Collection of math problems requiring exploration but no trig/calc?
January 2, 2022 8:20 PM Subscribe
I'm a test prep instructor; one of the challenges I face is getting students to spend more than 2 minutes engaging with problems. I'm always on the lookout for problems that will require my students to invest thought/energy/deliberation, but at the same time will NOT require higher math (my students by and large need neither calc nor trig, save for a basic understanding of SOHCAHTOA; they DO need algebra/geometry knowledge). Where can I find a collection of such problems to give as "extra credit?" Books, websites, anything would be helpful. Example problem inside.
Here's an example of a problem that I have used successfully (open book, calculator/Excel allowed): "What is the minimum number of people needed in a room for the probability that there is at least one pair with the same birthday to exceed 50%? (For simplicity, assume birthdays are randomly distributed and that no one is born on Feb. 29.)" This problem is a great stretch for my students after they have learned a few probability principles; whether they get the answer has much more to do with the effort they put in than with the amount of math they know. I'd love some problems like that, especially ones that are more algebra- or geometry-oriented, that go deeper than simple drills or example test items.
Here's an example of a problem that I have used successfully (open book, calculator/Excel allowed): "What is the minimum number of people needed in a room for the probability that there is at least one pair with the same birthday to exceed 50%? (For simplicity, assume birthdays are randomly distributed and that no one is born on Feb. 29.)" This problem is a great stretch for my students after they have learned a few probability principles; whether they get the answer has much more to do with the effort they put in than with the amount of math they know. I'd love some problems like that, especially ones that are more algebra- or geometry-oriented, that go deeper than simple drills or example test items.
You might look at the Critical Thinking Company's offerings. They offer books that they call "mathematical reasoning." I'm not sure anything there will be what you need, but we've used a number of their books in homeschooling and been pleased with the quality, and the focus on thinking strategies rather than pure problem-solving.
We've also used TOPS Science and Math; they're designed to be low-cost activities you can do with items you have around the house. I'm even less sure there's anything there for you, but some of the books feature "challenge cards" that invited students to answer questions about a certain situation. Perhaps some of them could work for you or give you ideas; the cost is nice and low.
posted by Well I never at 9:46 PM on January 2, 2022
We've also used TOPS Science and Math; they're designed to be low-cost activities you can do with items you have around the house. I'm even less sure there's anything there for you, but some of the books feature "challenge cards" that invited students to answer questions about a certain situation. Perhaps some of them could work for you or give you ideas; the cost is nice and low.
posted by Well I never at 9:46 PM on January 2, 2022
I know you asked for questions/resources, but taking a step back, here are a couple of higher level strategies. You don't have to use them together. The first strategy is quite common, but just in case you're not already doing it: set multi-part questions (part a is a gimme, part b requires thought/understanding and is often the "meat" of the question, and part c is legit difficult). The second strategy is more unusual and changes the way you (and they) think about grading more fundamentally, so may be out of bounds depending on norms/students: each question has a grading rubric out of say 10. You then square that score grade to come up with points to sum up for the overall test score. This means that getting two questions half finished and correct is worth 25 + 25 = 50 points. Getting once question completely finished is worth 100 points. It rewards focus and persistence. You can get a similar effect by not squaring the scores and weighting the later parts of the question more. But I think squaring feels more transparent to students.
One resource that might be useful is The Art of Problem Solving. Here's one of my favorite questions, although it does arguably require a "trick", and may be too similar to your example, but: given the probability of an outcome, what is the probability it happens once or more in N independent trials.
For example, the probability of a big earthquake in the next 12 months is one in a thousand. What's the probability at least one happens in the next 100 years. (It's not 100/1000!)
Or more topical: suppose the probability a false negative rapid test is 1%. 30 people attend a wedding after each getting negative rapid tests. What is the probability at least one of them has Covid? (It's not 30%!)
The trick (which becomes clearer if you think about the "once or more"/"at least once" aspect of the question) is: P(not outcome) = 1 - P, so P(zero outcome in N trials) = (1 - P)^N, so P(one or more outcomes in N trials) = 1 - (1 - P)^N.
posted by caek at 9:54 PM on January 2, 2022 [1 favorite]
One resource that might be useful is The Art of Problem Solving. Here's one of my favorite questions, although it does arguably require a "trick", and may be too similar to your example, but: given the probability of an outcome, what is the probability it happens once or more in N independent trials.
For example, the probability of a big earthquake in the next 12 months is one in a thousand. What's the probability at least one happens in the next 100 years. (It's not 100/1000!)
Or more topical: suppose the probability a false negative rapid test is 1%. 30 people attend a wedding after each getting negative rapid tests. What is the probability at least one of them has Covid? (It's not 30%!)
The trick (which becomes clearer if you think about the "once or more"/"at least once" aspect of the question) is: P(not outcome) = 1 - P, so P(zero outcome in N trials) = (1 - P)^N, so P(one or more outcomes in N trials) = 1 - (1 - P)^N.
posted by caek at 9:54 PM on January 2, 2022 [1 favorite]
Dover Books has a volume compiling problems from a USSR secondary school math competition, "The USSR Olympiad Problem Book", there may be some stuff that is not too hard in there, although maybe not. Here's one of the "Introductory Problems":
"In chess, is it possible for the knight to go (by allowable moves) from the lower left-hand corner of the board to the upper right-hand corner and in the process to light exactly once on each square?"
It is not. The start and end squares are the same color, and a Knight can only reach a square of the same color as its starting square in an even number of moves (it must land on a square of a different color every move), and it would take 63 moves, an odd number, to make a knight's tour where it lands on every square exactly once.
posted by thelonius at 2:23 AM on January 3, 2022
"In chess, is it possible for the knight to go (by allowable moves) from the lower left-hand corner of the board to the upper right-hand corner and in the process to light exactly once on each square?"
It is not. The start and end squares are the same color, and a Knight can only reach a square of the same color as its starting square in an even number of moves (it must land on a square of a different color every move), and it would take 63 moves, an odd number, to make a knight's tour where it lands on every square exactly once.
posted by thelonius at 2:23 AM on January 3, 2022
You'll find this level of question in UK GCSE papers. Most papers contain plenty of stats questions, basic geometry (triangles, circles), algebra, and SOHCAHTOA-level trig, percentages, proportions, basic graph questions, etc. There are many sources for these papers - we get ours with a subscription to a revision service called 'Save My Exams'. There's a monthly fee, but I'd imagine you wouldn't need a lengthy subscription. Not sure how copyright works in these cases though.
posted by pipeski at 2:59 AM on January 3, 2022
posted by pipeski at 2:59 AM on January 3, 2022
Math Problems for children from 5 to 15 (2004) [pdf] | Hacker News. I almost posted this once as a FPP but thought it a bit too much. (off to read the other comments)
posted by zengargoyle at 3:49 AM on January 3, 2022
posted by zengargoyle at 3:49 AM on January 3, 2022
I have used this Math Olympiad Contest Problems book, and the problems are accessible but challenging!
posted by mai at 4:57 AM on January 3, 2022
posted by mai at 4:57 AM on January 3, 2022
Two trains leave from opposite ends of the line. They will crash into each other. There's a fly who wants to warn them and flies from the front of one train to the other and back and forth until the the trains crash and the fly is crushed in the middle. How far did the fly fly? You know the initial distance the trains started from, their speeds, the speed of the fly (faster than the trains or it couldn't fly back and forth in the first place). Sounds calculus-y but isn't really.
Afraid my favorite "almost can do it" problem from a random "IQ" test sort of thing really does sorta require calculus or re-inventing it.
There's a boat in a lock that wants to unload cargo but the crane is broke. The lock is shaped thusly, the boat is shaped thatly, the cargo is this unit volume and density. How many barrels do you need to shove over the side before the boat is level with the dock and you can just roll the rest down a plank. Every barrel tossed off is going to raise the waterline in the lock and lighten the load on the boat causing it to float higher above the surface. That one wrecked my young brain, IIRC came from "OMNI magazine world's hardest IQ test". I'd probably give it a go again if I could find it.
What's the optimal strategy for flipping your mattress to promote even wear of both sides and both ends without having to remember what you did the last time? That's a bit of a zero knowledge straight path through a phase space of rotations that covers all possibilities equally.
posted by zengargoyle at 4:58 AM on January 3, 2022
Afraid my favorite "almost can do it" problem from a random "IQ" test sort of thing really does sorta require calculus or re-inventing it.
There's a boat in a lock that wants to unload cargo but the crane is broke. The lock is shaped thusly, the boat is shaped thatly, the cargo is this unit volume and density. How many barrels do you need to shove over the side before the boat is level with the dock and you can just roll the rest down a plank. Every barrel tossed off is going to raise the waterline in the lock and lighten the load on the boat causing it to float higher above the surface. That one wrecked my young brain, IIRC came from "OMNI magazine world's hardest IQ test". I'd probably give it a go again if I could find it.
What's the optimal strategy for flipping your mattress to promote even wear of both sides and both ends without having to remember what you did the last time? That's a bit of a zero knowledge straight path through a phase space of rotations that covers all possibilities equally.
posted by zengargoyle at 4:58 AM on January 3, 2022
At one time, and maybe still, colleges offered courses in "discreet math" for students who were not going on to calculus. I forget what it covers, bit it does include some combinatorics, i.e. questions that begin with "you have a bag containing 10 red balls and 10 yellow balls......". And coin flipping.
There are conundrums like the Monty Hall problem.
It may include some logic. There are whole books about truth tellers and liars. Look for books by Raymond Smullyan.
The Numberphile and Matt Parker YouTube videos are often on miscellaneous number theory problems or geometry problems. Your students might enjoy learning about the hierarchy of infinities.
posted by SemiSalt at 5:21 AM on January 3, 2022
There are conundrums like the Monty Hall problem.
It may include some logic. There are whole books about truth tellers and liars. Look for books by Raymond Smullyan.
The Numberphile and Matt Parker YouTube videos are often on miscellaneous number theory problems or geometry problems. Your students might enjoy learning about the hierarchy of infinities.
posted by SemiSalt at 5:21 AM on January 3, 2022
You may already know this, but "Fermi problem" is a search term that turns up a lot of fun lists of examples. I often spend a class or two on Fermi problems in college science classes for non-majors, and it's pretty much the only specific class anyone mentions positively afterward. I usually try to make it local (e.g., how long would it take the residents of Chicago to drink Lake Michigan), but I'm not sure it matters. Introducing it by explaining that they'll need to make estimates and that it's okay to be off by factors of hundreds if the method makes sense is unusual enough to catch the attention of some students who will tell you they hate math.
posted by eotvos at 5:31 AM on January 3, 2022
posted by eotvos at 5:31 AM on January 3, 2022
Search for "An introduction to topology for the high school student" by Nathaniel Ferron. Practically a lesson plan.
posted by SemiSalt at 6:20 AM on January 3, 2022
posted by SemiSalt at 6:20 AM on January 3, 2022
The Universe of Discourse : A little more about the pedagogy of what it means to be transcendental, Check out his '/math' / Mathematics category of posts. You might find something in there that meets your needs or is at least randomly novel.
Full disclosure, like watching 3blue1brown or mathologer or numberphile videos... that's the blog that I follow and enjoy. I even have a problem he gave me when I asked for a book signing that I'm waiting to solve until I think I can solve it at a glance. There's some good stuff in there for the maths inclined that might be reading.
posted by zengargoyle at 7:15 AM on January 3, 2022
Full disclosure, like watching 3blue1brown or mathologer or numberphile videos... that's the blog that I follow and enjoy. I even have a problem he gave me when I asked for a book signing that I'm waiting to solve until I think I can solve it at a glance. There's some good stuff in there for the maths inclined that might be reading.
posted by zengargoyle at 7:15 AM on January 3, 2022
Fermi problems hook my kid who is pretty uninterested in math as a whole. Questions like: How could we figure out the length of this long, echoey hallway? How long would your school have to be open to teach a million students? How long would your school have to be open for it to be entered a million times? If people laid down in a row, how many people could fit inside our town? How many across its width? How many if we wanted to fill our town as deep as it is wide? How many trees are in this heavily forested park? How many buildings are in our state? The less literally answerable, the better.
posted by tchemgrrl at 7:54 AM on January 3, 2022
posted by tchemgrrl at 7:54 AM on January 3, 2022
There are a bunch of good YouTube channels for this, including Mind Your Decisions and PreMath.
posted by jamjam at 8:33 AM on January 3, 2022
posted by jamjam at 8:33 AM on January 3, 2022
I think you could do worse than picking problems from Yakov Perelman's books.
My personal favourite (and the most fitting to your question) is Arithmetics for Fun. It contains some old school problems which are rarely encountered - so probably none of the kids saw them before. I didn't find it on archive.org, but a couple of his books are available here.
posted by kmt at 9:25 AM on January 3, 2022
My personal favourite (and the most fitting to your question) is Arithmetics for Fun. It contains some old school problems which are rarely encountered - so probably none of the kids saw them before. I didn't find it on archive.org, but a couple of his books are available here.
posted by kmt at 9:25 AM on January 3, 2022
back when we wore onions on our belts, schaum's outline of _____ were the go to problem solving catalogs: brief review of principles, problems, solutions, repeat for next topic.
jeff says they still make em fresh. here's algebra.
i would have never hurdled calc 2 or diff eq without one - where the trick is is to see the 'form' of the problem and solve repeatedly.
posted by j_curiouser at 9:48 AM on January 3, 2022 [1 favorite]
jeff says they still make em fresh. here's algebra.
i would have never hurdled calc 2 or diff eq without one - where the trick is is to see the 'form' of the problem and solve repeatedly.
posted by j_curiouser at 9:48 AM on January 3, 2022 [1 favorite]
Two trains leave from opposite ends of the line. They will crash into each other. There's a fly who wants to warn them and flies from the front of one train to the other and back and forth until the the trains crash and the fly is crushed in the middle. How far did the fly fly?
Good lord, I hated these sorts of problems in high school. Also the ones about filling up a bucket with a hole in it. The worst one (maybe in physics?) went something like "You are on safari when a bull elephant charges at you. When do you need to fire your elephant gun so as to hit it square between the eyes?" Like, wtf, question-writer??
There was a post on the blue a while ago about a math professor (I think at a community college?) who started setting different math problems -- and saw greater engagement from his students, especially those from backgrounds underrepresented in STEM. It's not a question bank, per se, but it is a way to think about math as a tool for understanding the world rather than a set of rules to be memorized and regurgitated. Which is I think what you are getting at with your question.
For geometry, for instance, maybe something about scaling recipes and different-sized/shaped tins? Bonus if the recipe is something the whole class can eat afterwards!
posted by basalganglia at 9:48 AM on January 3, 2022 [1 favorite]
Good lord, I hated these sorts of problems in high school. Also the ones about filling up a bucket with a hole in it. The worst one (maybe in physics?) went something like "You are on safari when a bull elephant charges at you. When do you need to fire your elephant gun so as to hit it square between the eyes?" Like, wtf, question-writer??
There was a post on the blue a while ago about a math professor (I think at a community college?) who started setting different math problems -- and saw greater engagement from his students, especially those from backgrounds underrepresented in STEM. It's not a question bank, per se, but it is a way to think about math as a tool for understanding the world rather than a set of rules to be memorized and regurgitated. Which is I think what you are getting at with your question.
For geometry, for instance, maybe something about scaling recipes and different-sized/shaped tins? Bonus if the recipe is something the whole class can eat afterwards!
posted by basalganglia at 9:48 AM on January 3, 2022 [1 favorite]
Great question! Here is a huge archive of leveled "Problems of the Week" from the University of Waterloo to pick through (C-D levels are where I'd start looking, the E's are good but sometimes hairy.)
I also like Stella's Stunners (and they are categorized by type.)
The AMC math contest also has some really nice rich problems, of varying approachability. Some released AMC 10's here.
posted by Wulfhere at 2:28 PM on January 3, 2022
I also like Stella's Stunners (and they are categorized by type.)
The AMC math contest also has some really nice rich problems, of varying approachability. Some released AMC 10's here.
posted by Wulfhere at 2:28 PM on January 3, 2022
More tangential to your question, but I have used VisualPatterns, Estimation180, and Which One Doesn't Belong as short nonroutine math warmups with success in many past classes.
posted by Wulfhere at 2:33 PM on January 3, 2022
posted by Wulfhere at 2:33 PM on January 3, 2022
With the disclaimer that the author was a professor of mine, there are number of interesting problems intended for non-mathematical types to puzzle over in Mathematics for Human Flourishing by Francis Su. The point of the book is a broader, inclusive study of math, not the problems, but there's at least a couple in every chapter.
posted by wnissen at 3:28 PM on January 3, 2022
posted by wnissen at 3:28 PM on January 3, 2022
Should be a lot of stuff in the maths olympiad.
posted by turkeyphant at 7:02 PM on January 3, 2022
posted by turkeyphant at 7:02 PM on January 3, 2022
This thread is closed to new comments.
If they get that have then do the sum of consecutive integers 367-645 (or whatever).
posted by If only I had a penguin... at 8:58 PM on January 2, 2022