# Lesson 113 – Doodles Do Algebra

Today your child learns to solve a first degree equation, well the first step anyway.

Solving simple equations involves three steps, the first of which is clearing the equation of fractions by multiplying through by the least common multiple.

The Least Common Multiple is a term your child learned a number of lessons ago and practiced quite a bit so she probably will remember it. If not, just explain that it is the smallest number that can be divided by two (or three or more) numbers. So the least common multiple of 2 and 3 is 6 and the least common multiple of 4 and 5 is 20 because 20 is the smallest number that is divisible by both 4 and 5 evenly. Things only get tricky when you find the least common multiple of three or more terms. So the least common multiple of 2 and 3 and 4 is 12 (not 24, that is a common multiple but not the least one) because 12 is divisible by 2 and 3 and 4 evenly.

So in the example by DoodlePoodle, the first step of solving the simple equation

$x/2+x/3=5$ is to multiply through by 6 (the least common multiple of 2 and 3)

Make sure that your child remembers to multiply through every single term in the equation (on both sides) because you need to maintain an Identical Equation (a term from Lesson 112’s vocabulary exercises).

So in our example, we multiply through by 6 and get $6x/2+6x/3=5*6$ which can be reduced to $3x+2x=30$ and that is where your child can stop for today. The goal for today is to get her to find the least common multiple and use it to clear the equation of any fractions. If she naturally continues on and solves the equation, don’t stop her – just emphasize the first step of solving the equation and let her skip some of the next lessons that deal with the final two steps of solving simple equations.

I have found that the moment you slow a child’s natural progress in learning a subject, she will instantly get bored and tune out and actually not learn what she was learning so rapidly before. In homeschooling, my biggest challenge is negociating the balancing act of teaching my twins the same material when one of them zooms through the information and the other one needs more practice with something. Flexibility is key to good teaching.

1. 5x-4x=60 (least common multiple was 20)

2. 3x-2x+x=5 (least common multiple was 6)

3. $2(2x-4)-15=10b$ or $4x-8-15=10b$ if she goes one step further. (least common multiple is 10)

4. $ad*a+ac*b-cd*c=acdx*d$ (least common multiple is acdx)

# Lesson 112 – Doodles Do Algebra

Finally we are ready to solve equations!

Today your child starts learning about how to solve simple equations of the first degree (these are equations where the unknown is not squared or cubed, or taken to any power beyond one).

But the first step, just like every other subject your child learned in this curriculum, is to learn the new vocabulary. So the worksheet today is simply matching the vocabulary word to its definition.

And, because my own kids love the opportunity to get creative when they do any lessons, the DoodleCat on this worksheet asks your child to draw her an attic to play in and a creative way to get up to it (also look for the mouse that is hiding on the sheet).

1. a

2. b

3. f

4. c

5. g

6. d

7. h

8. e

# Lesson 111 – Doodles Do Algebra

Today your child learns how to resolve a fraction into a series and that sometimes you use this technique to approximate numerical answers to complex and perhaps unsolvable questions in physics, biology, medicine, or even architecture.

DoodleTwo takes your child step by step through the conversion of $1/(1-x)$ to $1+x+x^2+x^3+...$. and then asks your child to try to show that $1/(1+x)=1-x+x^2-x^3+...$. This is really quite simple – it is just a matter of copying the format of the example that is worked out on the worksheet and changing the ’1+x’ term to a ’1-x’ as you go. If you have trouble, just ask in the comments below and I will be happy to help.

# Lesson 110 – Doodles Do Algebra

Today your child learns to reduce a complex fraction into a simple one. It sounds complicated but it is really simple. First you convert it from a mixed quantity into a fraction and then divide.

DoodleOne shows your child how to do the reduction using the example of 2 1/3 / 3 1/2 and walking your child through the example steps on the worksheet is all you need to do. If your child is confused, walk him through one or two of the problems as well – remember the important thing is that he really understands it, not that he does it all on his own when he is confused. Each new lesson builds on past lessons, so he will get a chance to show you he knows the material later if he needs the extra help now.

1. $ad/cb$

2. $21/2a$

3. $2ax/(2+ay)$

# Lesson 109 – Doodles Do Algebra

Today your child learns the second part of dividing algebraic fractions, when you divide one fraction by another. DoodlePig (the guinea pig) shows your child how on the worksheet. I don’t think it needs any additional explanation.

1. $2ay/3$

2. $(x+y)(a-b)$

# Lesson 108 – Doodles Do Algebra

Welcome to August! Today is the start of the new school year for us, so it always is a special day full of new supplies and new lessons to look forward to.

This lesson is dividing algebraic fractions by a single term. DoodlePoodle does a great job explaining in on the worksheet.

1. $2a/5b$

2. $(x+a)/4+3n)$

3. $(a+b)/(x+y)$

# Lesson 107 – Doodles Do Algebra

Today’s lesson is multiplying algebraic fractions. It is just like multiplying numerical fractions.

1. $(2x)(xb)/(ay)=2x^2b/ay$

2. $(2x^2)(10y)/5y=4x^2$

3. $(3x)(7b)/(4)(8)=21xb/32$

4. $(2x^2)(10y)/(5y)(3x)=4x$

# Lesson 106 – Doodles Do Algebra

Today’s lesson is subtracting algebraic fractions and it is just like adding fractions, only you have to keep track of the negative signs.

1. $5a/10-2a/10=3a/10$

2. $10x/15-9x/15=x/15$

3. $(x+y-(x-y))/3=2y/3$

4. $9a/12x-8a/12x=a/12x$

5. $((x+y-(x-y))/((x-y)(x+y))=2y/(x^2-y^2)$

# Lesson 105 – Doodles Do Algebra

Today’s lesson is adding fractions together. It is really simple and just like adding numerical fractions. DoodleOne gives a good explanation on the worksheet, so I won’t go into it here.

1. $b/2+b/5+b/10=5b/10+2b/10+b/10=(5b+2b+b)/10=8b/10=4b/5$

2. $(bc+ac+ab)/abc$

3. $(x+y+x-y)/4=2x/4=x/2$

4. $(2(x-y)+2(x+y))/((x+y)(x-y))=(2x-2y+2x+2y)/(x^2-y^2)=4x/(x^2-y^2)$

5. $2a+3a+3a/5+a+2a/3=6a+3a/5+2a/3=900a/15+9a/15+10a/15=919a/15$

# Lesson 104 – Doodles Do Algebra

Today is just more practice reducing fractions to mixed quantities.

1. $2x(2a-x)/(2a-x)-a^2/(2a-x)=2x-a^2/(2a-x)$
2. $a^2/(a-x)-2ax/(a-x)$
3. $(a+x)(a^2-ax^2-a^2x+x^2)/(a+x)-x^4/(a+x)=(a^2-ax^2-a^2x+x^2)-x^4/(a+x)$