Factor Song

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

A true artist is never appreciated in his own time. Or class.

Folks who follow the math problem calender on Twitter might've gotten a sneak peek last week at this little ditty. The plan was to use it here as well, but I wasn't exactly sure how I was going to use it. Or when, since the new school year keeps me busy until I get things settled into a routine.

The problem written in the top panel is how it appeared on the calendar. In the bottom panel, I rewrote it in a way more commonly seen in math class when covering combinations.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon.

Come back often for more funny math and geeky comics.

Factorial Facts

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You don't get down from an elephant, you get down from a duck!

This was originally supposed to be an English/Math fact tete-a-tete, with some piece of Shakespearean trivia. But before I could investigate any such Bardic lore, the above shower thought came to mind -- and I wasn't even in the shower at the time!

This is the way my mind works, with a bang and a whimper.

ObMath: 10!/6! = 7! is one of those lovely coincidences in math.
It occurs because 10 * 9 * 8 * 7 = 5040, as does 7 * 6 * 5 * 4 * 3 * 2 * 1.
This is because 10 = 5 * 2, 9 = 3 * 3, and 8 = 4 * 2. Then multiply that extra 3 and 2, and you get 6.
So 10 * 9 * 8 * 7 = 5 * 2 * 3 * 3 * 4 * 2 * 7 = 7 * (3 * 2) * 5 * 4 * 3 * 2 [* 1, of course].

Another example of this is 6!/3! = 5!, but this is pretty much a trivial case since 3! = 6.




Come back often for more funny math and geeky comics.

Five Things You Need to Know About Algebra, Part II

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(C)Copyright 2015, C. Burke.

True story: a student exiting the bathroom after a Regents exam, spotted me in the hall and asked, ''Mr. Burke, what does an exclamation point mean in a math problem?'' It didn't help that it was a meaningless probably and nothing to do with statistics or probability.

On another note: the number 5040 is very interesting. Not only is it 7 X 6 X 5 X 4 X 3 X 2 X 1, but it is also 10 X 9 X 8 X 7 (and that also means that 10!/7! = 6!). But here's another interesting fact: If the Moon was sitting on Earth's equator, the distance from the center of the Moon to the center of the Earth would be 5,040 miles. Unfortunately, if the Moon were sitting on Earth equator, it would be very bad because we would all die. Food for thought.

Blog: Jeopardy and Non-Common Core Math

Last week, Jeopardy had a Kids Week and on Friday night, one of the categories was Non-Common Core Math. As a math teacher and just someone who likes numbers, I was curious what the category would be. The kids, on the other hand, well they were curious, too, at first, but then ran away.

It started innocently enough, with the $200 answer being: "1 + 2 + 3 + 4 + 5". Quick mental math gave the question, "What is 15?" A simple exercise in triangle numbers, which are formed by summing consecutive numbers. It's one of those things which most kids will see and do even before they hear the phrase "triangle numbers", and long before they know they memorize the formula. Besides, a small sequence like this is quicker to add (if you don't have it already memorized) than computing a formula.

Things got trickier with the $400 answer, "1 - 2 + 3 - 4 + 5". There was some hesitation as the contestants (I almost typed "students") worked that one out before one of them arrived at "What is 3?" (I didn't tape it, so I can't review it to see if someone got it incorrect first. I don't remember.) There are two short cuts for this problem, and both have to do with pairing. If you noticed that each pair "1 - 2" and "3 - 4" yield a result of "-1", you have -1 + -1 + 5, which is 3. If you noticed that "-2 + 3" and "-4 + 5" yield a result of "+1", then you had 1 + 1 + 1, which is still 3. If you just oscillated your numbers, you took more time and you probably didn't buzz in in time.

The kids gave up on the third answer: "1 * 2 * 3 * 4 * 5". Given the ages of the kids, I would have thought that at least one of them had seen factorial before, and this was the definition of 5!, although the numeric equivalent was needed. Perhaps they got stuck on 24 * 5, not thinking to reverse the order (20, 60, 120, 120). Whatever the reason, no one got the answer, and they bailed on the category.

It proved so unpopular that when Trebek cautioned "less than a minute to go" (a.k.a < 1 min), the two Math clues remained, and were the final ones of the game.

The $800 answer: "-1 * 2 * -3 * 4 * -5". This actually bothered me that none of the kids gt it. First, for the reason Alex gave. Second, because I had to listen to Alex give it. The previous clue had a result of 120. The numbers multiplied are the same, only some of the signs have changed. Multiply a negative times a negative times a negative and the product will be negative (times two more positives, which won't affect it). The question should have been "What is -120?", which should have been easy considering the previous question gave them the number, and they only had to add on the sign.

The final reason to be annoyed? Alex took so long to explain what should have been obvious that we didn't get to see the last clue. Would it have had division? Exponentiation? Mathematical minds want to know!

But that clue won't be revealed, and it's likely that they avoid such mathematical categories during future Kids Weeks.

---

UPDATE: I wrote a little More about that Jeopardy category after the blog The Political Hat referenced this entry.

And Jeopardy had Another Math category, with adults, shortly after this.

Day 29 of 30: Recursive Functions

This is Day 29 of my 30-day blogging challenge. I can see the checkered flag in the distance.

Common Core Algebra seems more concerned about functions and families of functions than the previous Integrated Algebra in New York State was. It used to be that it was all linear equations and if you saw f(x), you just think "y". They had their uses, but we didn't get to talk much about them -- there was already too much to talk about in that course, and we had a hard time making all of it happen.

Well, Common Core is no different. They just give you a different "all that" to cover, and they really do want you to cover it all. When it comes to functions, they not only want the standard linear, quadratic, exponential and absolute function, but also cubic, square root, piecewise and recursive. Some of them need to be graphed, some only have to be used for evaluating, for example, f(3) or f(-5).

I can get to piecewise another time. My students are getting the hang of them. Well, some of them are. Some of the time. Okay, so maybe not. And I don't blame them. I didn't have to deal with those until later on.

Recursive functions, on the other hand, I don't remember from math class at all. Seriously. I remember them from computer programming. This isn't to say that I hadn't seen them in a math class first before I programmed a recursive function, but I know what left its mark on my memory and where it took.

A recursive function is one that calls itself. It uses itself in its own definition. The two most obvious examples (and Common Core won't use obvious examples are:

Factorial: f(n) = f(n - 1) * n, f(1) = 1

Triangular Numbers: t(n) = t(n - 1) + n, t(1) = 1

To calculate f(n) when n = 5, you do the following: f(5) = f(5-1) * 5 = f(4) * 5. But you need to know what f(4) is.
Well, that's easy. f(4) = f(4-1) * 4 = f(3) * 4. Okay, so now we need to know f(3) ... and then f(2) ... and so on down to f(1), which we are given.
As I told me students: you might as well start with f(2) and work your way up. We'll have to do all that work anyway. Then find f(3), f(4) and so on. If you are programming a computer ... well, I'd probably do it the same way for simple things, but use a recursive function when the professor tells me to do so. Most of the recursive things I've ever coded worked fine with a For...Next or Do...End loop.

Likewise, t(5) would work the same way. The only difference is that I used the letter t for triangular because I already used f for factorial in the same example. Be careful! Students can actually get confused by this. (I'm not kidding.)

Real-World Connection: Forgetting about where factorials and triangular numbers may occur in your everyday existence, recursive functions are part of the real world. Take the function clean(n). Suppose you wish to evaluate this function for the value of n = floor. There's a problem with this because your wife may tell you that you can't clean the floor until you evaluate clean(counters). And you can't do that until you evaluate clean(cabinets).

Now keep in mind that this is just supposed to be an illustration of a recursive function. However, it isn't exactly a true parallel example. For one thing, one you can't calculate 5! (5 factorial) without first calculating 4! (4 factorial). On the other hand, you can clean the floor without cleaning the counters or the cabinets. However, in both cases, according to my wife, at least, you'd be incorrect.

January 2014 Algebra Regents, Thread 2

It's Day 7, if anyone is keeping track

Continuing with the multiple-choice questions:

January 2014 Algebra Regents

11. The graph below illustrates the number of acres used for farming in Smalltown, New York, over several years.

Using a line of best fit, approximately how many acres will be used for farming in the 5th year?

The trend is downward. It will be below 300, but not so low as to hit zero. The only logical choice is 200.

12. When 16x3 - 12x² + 4x is divided by 4x, the quotient is

Divide each co-efficient by 4, subtract one from each exponent. The last term (4x) is divided by itself, leaving + 1. It's choice (4).

If you don't understand why it isn't choice (2), this comic may be relevant!

13. The width of a rectangle is 4 less than half the length. If ℓ represents the length, which equation could be used to find the width, w?

Translating the words into an equation W = (1/2) L - 4. Pretty straightforward.

14. Which data can be classified as quantitative?

Quantitative is a numeric value that count or add or average or compute. Something that isn't a description (In other words, not a zip code, phone number or shoe size).

Three choices are things. Only the sales tax rate is a measurable amount.

15. Two cubes with sides numbered 1 through 6 were rolled 20 times. Their sums are recorded in the table below.

What is the empirical probability of rolling a sum of 9?

Theoretically, the odds of a 9 are 4/36 = 1/9. However, that's not what they asked. An experiment was performed and we need to use the results of the experiment. (That's what "empirical" means.) Who knows? Maybe the dice are weighted slightly and, therefore, biased. Count the number of times 9 appears in the table. Then go back and do it again! Did you get the same number both times? If so, there's a good chance you didn't miss any. If you didn't, you might want to check a third time, just to be sure.

I counted five of them. Both times. Five out of twenty is 5/20 (which simplifies to 1/4, but they didn't simplify their answers).

16. What is the vertex of the graph of the equation y = 3x² + 6x + 1?

You can answer this by entering it in your graphing calculator and taking a look. But if you don't want to, here's another way:

The x-co-ordinate of the vertex can be found with the formula x = (-b)/(2a). So x = (-6)/(2*3)= (-6)/(6) = -1. Notice that two choices have been eliminated. Now substitute -1 for x and evaluate for y. y = 3(-1)^2 + 6(-1) + 1 = 3(1) + (-6) + 1 = -2. The vertex is (-1, -2), which is choice (1).

17. The length and width of a rectangle are 48 inches and 40 inches. To the nearest inch, what is the length of its diagonal?

Use the Pythagorean Theorem and hit the square root button rather than simplify any radical numbers. c^2 = 48^2 + 40^2 = 2304 + 1600 = 3904. The square of 3904 is 62.48, which rounds down to 62. Choice (2).

NUMBER SENSE ALERT!! Common sense will tell you that the diagonal is longer than the two sides, so 27 is out. Common sense will also tell you that the third side of a triangle MUST be shorter than the sum of the other two sides. So 88 and 90 are out. There's only one possible choice.

18. Which graph represents the solution set of 2x - 5 < 3?

Add 5 to both sides (3 becomes 8) and then divide by 2 (8 becomes 4), so x < 4. That's a ray with an open circle at one end and pointing to the left. Choice (1).

19. Jonathan drove to the airport to pick up his friend. A rainstorm forced him to drive at an average speed of 45 mph, reaching the airport in 3 hours. He drove back home at an average speed of 55 mph. How long, to the nearest tenth of an hour, did the trip home take him?

If it took 3 hours going 45 mph to get there, then Distance = Rate X Time = 45 X 3 = 135. The airport is 135 miles away.
T = D / R = 135 / 55 = 2.454545...., which is approximately 2.5 hours, to the nearest tenth. Choice (2).

20. The expression (2n)/5 + (3n)/2 is equivalent to

Forgive me for trying to minimize the number of images that I need to upload. Those were two fractions, without common denominators.
Multiply the first fraction by 2/2 and the second by 5/5. The numerators become 4n and 15n, respectively. They total to 19n over 10. Choice (3).

Note: If you said (5n)/7, you have bigger problems than Algebra. No, seriously. I'm not trying to be mean, You need to review the basics before you can move forward, or you will forever be scratching your head at incorrect responses.

21. When x = 4, the value of 2x⁰ + x! is

FLASHBACK! JUNE 2013!!: "Mr. Burke, what does an exclamation point in a math problem mean?" -- A frustrated student who ran into me after the Regents exam.

Mini-rant: I hate when factorials (the exclamation point !) are added to problems just because they can be. It has a purpose, and this isn't it.

For those who, like my former student, don't recall ever seeing an exclamation point before, it's called a "factorial", and what it means is this: multiply the number by the number before it times the number before that all the way down to 1. So 5! = 5 X 4 X 3 X 2 X 1 = 120. There are important uses for this is Statistics and Probability. For example, the number of ways that you can arrange five pictures in a row on a shelf is 5!.

This problem, however, is just stupid. On the other hand, it's "theoretical". In theory, there's no reason why we can't answer it -- even if it doesn't make sense to us.

Order of Operations: Substitute (4) for x. Take (4) to the 0th power, which is 1. Multiply 1 by 2 and you get 2. The addition has to wait because the factorial takes precedence because it is a form of exponentiation. (For anyone wanting to argue that point, consider this alternative explanation 2*3! is NOT equal to 6!, so the multiplication comes second!)

4! = 4 X 3 X 2 X 1 = 24. 2 + 24 = 26. Choice (3)

22. Which graph represents the solution of 2y + 6 > 4x?

First of all, it's a broken line because it's "greater than" and not "greater than or equal to". Eliminate two choices. Check if the Origin (0, 0) is in the solution. Is 0 + 6 > 4(0)? Yes, it is! So you have to shade the side of the line that includes (0, 0). That's choice (3).

Notice that you didn't have to put the problem into slope-intercept form, or graph it in any way to get the answer. Number Sense pays off again!

23. Which graph represents the exponential decay of a radioactive element?

Radioactive decay drops fast and then slows down, getting closer to zero (the x-axis) without ever meeting or crossing it. That's Choice (4).

24. Which fraction represents (x² - 25) / (x² - x - 20)expressed in simplest form?

Factor! Factor! Factor!

If you cancel out the two x², then get out of my classroom!!!!! No! You can't do that!

Factor the numerator into (x + 5)(x - 5). It's the Difference of Squares Rule. Factor the numerator into (x - 5)(x + 4).

Cancel out the two (x - 5) factors, and you are left with (x + 5) / (x + 4).

25. If abx - 5 = 0, what is x in terms of a and b?

Add 5 to both sides. abx = 5. Divide by the co-efficient of x, which is ab, and x = 5 / (ab).

Remember: Subtract the constant and divide! You'll find your x and you'll know why! or so the song goes.

26. Given: U = {x|0 < x < 10 and x is an integer}, S = {x|0 < x < 10 and x is an odd integer}
The complement of set S within the universal set U is

The complement of all the odd integers between 0 and 10, exclusive, within the Universal set of integers between 0 and 10, exclusive, would be the even integers between 0 and 10, exclusive. I say "exclusive" because of the "less than" signs, which indicate that 0 and 10 are NOT part of the solution. So it's {2, 4, 6, 8}, pick Choice (4), that's really great!

27. The roots of the equation 2x² - 8x = 0 are

Divide by two to get: x² - 4x = 0, which factors into (x)(x-4) = 0. So either x = 0 or x - 4 = 0. Therefore, x = 0 or x = 4. Choice (4).

28. Which equation illustrates the multiplicative inverse property?

Another definition question. You either know it or you don't. The inverse of a multiplication problem is its reciprocal. A number times its reciprocal equal 1. Choice (3).

29. What is the result when 4x² - 17x + 36 is subtracted from 2x² - 5x + 25?

First of all, subtracted from means taking the first from the second. As in "15 subtracted from 20" means "20 - 15". Don't do the problem backward!

Look at the co-efficients: 2 - 4 = -2; -5 - (-17) = +12. You know enough to get your answer, but check: 25 - 36 = -11. Choice (4)

30. Julie has three children whose ages are consecutive odd integers. If x represents the youngest child’s age, which expression represents the sum of her children’s ages?

Consecutive odd integers are x, x + 2, x + 4, etc. If there are three numbers, then their sum is x + x + 2 + x + 4, which is 3x + 6. Choice (4)

to be continued . . .

More on the Algebra Regents

It's been almost a week, and the tests are graded, so I don't suppose anyone still cares about this, but I'll go ahead anyway.

First off, if you weren't familiar with the word bivariate, you could have broken it down into bi-, meaning "two", and -variate, which looks like "variable", right? So bivariate: two variables. Which of the tables is measuring two variables and will give a scatter plot, as opposed to a bar graph. The answers, unfortunately, doesn't matter because the question was thrown out. A "lack of specificity" was the reason.

Likewise, if you took the test in Chinese, two answers were accepted to question number 1 because of a translation error. That happens a lot.

As for the Factorial question, a.k.a. "the question with the exclamation point", I was able to guess that answer without doing any work for one simple reason: the last step was to subtract 10, but only one answer was 10 less than another. That answer was the correct one. (I checked my guess afterward, of course.) For the record: 6! + 5!(3!)/(4!) - 10 can be done with the scientific calculator, if you know where to look, but it isn't necessary.
6! = 720, 3! = 6, 5!/4! = 5, so 720 + 5 * 6 - 10 = 720 + 30 - 10 = 740

I wanted to review some of the open-ended questions.

Question 31. An inequality with a negative multiplier. The trick was to remember to reverse the direction of the inequality symbol.

That is, -5(x - 7) < 15, when divided by -5 becomes (x - 7) > -3.
The final answer is x > 4.

Question 32. A volume question on the Algebra test. Silly. If they at least gave the Volume and asked to find, say, the height, you could argue it was an Algebra task, but, as is, it's a middle-school problem.

The formula for volume of a cylinder was in the back of the booklet: V = (pi)r^2*h.
The trick here is that the gave the diameter instead of the radius, so you had to divide 13 by 2 to get 6.5. If you didn't, your answer was four times larger than it should've been, but you most likely got 1 out of 2 points.

The final answer is 1,014*pi. Note: The question said "in terms of [pi]", so if you multiplied by 3.14 or used the pi key on your calculator (i.e., you did extra work!), you lost a point for not answering the question that they asked.

Question 33. A distance question with big numbers, with a conversion added on. Two questions on the test involved converting between hours and days and weeks. This was one of them.

The distance from Earth to Mars is 136,000,000 miles. A spaceship travels at 31,000 miles per hour. Determine, to the nearest day, how long it will take the spaceship to reach Mars.

Divided 136,000,000 by 31,000 to get the number of hours (4387.096774...) and then divide by 24 to get the number of days (182.795698...).
The final answer is 183 days.

Question 34. The Counting Principle. How many options are on the menu? They've given this question many times before, but this is the largest number of items that they've ever used. The Principle remains the same.

There are five main courses, three vegetables, five desserts, and three beverages. To find the number of possible means, multiple the four of them: 5 * 3 * 5 * 3 = 225.
How many have chicken tenders? That's 1 * 3 * 5 * 3 = 45, which is one-fifth of the total.
How many have pizza (1), corn or carrots (2), a dessert (5) and a beverage (3): 1 * 2 * 5 * 3 = 30

If you showed your work, you likely got one point for each correct answer.

Question 35. Trigonometry. Find the angle of elevation.
You have a right triangle with a height of 350 feet and a base of 1000 feet, and you want to find the angle on the ground. You have the opposite (350) and the adjacent (1000), but you don't know the hypotenuse, so that means that you need to use tangent to solve the problem.

So tan(x) = (350)/(1000) and, therefore x = tan-1(350/1000), which is approximate 19.29.
The final answers is 19 degrees
Partial credit likely for using sine or cosine, or if you answer is expressed in radians or if rounded incorrectly.

Question 36. Summation of radicals. I cover this in Geometry. I have simplified them when dealing with Pythagorean Theorem but I don't usually cover this as it requires an extra day or so that I don't have.

Bear with me as I try to type this without any graphics:

(25)^.5 - 2(3)^.5 + (27)^.5 + 2(9)^.5
"^" means raise to a power, "^.5" means raise to 1/2 power, which means square root.
Calculators and spreadsheets understand this.

The square root of 25 is 5 and twice the square root of 9 is 2*3 = 6. That leaves the root 3 term, which is in simplest form, and root 27, which simplifies to (9*3)^.5 = 3(3)^.5. So 5 + 6 = 11 and -2(3)^.5 + 3(3)^.5 = 1(3)^.5.
The final answers is 11 + (3)^.5. In other words, 11 + the square root of 3.

Question 37. Algebraic fractions.

This is the one time where your teacher may have given you bad advice. Actually, it was good advice, if you know when to use it, but this isn't the time.
I have colleagues who will tell students (the ones who hate fractions or just "can't do" them) to multiply by the denominators to get rid of them. In this case, that would make a big mess. Don't do that.
This is one approach you could take:

2 / 3x + 4 / x = 7 / (x + 1)
2 / 3x + 12 / 3x = 7 / (x + 1)
14/ 3x = 7 / (x + 1)
At this point, cross-multiplying will yield the equation 14(x + 1) = 21x
The final answers is x = 2.

Question 38.Probability. I was expecting more of a twist for a four-point problem. This one isn't too bad if you know what you're doing.
Five red marbles and three green marbles make eight marbles total
These are dependent events, and their probabilities will be multiplied.
P(red then green) = 5/8 * 3/7 = 15 / 56
P(both red) = 5/8 * 4/7 = 20 / 56 = 5 / 14
P(both red or both green) = P(both red) + P(both green) = (20 / 56) from above plus 3/8 * 2/7 = 6 / 56. The total is 26/56, which reduces to 13/28.
I recommend to my students to reduce fractions, but I can't say if it's required for full credit in this problem.

Question 39. An Area problem, which I have never covered in Algebra.
You needed to find the area of the rectangle, then the area of the semicircle, and then subtract the latter from the former.
The tricky part here was understanding the frickin' question! Seriously, they went out of the way to be obtuse about it, making this more of a reading comprehension problem than an actual math problem.
In the diagram, AB = 5, and AB = BC = DE = FE, but CD = 6, which means the radius, which can't be named because they didn't name the point at the center of the circle, is 3.

The rectangle is 5 X 16, so the area is 80.
The semicircle is 1/2(pi)(3)^2 = 4.5(pi), which is approximately, 14.13716694.
The shaded area is 80 - 14.13716694 = 65.86283306, which rounds to 65.86.
The final answers is 65.86 square inches.

STUPID QUESTION ALERT! The folks who write these tests know very well that students have been taught for years to approximate pi as 3.14, whether or not they have a calculator which has a "pi" key on it. Using 3.14, the numbers will change in this problem. The semicircle will become 14.13, and the final answer 65.87. Whether or not a student loses a point on this may very well depend upon the teacher scoring it.

Anyone (in New York) Have Any Reaction to the Algebra Regents exam?

The last Algebra Regents exam, in its current form, was given today. Next year's test will have to align with the Common Core Standards being implanted across the state and around the country. Problematically, some schools started their implementation this year, meaning that their students would be ill-prepared for this exam, which hardly lined up with those new standards.

Any thoughts on the exam? Please, share them. Let's have a dialogue.

EDIT: Thursday morning

In my opinion the dumbest question on the Algebra Regents was 6! + [5!(3!)]/4! - 10.

The question serves no purpose. It's order of operations... but with a twist! Sure, factorial gets covered with permutations, but this isn't any kind of combination/permutation question -- which, by the way, can be figured out without using an exclamation point.

There were too many statistics and probability questions, and there was no graphing. (Yes, there were a couple of multiple-choice questions which involved a graph, but students didn't need to make any graphs.) Statistics and probability are what get crammed in at the end of the year because there are so many topics in the Integrated Algebra curriculum, many of which are interconnected. Statitstics is a separate topic, and probability has been moved out of the curriculum under the Common Core standards. I feel sorry for any student whose school implemented the new curriculum a year early.

What else is there to complain about? (Anything to compliment?)

Problem of the Day: Factorials

I was going through old papers and torn magazine pages stored in folders that I hadn't looked at in years, finding stuff to recycle. I happened to glanced at a calendar of daily problems, most of which were either too advanced for the classes I was teaching at the time, or just a little too involved. However, one simple problem jumped out at me, and I decided to file that one away for next year. It's definitely a question that needs the solver to explain how he arrived at the answer, and would give me insight into their thinking.

I made a second problem based on the original. Here they both are:

1. Find the largest prime factor of (87!)(88!).

2. Find the largest prime factor of 87! + 88!

Why I love the problems: first, students needed to know something about factors, prime numbers and factorials. Second, seeing a number like 88!, that student whose first instinct is to reach for the calculator will have to put it down and find a new approach.

Answers below. Stop reading here if you didn't figure them out yet.

The answer to problem 1 is fairly straightforward. The factors of 87!*88! are
1 * 2 * 3 * ... * 87 * 1 * 2 * 3 * ... * 87 * 88.
There is no need to make factor trees to find the prime factorization (an approach students might take). The largest prime factor would be the largest prime number not greater than 88. That would be 83.

(If a student guessed 87, show them that 8+7 = 15, which is divisible by 3, so 87 is divisible by 3. Or just have them divide 87 / 3 and see that they'll get 29.)

The answer to problem 2 requires a little work. We want a factor of the sum, but factors are for products, not sums. No problem. Let's make a multiplication problem out of it.

87! + 88!
= 87! + 87!(88)
= 87!(1 + 88)
= 87!(89)

The largest prime factor of 87! is still 83. But 89 is a prime number, so it's the largest prime factor, which is another reason why this is such a neat little problem: 89 isn't in the initial problem, but every whole number less than 89 is!

I think that this is a great journal-type question to assess their understanding of concepts and their ability to communicate a solution. And it will fall in nicely with whatever they're calling "differentiating of instruction" this semester. (Yes, they mentioned a new term at the last department meeting, but I neglected to write it down or even really care.)

P.S. The first problem is mine. The second is the original.