Learn to Multiply and Divide Roots – Lesson 141 of Doodles Do Algebra

Today your child will learn to multiply and divide roots. The basic concept is the same as multiplying and dividing in algebra: you can multiply like terms. Remember when we talked about laying the apples, oranges, and raisins out on the kitchen table and having your child group them into like terms as a way to show him that you really cannot combine “terms” (or items) that are in different categories? It was a long time ago, if you have been working your way through this course, so it may be that you need to remind your child of the concept. This is where hands-on and real-life examples like the fruits at the kitchen table come in handy. Those types of memories are generally “sticky”. “Sticky” is a term I first came across in marketing classes, but it really applies to teaching your child just as well, and perhaps better. If you can associate a learned concept with a hands-on activity, all you have to do in the future is say, “remember when we sat at the table and grouped the apples and the oranges into piles and talked about grouping like terms – and then you ate the raisins?” That is a “sticky” memory and the more of those you can make for your child, the better he will retain what he learns.

Back to the algebra lesson. The explanation on the worksheet for today is quite detailed and walks your child (or you) through the process of multiplying or dividing monomials with roots in them. The only tricky bit is dividing or multiplying two monomials that have roots of different order (like a square root times a cube root). Then you have go back to the tools your child learned in lesson 137: x to the “a”th power times x to the “b”the power is x to the “a+b”th power. So as an example, the square root of 2 times the cube root of 2 is the same as 2 to the 1/2 power times 2 to the 1/3 power, which is 2 to the (1/2 plus 1/3) power, or 2 to the 5/6 power. Not hard, right?

Download:

Answers:

1. $6c^2d$ times the square root of x

2. 15a times the square root of b divided by the square root of c

3. 6 times the cube root of 3 divided by the square root of 2 (here you cannot combine the radicals)

4. 3 times 2 to the 1/6 power (because you end up with $2^(1/2)$ divided by $2^(1/3)$ which is the same as $2^(1/2)$ multiplied by $2^(-1/3)$ so that is (by way of the rule your child learned in lesson 137) $2^((1/2)-(1/3))=2^((3/6)-(2/6))=2^(1/6)$

Now @ Amazon: DoodlesDoAlgebra Book 3- The Basic Math of Algebra

I am releasing a new title in the Doodles Do Algebra series:  “The Basic Math of Algebra” on Amazon today.

I originally wrote the book for my own children and now I am sharing it so that you can teach Algebra to your own children at home. You can begin with the first book in the series, “Starting Out With Mental Algebra” (a book which received 5 stars on Amazon) or you can jump straight in to this third book in the series. The book is available on Amazon in Kindle format and early next year I will be publishing the series as hard-copy workbooks as well.

How Much of The Basic Math of Algebra Is Covered In This Book?

This third book in the Doodles Do Algebra series teaches your child how to evaluate equations followed by a comprehensive tour through addition, subtraction, multiplication, and division of monomials and polynomials, including long division of polynomials. This leads the way towards the next books in the series that cover factoring and fractions, followed by learning to solve simple equations, and more.

Do I Have To Start With The First Book In The Series, Or Can I Pick And Choose Subjects?

The book series is designed as a complete curriculum and also as a supplement so that if your child is having difficulties with a specific area of Algebra, you can use just the relevant Doodles Do Algebra book.

What Is In This Book? (The Basic Math of Algebra)

• Evaluating Equations
• Writing Equations
• Algebraic Addition (of both positive and negative terms)
• Algebraic Subtraction
• Distributive Property of Multiplication
• Commutative Property of Multiplication
• Multiplication of both monomials and polynomials
• Division of monomials
• Long Division of Polynomials

This book includes the student work pages, teacher’s notes, and answer key. Unlike most curricula for homeschooling that seems to include teacher’s notes as an afterthought, this series is focused heavily on notes to the teacher. We provide alternatives for teaching each lesson so that you can adjust the material to fit your child. No matter how your child learns and understands math best, we have a suggestion.

The Approach Answers The Question We All Hear: “Why Do I Have To Learn This?”

All of us, at one time or another, have asked, “But why do I have to know this?” This curriculum is designed to eliminate those questions. Children begin solving real life problems that get progressively harder, perhaps even pushing your own limits of concentration but I guarantee your child will breeze through the material. At the end of this book, we introduce the concept of the unknown as a way to keep track of the bits and parts of a problem. Then your child will fully understand why they are learning algebra, not just how to do the problems.

A Lesson A Day Is The Best Way!

Each lesson is meant to be done in one day and is designed to be flexible. If your child understands right away, then encourage them and move on. If, however, your child doesn’t understand a topic, then I provide alternative teaching methods for you to try in the teacher’s guide section at the end of this book.

Curricula Designed For Both Independent Learning Or Working With You, Whatever Is Best For Your Child.

The lessons are laid out in a fashion that allows your child to work independently as much as possible. You generally need to spend a few minutes with your child prior to any independent work in order to set the stage for the day’s learning. Depending on your child’s age and ability to work independently, you may feel most comfortable working through the entire lesson each day with your child. I have found with my own kids that on some days they really want to work by themselves, and on others they really want to do the lesson together. This curriculum is designed to handle both scenarios and allows you to be completely flexible.

The Doodles Do Algebra series includes the algebra topics encountered in modern day Algebra I and Algebra II, as well as topics that are no longer covered until college (such as calculating the square root of large numbers without a calculator, or a computer). The curriculum is based on the teaching methodology of algebra texts written in the late 1600’s to the early 1800’s and used by English and American children.

Doodles Do Algebra – Lesson 140

Today your child learns how to add and subtract second degree radicals. I think that the basic explanation on the worksheet is enough to help your child through the process. But the basic idea is that there are two steps: first to reduce the radicals that you are adding or subtracting to their simplest form, and then second to add like terms. “Like terms” when you are adding or subtracting radicals is just like the “add like terms” you taught your child back when he first began learning algebra and perhaps you would say to him, “one ‘x’ plus two ‘xes’ equal three ‘xes’.” Sound familiar? If you have worked your child through this curriculum, you will remember saying ‘xes’ many, many times while explaining algebra to him.

Well, adding and subtracting radicals is the same idea, once they have been reduced to their simplest form, only you say “one ‘square root of 2’ plus two ‘square root of 2es’ equals three ‘square root of 2es’,” instead of ‘xes’.

The last question today pushes your child to think about extending the adding and subtracting of second degreee radicals into the realm of constants and unknowns. It is the same principle, and maybe even easier that doing it with numerical values.

If my explanation does not make sense, just let me know in the comments below and I will help you. It is one of those Monday mornings where you cannot seem to drink enough coffee…

Answers:

1. 5 square root of 3

2. 8 square root of 2

3. 2 square root of 5

4 (a-2c) square root of b

Lesson 139 of Doodles Do Algebra

Today’s lesson teaches your child to change rational quantities to radicals that have the same value. This is actually manipulating quantities in the reverse direction of the last couple of lessons, so once he realizes that, your child should have no problem with the lesson. The explanation and example on the worksheet for today explains the process very well so I won’t repeat it here.

This is one of those examples in algebra, and even more broadly in math, when there is no need to worry about review because topics a built one upon another. You child will learn a math skill and then use it again for more complicated math later. This is one of the reasons that I focused everything in our homeschool on getting my own kids through algebra. Then after algebra, they were equipped to understand not only geometry and trig and calculus, but also applications of algebra like economics and physics and chemistry and biology and marketing and accounting and even drafting dress patterns and pretty much everything.

So the ‘take-home’ message here is that the key to understanding the world lies in a solid understanding of math (at least through algebra and actually through calculus). And the reason we homeschool our children is to give them the best shot at understanding their world, in spite of the sacrifices that come out of that choice. But for us, the sacrifices are not really sacrifices at all. They are blessings and a general manifiestation of our duty as parents to do the best for our children. And algebra is key to it all.

So if you have questions about teaching algebra to your children, or if you just want to add your two cents, please comment below and I will respond.

Answers:

1. Square root of 36

2. Cube root of 8

3. This is really a two step problem. First to express 3 as a square root (3 is the square root of 9), and second to combine the square root of 9 and the square root of 6. If your child has trouble here, remind him of the lesson a few weeks ago when he learned how to do this: multiplying the square root of one number (or quantity) by the square root of another is the same as the square root of the product of the two quantities. That means the answer is the square root of 9 times 6, or the square root of 54.

I apologize for having to write out these answers in English instead of using math symbols. I am having trouble finding a wordpress plug in that lets me display symbols like square roots, so bear with me for now.

Book 2: Unknowns and The 29 Articles of Algebra Published Today

Get the book here:Unknowns and The 29 Articles of Algebra, Book 2 of Doodles Do Algebra

How Much of Algebra Is Covered In This Book?
This second book in the Doodles Do Algebra series teaches the concept of the unknown and then covers the basic vocabulary of algebra (called the 29 Articles). The concept of the unknown, or typically x, is introduced at the start of the book as a transition between the mental math your child did in the first book in this series and the abstract vocabulary in the second part of this book that is necessary for your child to really become proficient with algebra.
The entire series of lessons covers Algebra I and Algebra II as well as advanced topics most children today don’t learn until college.

What Is In This Book?
This book includes the student work pages, teacher’s notes, and answer key. Unlike most curricula for homeschooling that seems to include teacher’s notes as an afterthought, this series is focused heavily on notes to the teacher. We provide alternatives for teaching each lesson so that you can adjust the material to fit your child. No matter how your child learns and understands math best, we have a suggestion.

The Approach Answers The Question We All Hear: “Why Do I Have To Learn This?”
All of us, at one time or another, have asked, “But why do I have to know this?” This curriculum is designed to eliminate those questions. Children begin solving real life problems that get progressively harder, perhaps even pushing your own limits of concentration but I guarantee your child will breeze through the material. At the end of this book, we introduce the concept of the unknown as a way to keep track of the bits and parts of a problem. Then your child will fully understand why they are learning algebra, not just how to do the problems.

A Lesson A Day Is The Best Way!
Each lesson is meant to be done in one day and is designed to be flexible. If your child understands right away, then encourage them and move on. If, however, your child doesn’t understand a topic, then I provide alternative teaching methods for you to try in the teacher’s guide section at the end of this book.

Curricula Designed For Both Independent Learning Or Working With You, Whatever Is Best For Your Child.
The lessons are laid out in a fashion that allows your child to work independently as much as possible. You generally need to spend a few minutes with your child prior to any independent work in order to set the stage for the day’s learning. Depending on your child’s age and ability to work independently, you may feel most comfortable working through the entire lesson each day with your child. I have found with my own kids that on some days they really want to work by themselves, and on others they really want to do the lesson together. This curriculum is designed to handle both scenarios and allows you to be completely flexible.

Lesson 138 – Reducing Higher Order Roots

Today your child continues working with radicals and learning to reduce them.

As a bit of preparatory work today, it will probably help your child for you to sit down with him and review two of the rules of powers that were summarized in lesson 137.

First review the concept of

$(x^a)^b=x^(a*b)$

and

$x^1/2$ is the same as the square root of x

Write the the two concepts down on a piece of paper for your child and explain the concepts, or review then as you write. Then show your child that you can use these two concepts to get to the concept he will need today.

First: if $x^1/2$ is the same as the square root of x, then $x^1/3$ is the same as the cube root of x and $x^4$ is the same as the fourth root of x.

Then you can reduce a root order using the first concept. This means that if you have the fourth root of x, then that is the same as $x^1/4$ which is the same as $x^1/2*x^1/2$ because $1/2*1/2=1/4$, or you can think of it as square root of the square root is the fourth root.

The examples on the worksheet also need your child to understand that the third root of the third root is the 9th root (cause 3*3=9, or thinking of it differently $1/3*1/3=1/9$) and the third root of the second root is the sixth root (cause 3*2=6, or $1/3*1/2=1/6$).

Please feel free to comment below if you have any questions. If your child doesn’t really understand I can help you figure out a different way of explaining the concept.

I always approach homeschooling my own kids in terms of finding the right explanation or method for them to understand a subject. I tell them that they are smart and able and if they have trouble with fundamentally understanding a subject, we just have to work together to find a way of explaining things so that they understand. It is not at all related to how smart they are, only dependent on how they learn. This is one of the real benefits of homeschooling – your child doesn’t ever have to feel like he is not smart. If he works at it, he will absolutely be able to learn anything and if a subject is hard it is most likely due to a mismatch between how he learns and how the material is presented to him. In that regard, the best thing I ever did was to use a learning style test on both my kids and on me when we began homeschooling. This gave me insight into my kids, who are dramatically different in spite of being twins. I also took the test so that I could see the difference between how I learn and how they do individually. This was very, very useful as I can translate material for each of them from the form I would most naturally understand into a form that suits each of them best. I highly recommend you do something like this with your own children.

Answers:

1. The sixth root of $4a^2$ is the same as the sixth root of 4 times the sixth root of $a^2$. The sixth root of 4 is the same as the cube root of the square root of 4 (I am sorry I can’t write this out right now in math symbols – I am working on getting the write scripting language to be able to display these kinds of answers on the web site, just bear with me for now please) and that is the cube root of 2 (cause the square root of 4 is 2. Then the sixth root of $a^2$ is the cube root of the square root of $a^2$ which is the cube root of a. So the answer is the cube root of 2 times the cube root of a, or the cube root of 2a.

2. Here you go throught the same process, only there is one more component in the problem. The answer is the cube root of [math]

Lesson 137 – Reducing Radicals & The Perfect Monomial

Today your child begins learning to reduce radicals. No, this is not advanced nuclear chemistry or using a shrink ray on protesters in the streets. It really is math, well, part of algebra for our purposes.

The new idea your child needs to learn today focuses on the concept of a monomial that is a perfect square. Now my test subjects were divided on the issue of whether the explanation on the worksheet was sufficient today, so I am adding a more visual explanation of a monomial that is a perfect square here for you to do with your child as an optional activity if you think it is necessary. Get some paper and cut it into different sized squares. For that matter, you could usee any item you have around the house. You just need at least three different items and they need to be square.

After you have your three squares together, take some tape, or something else that is sticky and that you can write on and put a piece on each side of each of the squares (actually you only need it on adjoining sides, but it may be easier for your child this way). Then sit down with your child. Grab the smallest square object and explain that this is a perfect square – each side is the same length. So you could call it 2 inches by 2 inches (or roughly whatever size the object is that you have in your hand) OR you could call it ‘x’ inches by ‘x’ inches or ‘half an apple’ by ‘half an apple’ or ’6 marbles’ by ’6 marbles’. It really doesn’t matter what you call it, the important thing is that each side is the same. So pretend it is 4 marbles by 4 marbles. The area of that square would then be 4 marbles times 4 marbles, or 16 marbles. 16 is called a perfect square because 4 times 4, or 4 squared, is 16. (your should see her eyes light up with recognition right about now). Go back to that same square. Explain that if instead of 4 marbles on a side, you had no idea how long it was so you called it “x” marbles by “x” marbles (unknown number of marbles on a side) then the area of the square would be x times x, or x squared marbles. And that, too is a perfect square, just like 16.

Now armed with the idea of a perfect square in terms of numbers (like 2*2=4, 3*3=9, and 4*4=16), and in terms of letters or unknowns (like x*x=x squared), you can reach for the other two sized squares you have gathered. On the smallest, write 2 on each side, on the largest write “3” on each side and on the middle sized square write “2 times square root of 2” on each side. Now explain that the area of the small square is 2 times 2 or 4, the area of the largest square is 3 times 3 or 9, and that the area of the middle square is 2 root 2 times 2 root 2, or 2 times 2, times root 2 times root 2 which equals 4 times 2, or 8. Make sure she sees this (you might write it out in the center of the middle square or on a separate piece of paper.

Now go back to the littlest square and show her that is a perfect square because each side is an integer (2), and then show her that the largest square is also a perfect square because each side is also an integer (3). Now pick up the middle square again and show her that this middle square is not a perfect square because each side is not an integer. 8 is not a perfect square but 8 contains a perfect square times another number. Show her that 8=4*2 and that 4 is a perfect square (you just need to grab that smallest square with the area of 4 and 2 on a side to show her now). So the square root of 8 is the square root of 4 times the square root of 2, or 2 times the square root of 2.

Now explain to your child that you can extend that idea beyond numbers to x’s and a’s and other letters. Then you can walk her through the example on the worksheet.

This was a long explanation, but if you have a child who is spatially-focused (especially those kids who are dyslexic) this might be a very good way to explain the topic.

Answers:

1. 12 is 4 (4 is a perfect square) times 3, so the square root of 12 is the square root of 4 times the square root of 3, or 2 times the square root of 3. This means that the answer is $2x$ times the square root of 3

2. $3x$ times the square root of 3x

3. $4ax$ times the square root of a

4. $2ab^2y$ times the square root of 10by

Now Available on Amazon: Doodles Do Algebra Book 1 – Starting Out With Mental Algebra

Now Available on Amazon!

I finally published the first book in the Doodles Do Algebra series and it is now available on Amazon.

To Celebrate, You can get it Free Today, September 19, on Amazon!

Get the book here: Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra

What Is In This Book? This book includes the student workpages, teacher’s notes, and answer key. Unlike most curricula for homeschooling that seems to include teacher’s notes as an afterthought, this series is focused heavily on notes to the teacher. We provide alternatives for teaching each lesson so that you can adjust the material to fit your child. No matter how your child learns and understands math best, we have a suggestion.

How Much Of Algebra Is Covered In This Book? This first in the Doodles Do Algebra series is focused on Mental Math and is designed as the first step to learning Algebra for your child. The philosophy is focused on helping your child to think about abstracting and grouping items in order to make calculations of cost, price, age, and amount. The concept of the unknown, or typically x, is introduced at the end of this book and will be the first topic in the next book in the series.

The entire series of lessons covers Algebra I and Algebra II as well as advanced topics most children today don’t learn until college.

Why Is Mental Math The First Step? The Doodles Do Algebra series is based on the teaching methodology and sequence that teachers used in the 1700’s and 1800’s in America, a style of teaching that led to some of the smartest and greatest thinkers our country has known. I developed this math book for my own kids, who learn in completely different ways from each other. It worked wonderfully for them, so hopefully it will work for your child too.

The Approach Answers The Question We All Hear: “Why Do I Have To Learn This?” All of us, at one time or another, have asked, “But why do I have to know this?” This curriculum is designed to eliminate those questions. Children begin solving real life problems that get progressively harder, perhaps even pushing your own limits of concentration but I guarantee your child will breeze through the material. At the end of this book, we introduce the concept of the unknown as a way to keep track of the bits and parts of a problem. Then your child will fully understand why they are learning algebra, not just how to do the problems.

A Lesson A Day Is The Best Way! Each lesson is meant to be done in one day and is designed to be flexible. If your child understands right away, then encourage them and move on. If, however, your child doesn’t understand a topic, then I provide alternative teaching methods for you to try in the teacher’s guide section at the end of this book.

Curricula Designed For Both Independent Learning Or Working With You, Whatever Is Best For Your Child. The lessons are laid out in a fashion that allows your child to work independently as much as possible. You generally need to spend a few minutes with your child prior to any independent work in order to set the stage for the day’s learning. Depending on your child’s age and ability to work independently, you may feel most comfortable working through the entire lesson each day with your child. I have found with my own kids that on some days they really want to work by themselves, and on others they really want to do the lesson together. This curriculum is designed to handle both scenarios and allows you to be completely flexible.

Lesson 135 – Doodles Do Algebra

Today your child extends the concepts of square roots to unknowns. The only point you really need to make sure your child understands is that 2 times 2 is the same as negative 2 times negative 2, the answer is 4. So that means that the square root of 4 could be a positive or a negative 2, we cannot tell so you have to write both possibilities down as the answer.

Answers:

1. $+-2ax$

2. $+-4bx^2$

3. $+-3a^2bx$

4. $+-9m^2by^2$

Lesson 134 – Doodles Do Algebra

Today your child learns how to take the square root of a fraction. Basically the square root of a fraction is the same as the square root of the numerator divided by the square root of the denominator. The problems for your child to work are simple and build to the idea of taking the square root of higher order unknowns… a preview of the next few days.

Answers:

1. 3/2

2. 4/5

3. a/x