means that it happened 15-6 2/3, or 9 1/3 years ago since son is currently 15, or the answer is that son will be 1/4 as old as dad in -9 1/3 years (recall the empty ice tray).
Today your child will learn about a special case of solving for 2 unknowns: when you are given the sum and the difference between two numbers.
DoodlePoodle explains the process very well on the worksheet so I will not go into the details here.
Download Lesson 125 of Doodles Do Algebra HERE
1. In this problem we help get your child started by setting up the two equations that come from the word problem.
The equations are
and when you solve for the unknowns, you get x=62.5 and y=37.5
2. The equations are
and when you solve for the unknowns, you get x=3.125 and y=2.375 (you can just as well do these problems as fractions instead of as decimals)
Today’s exercise is to learn to recognize and translate a word problem into 3 independent equations with 3 unknowns, or 4 equations and 3 unknowns. As a bonus, and for practice your child gets to pick one of the two problems and solve it all the way after writing down the equations.
Download Lesson 124 of Doodles Do Algebra HERE
1. First equation:
Solution: rewrite second equation in terms of y equals something and the third equation in terms of z equals something. Then plus those both into the first equation and solve for x. Then plug that value of x into the second equation to solve for y and into the third equation to solve for z and you are done.
First equation: where r is price of one roll and c is price of one cookie.
Second equation: where c is price of one cookie and l is price of one loaf of bread.
Third equation: where c is price of one cookie and p is price of one pretzel.
Since I did not include the cost of the final trip to the store, you do not have 4 independent equations and 4 unknowns, only 3 equations and 4 unknowns. This is not a solvable problem, so if your child points that out then give her a gold star and lots of praise!