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Archive for the ‘Master Equations’ Category

“Algebra Survival” Program, v. 2.0, has just arrived!

The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.


Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.








How to Decrease Algebraic Mistakes – Part 1

Anyone who has worked with students learning algebra knows the truth to the maxim:  MISTAKES HAPPEN.

This is the first in a series of posts offering PRACTICAL SUGGESTIONS for decreasing the number of algebraic mistakes students make.

No Mistakes

Let's Reduce Mistakes in Algebra!

First, it’s useful to recognize a key fact:  we can’t help students with mistakes if we don’t know what causes those mistakes.

Years of tutoring have taught me a lot about why students make mistakes. And one major cause of mistakes in algebra is that students combine terms that should not be combined. Not all their fault, though. Students are often confused about what they may and may not combine. And it is tricky!

Take a problem like this:  8 – 2(3x – 7)

Certainly some kids can simplify this expression with no trouble. But in my experience, many struggle with a problem like this (when first learning it), and quite a few stay befuddled for quite some time.

The biggest mistake is that students think they can and should combine the 8 and the 2 through subtraction, proceeding like this:

8 –  2(3x – 7)

=            6(3x – 7)

=            18x – 42

Q:  How can we help students avoid this mistake?

A:  Use a mark that show students what gets combined and what stays separate.

I will start to elaborate on how I do this in tomorrow’s post.

Extra, extra!   I thought it would be interesting for you readers to send in comments on the kinds of algebraic mistakes that “drive you up the wall” the most. When I get a number of comments in, I will conduct a poll to see which mistakes people find most vexing. Should be “fun.”

Not all variables are created equal

Are all variables the same?

Does every variable serve the same purpose?

When you think about it, you’ll see that the answer is “no.” Variables serve different purposes. When we explain this to students, we help them understand how variables work. Explaining this helps students understand how algebra “works.” You’ll see what I mean in a moment.

Consider the famous slope-intercept equation:  y = mx + b

A student recently asked me:  Are the  x and y variables the same as the m and b variables? What a great opportunity to explain something important!

I explained that the x and y variables serve completely different purposes than  the m and b variables. Here’s how.

The variables m and b are what I call “identifier” variables. By which I mean that they help us identify a specific line. To explain that, I asked the student a set of questions about something everyone understands — home addresses.

What would happen, I asked, if someone wanted to know where I live, and I told him that I live at 942? The student replied that this would not be enough info.

Then I asked, what if I told this person only that I live on Vuelta del Sur (a street name where I live in Santa Fe, NM)? Again the student said that this would not be enough info.

But what if I told this person that I live at 942 Vuelta del Sur. This, the student realized, would be enough information to enable someone to find my house. (All they have to do is Google me, and they’ll have my house AND directions!)

I pointed out that a similar situation applies to lines.

If I have a specific line in mind, and I want someone else to know the line I’m thinking of, is it enough to give this person just the line’s slope? No, for it could be any line with this slope, of which there are infinitely many parallel lines. What if I don’t give the slope but I do give the line’s y-intercept? Still not enough, as there are infinitely many lines that run through this y-intercept. But what if I tell the person both the slope and the y-intercept. Aha! The student could see — through drawings I made of this situation on a coordinate plane — that when you provide both slope and the y-intercept, there is one and only one line that could be indicated.


Three lines — the red and blue lines have the ...

Red & blue lines have same slope, so slope alone does not indicate a specific line; Red and green lines have same y-intercept, so y-intercept alone does not identify a specific line.


I explained that variables like m and b, which help identify a specific line, are “identifier” variables; their job is to identify a specific line. If your students are more advanced, you can explain that there are other identifier variables in different kinds of equations. For example, in the equation of a parabola:   y – k  = a(x – h)^2, the identifier variables would be the variables a, h, and k.

But what about variables like x and y? What do they do? What is their purpose?

These variables, I explained, have a completely different purpose. I call variables like x and y “ordered-pair generators.”

To explain this, I show students a simple linear equation like  y = 2x, and demonstrate how, using a “T-table,” you can use this equation to generate as many ordered pairs as you’d like, ordered pairs like (0,0), (1,2), (2,4), (3,6), etc. Point out that you can keep going and going. And then explain that the purpose of the x and y variables is to generate the infinitely many points that make up the line.

So the m and b variables tell us where the line is, and the x and y variables allow us to find the infinitely many actual points on the line. The two sets of variables, while different in purpose, work together toward a common goal:  to give us the equation of a line.

There are other purposes that variables serve, of course. And I’ll probably describe some of the other purposes in future posts. But the main point is that it helps students to recognize that variables do serve different purposes. Armed with that understanding, they can make much more sense of algebra’s formulas and equations.

Conquering Mixture Problems — Answers

Answers to Mixture Problems

In my last post I provided three mixture problems for all of you to do.

Here again are the problems, with the answers to them italicized.

1.  Kendra starts with 10 liters of a 40% antifreeze solution. How many liters of pure antifreeze would she need to add to end up with a solution that is 60% antifreeze?

Kendra would need to add 5 liters of pure antifreeze.

2.  Keith the chemist has a solution that is 25 quarts of 20% Boric Acid. How many quarts of 70% Boric Acid would Keith need to add to end up with a solution that is 50% Boric Acid?

Keith would need to add 37.5 quarts of 70% Boric Acid.

3.  Erin has a 2-liter solution that is 15% alcohol. How much pure alcohol would she need to add to it to end up with a solution that is 40% alcohol?

Erin would need to add 5/6 of a quart.

Conquering Mixture Problems — Practice

In my last two blogs I showed how to solve mixture problems. So now I want to give you some practice, so you can become an expert at solving these kinds of problems.

The answers will be stated in the next blog.

1.  Kendra starts with 10 liters of a 40% antifreeze solution. How many liters of pure antifreeze would she need to add to end up with a mixture that is 60% antifreeze?

2.  Keith the chemist has a mixture that is 25 quarts of 20% Boric Acid. How many quarts of 70% Boric Acid would Keith need to add to end up with a mixture that is 50% Boric Acid?

3.  Erin has a 2-liter mixture that is 15% alcohol. How much pure alcohol would she need to add to it to end up with a solution that is 40% alcohol?

Conquering “Mixture” Problems, Part 1

In the last blog you learned how to use a cool tool, “the master equation,” to slay (rate) x (time) = (distance) problems, R x T = D.

Now that you are initiated into the wonders of master equations, you might like to know that you can also use them for problems that many find even trickier:  those dreaded “mixture” problems.

Think for a sec, if you dare, and you’ll recall these little beasts, problems like this:

You start out with 5 liters of a 40% antifreeze solution. How many liters of pure antifreeze would you need to add to wind up with a mixture that is 73% anti-freeze.

The nightmares coming back to you now?

But as I mentioned, you can now use a “master equation” to solve these problems, just as we  did with R x T  =  D problems.

First, though, you need to understand something fundamental about mixture problems. And it helps if you can relate it to what we just learned about R x T = D problems.

With R x T  = D problems, a key was seeing that any distance can be represented by a rate multiplied by a time. For example, if a car travels 60 mph for 4 hours, we can express the distance it travels as the (rate)  x the (time):  (60 mph)  x  (4 hours) = 240 miles. The distance IS the product.

With “mixture problems,” there is a similar situation. For any mixture, we can express the amount of stuff that we care about through this basic but all-important equation:  Stuff =  (Concentration) x  (Volume of liquid). Or, still more shorthand: Stuff  =  (Concentration)  x  (Volume), which I like to abbreviate as
S  =  C  x  V.

What does this mean?  Well, here’s an example. Suppose in a word problem you’re told that you have 4 liters of a 50% antifreeze solution. You need to know how much actual antifreeze is in that solution. The antifreeze is the “stuff” we care about here. Use your new equation:  Stuff  =  (Concentration)  x  (Volume). So just multiply the (concentration) by the (volume) of liquid. That means you multiply  (50% concentration)  x  (4 liters), which is the same as (.5)  x  (4.0)  =  2.0. This means that in those four liters of solution there are exactly 2 liters of antifreeze. Wondering why this is true?  Just remember that 50% means HALF. So a  50% antifreeze solution means that half the liquid is antifreeze. Since you have 4 liters, half of that, 2 liters, is antifreeze.

What’s great is that you use this same principle and equation no matter how complicated the numbers might become (and you know that they don’t always stay easy, right?). So suppose you’re dealing with 12 liters of a 35% antifreeze solution. No problem. To see how much antifreeze is in those 12 liters, just use your new equation:  S  =  C  x  V. Antifreeze =  (.35) x (12)  =  4.2. This means that in those 12 liters of solution there are exactly 4.2 liters of antifreeze.

Taking this one step further, suppose that you need an algebraic expression to stand for a certain volume of liquid, an expression like (12 – x). And suppose you know that this liquid is 65% antifreeze. To express the amount of antifreeze in this solution, you still multiply the concentration by the volume, but now it looks like this:
Antifreeze  =  (.65) (12 – x).

That is all there is to it …  S  =  C  x  V. Burn that idea into your mind, right next to  R x T  =  D, and the rest will be “cake.”

One other thing to know about “mixture” problems. All you really care about in these problems is the amount of the solution whose % concentration you are given. So, for example, in a problem about antifreeze, the “master equation” you would use is this:

(Original Amount of Antifreeze) + (Antifreeze Added) =  (Amount of Antifreeze at End)

In my next blog I will show how you put these ideas together to actually interpret and solve a mixture problem. Trust me, now that you know S  =  C  x  V, it won’t be difficult.

Using “Master Equations”

In my last blog I described what  master equations are and how you can use them to solve word problems. I then promised to show you how to use master equations to actually solve word problems.

Here is the blog that shows how you use them to solve equations.

The two master equations I described were:

1)  Distance 1  =  Distance 2


2)  Distance 1 +  Distance 2  =  Distance Total

Let’s see how you solve a word problem with one of these master equations.

Here’s the word problem that we will be solving:

Tino and Gino get into an argument and drive away from one another. Tino leaves first, heading north at 65 kilometers per hour. Two hours later Gino heads south, traveling 45 kilometers per hour. The question:  at what time will Tino and Gino be 460 kilometers apart?

Step 1: Decide which master equation to use. Since Tino and Gino are traveling in opposite directions, they are covering different distances. Since their distances are different, we would not use the master equation Distance 1 = Distance 2 . The only other option at this time is Distance 1 + Distance 2 = Distance Total.

We can use this master equation, calling the distance that Tino travels  Distance 1, and  calling the distance that Gino travels Distance 2. Then the Distance Total would be the 460 kilometers.

At this point we can specify the master equation for this problem like this:

(Distance of Tino)  +  (Distance of Gino)  =  460

The next step is extremely useful, and it makes everything start coming together. To use this step we rely on the fact that distance = (rate) x (time). That being the case, we can express (Distance of Tino) as (rate Tino) x (time Tino), and we can similarly express (Distance of Gino) as (rate Gino) x (time Gino).  Using this step, the original master equation morphs into:

(rate Tino) x (time Tino)  +  (rate Gino) x (time Gino)  = 460

Once we reach this level of specificity, we can start filling in the blanks, as follows:

(rate Tino)  =  65

(rate Gino)  =  45

Of course we also need to come up with expressions for (time Tino) and (time Gino), and this is a bit more tricky, but not too bad. Notice that the problem says that Tino leaves first, and that Gino leaves two hours later. That means that Gino drives two hours LESS than Tino. In algebra-ese, we can express this idea by letting t = the time Gino drives, and then (t + 2)  for the time that Tino drives.

So now we have:

(time Tino)  =  t + 2

(time Gino)  = t

Putting it all together we make this grand substitution:

(rate Tino) x (time Tino)  +  (rate Gino) x (time Gino)  =     460
(65)         x          (t + 2)        +       (45)        x         (t)           =      460

Do you see what is great about this equation? We have one equation and just one variable. In the world of algebra that means “Hallelujah” because it tells us that we can solve for the variable — which we do as follows:

65t  +  130                       +                   45t                          =    460

110t  +  130   =   460

110t                 =  330

t  =  3

Since we let the variable t stand for Gino’s time driving, this means that Gino has been on the road for three hours when he is 460 kilometers from Gino.

Since Tino drove two hours more than Gino, Tino must have been on the road for five hours when he and Gino were 460 kilometers apart.

Problem solved.

Again, the main point is simply that understanding master equations gives you a guideline that makes it simple to understand problems that otherwise would have left us scratching our heads.

I’ll probably write a bit more about master equations, as they are so useful that everyone should really know what they are and how to use them.