# Lesson 103 – Doodles Do Algebra

Today your child learns that sometimes all you can do is reduce a fraction into a mixed quantity and DoodleCat provides an example.

1. $(ab+b^2)/a=ab/a+b^2/a=b+b^2/a$

2. $(cd-d^2)/d=cd/d-d^2/d=c-d$

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

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

5. $(a^2-x^2+3)/(a+x)=(a^2-x^2)/(a+x)+3/(a+x)=(a+x)(a-x)/(a+x)+3/(a+x)=(a-x)/(a+x)+3/(a+x)$

# Lesson 102 – Doodles Do Algebra

Today’s lesson makes the connection between dividing one term by another as being the same as making a fraction and then reducing it, as your child has done over the last few days.

1. $5x^2y/3xy^2=5x/y$

2. $15a^2b^2c/25a^3bc=3*5*a*a*b*b*c/5*5*a*a*a*b*c=3b/5a$

3. $25abc/5ac^2=5b/c$

4. $amn^2/a^2m^2n=n/am$

5. $25ax^2/(5ax^2-5axy)=25ax^2/5ax(x-y)=5x/(x-y)$

6. $(3m^2+3n^2)/(15m^2+15n^2)=3(m^2+n^2)/15(m^2+n^2)=1/5$

7. $(x^3y^2+x^2y^3)/(ax^2y+axy^2)=x^2y^2(x+y)/axy(x+y)=xy/a$

8. $(x^2+2x-3)/(x^2+5x+6)=(x+3)(x-1)/(x+3)(x+2)=(x-1)/(x+2)$

# Lesson 101 – Doodles Do Algebra

Today your child starts to put more of the factoring techniques he learned last month to use. DoodleTwo explains how. Basically, first you factor the polynomials in the numerator and denominator, and then you cancel like terms. If your child has trouble remembering the concepts of factoring polynomials, just help her through a few problems together to get her started. Not to worry, this concept (like all the others) will be reviewed in the future when we use it to build a more complex concept.

1. $3(z^2-8z+3)/4(z^2-8z+3)=3/4$

2. $5a(a+x)/(a+x)(a-x)=5a/(a-x)$

3. $(n-1)(n-1)/(n+1)(n-1)=(n-1)/(n+1)$

4. $7a(2a-b)/5c(2a-b)=7a/5c$

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

6. $(a^2+b^2)/(a^2+b^2)(a^2-b^2)=1/(a^2-b^2)$

7. $(x+y)(x-y)/(x+y)(x+y)=(x-y)/(x+y)$

8. $x^2(x-a)/(x-a)(x-a)=x^2/(x-a)$

9. $2x(x-3)/(x-3)(x+2)=2x/(x+2)$

10. $(x+5)(x-3)/(x+5)(x+3)=(x/3)/(x+3)$

# Lesson 100 – Doodles Do Algebra

Today your child plays with reducing a fraction to its lowest terms. This means that a fraction gets reduced to prime factors in the numerator and the denominator. DoodleOne explains how to do it very thoroughly on the worksheet.

1. $4a^3x^2/6a^4=2*2*a*a*a*x*x/2*3*a*a*a*a=2*x*x/3*a=2x^2/3a$

2. $3a/4x$

3. $3a^2x/4y^4$

4. $3x/4y$

5. $3y^2z/2$

6. $2*2*2*a*a*b/(2*2*3*a*b*b+2*2*a*b*c)=2*2*2*a*a*b/2*2*a*b*(3*b+c)=2*a/(3*b+c)=2a/(3b+c)$

in this problem you need to factor like terms out of the denominator using the techniques you learned in the lessons on factoring.

7. $2acx(ax+1)/2*5*a*c*c*x=(ax+1)/5c$

8. $5ab(a+b)/5ab(c+d)=(a+b)/(c+d)$

# Lesson 99 – Doodles Do Algebra

Today your child learns the last proposition – if you multiply (or divide) the top (numerator) and bottom (denominator) of a fraction by the same thing, the fraction keeps the same value.

Multiply Numerator $(4*2)/16=8/16=1/2$

Multiply Denominator $4/(16*2)=4/32=2/16=1/8$

Divide Numerator $(4/2)/16=2/16=1/8$

Divide Denominator $4/(16/2)=4/8=2/4=1/2$

Multiply both Numerator and Denominator $(4*2)/(16*2)=8/32=4/16=1/4$

Divide both Numerator and Denominator $(4/2)/16/2)=2/8=1/4$

# Lesson 98 – Doodles Do Algebra

Today your child learns a few propositions that define how you multiply and divide fractions. It is also a review of what numerator and denominator mean.

Multiply Numerator $(2*2)/18 = 4/18 = 2/9$

Multiply Denominator $2/(18*2)=2/36=1/18$

Divide Numerator $(2/2)/18=1/18$

Divide Denominator $2/(18/2)=2/9$

Note: you can point out to your child the symmetry between multiplying the numerator and dividing the denominator here after she has completed the math table on the worksheet.

# Lesson 97 – Doodles Do Algebra

Today we begin talking about fractions. And just like every other new topic in algebra, we begin by learning some vocabulary. And so today’s exercise is to match definitions with vocabulary, which is a nice break after the more intensive topics you and your child waded through the last few days.

The answers are given within the worksheet itself.

Have some fun today – maybe give DoodlePoodle some sunglasses or a funny hat or just color him in like he just fell into a paint bucket (that is a good part of what the doodles on the worksheets are for).

# Lesson 96 – Doodles Do Algebra

Today your child learns about the relationship between least common multiple and greatest common factor. The least common multiple is the least quantity that contains the quantities exactly and the greatest common factor contains all the factors common to the quantities. So if you multiply together all the quantities and divide them by their greatest common factor, you will be left with the least common multiple.

1. $24a^4x^2y(a+x)$

2. $12a^2(a-x)(a+x)x^4$

3. $24a^4x^3y^2(x-y)$

4. $60a^2x^5(x+y)(x-y)^2$

# Lesson 95 – Doodles Do Algebra

Today we start learning about Least Common Multiple.

The Least Common Multiple is the least quantity that will exactly contain two or more other quanities. This means you can divide the least common multiple by any of the other quanities and there will be no remainder.

DoodleOne gives examples of 2, 3, and 6 that have a least common multiple of 6. You find it by multiplying together all the primes in all the quanities.

The best way to keep track is in a table.

Example:

So the least common multiple of ax, bx, and abc look like

prime              first quantity              second quantity              third quantity

ax              bx              abc

a              factor this out where you can and you get…

x              bx              bc

x              factor this out where you can and you get…

1              b              bc

b              factor this out where you can and you get…

1              1              c

c              factor this out where you can and you get…

1              1              1

now you have the first column with all the primes. The Least Common Multiple is all those primes multiplied together: axbc.

1.

prime              first quantity              second quantity              third quantity

$4a^2$              $3a^3x$              $6ax^2y^3$

3              factor this out where you can and you get…

$4a^2$              $a^3x$              $2ax^2y^3$

2              factor this out where you can and you get…

$2a^2$              $a^3x$              $ax^2y^3$

2              factor this out where you can and you get…

$a^2$              $a^3x$              $ax^2y^3$

a              factor this out where you can and you get…

$a$              $a^2x$              $x^2y^3$

a              factor this out where you can and you get…

$1$              $ax$              $x^2y^3$

a              factor this out where you can and you get…

$1$              $x$              $x^2y^3$

x              factor this out where you can and you get…

$1$              $1$              $xy^3$

x              factor this out where you can and you get…

$1$              $1$              $y^3$

y              factor this out where you can and you get…

$1$              $1$              $y^2$

y              factor this out where you can and you get…

$1$              $1$              $y$

y              factor this out where you can and you get…

$1$              $1$              $1$

So the Least Common Multiple is $3*2*2*a*a*a*x*x*y*y*y=12a^3x^2y^3$

2. $2*2*2*3*a*a*a*x*x*x*x*y*y=24a^3x^4y^2$

3. $2*2*3*3*c*c*c*n*n*n*n*x*x*x*z=36c^3n^4x^3z$

4. $2*3*3*5*c*x*x*x*z*z*z*z=60cx^3z^4$

# Lesson 94 – Doodles Do Algebra

Today your child practices greatest common divisor some more.

1. Find the prime factors of $x^2+2x-3$ which are (x+3), (x-1). Then find the prime factors of $x^2+5x+6$ which are (x+2), (x+3). So the greatest common divisor is (x+3).

2. This problem you have to just divide $6a^2+7ax-3x^2$ by $6a^2+11ax+3x^2$ and find the greatest common divisor is 2a+3x.

3. The factors in $a^4-x^4$ are $(a^2+x^2)(a^2-x^2)$. The factors in $a^3+a^2x-ax^2-x^3$ are $(a+x)(a^2-x^2)$. So the greatest common divisor is $(a^2-x^2)$.

4. The greatest common divisor is (a-x).

All of the problems we did had no remainder in the division.

And that was the end of the lessons on Greatest Common Divisor.