Let’s suppose you are working on problems involving distance, speed and time. You might consider posing these problems one at a time.

**Q1:** Suppose I travel 2 hours at 40 mph and then 2 hours at 20 mph. What is my average speed for the trip?

**Q2:** Suppose I travel 2 hours at 40 mph and then 4 hours at 20 mph. What is my average speed for the trip?

**Q3:** Suppose the cities Abat and Boto are 80 miles apart and I travel from Abat to Boto at 40 mph and from Boto to Abat at 20 mph. What is my average speed for the trip?

——–

I should point out that I am now a units freak since my engineering husband always insisted that I include units when talking to him. In becoming such, I have learn to avoid many traps such as answering 30 mph to Q3.

——–

Average speed = total distance/total time. Notice that the unit is correct!

**A1: ** The total distance is: 2 hrs*40 mph+2 hrs*20 mph= 120 miles. The total time is: 2 hrs+2 hrs=4 hrs.

average speed = 120 miles/4hrs = **30 mph**

**A2: ** The total distance is: 2 hrs*40 mph+4 hrs*20 mph= 160 miles. The total time is: 2 hrs+4 hrs = 6 hrs.

average speed = 160 miles/6 hrs **= 26.7 mph**

**A3: ** The total distance is: 80 miles+80 miles=160 miles.

The total time must be calculated and is: 80 miles/40 mph + 80 miles/20 mph =2 hrs+4 hrs=6 hrs.

So this is the same problem as Q2. I drive twice as long a time coming back at 20 mph as going at 40 mph .

average speed = **26.7 mph**

—

**Resource: **These questions arose when I was thinking about the 2nd comment to my last blog about **harmonic averages** (thank-you Math Maker!) and so I looked around for more information at wikipedia (more about this to come in future blogs).

## 12 May 2009

### Thinking questions for rates

## 25 April 2009

### Goodie for April 2009

I loved this question. You can ask it in almost any class.

If Johnny has a 82% average for the 1st 9weeks and a 75% average for the 2nd 9weeks, what grade would he have to get on the final to receive 80% semester grade in this class. The grades are weighted as follows: each 9 weeks average counts 40% and the final test counts 20%.

Source: MathForum@Drexel

How did I find it? I was working on a Build your Own Simulator Kit using the freeware GeoGebra. To build the simulator, the student must make the ball go from point A to point B as * s* goes from 0 to 1. How to explain that this is: P=A*(1-

*)+B**

**s***? You can see here. Then a colleague of mine said “This is*

**s***weighted averages*and can be used in algebra, probability and geometry and can be expanded to n weights. Too bad we don’t talk about that anymore. Those problems are really rich.” So I looked around to see what he was talking about and found this problem. There is so much logic and usefulness in this problem – but not really any kind of special mathematics. Too bad we label it with such an awful name: weighted averages. It is just a good question.

Solution:

1. Check that the grading system makes sense: 2 × 9 week + 1 × final test = 2 × 40% +1 × 20% = 100%. Right.

2. Let * x*=

*grade on final test*.

Each 9 week grade counts 40% and final test 20% so we have: 82% × 40% + 79% ×40% +

*×20%*

**x**What does this expression equal? Well, we want an 80% total average at the end.

But we cannot write

*Expression*=80% because it is obvious that we need another “%”.

(This is actually the hard part.) We have not used “at the end“. What does “at the end” mean? It means 100%.

So our equation is: 82% × 40% + 79% ×40% + * x* × 20% = 80% × 100%.

Cancel all percentages and solve to get ** x=79%**.

## 8 March 2009

### Baddie of the Month – Teaching complex numbers with the quadratic formula

**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**March 2009 Baddie of the Month** – Teaching complex numbers with the quadratic formula.

Who was the dingbat who first decided to work complex numbers when teaching the quadratic formula?

**And why?** Because you **can** get complex conjugate numbers from the quadratic formula?

**A car has brakes. Do we teach hydraulics to someone learning to drive a car?**

**Storyline: **We are teaching quadratics. Everything we discuss is totally real. (In fact, we usually "fix" our problems to be with integers, but that is subject of previous rants.)

We are factoring, finding intercepts, drawing graphs in the Cartesian plane – all real numbers.

**In the middle of this, we start teaching a totally different subject – namely complex numbers**.

And then we go back to real numbers and real applications of quadratic functions.

**Let’s face facts.**

(a) Complex numbers have NO relation to quadratic functions or their applications that we will work on.

(b) Complex numbers have NO visual representation on the graph of a quadratic function*.

Ergo – complex numbers do NOT help us understand quadratics.

**Conclusion: Do NOT mention complex numbers at all when teaching the quadratic formula.**

Simply state that when D<0 (negative discriminant) the quadratic function has no roots and therefore does not cross the x-axis. Don’t mention "real roots". Don’t go anywhere else with this discussion at all.

**STICK TO THE SUBJECT MATTER AT HAND – Quadratic functions and their applications. **

—————–

This is the material in a typical textbook in the chapters for quadratic functions. (This is actually a reasonable toc. Some left me gasping for breath.)

Let’s go. We only have 2 months. **No problem – we will fix everything to be integers. **

- Here is a quadratic function
*y*=*x*². Let’s make a table of points. (Hmm – I’ve only ever graphed a line. I know I can graph a line using any 2 points. What the heck are all these points?) - The graph of a quadratic function is a parabola. (Hmm – So a parabola is the graph of a quadratic function?)
- Here is the general expression of a quadratic
*y*=a*x*²+b*x*+c, where*x*and*y*are variables and a, b and c are constants. (Hmm – They all look like letters to me.) Also,*m*²-2k*m*+k² is a quadratic. (Huh?) - Let’s factor a
*x*²+b*x*+c. Here are a bunch of rules. (Dang, I can’t factor this quadratic*x*²-3*x*-2. Teacher said it was because I copied wrong*x*²-3*x*+2. Do these rules work?)

Here are some more rules. (Hmm – Why do we want to factor? What are those numbers anyway?) - Let’s complete the square. Here is the plan. The point (h,k) is the vertex. (Hmm – Why is it the vertex? Why is there a minus in front of h and a plus in front of k?)
- Let’s graph quadratics by completing the square and transforming the graph of
*y*=*x*². (Say what? You go left when? First upside down? And then stretch?) - Here is the quadratic formula. We prove it using completing the square. (Wow, look at all those letters and equations. Now – square root. Plus and minus sign. Never seen that
*operation*before – cool.) - Using quadratic formula, let’s find the roots of a quadratic. (Roots? Is there a function here? Is there an equation here? Linear functions have roots?) The roots are the factors. (Hmm. Roots look easy to find. Couldn’t we just factor that way and skip all that factoring stuff?)
- Let’s graph a quadratic by completing the square. Now use the quadratic formula to find the roots. The roots are the x-intercepts of the function. (Hmm. Roots look easy to find. Aren’t parabolas symmetric? Why can’t I just find the vertex by going halfway between the roots and skip all that completing the square and transforming the function?)

Okay – I am sure they got all that. **Let’s pause and do a totally different subject. **

- D is called the discriminant. D can be positive, zero or negative. If D is negative, the quadratic function doesn’t have real roots. (Real roots? Are there fake roots?)
- If D is negative, the quadratic function has complex roots, which are complex numbers. You remember:
*z*=*x*+i*y*. Complex roots come in pairs called complex conjugates.

***Wait – we can make this worse**. Let’s graph complex numbers and their conjugates in the plane. (No kidding – my son did this in the middle of learning to graph quadratics.)

(Whoa – I thought we were talking about quadratic functions and graphing parabolas. What do I get with complex conjugates? Where do I put these on the graph? Whaaaaaaaaaat?)

Now back to quadratics. **Back to the reals – are we totally confused yet?**

- Now let’s look at applications of quadratic functions.

**Related topics: **

Past: February 2009 Goodie of the month – Good questions for Quadratic Equations/Functions

Past: February 2009 Baddie of the month – Teaching completing the square to graph a quadratic.

Past: January 2009 Baddie of the Month – Hand-factoring a quadratic with a≠1.

## 22 February 2009

### Goodie of the Month – Fun and Learning with Quadratics

**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**February 2009 Goodie of the Month** – Real Fun and Learning with Quadratics

In the second half of Algebra 1:

A typical standard is: Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

A typical question for this is: A ball is thrown straight down with a speed of 20 [ft/s] from a height of 80 [ft]. When will it hit the ground?

A typical application of technology is:

Tell the student that the function is y(t)=-16t^{2}-20t+80. They know they can’t graph with *t* so they switch to *x*, which they graph on their graphing calculator.

They see that parabola crosses the *x*-axis. They find the intersection and write *x*=1.7 and get their points.

Now ask them "Where does the ball hit the ground?". They will point to the intersection point – totally forgetting that this is *vertical motion* and that the ball hits the ground at (0,0)!

Ask them "What is the units on your answer?". You will be lucky if they give you [seconds] and not [feet]!

So why is this a “Goodie of the Month”?

The problem isn’t the standard. Nor is it the question. Both are excellent. The problem is the technology – it is undoing the learning.

Let’s change the technology! If the animation below doesn’t work – open this link: Vertical Motion

**Here are some “good problems”.**

- Set ho = 80[ft] and vo = -20[ft/s]. Run the animation. Point with your finger to the place
**where**the ball hit the ground. Now find the place on the graph where it says**when**it hit the ground. - Set ho = 80[ft] and vo= 20[ft/s]. Run the animation. Notice that the ball goes up before it goes down. Why is this? Reset the animation and using the step forward + and step backward – buttons, stop the animation when the ball is at its highest point. Point with your finger to the place
**where**the ball is at its highest point. Now find the place on the graph where it says**when**it is at its peak. What time is this? - Set ho = 0[ft] and vo= 100[ft/s]. Find when the ball hits the ground. Do this via the animation and algebraically using the function. When is the ball at its highest point (remember – parabolas are symmetric!)? What is this highest point? Do not forget units!

**Here are some “good questions” for the function: ** *h*(*t*)=ho+vot-16t^{2}

- The function h(t) gives height in [ft]. So each member of this function must give [ft].
- ho is (initial) height. So its unit is [ft]. It is all by itself so this member is in [ft].
- vo is (initial) velocity. So its unit is [ft/s]. How does this member give [ft]?
- What do you think the unit of “16” is so that this last member gives [ft]?

- In what part of the function is gravity playing a part? In which of the above problems is the only force gravity?
- Why do you think there is a plus sign in front of vo and a minus sign in front of 16? That is, what does it mean in mathematics/physics for an object to have a positive velocity? Does gravity increase this velocity?
- Make up a problem that describes this situation: ho = 0[ft] and vo= 100[ft/s].

**Links for this interActivity (worksheets, downloads, etc.): Open Metadata**

## 8 February 2009

### Baddie of the Month – Teaching “completing the square” for quadratics

**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**February 2009 Baddie of the Month** – Teaching “completing the square” for quadratics

Yes, she is stuck on quadratics. But quadratics are so useful and can be fun and we keep teaching techniques that don’t give much value for the time and effort.

**Why?** These are reasons I have found online or been given **"for"** studying “completing the square”:

(1) one can then prove the quadratic formula , (2) one can find the vertex of the quadratic,

(3) one can graph quadratics using graph transforms and (4) one can solve integrals of the form dx/(x^2+bx+c)

**My response: **

- Students at this level cannot understand a mathematical proof. All they see is a manipulation of symbols/letters/numbers. Ask any non-math student. Ask any non-math adult. They didn’t get it. Period. So we teach them a technique in order for them to see a proof they don’t understand. (I am happy if they get accurate results from the quadratic formula with any numbers for a, b and c and can relate them to the graph of the quadratic and/or the answers to the question that was posed and judge the reasonableness of all.)
- It is much easier and useful to find the vertex of a quadratic by first realizing that every parabola is symmmetric and thus the vertex must be the value of the function at the half-way point between the roots*. Finding the vertex this way requires them to relate solutions from the quadratic formula to roots/zeros/x-intercepts, reinforces learning about midpoints, relates the vertex to the quadratic formula – more reinforcement and it teaches them to find function values.
**Win-win-win-win**. Using "completing the square" teaches them to manipulate numbers. - Does anyone actually graph a function using a graph transform – ever? After trying to teach graph transforms for over 20 years – I have decided that the best I can hope for is "a vague understanding" that "x^2+4" is "x^2 up 4" and 3sinx is "3 times taller than sinx" and cos2x is "2 times faster than cosx". Trying to combine all these is a total waste of time.
- Solve integrals – good grief. Maybe we should teach them partial fractions in algebra 1 too.

**How to find the vertex:** **The x-value of the vertex is x= -b/2a. Substitute this value into the quadratic to get the y-value.**

(BTW: nowhere did I find the semi-plausible defense that “completing the square” is also used to find the center of a circle and identify conic sections in the 2nd half of Algebra2. By then, I agree that completing the square is not an unreasonable technique to teach.

*roots found using the quadratic formula I am sure .

**Related topics: **

Past: January 2009 Baddie of the Month: Hand-factoring a quadratic with a≠1.

Future: Baddie – Teaching complex numbers in the same 2 month span as you teach graphing of quadratics.

Future: Goodie – Good questions for Quadratic Equations/Functions

## 25 January 2009

### Goodie of the Month – A Good Question for Algebra 1

**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**January 2009 Goodie of the Month** – A Good Question for Algebra 1

Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer’s rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships’ positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.

Assume that both ships continue to move at a constant speed on their respective linear courses. Using graphs and equations, find out if the two ship will collide.

Why do I like this question?

- Students can understand it and it is fun.
- They can graph it on paper or using a graphing program.
- It involves finding the equation of a line through 2 points (twice) – good reinforcement.

**Why do I think it is a “good question”?**

- The graph
*looks like*every 2×2 system of linear equations they have solved in Algebra 1.- It looks like the boats collide at the intersection point (see below).
- It seems like all they need to do is solve the system and be done.
- … until you say “Where is time on the graph?”.

- The student can build a animated simulator that “shows time” –
**easily**!- Then they can see that the boats do not collide.
- Below is a simulator I built using the freeware GeoGebra.
- Here are step-by-step directions.

- The kids can make the boats collide – what fun!.
- They can move the starting points until they get the boats to collide.
- They can also adapt the simulator so that they can change the slopes and get the boats to collide. Directions here.
- My thanks to David Cox for seeing this!

- You can get all kinds of mathematics out of them.
- You can get them to calculate when each of the boats reaches the intersection point in the original question.
- You can get them to check the math on their “colliding simulator” to see if the boats really do collide, where and when.
- You can ask them about a 3D graph and what this would look like when the boats don’t collide and when they do.

To animate, click on the play button at bottom left of **graph**.

To animate manually, right-click on slider and deselect “Animation on”. Then, click and drag the point on the slider.

Source: I found this question asked on answers.yahoo.com. My webpage for this question is: mathcasts.org/mtwiki/Gq/BoatCollide

## 10 January 2009

### Baddie of the Month – Factoring a Quadratic with a≠1

**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

I am going to try to blog a baddie and a goodie per month. We shall see and of course – this is my opinion.

**January 2009 Baddie of the Month** – Factoring a quadratic with a≠1 “by hand”.

Okay, I can mostly understand learning to factor “by-hand”: x²+3x+2 or x²+x-2.

Once you understand the principles and get the technique (my scheme), factoring a quadratic by hand with a=1 is faster than using the quadratic formula.

But, I absolutely and totally do not understand the reasoning behind other factoring-by-hand techniques!

**Why not: **Factoring techniques

- serve no useful purpose – once factored, with a≠1 you must still solve the individual factors.
- don’t always work – MOST quadratics even with a=1 and real roots CANNOT be factored by hand.
- are hard to learn, there are many “special cases”, they take alot of time to teach, …

**What to do and why:** **Use the quadratic formula for all your factoring needs**.

- We are going to teach them the quadratic formula anyway.
- It always works – either we get real roots and can factor or we get non-real roots and know we cannot factor.
- By using the quadratic formula all of the time, the connection between quadratics, roots, x-intercepts, graphs of quadratics becomes clear.
- Repetition of a single technique is much more likely to stay in their heads.

**Conclusion:** Don’t teach factoring by hand except when a=1. Use the quadratic formula.

**Here’s how:** ax²+bx+c=a(x-x1)(x-x2) where x1=(-b+D))/2a, x2=(-b-D))/2a, D=√(b²-4ac)

(Here “by hand” means looking for the factors without a formula like when you say “The factors of 2 are 1 and 2 and oh yes, they add to 3 (first expression) or the factors of 2 are 1 and 2 and oh yes, they subtract to 1 (second expression)”.)

**Related topics to come in future blogs**

Please, please don’t teach complex numbers in the same 2 month span as you teach graphing of quadratics.

## 12 September 2008

### Race Car Activity – Exploring Slope and Intercepts in the Real World

Click and drag the slider points to adjust the cars speeds and positions. Then use the animation buttons.

(The animation buttons may no longer work because there are multiple animated pages on this blog. If they don’t, please go to the webpage – they will work there.)

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Here is the webpage: http://mathcasts.org/mtwiki/Activity/CarRace

My thanks to Jon Ingram for showing me how to do this! See how! (after September 15)

## 30 August 2008

### GeoGebra on a Blog

Successfully embedding a geogebra applet!

You should be able to click and drag the blue point. Please let me know if you have any problems!

My thanks to Jon Ingram for showing me how to do this!

To see how to do this, go to: http://mathcasts.org/mtwiki/GgbHelp/Embed

(after September 1)

## 31 March 2008

### Algebra II – The Key to Success?

The report: The Final Report of the National Mathematics Advisory Panel highlights this comment:

“Students who complete

Algebra IIare more thantwice as likely to graduate from college compared to students with less mathematical preparation.”

Oh my.

• What other subject gives this statistic?

• What is about Algebra II that gives this result?

I also ask.

• How many of us pay “double taxes” for mathematics education for our children?

Taxed once for schools and again for tutoring. I do.

Again from this report:

Moreover, there are

large, persistent disparities in mathematics achievementrelated torace and income—disparities that are not only devastating for individuals and families but also project poorly for thenation’s future, given theyouthfulness and high growth ratesof the largest minority populations.

And this is true where I live too – not just in the US. So I bet it is true in many, many places.

• **Now – what are the answers to these questions and ****what do we do to change**** the situation? **