Day One of my Calculus I class: Part 4

In this series of posts, I’d like to describe what I tell my students on the very first day of Calculus I. On this first day, I try to set the table for the topics that will be discussed throughout the semester. I should emphasize that I don’t hold students immediately responsible for the content of this lecture. Instead, this introduction, which usually takes 30-45 minutes, depending on the questions I get, is meant to help my students see the forest for all of the trees. For example, when we start discussing somewhat dry topics like the definition of a continuous function and the Mean Value Theorem, I can always refer back to this initial lecture for why these concepts are ultimately important.

I’ve told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

I then applied these two themes to find the speed of a falling object at impact.

I now switch to a second, completely unrelated (or at least it seems completely unrelated) problem.

Problem #2. Find the area under the parabola $f(x) = x^2$ between $x=0$ and $x=1$ .

I draw the picture and ask, “OK, what formula from geometry can we use for this one?” Stunned silence.

I say, “Of course you can’t do this yet. This is a curved thing. Back in high school geometry, you learned (with one exception) the areas of straight things. What straight things had area formulas in high school geometry?” I’ll always get rectangles and triangles as responses. Occasionally, someone will volunteer parallelogram or rhombus or kite.

So I ask the leading question, which of these shapes is easiest? Students always answer, “Rectangles.” Which then leads me to the next question: How can we approximate the area under a parabola with a bunch of rectangles?

Again, stunned silence. I let my students think about it for at least a minute, sometimes two minutes. Hopefully, one student will volunteer the answer that I want, though occasionally I’ll have to coax it out of them.

Eventually either a student volunteers (or else I tell the class) that we ought to use a bunch of thin rectangles. For starters, I’ll use five rectangles and a very rough sketch on the board.

I’ll start with the right-most rectangle… what is its area? Students immediately see that the width is $1/5$ , but the length takes a little bit more thought. And I make my students figure it out without me giving them the answer. Eventually, someone notices that the height is simply $f(1) = 1$ , so that the rightmost rectangle has an area of $0.2$ .

I then move to the rectangle that’s second to the right. This also has a width of $1/5$ , but the height is $(0.8)^2 = 0.64$ . So the area is $0.128$ .

Eventually, we get that the sum of the areas is $0.008 + 0.032 + 0.072 + 0.128 + 0.2 = 0.44$ . Students can easily see that this is a decent approximation to the area under the parabola, but it’s a bit too large.

I then ask the same question that I had before: how can we get a better approximation? Students will usually volunteer either “More rectangles” or “Thinner rectangles,” which of course are logically equivalent. I then proceed with 10 equal-width rectangles. Occasionally, a student volunteers that perhaps we should use thinner rectangles only on the right side of the figure, which of course is a very astute observation. However, I tell my class that, for the sake of simplicity, we’ll stick with rectangles of equal width.

With ten rectangles (and I redraw the picture with ten thin rectangles), the approximation is quickly found to be

$0.1 [ (0.1)^2 + (0.2)^2 + \dots + (0.9)^2 + 1^2] = 0.385$

I like using ten rectangles, as that’s probably the largest number that can be handled in class without a calculator (until the very last step of adding up the areas).

By now, the class sees what the next steps are: take more and more rectangles. At this point, I’ll resort to classroom technology to make the process a little quicker. I personally prefer Microsoft Excel, though other software packages can be used for this purpose. For $100$ rectangles, the class quickly sees that the sum of the rectangles is

$0.01 [ (0.01)^2 + (0.02)^2 + \dots + (0.99)^2 +( 1.00)^2] = 0.33835$

My class can see that the answer is still too large, but it’s certainly closer to the correct answer.

I’ll then tell the class that this is another example of passing to limits, the second theme of calculus. I’ll describe this more fully in the next post.

Day One of my Calculus I class: Part 3

I’ve just told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

We are now studying the following problem.

Problem #1. A building on campus is 144 feet tall. A professor takes a particularly annoying student to the top of the building, and throws him (or her) off to his (or her) certain demise. (Usually I pick a student that I know and like as the one to throw off the building. This became a badge of honor over the years.) The distance that the student travels (in feet) after $t$ seconds is $f(t) = 16t^2$ . How fast is the student going when he (or she) hits the concrete sidewalk?

At this point in the lecture, we have done some experimental numerical work with successfully smaller time intervals to find better and better approximations to the speed at impact.

With a time interval of length $3$ seconds, the approximation is $48$ ft/s.
With a time interval of length $1$ seconds, the approximation is $80$ ft/s.
With a time interval of length $0.5$ seconds, the approximation is $88$ ft/s.
With a time interval of length $0.1$ seconds, the approximation is $94.4$ ft/s.
With a time interval of length $0.01$ seconds, the approximation is $95.84$ ft/s.

By this point, students realize that we’re getting better and better approximations… however, we’re probably not going to get the correct answer by just plugging in numbers. And we certainly can’t just take a time interval of $0$ seconds since dividing by zero is a no-no.

Depending on my read of the class — on whether or not they’re ready for a little more abstraction — I’ll then ask the class, “How can we make these fractions without plugging in all of these numbers?” Usually students are at a loss at first. Perhaps someone will volunteer that we ought to introduce a variable… but, in my experience, even bright students at the start of calculus do not have this step of abstraction at the tips of their fingers. So I’ll lay out the fractions that we’ve studied so far, like

$\displaystyle \frac{f(3) - f(2.9)}{0.1} \qquad$ and $\qquad \displaystyle \frac{f(3)-f(2.99)}{0.01}$ ,

and ask, “How could we do this more systematically? Does anyone see a pattern in these fractions?” Hopefully someone will notice that the input of the second function call is 3 minus the denominator; if not, I’ll volunteer this observation to the class. So both of these fractions can be written as

$\displaystyle \frac{f(3) - f(3-h)}{h}$ ,

where $h$ is a small positive number. Let’s now simplify this fraction:

$\displaystyle \frac{f(3) - f(3-h)}{h} = \displaystyle \frac{16(3)^2 - 16(3-h)^2}{h}$

$= \displaystyle \frac{144 - 16(9-6h+h^2)}{h}$

$= \displaystyle \frac{144 - 144 + 96h - 16h^2}{h}$

$= \displaystyle \frac{96h - 16h^2}{h}$

$= \displaystyle 96 - 16h$ .

The last step is permitted because $h$ is assumed to be a nonzero number. I then check to see if the previous work matches this algebraic expression:

If $h=1$ , then $96 - 16h = 80$ , matching the previous answer.
If $h=0.1$ , then $96-16h = 94.8$ , matching the previous answer.

I then ask the class, what’s the ultimate goal with $h$ ? The answer: send $h$ to zero. So we conclude that the velocity at impact is $96 - 16(0) = 96$ ft/s, which is the final answer.

Reviewing, the curved thing was the changing speed of the falling object, which was approximated by the straight thing, the ordinary distance-rate-time formula. Finally, we passed to limits to find the real velocity at impact.

All of the above is eventually done more systematically later in the semester after the properties of derivatives have been more fully developed. However, I think that doing this calculation on the very first day of class gives my students a taste of what’s going to be happening in the days and weeks to come. Again, I emphasize that I probably cover this material in maybe 15-20 minutes, and that I don’t hold students immediately responsible for repeating such a calculation on their own. (I do hold them responsible for this, of course, after they know how to differentiate $f(t) = 16 t^2$ .

Day One of my Calculus I class: Part 2

I’ve just told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

I then transition to applying these two themes to two different problems. Here’s the first.

Problem #1. A building on campus is 144 feet tall. A professor takes a particularly annoying student to the top of the building, and throws him (or her) off to his (or her) certain demise. (Usually I pick a student that I know and like as the one to throw off the building. This became a badge of honor over the years.) The distance that the student travels (in feet) after $t$ seconds is $f(t) = 16t^2$ . How fast is the student going when he (or she) hits the concrete sidewalk?

And then I ask my students how to solve this. Usually, they can come up with the first few ideas.

1. When the student hits the sidewalk and meet his/her demise? So we must solve $16t^2 = 144$ , so that $t = \pm 3$ . (And I make sure that they remember that this quadratic equation has two roots.) The solution $t = -3$ is clearly extraneous, so the time elapsed until the student meets his/her demise is $3$ seconds.

2. How fast is the student going after $3$ seconds? Most students realize the inherent difficulty of this question because the student’s speed is increasing as he/she gets closer to the ground. Some students will volunteer the word “accelerate.”

At this point, I’ll volunteer that the changing speed is a curved thing. Back in pre-algebra, students were taught

$\hbox{rate} = \hbox{distance} / \hbox{time}$

under the assumption that the rate was constant. However, if the rate is changing, all bets are off.

Still, the question remains: how fast is the student moving after 3 seconds? How should we measure this? Usually, someone will suggest that we just divide $144$ feet by $3$ seconds, for a rate of $48$ ft/sec. I then point out that this is an example of approximating a curved thing by a straight thing. The straight thing is the usual distance-rate-time formula, while the curved thing is the changing speed of the student as he/she falls. So the answer of $48$ ft/sec is not the correct answer, but it’s an approximate answer.

This leads to the next question: is this estimate too high or too low? Unequivocally, students answer “too low” since the student travels the slowest at the start of the fall and the fastest at the end of the fall. So since this interval of 3 seconds includes the slower speeds at the start of the fall, the answer of $48$ ft/sec will underestimate the speed at impact.

Which then leads to the next obvious question: How can we get a better approximation? I leave the question open-ended like this and take suggestions from the class. This often takes a while, and I’ll get a lot of creative (but bad) ideas. And that’s OK… the next step is hardly the most intuitive thing that immediately jumps to mind. I think that the process of keeping the answer unknown until someone volunteers the correct next step is worth it.

Eventually (though it might take a couple of minutes), somebody will suggest using a shorter time interval, like the distance traveled between $t=2$ and $t=3$ . We see that $f(2) = 64$ and $f(3) = 144$ , and so the new approximation is $(144-64)/1 = 80$ ft/s. I store these two approximations ( $48$ ft/s with a time interval of $3$ seconds and $80$ ft/s with a time interval of $1$ second in a table on the side of the chalkboard. The values derived below are entered in the table as they’re found.

I then note that the previous approximation was $48$ ft/s, and then ask the class, “Do you think that $80$ ft/s will be a better or worse approximation than $48$ ft/s?” Invariably, they’ll say it’s a better approximation because the change in speed isn’t as great from $t=2$ to $t=3$ as from $t=0$ to $t=3$ . I’ll then ask if they think that $80$ ft/s is too high or too low. Again, they’ll answer too low for the same reason as before.

Then I ask the obvious next question, “How do we find a better approximation?” The class typically responds something to the effect of, “Take a smaller interval.” I ask for a suggestion, and I’ll usually get something like $t=2.5$ to $t=3$ . We see that $f(2.5)=100$ ft/s and $f(3)$ is still $144$ ft/s, so that the new approximation is $(144-100)/0.5 = 88$ ft/s. Students will volunteer that this should be better than the previous two approximations but still less than the correct answer.

Then I do it again: “How do we find a better approximation?” The class typically respond, “Take an even smaller interval.” I suggest $t=2.9$ to $t=3$ . We see that $f(2.9)=134.56$ ft/s (by this point, a calculator is certainly needed) and $f(3)$ is still $144$ ft/s, so that the new approximation is $(144-134.56)/0.1 = 94.4$ ft/s. If we do it again with $t =2.99$ , we see that $f(2.99) =143.0416$ ft/s, for an approximation of $(144-143.0416)/0.01 = 95.84$ ft/s.

I turn to the class and ask, “Have we found the right answer yet?” They’ll answer “No” in unison, but they’ll note that the approximations are probably pretty good right now. Astute students will notice that the approximations appear to be “leveling off” to some final value.

I’ll then tell the class that this is an example of passing to limits, the second theme of calculus. By making the time intervals smaller and smaller, we get better and better approximations to the true speed at impact. In the next post, I’ll describe how I informally introduce the concept of a limit with this example.

Day One of my Calculus I class: Part 1

I begin by noting the different topics that appear in Precalculus, which they should have taken in the recent past:

The definition of a function and an inverse function
Graphing polynomials and rational functions
Properties and applications of exponential and logarithmic functions
Trigonometry
Sequences and series

These different topics, when taught in Precalculus, really don’t talk to one another. With a couple of exceptions, it feels like five different units being squeezed into the same course. I’ll present a visual image of laying down an imaginary brick on the floor, and then laying down a second brick next to the first one, and so on. The above topics (with a couple of exceptions) really don’t build upon each other; they’re lateral to one another. In other words, these topics made the foundation necessary for the study of calculus. After all, the class was called Pre-Calculus.

Now that we’re in calculus, I tell my students, we’re going to have topics that build on this foundation, and the topics will build on each other. Continuing the building image, I’ll start laying imaginary bricks on the initial foundation, building vertically higher and higher, noting that the topics that we’ll see in Weeks 13 and 14 will ultimately be built upon the topics that we’ll talk about in Weeks 1 and 2. Unlike Precalculus, the topics in Calculus are explicitly interconnected, building up a body of thought from the foundation of Precalculus.

So the good news is that, unlike Precalculus, Calculus I will be an incrementally developed course from start to finish. The bad news, of course, is that Calculus I will be an incrementally developed course from start to finish. In Precalculus, if you didn’t particularly like one topic (say, logarithms), that really would not affect your success later on with a future topic (say, trigonometry). However, in Calculus, the whole course is put together from start to finish.

The good news is that while there are many interconnected topics in calculus, there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

And we’re going to be applying these two themes again and again throughout the semester. (I wish I could take credit for synthesizing the topics of calculus into these two themes, but I learned this idea from my own calculus professor back in the mid-1980s.)

For the remainder of this first lecture, I show how these two themes apply to two completely different problems:

Finding the speed of a falling object when it hits the ground.
Finding the area under the curve $y = x^2$ between $x = 0$ and $x = 1$ .

I’ll describe how I present these to new calculus students in the coming posts.

Engaging students: Central and inscribed angles

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Theresa (Tress) Kringen. Her topic, from Geometry: central and inscribed angles.

What interesting word problems using this topic can students do now?

After defining the terms central angle and inscribed angle, students can use a central angles to draw a pie graph or pie chart. They can depict the data using a visual. Based in the percentage of any part of a whole, they will crate a fraction of the whole circle by dividing 360 degrees by that percentage to give the piece of the pie in which they needed to find.

Say a student is given the data below and asked to graph the data into a pie chart:

Students’ favorite colors:

Blue 10

Yellow 3

Red 7

Orange 3

Green 10

Purple 6

Pink 9

Other 2

Students would be required to give percentages based on the 50 students with the percentages listed as: Blue 20%, Yellow 6%, Red 14%, Orange 6%, Green 20%, Purple 12%, Pink 18%, other 4%. This would correspond to the percentage of the 360 degree central angle.

To tie in inscribe angles, I would have to students explain why a pie chart would not work with inscribed angles.

How does this topic appear in high culture?

In order to engage students I could help them understand inscribed angles by relating it to the camera angle in their video games. Describing an inscribed angle as a camera angle on their video game would help them understand it better. As they move throughout the game, their camera angle changes. Based on the camera’s location, you are able to see a certain portion of the screen. If there isn’t much of an angle, the range of view is small or zoomed in. This could be explained as the radius of the circle. The smaller the radius, the less view there is. Thus, the opposite is true. If the radius is large, the camera has a larger view of the object. If the camera has a larger angle of view, more is visible in the camera. I would then relate this to the arc length that the angle creates. I would explain that if the angle of the camera is small, the area of the arc length, or view of the camera would also be small. If the angle of the camera is larger, the arc length or view of the camera is much larger.

How can technology be used to effectively engage students with this topic?

Once students are given the application problem listed above, I could then engage them further by asking them to use word or excel to graph the information given into a document. They would be required to make a chart of the data with the listed percentages of each parameter along with the degree of the angle that the parameter requires to make the pie graph. I would require this since the technology would calculate this on its own without the student having to put in the effort. To make it fun, I would give the students a few extra minutes to make their pie graph their own by customizing it to reflect their personality and style.

To further engage them, I could also ask that each student create a questionnaire that asked each student what their favorite choice of any given set of choices were. They would be required to have at least 7 responses as to make a 7 piece pie chart, but they would be able to choose the topic, and find the information for their parameters on their own. Once they did this, they would be required to make an additional pie chart with their results to present to the class.

Engaging students: Finding least common multiples

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Theresa (Tress) Kringen. Her topic, from Pre-Algebra: finding least common multiples.

What interesting word problems using this topic can your students do now?

While having students working on finding the least common multiples I could engage them by having them solve some word problems that would bring up real world problems in a way that they can relate what they learned to problems that deal more than with just numbers. One problem that could be presented ot the students is the following:

If you’re given packages of notebooks that contain 6 each and you are required to repackage them to send them to a school in need in groups of 22, what it the least amount of groups and original packages of notebooks that you can get without any notebooks left over?

In this problem, the students would be required to find the least common multiple of both 6 and 21. Since six doesn’t not go into 22 without a remainder, they would have to find lcm(6,22). Since the least common multiple of both 6 and 22 is 66, the students would have to apply what they know about least common multiples of numbers to figure out the word problem.

To continue with this, the students could then be asked to do the same thing for three numbers.

How does this topic extend what your students should have learned in previous courses?

Students should have covered factors and multiples of numbers around fifth grade. Therefore finding the least common multiple of a number extends the topic from these previous topics. Since students can figure out the factors of a number, they should also know if one number is a factor of the second. If it is, then they will know that the second number is the least common multiple of the two given numbers. Say the students are given 3 and 9. The students should be able to tell right away that 3 goes into 9. Since 3×3=9 and 9×1=9 and since no number smaller than 9 can also be a multiple of nine, the least common multiple of 3 and 9 is 9.

When also looking at the least common multiples of a number, students know what multiples of a number are from previous courses. They will know that 18 is a multiple of nine as well as 27, 36, and 45. Students know that 3 times 9 is 27, but they will also know that since the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30, etc. they will also know that even though 3 times 9 is 27, that there is a number smaller than 27 that is also a common multiple of 3 and 9.

How can technology be used to effectively engage students with this topic?

Students like games and it’s even better for the teacher if they are able to play while they learn or practice a given subject that they have learned. In order to engage each student, there a number of online games students can play to help them practice finding the least common multiples of given numbers. I have found a number of online games that students could go to for an activity. It pushes them, allows the students to go at their own pace, and allows students to be less worried about how fast or slow they are compared to other students.

One game is a timed game that gives the students two numbers to find the least common multiple of. They are given two minutes to see how many they can compute in that amount of time. They are still permitted to go at their own pace, but they are also pushing themselves to do better than the time before.

http://www.basic-mathematics.com/least-common-multiple-game.html

A second game give the students two numbers and asks for the least common multiple. It is basically multiple choice since they are to select a number our of five or six different numbers. If they select the correct answer, they are permitted to “throw a snowball.” Each correct response helps them win the snowball fight.

http://www.fun4thebrain.com/beyondfacts/lcmsnowball.html

Engaging students: Slope-intercept form of a line

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Theresa (Tress) Kringen. Her topic, from Algebra I: the point-slope intercept form of a line.

What interesting word problem using this topic can your students do now?

When learning about slope-intercept from of a line, word problems would help my students engage and help process the information in a real world situation. I would present an equation for the speed of a ball that is thrown in a straight line up into the air. The equation given: $v= 128-32t$ . I would explain that because we’re working with time and speed, height is not a variable in the equation. With $v$ representing the speed or velocity of the ball in feet per second and t representing the time in seconds that has passed. I would include the following questions:

1. What is the slope of the given equation? Since the equation is given in slope intercept form, the students should be able to give the answer quickly if they understood the lesson. The answer is $-32$ .

2. Without graphing the equation, which way would the line be headed, up and to the right or down and to the right? Because the students know that the slope is negative and given that they understood the lesson, they should be able to answer that the line is decreasing and is headed down and to the right.

http://www.purplemath.com/modules/slopyint.htm

How can this topic be used in your students’ future courses in mathematics or science?

Students can use this topic for many math or science courses. When dealing with a linear equation, slope-intercept form of a line can help the student understand what the graph looks like without actually graphing it. This is useful when needing to find the $y$ intercept (when $x$ is equal to zero) and what the slope of the line is. This is also useful to know for understanding what slope is. When students understand that a slope of a particularly large number (a large whole number such as $1,000$ or an improper fraction that equates to a large number such as $30,999/2$ ) is rising quickly as opposed to a slope of a smaller number (a smaller whole number such as two or a fraction that represents a very small portion of one such as $1/30,000$ ) which is not rising quickly. It is helpful for the students to understand that a very large slope will look almost vertical and a small slope will look almost horizontal, with both depending on the degree of largeness or smallness.

How can technology be used to effectively engage students with this topic?

When working with slope-intercept form, a student can actively be engaged through technology by attempting to make connections of how a graph looks on the graphing calculator and what the equation looks like in slope-intercept form. When allowing the students to make connections between them in small groups, they will have discovered the information form themselves. This will allow the students to more effectively program the information into their memories. To set this up, I would give each group a graphing calculator and a list of equations in slope-intercept form. On the paper with the list, I would have the students fill out information pertaining to the graph that they see. This information would include the slope and the y-intercept. I would split up the students into their cooperative learning groups two and ask them to draw a conclusion between where the line ends up compared to what the equation looks like. Once the students have typed their equation into the graphing calculator the students should fill out the paper provided. Once they have finished, I would ask them to see if they see any patterns between the equations and their answers.

From high school math teacher to quarterback for the Dallas Cowboys

I’ve never been a fan of the Dallas Cowboys, but Jon Kitna remains one of the good guys of the NFL. After retirement, he went to work as a math teacher and football coach at his high school alma mater, getting students with learning disabilities to understand algebra (and thus be prepared for higher-level math classes in later years).

After the unfortunate injury to starting quarterback Tony Romo, the Dallas Cowboys called upon Kitna for emergency service. He plans to donate his one-game salary back to the high school.

Reference: http://bleacherreport.com/articles/1901693-jon-kitnas-salary-decision-proves-his-return-is-noble

Log-log and log-linear plots

Source: http://www.xkcd.com/1007/

Engaging students: Order of operations

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Theresa (Tress) Kringen. Her topic, from Pre-Algebra: order of operations.

How can this topic be used in your students’ future courses in mathematics or science?

Order of operations is commonly used in most mathematics problem that involve more than one operation or when parenthesis are involved. It would be easy to show the students what the answer to a given problem, say 5+20/5, would be when using the proper order of operations, then solve the problem by solving left to right as you would read a book. It is clear, to a math major, that the answer is 9. For someone who does not know the order of operations, they most likely would come up with the answer of 5. The difference in the correct answer and the incorrect answer is only 4, but the problem is only working with numbers less than or equal to twenty. It would then be beneficial to point out that when dealing with more complex problems, that this answer may become even larger. If the class was working on given problems, I would give them a few word problems to solve. Once they solved them on their own, I would show them that the difference between the correct way to answer the given problem and the incorrect way to answer the problem to help them connect the concept to why it is important to compute answers in the way.

How does this topic extend what your students should have learned in previous courses?

This topic extends what students should have previously learned by allowing them to use their skills of multiplication, division, exponents, addition, and subtraction to solve more complex problems. When learning how to solve problems more complicated than what they have been given in the past, they use this topic to guide them through to the next step. They must already be familiar with all of the operations by themselves prior to using the order of operations to solve a problem. Once they are accustomed to using the order of operations, the will be given more challenging problems and their math skills will build upon itself. It is clear that if a student is unable to solve a simple problem, such as an exponent problem or a more complicated division problem, they will not be able to use the order of operations for problems that contain what they have not learned.

How did people’s conception of this topic change over time?

It is believed that the idea of using multiplication before addition became a concept adopted around the 1600s and was not disagreed about. The other operations took their place in the order over time, beginning in the 1600s. It seems that although it was not documented well, most mathematicians agreed upon the same order. It wasn’t until books stated being published that it was important to document the order of operations. The notation may have been different depending on who was writing on the subject, but the concept was the same. It seems that although it was not documented well, most mathematicians agreed upon the same order. Once books were being published, the order, PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction), was put into print. Now, teachers use the phrase Please Excuse My Dear Aunt Sally as a way for students to remember the acronym and are able to put it to use.

http://jeff560.tripod.com/operation.html

http://mathforum.org/library/drmath/view/52582.html