This Blog exists for the collective benefit of ALL Algebra students. While the posts are usually specific to Mr. Chamberlain's class, any and all "algebra-ticians" are welcome. The more specific your question (including your own attempts to answer it) the better.

Mr Chamberlain, I am going over my notes and I don't really understand how to use the counters.I know the answer, but don't know how I can get the answer using the counters.Can you explain it?

Mr. Chamberlain, I forgot which number line method is directionally, symmetrically, and a difference maker. Could you please inform me on which is which?

THANKS FOR ASKING... I'm sure a lot of folks have this question!! STAY WITH IT!! The more you can UNDERSTAND the multiple ways of visualizing a math problem, the more success you will have solving the diverse and more complex set of problems that lie ahead in your budding mathematical career.

The BIG IDEA here is that if you can UNDERSTAND how to represent math problems visually, it is more likely that you can use "simple math" problems to solve them.

Suggestion: You'll need to draw pictures as you follow along with the logic below.

"Directionally" means that you use the "arrows" or "vectors" to move in one direction or another. Then you use your UNDERSTANDING to decide which simple math problem will help you out.

If you were doing 30 - 70, you would start at zero and draw an arrow to 30, and then draw an arrow of length 70 to the left. Clearly you go past zero on your "trip" to the left. How far past zero??? Clearly, it took you 30 units to get back to zero, so since you are traveling a total of 70 units, the math problem of 70 - 30 informs you that you need to go another 40 units past zero, which clearly is -40.

Symmetrically means that if you perform an operation on the left side of zero, you can REFLECT it to the right side of zero, where it typically makes more sense, since you are then working with "comfortable" positive numbers.

Let's take -85 + 40. Make a mark on -85. Make an arrow that is roughly 40 long traveling to the right. Next, you can reflect that point and line to the other side of zero. Now, you have a point on +85 and a line traveling 40 to the left. What is the "simple math" problem that matches up with the picture on the right side of zero? Clearly it is 85 - 40 which (even your friendly 4th grader) would have an answer of +45. Just as clearly, you can now reflect this solution back to the left side of zero, such that the clear answer to -85 + 40 = -45.

The "Difference Maker" approach is as follows: Let's take 55 - 180 for example. This sentence can be read as "What is the difference between 55 and 180." An even more accurate way to read this sentence is "How much more or less than is 55 compared to 180." (note: remember, this is subtraction, so order matters!).

So, make a mark on 55 and a mark on 180. Draw a segment from 55 to 180. Ask a 4th grader to look at that picture and tell you how to figure out the difference... they will tell you that 180 - 55 = 125. You will intelligently consume that answer and say "Yes, 125 is the absolute value of the difference, but since 55 is clearly 125 LESS than 180, the answer to the original math problem is that 55 - 180 = -125.

It is not really important that you remember and memorize the terms "directionally," "symmetrically," or "difference maker." The important learning here is that if you can use counters and line numbers to represent simple math sentences, you will become more skilled at visual representations of more complex problems, rather than trying to memorize formulas and procedures. YOU WILL UNDERSTAND!! As your math "coach" I encourage you to play and struggle and play and see how it goes.

If you are struggling too much, please come in for some morning math madness!

Excuse me, but I don't understand what the terms "directionally", symmetrically, and "difference maker" in terms of the number line math. I took brief notes on the topic, but I can't tell which is which. Please help?

basically, see my answer above... sometimes you can combine the methods on one number line... for example you can draw the arrows (that's directionally) and then reflect them (that's symmetrically) to find your simple math problem.

Fair enough! Directional simply means that you are traveling with the arrows (aka vectors).

When you travel with the arrows in the negative side of town, you can always reflect "things" over to the positive side of town and do the work there. Why?? Cuz most of us are positive people and we find it easier to work on the positive side of zero!

We could do a difference making operation on the negative side of town, i.e. -7 - (-17) and flip it over to positive by using 7 and 17.

You KNOW that 17-7 = 10 so that is your absolute value You KNOW that -7 is greater than -17 (it's further to the right, right?) Therefore -7 - (-17) must = +10

Note: the "+10" is for emphasis, you would write the answer as simply: -7 - (-17) = 10

Mr. Chamberlain, I have a question concerning problem 2 on homework 1-5 with the counters. I know the answer, but I am not sure how to explain how to use the counters to get to the answer.

Mr Chamberlain, I am going over my notes and I don't really understand how to use the counters.I know the answer, but don't know how I can get the answer using the counters.Can you explain it?

ReplyDeleteFor help with the integer counters, visit this link and click on the videos in the first column "New Language of Mathematics"... lmk if it helps.

DeleteOf course, you're welcome to come in for morning math madness, too.

Cut-and-paste this link into your browser:

http://mathchamberacademy.pbworks.com/w/page/51673284/Unit%201%20-%20Foundations%20for%20Algebra

Thanks, it really helped.

DeleteYAY!! THREE CHEERS FOR THE BLOG!! THANKS FOR THE FEEDBACK!!

DeleteI had a similar problem as Presley, and after using video #2, I remembered how to do it. Thanks Mr. C. :)

DeleteMr. Chamberlain, I forgot which number line method is directionally, symmetrically, and a difference maker. Could you please inform me on which is which?

ReplyDeleteTHANKS FOR ASKING... I'm sure a lot of folks have this question!! STAY WITH IT!! The more you can UNDERSTAND the multiple ways of visualizing a math problem, the more success you will have solving the diverse and more complex set of problems that lie ahead in your budding mathematical career.

DeleteThe BIG IDEA here is that if you can UNDERSTAND how to represent math problems visually, it is more likely that you can use "simple math" problems to solve them.

Suggestion: You'll need to draw pictures as you follow along with the logic below.

"Directionally" means that you use the "arrows" or "vectors" to move in one direction or another. Then you use your UNDERSTANDING to decide which simple math problem will help you out.

If you were doing 30 - 70, you would start at zero and draw an arrow to 30, and then draw an arrow of length 70 to the left. Clearly you go past zero on your "trip" to the left. How far past zero??? Clearly, it took you 30 units to get back to zero, so since you are traveling a total of 70 units, the math problem of 70 - 30 informs you that you need to go another 40 units past zero, which clearly is -40.

Symmetrically means that if you perform an operation on the left side of zero, you can REFLECT it to the right side of zero, where it typically makes more sense, since you are then working with "comfortable" positive numbers.

Let's take -85 + 40. Make a mark on -85. Make an arrow that is roughly 40 long traveling to the right. Next, you can reflect that point and line to the other side of zero. Now, you have a point on +85 and a line traveling 40 to the left. What is the "simple math" problem that matches up with the picture on the right side of zero? Clearly it is 85 - 40 which (even your friendly 4th grader) would have an answer of +45. Just as clearly, you can now reflect this solution back to the left side of zero, such that the clear answer to -85 + 40 = -45.

The "Difference Maker" approach is as follows:

Let's take 55 - 180 for example. This sentence can be read as "What is the difference between 55 and 180." An even more accurate way to read this sentence is "How much more or less than is 55 compared to 180." (note: remember, this is subtraction, so order matters!).

So, make a mark on 55 and a mark on 180. Draw a segment from 55 to 180. Ask a 4th grader to look at that picture and tell you how to figure out the difference... they will tell you that 180 - 55 = 125. You will intelligently consume that answer and say "Yes, 125 is the absolute value of the difference, but since 55 is clearly 125 LESS than 180, the answer to the original math problem is that 55 - 180 = -125.

It is not really important that you remember and memorize the terms "directionally," "symmetrically," or "difference maker." The important learning here is that if you can use counters and line numbers to represent simple math sentences, you will become more skilled at visual representations of more complex problems, rather than trying to memorize formulas and procedures. YOU WILL UNDERSTAND!! As your math "coach" I encourage you to play and struggle and play and see how it goes.

If you are struggling too much, please come in for some morning math madness!

Thanks Mr. C, that really helped me.

DeleteYes me too. Thank you, I had the same question. This helped alot!

DeleteI had the same question go figure. Thanks for the explanation!

DeleteThanks Mr. C! Also had the same question.

DeleteJust for the blog's sake, I definitely had that same question also.

DeleteThank you also this post helped me out a lot with some of the questions

DeleteWelp. Looks like this was a common question... XD

DeleteExcuse me, but I don't understand what the terms "directionally", symmetrically, and "difference maker" in terms of the number line math. I took brief notes on the topic, but I can't tell which is which. Please help?

ReplyDeletesee above!

DeleteOk, I read it, and I understand now. Thanks for taking the time to type that really long explanation. =)

DeleteHi Mr Chamberlain,

ReplyDeleteFor the number line problems on the packet where it says solve symmetrically and directionally, should I draw two number lines?

basically, see my answer above... sometimes you can combine the methods on one number line... for example you can draw the arrows (that's directionally) and then reflect them (that's symmetrically) to find your simple math problem.

DeleteI understand how to do directionally and symmetrically, but how do we do both in the same problem like it's asking us too.

ReplyDeleteI red your response to Kia's question, but I still don't get how we would do it.

ReplyDeleteFair enough!

DeleteDirectional simply means that you are traveling with the arrows (aka vectors).

When you travel with the arrows in the negative side of town, you can always reflect "things" over to the positive side of town and do the work there. Why?? Cuz most of us are positive people and we find it easier to work on the positive side of zero!

We could do a difference making operation on the negative side of town, i.e. -7 - (-17) and flip it over to positive by using 7 and 17.

DeleteYou KNOW that 17-7 = 10 so that is your absolute value

You KNOW that -7 is greater than -17 (it's further to the right, right?)

Therefore -7 - (-17) must = +10

Note: the "+10" is for emphasis, you would write the answer as simply: -7 - (-17) = 10

Mr. Chamberlain, I have a question concerning problem 2 on homework 1-5 with the counters. I know the answer, but I am not sure how to explain how to use the counters to get to the answer.

ReplyDeleteWell, at this point in the evening, just do the best you can.

DeleteDid you check out the info from earlier in this post? Ill repeat it here for you...

For help with the integer counters, visit this link and click on the videos in the first column "New Language of Mathematics"... lmk if it helps.

Of course, you're welcome to come in for morning math madness, too.

Cut-and-paste this link into your browser:

http://mathchamberacademy.pbworks.com/w/page/51673284/Unit%201%20-%20Foundations%20for%20Algebra

We will be reviewing counters again in class tomorrow AND learning how to multiply.