Decimal Approximations of Logarithms (Part 4)

While some common (i.e., base-10) logarithms work out evenly, like $\log_{10} 10,000$, most do not. Here is the typical output when a scientific calculator computes a logarithm:

To a student first learning logarithms, the answer is just an apparently random jumble of digits; indeed, it can proven that the answer is irrational. With a little prompting, a teacher can get his/her students wondering about how people 50 years ago could have figured this out without a calculator. This leads to a natural pedagogical question:

Can good Algebra II students, using only the tools at their disposal, understand how decimal expansions of base-10 logarithms could have been found before computers were invented?

Here’s a trial-and-error technique — an exploration activity — that is within the grasp of Algebra II students. It’s simple to understand; it’s just a lot of work. The only tools that are needed are

• The Laws of Logarithms
• A hand-held scientific calculator
• A lot of patience multiplying $x$ by itself repeatedly in a quest to find integer powers of $x$ that are close to powers of $10$.

In the previous post in this series, we found that

$3^{153} \approx 10^{73}$

and

$3^{323,641} \approx 10^{154,416}$.

Using the Laws of Logarithms on the latter provides an approximation of $\log_{10} 3$ that is accurate to an astounding ten decimal places:

$\log_{10} 3^{323,641} \approx \log_{10} 10^{154,416}$

$323,641 \log_{10} 3 \approx 154,416$

$\log_{10} 3 \approx \displaystyle \frac{154,416}{323,641} \approx 0.477121254723598\dots$.

Compare with:

$\log_{10} 3 \approx 0.47712125471966\dots$

Since hand-held calculators will generate identical outputs for these two expressions (up to the display capabilities of the calculator), this may lead to the misconception that the irrational number $\log_{10} 3$ is actually equal to the rational number $\displaystyle \frac{154,416}{323,641}$, so I’ll emphasize again that these two numbers are not equal but are instead really, really close to each other.

We now turn to a question that was deferred in the previous post.

Student: How did you know to raise 3 to the 323,641st power?

Teacher: I just multiplied 3 by itself a few hundred thousand times.

Student: C’mon, really. How did you know?

While I don’t doubt that some of our ancestors used this technique to find logarithms — at least before the discovery of calculus — today’s students are not going to be that patient. Instead, to find suitable powers quickly, we will use ideas from the mathematical theory of continued fractions: see Wikipedia, Mathworld, or this excellent self-contained book for more details.

To approximate $\log_{10} x$, the technique outlined in this series suggests finding integers $m$ and $n$ so that

$x^n \approx 10^m$,

or, equivalently,

$\log_{10} x^n \approx \log_{10} 10^m$

$n \log_{10} x \approx m$

$\log_{10} x \approx \displaystyle \frac{m}{n}$.

In other words, we’re looking for rational numbers that are reasonable close to $\log_{10} x$. Terrific candidates for such rational numbers are the convergents to the continued fraction expansion of $\log_{10} x$. I’ll defer to the references above for how these convergents can be computed, so let me cut to the chase. One way these can be quickly obtained is the free website Wolfram Alpha. For example, the first few convergents of $\log_{10} 3$ are

$\displaystyle \frac{1}{2}, \frac{10}{21}, \frac{21}{44}, \frac{52}{109}, and \frac{73}{153}$.

A larger convergent is $\frac{154,416}{323,641}$, our familiar friend from the previous post in this series.

As more terms are taken, these convergents get closer and closer to $\log_{10} 3$. In fact:

• Each convergent is the best possible rational approximation to $\log_{10} 3$ using a denominator that’s less than the denominator of the next convergent. For example, the second convergent $\displaystyle \frac{10}{21}$ is the closest rational number to $\log_{10} 3$ that has a denominator less than $44$, the denominator of the third convergent.
• The convergents alternate between slightly greater than $\log_{10} 3$ and slightly less than $\log_{10} 3$.
• Each convergent $\displaystyle \frac{m}{n}$ is guaranteed to be within $\displaystyle \frac{1}{n^2}$ of $\log_{10} 3$. (In fact, if $\displaystyle \frac{m}{n}$ and $\displaystyle \frac{p}{q}$ are consecutive convergents, then $\displaystyle \frac{m}{n}$ is guaranteed to be within $\displaystyle \frac{1}{nq}$ of $\log_{10} 3$.)
• As a practical upshot of the previous point: if the denominator of the convergent $\displaystyle \frac{m}{n}$ is at least six digits long (that is, greater than $10^5$), then $\displaystyle \frac{m}{n}$ must be within $\displaystyle \frac{1}{(10^5)^2} = 10^{-10}$ of $\log_{10} 3$… and it’ll probably be significantly closer than that.

So convergents provide a way for teachers to maintain the illusion that they found a power like $3^{323,641}$ by laborious calculation, when in fact they were quickly found through modern computing.

Decimal Approximations of Logarithms (Part 3)

While some common (i.e., base-10) logarithms work out evenly, like $\log_{10} 10,000$, most do not. Here is the typical output when a scientific calculator computes a logarithm:

To a student first learning logarithms, the answer is just an apparently random jumble of digits; indeed, it can proven that the answer is irrational. With a little prompting, a teacher can get his/her students wondering about how people 50 years ago could have figured this out without a calculator. This leads to a natural pedagogical question:

Can good Algebra II students, using only the tools at their disposal, understand how decimal expansions of base-10 logarithms could have been found before computers were invented?

Here’s a trial-and-error technique — an exploration activity — that is within the grasp of Algebra II students. It’s simple to understand; it’s just a lot of work.

To approximate $\log_{10} x$, look for integer powers of $x$ that are close to powers of 10.

In the previous post in this series, we essentially used trial and error to find such powers of 3. We found

$3^{153} \approx 9.989689095 \times 10^{72} \approx 10^{73}$,

from which we can conclude

$\log_{10} 3^{153} \approx \log_{10} 10^{73}$

$153 \log_{10} 3 \approx 73$

$\log_{10} 3 \approx \displaystyle \frac{73}{153} \approx 0.477124$.

This approximation is accurate to five decimal places.

By now, I’d imagine that our student would be convinced that logarithms aren’t just a random jumble of digits… there’s a process (albeit a complicated process) for obtaining these decimal expansions. Of course, this process isn’t the best process, but it works and it only uses techniques at the level of an Algebra II student who’s learning about logarithms for the first time.

If nothing else, hopefully this lesson will give students a little more appreciation for their ancestors who had to perform these kinds of calculations without the benefit of modern computing.

We also saw in the previous post that larger powers can result in better and better approximation. Finding suitable powers gets harder and harder as the exponent gets larger. However, when a better approximation is found, the improvement can be dramatic. Indeed, the decimal expansion of a logarithm can be obtained up to the accuracy of a hand-held calculator with a little patience. For example, let’s compute

$3^{323,641}$

Predictably, the complaint will arise: “How did you know to try $323,641$?” The flippant and awe-inspiring answer is, “I just kept multiplying by 3.”

I’ll give the real answer that question later in this series.

Postponing the answer to that question for now, there are a couple ways for students to compute this using readily available technology. Perhaps the most user-friendly is the free resource Wolfram Alpha:

$3^{323,641} \approx 9.999970671 \times 10^{154,415} \approx 10^{154,416}$.

That said, students can also perform this computation by creatively using their handheld calculators. Most calculators will return an overflow error if a direct computation of $3^{323,641}$ is attempted; the number is simply too big. A way around this is by using the above approximation $3^{153} \approx 10^{73}$, so that $3^{153}/10^{73} \approx 1$. Therefore, we can take large powers of $3^{153}/10^{73}$ without worrying about an overflow error.

In particular, let’s divide $323,641$ by $153$. A little work shows that

$\displaystyle \frac{323,641}{153} = \displaystyle 2115 \frac{46}{153}$,

or

$323,641 = 153 \times 2115 + 46$.

This suggests that we try to compute

$\displaystyle \left( \frac{3^{153}}{10^{73}} \right)^{2115} \times 3^{46}$,

and a hand-held calculator can be used to show that this expression is approximately equal to $10^{21}$. Some of the last few digits will be incorrect because of unavoidable round-off errors, but the approximation of $10^{21}$ — all that’s needed for the present exercise — will still be evident.

By the Laws of Exponents, we see that

$\displaystyle \left( \frac{3^{153}}{10^{73}} \right)^{2115} \times 3^{46} \approx 10^{21}$

$\displaystyle \frac{3^{153 \times 2115 + 46}}{10^{73 \times 2115}} \approx 10^{21}$

$\displaystyle \frac{3^{323,641}}{10^{154,395}} \approx 10^{21}$

$3^{323,641} \approx 10^{154,395} \times 10^{21}$

$3^{323,641} \approx 10^{154,395+21}$

$3^{323,641} \approx 10^{154,416}$.

Whichever technique is used, we can now use the Laws of Logarithms to approximate $\log_{10} 3$:

$\log_{10} 3^{323,641} \approx \log_{10} 10^{154,416}$

$323,641 \log_{10} 3 \approx 154,416$

$\log_{10} 3 \approx \displaystyle \frac{154,416}{323,641} \approx 0.477121254723598\dots$.

This approximation matches the decimal expansion of $\log_{10} 3$  to an astounding ten decimal places:

$\log_{10} 3 \approx 0.47712125471966\dots$

Since hand-held calculators will generate identical outputs for these two expressions (up to the display capabilities of the calculator), this may lead to the misconception that the irrational number $\log_{10} 3$ is actually equal to the rational number $\displaystyle \frac{154,416}{323,641}$, so I’ll emphasize again that these two numbers are not equal but are instead really, really close to each other.

Summarizing, Algebra II students can find the decimal expansion of $\log_{10} x$ can be found up to the accuracy of a hand-held scientific calculator. The only tools that are needed are

• The Laws of Logarithms
• A hand-held scientific calculator
• A lot of patience multiply $x$ by itself repeatedly in a quest to find integer powers of $x$ that are close to powers of $10$.

While I don’t have a specific reference, I’d be stunned if none of our ancestors tried something along these lines in the years between the discovery of logarithms (1614) and calculus (1666 or 1684).

Decimal Approximations of Logarithms (Part 2)

While some common (i.e., base-10) logarithms work out evenly, like $\log_{10} 10,000$, most do not. Here is the typical output when a scientific calculator computes a logarithm:

To a student first learning logarithms, the answer is just an apparently random jumble of digits; indeed, it can proven that the answer is irrational. With a little prompting, a teacher can get his/her students wondering about how people 50 years ago could have figured this out without a calculator. This leads to a natural pedagogical question:

Can good Algebra II students, using only the tools at their disposal, understand how decimal expansions of base-10 logarithms could have been found before computers were invented?

Here’s a trial-and-error technique — an exploration activity — that is within the grasp of Algebra II students. It’s simple to understand; it’s just a lot of work. While I don’t have a specific reference, I’d be stunned if none of our ancestors tried something along these lines in the years between the discovery of logarithms (1614) and calculus (1666 or 1684).

To approximate $\log_{10} x$, look for integer powers of $x$ that are close to powers of 10.

I’ll illustrate this idea with $\log_{10} 3$.

$3^1 = 3$

$3^2 = 9$

Not bad… already, we’ve come across a power of 3 that’s decently close to a power of 10. We see that

$3^2 = 9 < 10^1$

and therefore

$\log_{10} 3^2 < 1$

$2 \log_{10} 3< 1$

$\log_{10} 3< \displaystyle \frac{1}{2} = 0.5$

Let’s keep going. We just keep multiplying by 3 until we find something close to a power of 10. In principle, these calculations could be done by hand, but Algebra II students can speed things up a bit by using their scientific calculators.

$3^3 = 27$

$3^4 = 81$

$3^5 = 243$

$3^6 = 729$

$3^7 = 2,187$

$3^8 = 6,561$

$3^9 = 19,683$

$3^{10} = 59,049$

$3^{11} = 177,147$

$3^{12} = 531,441$

$3^{13} = 1,594,323$

$3^{14} = 4,782,969$

$3^{15} = 14,348,907$

$3^{16} = 43,046,721$

$3^{17} = 129,140,163$

$3^{18} = 387,420,489$

$3^{19} = 1,162,261,467$

$3^{20} = 3,486,784,401$

$3^{21} = 10,460,353,203$

This looks pretty good too. (Students using a standard ten-digit scientific calculator, of course, won’t be able to see all 11 digits.) We see that

$3^{21} > 10^{10}$

and therefore

$\log_{10} 3^{21} > \log_{10} 10^{10}$

$21 \log_{10} 3 > 10$

$\log_{10} 3 > \displaystyle \frac{10}{21} = 0.476190\dots$

Summarizing our work so far, we have

$0.476190\dots < \log_{10} 3 < 0.5$.

We also note that this latest approximation actually gives the first two digits in the decimal expansion of $\log_{10} 3$.

To get a better approximation of $\log_{10} 3$, we keep going. I wouldn’t blame Algebra II students a bit if they use their scientific calculators for these computations — but, ideally, they should realize that these calculations could be done by hand by someone very persistent.

$3^{22} = 31,381,059,609$

$3^{23} = 94,143,178,827$

$3^{24} = 282,429,536,481$

$3^{25} = 847,288,609,443$

$3^{26} = 2,541,865,828,329$

$3^{27} = 7,625,597,484,987$

$3^{28} = 22,876,792,454,961$

$3^{29} = 68,630,377,364,883$

$3^{30} = 205,891,132,094,649$

$3^{31} = 617,673,396,283,947$

$3^{32} = 1,853,020,188,851,841$

$3^{33} = 5,559,060,566,555,523$

$3^{34} = 16,677,181,699,666,569$

$3^{35} = 50,031,545,098,999,707$

$3^{36} = 150,094,635,296,999,121$

$3^{37} = 450,283,905,890,997,363$

$3^{38} = 1,350,851,717,672,992,089$

$3^{39} = 4,052,555,153,018,976,267$

$3^{40} = 12,157,665,459,056,928,801$

$3^{41} = 36,472,996,377,170,786,403$

$3^{42} = 109,418,989,131,512,359,209$

$3^{43} = 328,256,967,394,537,077,627$

$3^{44} = 984,770,902,183,611,232,881$

Using this last line, we obtain

$3^{44} < 10^{21}$

and therefore

$\log_{10} 3^{44} < \log_{10} 10^{21}$

$44 \log_{10} 3 < 21$

$\log_{10} 3 < \displaystyle \frac{21}{44} = 0.477273\dots$

Summarizing our work so far, we have

$0.476190\dots < \log_{10} 3 < 0.477273\dots$.

A quick check with a calculator shows that $\log_{10} 3 = 0.477121\dots$. In other words,

• This technique actually works!
• This last approximation of $0.477273\dots$ actually produced the first three decimal places of the correct answer!

With a little more work, the approximations

$3^{109} \approx 1.014417574 \times 10^{52} > 10^{52}$

$3^{153} \approx 9.989689095 \times 10^{72} < 10^{73}$

can be found, yielding the tighter inequalities

$\displaystyle \frac{52}{109} < \log_{10} 3 < \displaystyle \frac{73}{153}$,

or

$0.477064\dots < \log_{10} 3 < 0.477124$.

Now we’re really getting close… the last approximation is accurate to five decimal places.

Decimal Approximations of Logarithms (Part 1)

My latest article on mathematics education, titled “Developing Intuition for Logarithms,” was published this month in the “My Favorite Lesson” section of the September 2018 issue of the journal Mathematics Teacher. This is a lesson that I taught for years to my Precalculus students, and I teach it currently to math majors who are aspiring high school teachers. Per copyright law, I can’t reproduce the article here, though the gist of the article appeared in an earlier blog post from five years ago.

Rather than repeat the article here, I thought I would write about some extra thoughts on developing intuition for logarithms that, due to space limitations, I was not able to include in the published article.

While some common (i.e., base-10) logarithms work out evenly, like $\log_{10} 10,000$, most do not. Here is the typical output when a scientific calculator computes a logarithm:

To a student first learning logarithms, the answer is just an apparently random jumble of digits; indeed, it can proven that the answer is irrational. With a little prompting, a teacher can get his/her students wondering about how people 50 years ago could have figured this out without a calculator. This leads to a natural pedagogical question:

Can good Algebra II students, using only the tools at their disposal, understand how decimal expansions of base-10 logarithms could have been found before computers were invented?

Students who know calculus, of course, can do these computations since

$\log_{10} x = \displaystyle \frac{\ln x}{\ln 10}$,

$\ln (1+t) = t - \displaystyle \frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \dots$,

a standard topic in second-semester calculus, can be used to calculate $\ln x$ for values of $x$ close to 1. However, a calculation using a power series is probably inaccessible to bright Algebra II students, no matter how precocious they are. (Besides, in real life, calculators don’t actually use Taylor series to perform these calculations; see the article CORDIC: How Hand Calculators Calculate, which appeared in College Mathematics Journal, for more details.)

In this series, I’ll discuss a technique that Algebra II students can use to find the decimal expansions of base-10 logarithms to surprisingly high precision using only tools that they’ve learned in Algebra II. This technique won’t be very efficient, but it should be completely accessible to students who are learning about base-10 logarithms for the first time. All that will be required are the Laws of Logarithms and a standard scientific calculator. A little bit of patience can yield the first few decimal places. And either a lot of patience, a teacher who knows how to use Wolfram Alpha appropriately, or a spreadsheet that I wrote can be used to obtain the decimal approximations of logarithms up to the digits displayed on a scientific calculator.

I’ll start this discussion in my next post.

Engaging students: Equations of two variables

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 again comes from my former student Trent Pope. His topic, from Algebra: equations of two variables.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

I found a website that has many word problems where students can solve for two variables. An example of one of these problems is “If a student were to buy a certain number of $5 scarfs and$2 hats for a total amount of \$100, how many scarfs and hats did they buy?”. This example would give students a real world application of how we use two variable equations. It would show students that there are multi variable problems when we go to the store to shop for things, like food or clothing. An instance for food would be when a concession stand sells small and large drinks at a sporting event and want to know how many drinks they have sold at the end of the night. After using a two variable linear equation and knowing the price of the cups, total amount earned, and total cups sold, students would be able to solve for the number of small cups as well as large cups sold.

B2. How does this topic extend what your students should have learned in previous courses?

This topic extends on the students’ ability to graph and solve a linear equation, which should have been taught in their previous classes. The only difference is that the variable, y, that you solved for in Pre-Algebra is now on the same side as the other variable. For instance, the equation y =(-1/4) x + 4 is the same as x + 4y = 16. We see that we solve for the same variables, but they are both on the same side. This is because you are solving the same linear equation. A linear equation can be written in multiple forms, as long as the forms have matching solutions. This is something that students could prove to you by graphing and solving the equations. They would solve the equations to see that they have the same variables. This makes students more aware that they need to be able to compute for other variables besides x if the question asks for it.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The most effective way to engage a student about this topic is by using a graphing calculator. This is to help students make the visual connection with the topic and check to see if they have graphed the equations the correct way. Students learn more effectively through visual demonstration. Because students are the ones to solve for the equation and plug it into the calculator to check their work, they are going to be able to make that connection, and we will be able to verify that they understand the material. As teachers, we need to incorporate more technology into the ways of learning because we are surrounded by it daily. Using graphing calculators would be a great way to show and check the work of a two variable equation. This gives students a chance to see what mistakes they have made and what lose ends need to be tied up.

References

Solving Word Problems using a system with 2 variables. n.d. <https://sites.google.com/site/harlandclub/Home/math/algebra/word2var&gt;.

My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

My Favorite One-Liners: Part 112

This was also the story of my childhood.

My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

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 again comes from my former student Megan Termini. Her topic, from Algebra: the quadratic formula.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The Quadratic Formula came about when the Egyptians, Chinese, and Babylonian engineers came across a problem. The engineers knew how to calculate the area of squares, and eventually knew how to calculate the area of other shapes like rectangles and T-shapes. The problem was that customers would provide them an area for them to design a floor plan. They were unable to calculate the length of the sides of certain shapes, and therefore were not able to design these floor plans. So, the Egyptians, instead of learning operations and formulas, they created a table with area for all possible sides and shapes of squares and rectangles. Then the Babylonians came in and found a better way to solve the area problem, known as “completing the square”. The Babylonians had the base 60 system while the Chinese used an abacus for them to double check their results. The Pythagoras’, Euclid, Brahmagupta, and Al-Khwarizmi came later and all contributed to what we know as the Quadratic Formula now. (Reference A)

A2. How could you as a teacher create an activity or project that involves your topic?

A great activity that involves the Quadratic Formula is having the students work in groups and come up with a way to remember the formula. It could be a song, a rhyme, a story, anything! I have found a few examples of students and teachers who have created some cool and fun ways of remembering the Quadratic Formula. One that is commonly known is the Quadratic Formula sung to the tune of “Pop Goes the Weasel” (Reference B). It is a very catchy song and it would be able to help students in remembering the formula, not just for this class but also in other classes as they further their education. Now, having the students create their own way of remembering it will benefit them even more because it is coming from them. An example is from a high school class in Georgia. They created a parody of Adele’s “Rolling in the Deep” to help remember the Quadratic Formula (Reference C). It’s fun, it gets everyone involved, it engaging, and it helps student remember the Quadratic Formula.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is a great way of engaging students in today’s world. Many students now have cell phones or the school provides laptops to be used during class. Coolmath.com is a great website for students to use to learn about the quadratic formula and great way to practice using it. They show you why the formula works and why it is important to know it because not all quadratic equations are easy to factor. There are a few examples on there and then they give the students a chance to practice some random problems and check to see if they got the right answer. This website would be good for student in and out of the classroom (Reference D). Khan Academy is another great way for students to learn how to use the quadratic formula. They have many videos on how to use the formula, proof of the formula, and different examples and practices of applying the quadratic formula (Reference E). Students today love when they get to use their phones in class or computers, so technology is a great way to engage students in learning and applying the quadratic formula.

References:

A. Ltd, N. P. (n.d.). H2g2 The Hitchhiker’s Guide to the Galaxy: Earth Edition. Retrieved September 14, 2017, from https://h2g2.com/approved_entry/A2982567
B. H. (2011, April 04). Retrieved September 14, 2017, from https://www.youtube.com/watch?feature=youtu.be&v=mcIX_4w-nR0&app=desktop
C. E. (2013, January 13). Retrieved September 14, 2017, from https://www.youtube.com/watch/?v=1oSc-TpQqQI

Engaging students: Graphs of linear equations

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 again comes from my former student Saundra Francis. Her topic, from Algebra: graphs of linear equations.

B1. How can this topic be used in your students’ future courses in mathematics and science?

Learning how to graph linear equations is the basis for many topics that students will learn later in Algebra and future mathematics and science courses. Students will now be able to solve word problems using graphs to model the situation describe in the problem. Being able to graph linear equations will help students graph non-linear equations since they will be able to apply the steps they learn on how to graph to different types of equations, Students will also be able to graph inequalities to find solutions for an equation since graphing equations is the first step in graphing inequalities. Another application of graphing linear equations is when students need to make graphs when completing science labs, many times students need to graph their data collected and find an equation that represents the data.

C3. How has this topic appeared in the news?

Graphs of linear equations are displayed in the markets sections on The New York Times. Segments of different linear equations can be put together match the graphs that display the rise and fall of different markets and stocks. Time is displayed on the x-axis while the y-axis list the price of the stock. The slope of the line is the percent change in the price of the stock and can be positive or negative depending if the price rose or fell. The y-intercept would be the price that the stock or market was at before the percent change. This will engage students because it is an example of how graphs of linear equations is displayed in the real world and they get a chance to see how they can use this concept in the future. This could also be made into an activity where students discover the linear equations that are combined to make a certain market or stock graph.

D1. What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

René Descartes was born in 1596 and was a French scientist, philosopher, and mathematician. He is thought to be the father of modern philosophy. Descartes started his education at age nine and by the time he was twenty-two he had earned a degree in law. Then Descartes tried to understand the natural world using mathematics and logic, which is when he discovered how to visually represent algebraic equations. Descartes was the first to use a coordinate system to display algebraic equations. In 1637 Descartes published La Géométrie, which was where he first showed how to graph equations. He linked geometry and algebra in order to represent equations visually. While thinking about the nature of knowledge and existence Descartes stated, “I think; therefore I am”, which is one of his most famous thoughts. Students will gain interest in graphing equations when they are told about Descartes since he was an interesting person and he discovered things not only in the field of mathematics but philosophy too.