Doodles Do Algebra – Lesson 13

unknowns-and-the-29-articles-of-algebra

Update: This Lesson Is Part of Book 2 of Doodles Do Algebra: “Unknowns And The 29 Articles of Algebra” available on Amazon.

 

Teacher’s Notes:

DoodleCat takes your child through the concept of ‘less than’ (from an algebraic standpoint) today.

There are two basic operations at work here, the first is dealing with addition/subtraction across a complete equation, and the second is multiplication/division across a complete equation, both in order to solve an equation for an unknown. The goal is for your child to intuitively understand what is going on and why when he is solving these problems. When I grew up, we learned that you need to remember to “do the same thing to both sides of an equation,” but it wasn’t until I tried teaching this to my own kids that I realized how arbitrary and ultimately useless this method was. Algebra books used by people in the 1600’s, 1700’s, and into the 1800’s (from Cocker’s to Ray’s) teach in a much more straightforward and clear manner by showing why you are learning at each step of the way through algebra. You reason your way through instead of memorizing facts. This works for my family as one of the basic principles of the classical education methods we use is that the memorization stage (or grammar stage) of learning is only the first step in the process of learning and by the time most children are ready for algebra, they are more than ready to learn the why instead of blindly memorizing the what.

 

Worksheet:

Available in the book, “Unknowns And The 29 Articles of Algebra, Book 2 of Doodles Do Algebra

 

If your kiddo is having trouble seeing how to solve these types of problems, you can get back out the pennies, or jellybeans. For solving a problem like x-1 = 3, you can make one pile of 3 beans/pennies and ask your child to count out an equivalent pile of their own (your child now counts out 3 beans). “But,” you say, “I am going to need to take one of your beans away. So now how many do you need to put out in the first place so that after I take one away, there will be three left?”

You can follow the same process so that your child can easily see the answer to 2x = 6: Make a pile of 6 pennies. Then ask your child to make their own pile of pennies that is equal to yours. Now tell your child that you are going to double their pile before you check to see if they have the same number in their pile as you have in yours, so how many pennies do they need to put out in that case? Right, 3.

Answers:

1. This one is possibly a bit tricky: ‘a number less 1’ means the number minus 1 equals 3, or x-1=3. Once you show this to your child, assuming they are having issues getting it, it is easier to see that x=4.

2. This problem is the same as the first one, only it isn’t a word problem. The idea is to show your child how the word problem describes the algebraic problem, and vice versa. So the answer is 4.

3. 2x=6, and x=3

4. 3x=12 and x=4

By now your child will be understanding how to manipulate and solve real problems using basic algebra. In a few lessons, we will switch gears and spend a while learning the vocabulary of algebra using some poetry and games. After that your child will be ready for more complex problems.

As always, please leave any feedback you have.

Thanks!

doodlemom

Doodles Do Algebra – Lesson 12

 

Update: This Lesson Is Part of Book 2 of Doodles Do Algebra: “Unknowns And The 29 Articles of Algebra” available on Amazon.

 

unknowns-and-the-29-articles-of-algebraTeacher’s Notes:

Today’s lesson is really self-explanatory. Your child is learning and then practicing how to translate the concept of “twice” from an algebraic standpoint. The idea is that “twice x+1” is the same as “twice x” plus “twice 1”, which works out to 2x+2.

 

Worksheet:

Available in the book, “Unknowns And The 29 Articles of Algebra, Book 2 of Doodles Do Algebra

 

Answers:

  1. 3x+3
  2. 4x+4
  3. 5x+5
  4. 6x+3
  5. 8x+4
  6. 10x+5
  7. 3x+3

As always, please leave feedback when you can!

Thanks,doodlemom

 

Doodles Do Algebra – Lesson 11

unknowns-and-the-29-articles-of-algebra

Update: This Lesson Is Part of Book 2 of Doodles Do Algebra: “Unknowns And The 29 Articles of Algebra” available on Amazon.

 

Teacher’s Notes:

Today’s lesson walks your child through the process of solving the problem on their own. Hopefully your kid can do today’s sheet all by themselves, but if not, just walk them through the steps by reading aloud.

 

Worksheet:

Available in the book, “Unknowns And The 29 Articles of Algebra, Book 2 of Doodles Do Algebra

 

Answers:

number of worms:   x (James)              x+2 (John)              x + 2 + 3 (Tom)

Total = 28 = x + x + 2 + x + 2 + 3

28 = 3x + 2 + 2 + 3

28 223 = 3x => 21 = 3x => x=7 (that is how many worms James caught).

So James caught 7 worms, John (who caught two more than James) caught 9 worms, and Tom (who caught 3 more than John) caught 12 worms.

doodlemom

Doodles Do Algebra – Lesson 10

kindle-book-cover

Update: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today is a review day for your child to practice the last 4 lessons, brought to you by DoodlePig.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

  1. This problem walks your child through the process of solving how many oranges, peaches, and lemons Tom purchased. So if he bought an equal number of each type of fruit, and we say that is “x”, or the unknown variable, then the total price of peaches was 2 cents times “x” peaches. Then the total price of oranges was 4 cents times “x” oranges, and the total prices of lemons was 3 cents times “x” lemons. You add 2x + 4x + 3x together to get 9x, and that equals the total price of 45 cents paid for the whole pile of fruit. So if 9x = 45, x = 5. That means that Tom bought 5 of each type of fruit.
  2. Now your child gets to do a problem that is abstract. It is not about buying fruit or how old people are, but just simply numbers. If the first number is x, and the larger exceeds the smaller by 15, then the second number is x+15. The Sum of the two numbers is 25 and equals x plus x+15. So solving for x, 25 = x + x + 15, or 25 = 2x + 15, or 25-15 = 2x, or 10 = 2x, and finally x=5.

 

I hope that you and your child are enjoying these algebra pages. The next book will be out soon. In the meantime you can get all the information in the upcoming lessons as well as worksheet downloads for pages that have not yet been published on this site.

As always, I welcome feedback.

Doodles Do Algebra – Lesson 9

kindle-book-coverUpdate: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

DoodleCat pretty well explains the problems for today so I think I just need to provide the answers. The problems are essentially like the ones in Lesson 8, but your child just gets more practice.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

  1. first number = x, and second number = 2x + 5, then Sum = 35 = x + 2x + 5. Now Solve for x: 35 = 3x + 5, then 35-5 = 3x, then 30 = 3x, then x=10. So, first number = 10 and second number = 25.
  2. smaller flock = x, and larger flock = 5x + 5, then total = 83 = x + 5x + 5. Now Solve for x: 83=6x + 5, then 83-5 = 6x, then 78 = 6x, then x=78/6, or x=13 (it is very helpful here to know your 13 times tables). So the smaller flock has 13 sheep, and the larger flock has 70 sheep.

 

Doodles Do Algebra – Lesson 8

kindle-book-cover

Update: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today DoodlePoodle gets in on the action and we add more complexity to the problems covered in previous lessons. The underlying goal is for your child to start thinking in terms of “x” as a placeholder for some unknown quantity.

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

1. This problem takes your child step by step through the process of figuring out how many boys and how many girls there are in the Smith family. We choose the unknown, x, to represent the number of girls (because there are 3 more boys than girls, thus the smallest number is the number of girls). Based on that the number of boys is x+3 (or three more boys than girls). Then the total (boys and girls together) is (x+3) + x = 13 (the total number of kids). Solving for x means adding x + 3 + x to get 2x + 3. Then to solve for x, you follow along the example given by DoodlePoodle. So 2x + 3 = 13 is the same as 2x = 13 – 3 (you can explain this with the idea that if you take (or add) something from one side of an equation, you have to do the same to the other side of the equation for the equation to stay the same). This is one of those steps where your child just needs to take it on faith right now and learn how to solve these types of problems. The deeper understanding of why it is so comes later. So back to the problem. 2x = 10 (since 13-3 is 10), and then 1x = 5 (when you divide both sides of an equation by the same number the unknown keeps the same value). And finally, x=5. So there are 5 girls (x) and 8 boys (x+3) in the Smith family.

2. This problem is completely abstract. Point out to your child the similarity with the first problem, and things will become clear. The sum of the two numbers in this problem is just like The Smiths having 13 children. They are both talking about the total, when you add up both kinds of unknowns (numbers in this case, and children in the previous problem). So Number One + Number Two = 35. Now, the problem also states that the difference between those two numbers is 5. That is the same thing as saying that one number is 5 more than the other, and that is just like the previous problem where there were 3 more boys than girls. So let’s pick Number One = x (this will be the smaller number). Then Number Two = x+5. So Number One + Number Two = The Total, or x + (x+5) = 35. Now we simplify (add in this case) to get 2x + 5 = 35. Then we subtract 5 from both sides, 2x = 30. And finally we divide both sides by 2 (the coefficient of the unknown – a good concept to start embedding in your child’s mind now) and x=15. So One Number is 15 and the Second Number is 20.

So you can see that we are slowly shifting your child away from a simple conceptual understanding of algebra and towards a practical ability to solve more complex, and sometimes abstract, problems. The goal is for your child to be familiar with how algebra basically works, and then to take a trip through a series of vocabulary and definitions (in order to start using Algebra-speak) before returning to Algebra in a more abstract fashion.

Please give me any feedback you have (good or bad) in the comments below!

 

Doodles Do Algebra – Lesson 7

kindle-book-cover

Update: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today your child begins using the concept of an unknown, or “x”, in order to simplify problems. DoodleOne, or darling daughter doodle, explains this in the cartoon and walks your child through a sample problem. For this math lesson, it really helps to walk your child through the steps, even if you are reading from the answer. What matters most is that he really understands how to do the problem, not if he can do it independently at this stage.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

The Doodle Daily today walks your child through the process of assigning x as an unknown and relating that unknown to the number of kids in the various classes Penny and her sisters attend at Co-Op.

If x represents the number in Grammar              x

what represents the number (of kids) in Geography?              3x (because there are 3 times as many kids studying Geography as Grammar)

in Math?              2 times 3x, or 6x (because there are twice as many kids studying Math as Geography, which can be represented by 3x)

6x – x is represented by the number of kids in Math minus the number of kids in Grammar, (this is just to check your child’s comprehension of how the x’s relate to the number of kids in each class and to make the next step easier to understand)

Now, There are 10 more kids in Math than in Grammar so (keeping in mind that there are more kids in Math) if we subtract the kids in Grammar from the kids in Math, we will be left with 10 kids. [You can show this with a pile of pennies (say for Math kids) and nickels (for Grammar kids). Make sure to point out that there are ten more pennies than nickels, and then have your child take one away from each pile (pennies and nickels) until none are left. This is a hands on way of showing the subtraction of kids in Grammar from kids in Math, in case you get that blank stare that comes shortly before tears of frustration as you explain the problem.] Getting back to the problem, this means that 6x – x is actually 10,

What is x equal to?              If 6x’s minus one x is 10, then 5 x’s (6 x’s minus one x) is 10, and x is 2.

So there are 2 kids in Grammar (that was equal to our x)

There are 6 kids in Geography (that was equal to our 3x)

And there are 12 kids in Math (that was equal to our 6x, or twice the number in Geography).

 

Please give me any feedback you have (good or bad) in the comments below!

 

Doodles Do Algebra – Lesson 6

kindle-book-cover

Update: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today’s algebra lesson expands on the last 5 and adds a new complexity: figuring out quantities of two items when they are not the same but are related by some constant (like one being twice the other or one being three less than the other…)

This is where it becomes helpful to think in terms of x’s, but the problems can still be generally done in your (or your child’s head). For those of you who remember algebra class, this is the if you have two independent unknowns, you need two independent equations to solve for the unknowns” maxim at work. Last week all the problems were “one independent unknown with one independent equation” type, even if it didn’t always seem like it, since each problem collapsed down into one unknown even if we were talking about two people doing something. They were always equal distances away from each other, or an equal number of fruits were purchased, or the stick was broken in the middle. Today we look at how to solve the problems where things are not equally divided. It isn’t hard, as you might imagine, just different.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

  1. We have two numbers, that are not equal but one is twice the other, that when you add the larger to twice the smaller, the sum is 28. If you are reading this in a quiet moment, and are fully awake and caffeinated, you will notice that the larger number is 2 times the smaller and you add the larger (2 times the smaller) to the twice the smaller (2 times the smaller again!) to get a total of 28. That means you are adding 2 times the smaller to 2 times the smaller, so you get 2+2 (think in terms of 2 x’s plus 2 x’s makes 4 x’s, or apples or pennies just like last week), or 4 times the smaller equaling 28, or the smaller number being 7 and the larger, twice the smaller, or 14. So the smaller number is 7, larger number is 14.
  2. Now we do the same thing as in the first problem but instead of numbers we are talking about the price of oranges and lemons. Auntie 2 oranges and 5 lemons, and spent 27 cents. The second and most important fact you are given is that an orange costs twice as much as a lemon (last week, this problem would have been simplified to lemons and oranges costing the same, but this week we are taking off the training wheels!) So whatever you spent on a lemon, let’s call it x cents, you spent twice that, or 2x on an orange. Now you add up 5 lemons at x cents (or 5x) plus 2 oranges at 2x cents (or 4 x) equals 27 cents total. So 4x plus 5x equals 27, or 9x equals 27, or x equals 3 cents. Now you go back to the definition of lemons costing x cents and oranges costing twice that or 2x cents. So a lemon costs 3 cents and an orange costs 6 cents.
  3. Alright, now we do it one more time only this time the unknown is the number of fruit, not the price. Crockett (who is our big black standard poodle) bought apples and peaches. He bought twice as many apples as peaches so x is the number of peaches and 2x is the number of apples. He spend a whopping 24 cents on the whole pile of fruit, and apples cost 2 cents and peaches cost 4 cents each. So Crockett bought x peaches at 4 cents (or 4x) plus 2x apples at 2 cents (or 4x) and that added up to a total price of 24 cents. So 4x plus 4x equals 24, or 8x equals 24, and so x equals 3 fruits. Now we go back to our definition of twice as many apples purchased as peaches, so Crockett bought 3 peaches and 6 apples.

Please give me any feedback you have (good or bad) in the comments below!

 

Doodles Do Algebra – Lesson 5

kindle-book-cover

Update: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today’s lesson summarizes and reviews the ideas learned in the last 4 lessons. You have now met all the players in our math book. Each week, or five days, your child will get a lesson from each of our characters.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

  1. The crow breaks the 30 foot pole in two equal parts (half) so each part is one half of the total, or one half of 30 feet. Answer: each piece is 15 feet long.
  2. Your basement has a total of 20 barrels of root beer, the number of full barrels are four times greater than the number of empty barrels, so the total number of barrels is divided 5 “ways” – 4 for the full barrels and 1 for the empty. And that means that the number of barrels in each “way” is 20 (total number) divided by 5 (the number of “ways”), or 4. So there are 4 empty barrels and 4 times 4, or 16 full barrels.
  3. Here the farmer spends a total of $60. He buys a sheep, a cow, and a moose (this must be Canada or Montana). The cow is worth 3 sheep, and the moose is worth 2 cows, or 2 times 3 which equals 6 sheep. So the total money spent in sheep is 1 (for the sheep) plus 3 (for the cow) plus 6 (for the moose) which equals 10. So a sheep costs $60 divided by 10, or $6. The the final answer is Sheep: $6, Cow: $18, and Moose: $36.
  4. The two overly optimistic men on pogo sticks are hopping towards each other at the same speed. This means that they will meet in the middle. Since they start a whopping 48 miles apart, they will each travel 24 miles before they meet. If it takes them 6 hours to meet (in the middle), then each one travels 24 miles in 6 hours, or 24 divided by 6 miles per hour, which equates to 4 miles per hour, unrealistic at best but fun nonetheless.

By now, your child should be getting the basic hang of doing these types of problems in his or her head. The more fun you keep things, the easier it will go. And they will see it all again later, so just explain the problems that he or she gets stuck on and move along.

 

Doodles Do Algebra – Lesson 4

kindle-book-coverUpdate: This Lesson Is Part of Book 1 of Doodles Do Algebra: “Starting Out With Mental Algebra” available on Amazon.

 

Teacher’s Notes:

Today’s lesson continues the same ideas as lessons 2 and 3, but begins to think more about using the concept of an unknown, or “x” as a placeholder. DoodleCat begins by reinforcing the idea that 2x is two x’s and 3x is three x’s and so on. The questions that follow in the Doodle Dailies rely on using that idea.

 

Worksheet:

Available in the book, “Starting Out With Mental Algebra, Book 1 of Doodles Do Algebra”

 

Answers:

  1. Have your child think of the number of lemons and oranges as “x”, which we can do since Kendra bought an equal number of lemons and oranges. If this is hard, you can equate the concept of “x” to fruit as a general class. So Kendra bought 2x plus 3x dollars, or 5x, worth of fruit for a total of 30 dollars. Then the number of each type of fruit was 30 divided by 5, or six. So Kendra bought 6 lemons and 6 oranges.
  2. This is the classic train problem, only we use a Dad and a Gpa instead. Dad and Gpa are moving at the same speed towards each other, which means we can say that they are both 4x miles (miles per hour times the unknown hours) away from the middle where they will meet. That means they are a total of 4x miles plus 4x miles away from each other, or 8x miles which is equal to 40 miles (the distance given in the problem). This means that x is 40 divided by 8, or 5 – so Dad and Gpa will meet in 5 hours.
  3. This last problem is maybe the most practical. It is in some senses like problem one, but the two pipes that are emptying the cistern are doing it at different rates (one discharges four times faster that the other). So one pipe discharges at a rate of 4x gal/hour and the other at a rate of x gal/hour and when you add the two together, you empty the 100 gallon cistern in 2 hours. So 4x gal/hour plus x gal/hour equals 100/2 gallons/hour (see how we used the capacity of the cistern and the total time to empty it to calculate the constant that equals the overall rate of discharge? That is because rate means the number of units changed per time elapsed and in our case we have gallons and hours.) So 4x + x = 100/2, or 5x =50, and x =10. That means that one pipe has a flow of 10 gallons/hour and the other pipe has a flow of 40 gallons/hour (since one pipe discharged at a rate 4 times the other).

If this seems a bit daunting, not to worry just lead your child through it and don’t worry – we will come back to it again in future lessons. Your child should not get discouraged – the main point is to expose him (or her) to algebra-type thinking at this stage.

 

Please give me any feedback you have (good or bad) in the comments below!