Lessons from teaching gifted elementary school students (Part 2)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

$A \times A = B$

$B \times B = C$

$C \times C = D$

If the pattern goes on, and if $A = 2$ , what is $Z$ ?

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

Let’s calculate the first few terms to try to find a pattern:

$B = 2 \times 2 = 2^2$

$C = 2^2 \times 2^2 = 2^4$

$D = 2^4 \times 2^4 = 2^8$

etc.

Written another way,

$A = 2^1 = 2^{2^0}$

$B = 2^{2^1}$

$C = 2^{2^2}$

$D = 2^{2^3}$

Continuing the pattern, we see that $Z = 2^{2^{25}}$ , or $Z = 2^{33,554,432}$ .

If you try plugging that number into your calculator, you’ll probably get an error. Fortuniately, we can use logarithms to approximate the answer. Since $2 = 10^{\log_{10} 2}$ , we have

$Z = \left( 10^{\log_{10} 2} \right)^{33,554,432} = 10^{33,554,432 \log_{10} 2}$

Plugging into a calculator, we find that

$Z \approx 10^{10,100,890.5195}$

$\approx 10^{0.5195} 10^{10,100,890}$

$\approx 3.307 \times 10^{10,100,890}$

When this actually happened to me, it took me about 10 seconds to answer — without a calculator — “I’m not sure, but I do know that the answer has about 10 million digits.” Naturally, my class was amazed. How did I do this so quickly? I saw that the answer was going to be $Z = 2^{2^{25}}$ , so I used the approximation $2^{10} \approx 1000$ to estimate

$2^{25} = 2^5 \times 2^{10} \times 2^{10} \approx 32 \times 1000 \times 1000 = 32,000,000$

Next, I had memorized the fact that that $\log_{10} 2 \approx 0.301 \approx 1/3$ . So I multiplied $32,000,000$ by $1/3$ to get approximately 10 million. As it turned out, this approximation was a lot more accurate than I had any right to expect.

Lessons from teaching gifted elementary school students (Part 1)

Here’s a question I once received:

When playing with my calculator, I noticed the following pattern:

$256 \times 256 = 65,5\underline{36}$

$257 \times 257 = 66,0\underline{49}$

$258 \times 258 = 66,5\underline{64}$

Is there a reason why the last two digits are perfect squares? I know it usually doesn’t work out this way.

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

The answer is: This always happens as long as the tens digits is either 0 or 5.

To see why, let’s expand $(50n + k)^2$ , where $n$ and $k$ are nonnegative integers and $0 \le k \le 9$ . If $n$ is odd, then the tens digit of $50n+k$ will be a 5. But if $n$ is even, then the tens digit of $50n+k$ will be 0.

Whether $n$ is even or odd, we get

$(50n+k)^2 = 2500n^2 + 100nk + k^2 = 100(25n^2 + nk) + k^2$

The expression inside the parentheses is not important; what is important is that $100(25n^2 + nk)$ is a multiple of 100. Therefore, the contribution of this term to the last two digits of $(50n+k)^2$ is zero. We conclude that the last two digits of $(50n+k)^2$ is just $k^2$ .

Naturally, elementary-school students are typically not ready for this level of abstraction. That’s what I love about this question: this is a completely natural question for a curious grade-school child to ask, but the teacher has to have a significantly deeper understanding of mathematics to understand the answer.

A cute joke

Source: https://www.facebook.com/djheavygrinder/photos/a.85963791111.80105.6258151111/10152862899381112/?type=1&theater

Make It So, Make It So, Make It So

This week marks the one-year anniversary of this brilliant bit of tomfoolery.

Texans QB Ryan Fitzpatrick’s Son Shows Off Math Skills During Postgame Press Conference (Part 2)

From Bleacher Report:

Houston Texans quarterback Ryan Fitzpatrick… threw for 358 yards and six touchdowns in a 45-21 victory over the Tennessee Titans on Sunday [November 30, 2014]. However, [his son] Brady was the star of the postgame press conference.

Fitzpatrick put his son on the spot at the end of the press conference. In a matter of seconds, Brady was able to multiply 93 by 97 in his head.

Source: http://bleacherreport.com/articles/2284833-texans-qb-ryan-fitzpatricks-son-shows-off-math-skills-during-postgame-presser

After the thought bubble, I’ll reveal the likely way that young Brady did this.

Here’s a trick for multiplying two numbers in their 90s which is accessible to bright elementary-school students. We begin by multiplying out $(100-x)(100-y)$ :

$(100-x)(100-y) = 10,000 - 100x - 100y + xy$

$(100-100y) = 100(100 - [x+y]) + xy$

For $93 \times 97$ , we have $x = 7$ and $y = 3$ . So $x+y = 10$ , and $100 - [x+y] = 90$ . So the first two digits of the product is $90$ .

Also, $xy = 21$ . So the last two digits are $21$ .

Put them together, and we get the product $100 \times 90 + 21 = 9021$.

I don’t expect that young Brady knew all of this algebra, but I expect that he did the above mental arithmetic to put together the product. Well done, young man.

Texans QB Ryan Fitzpatrick’s Son Shows Off Math Skills During Postgame Press Conference (Part 1)

From Bleacher Report:

Houston Texans quarterback Ryan Fitzpatrick… threw for 358 yards and six touchdowns in a 45-21 victory over the Tennessee Titans on Sunday [November 30, 2014]. However, [his son] Brady was the star of the postgame press conference.

Fitzpatrick put his son on the spot at the end of the press conference. In a matter of seconds, Brady was able to multiply 93 by 97 in his head.

Source: http://bleacherreport.com/articles/2284833-texans-qb-ryan-fitzpatricks-son-shows-off-math-skills-during-postgame-presser

I’ll reveal the (likely) way that young Brady Fitzpatrick pulled this off tomorrow. In the meantime, I’ll leave a thought bubble if you’d like to try to figure it out on your own.

Advent calendar

Source: http://www.xkcd.com/994/

Why Science Teachers Should Not Be Given Playground Duty

Source: https://www.facebook.com/KMPHFOX26/photos/a.211376604011.169435.210413314011/10152781377649012/?type=1&theater

Creating a Culture of Inquiry in Mathematics Programs

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight “Creating a Culture of Inquiry in Mathematics Programs,” by Jill Dietz. Here’s the abstract:

We argue that student research skills in mathematics should be honed throughout the curriculum just as such skills are built over time in the natural and physical sciences. Examples used in the mathematics program at St. Olaf College are given.

The full article can be found here: http://dx.doi.org/10.1080/10511970.2012.711804

Full reference:Jill Dietz (2013) Creating a Culture of Inquiry in Mathematics Programs, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:9, 837-859, DOI: 10.1080/10511970.2012.711804

Engaging students: Synthetic Division

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 Chelsea Hancock. Her topic, from Precalculus: synthetic division.

The method of synthetic division is an alternative version of long division concerning polynomials. Synthetic division uses the basic mathematical skills of addition, subtraction, multiplication, and negative signs. They must also understand the definitions of polynomial, coefficient, and remainder. A polynomial is an expression with multiple terms, poly meaning “many” and nomial meaning “term.” A coefficient is a number used to multiply a variable. The remainder is the amount left over after division. Synthetic division involves multiplying, then adding or subtracting the coefficients of two polynomials. On some occasions, there will be a remainder after dividing the polynomials.

Mathematicians are lazy. That is a fact of life. One mathematician understood this, so in 1809 he created a cleaner, faster, and much simpler method for division. His name was Paolo Ruffini. In order to more efficiently divide polynomials, Ruffini invented the Ruffini’s Rule, known more commonly as synthetic division in today’s society. In 1783, he entered the University of Modena and he studied mathematics, medicine, philosophy and literature. Then, in 1798 he began teaching mathematics at the University of Modena. He was required to swear an oath of allegiance to the republic, but due to religious purposes, refused to do so. This resulted in the loss of his professorship and was prevented from teaching.

There are several videos on the Internet involving synthetic division, but there are two in particular that I personally think are excellent demonstrations of both the method itself and why it works. I have labeled these clips Video 1 and Video 2. Video 1 is a demonstration of the method in action, using a specific example involving numbers, walking the viewers through the process through the whole video. Video 2 explains why using synthetic division instead of using long division is the more efficient and less complicated method for dividing polynomials. The clip uses the same example used in Video 1, but this time the polynomials are divided using long division, walking the viewers through the process the entire time. As the narrator moves through the process, he makes connections between the synthetic division method and the long division method and draws conclusions between the two. By the end of the video, it is evident which is the cleanest method to use when concerning the division of polynomials. These videos not only give great tutorials on both methods of division, but allows the viewers to see the benefits and uses of synthetic division when it is possible to use it.

Video 1: