Problem of the Week Guidelines

This is supposed to be a harder problem that takes some real thought. Don’t wait until the night before it’s due to start work!The POTW is worth 10 points. Here’s the point breakdown:

State the problem correctly. You may copy it word-for-word or put into your own words. Just make sure it is an accurate version of the problem. 1 point

Answer the problem correctly. I expect this to be in one or two complete sentences. Even if you aren’t sure of the answer, give your best guess. What is the answer? 2 points

Explain the process you used to solve the problem. If you used scrap paper, attach it, but be sure to write out and explain what you did. If you tried something and it didn’t seem to work, explain that too! These problems will get hard enough that you won’t solve them all on the first go. This should be at least two or three paragraphs.How did you find your answer? 5 points

Justify your answer. Explain why you believe it is correct, or why you chose that answer instead of a different possibility. Is your answer reasonable? Does it answer the question? Are there any other possible answers?Why do you think this is the correct answer? 2 points

You can earn partial points, such as 1.5 points out of 2 for a decent but wrong answer.I expect that a well-written POTW will be at least one full page. You don’t have to use a word processor but it does help make your paper look nicer. Attach any diagrams, models or scrap paper along with your written page – show me how you figured it out.Notice that getting the answer right is worth only 2 out of 10 points. If you get a wrong answer but really explain it well and provide a good reason for why you believe it is correct, you could still get 8/10. If you get the right answer but don’t explain or justify it, you could get just 2/10. Explain! I am more interested in how you think, how you figure things out, and how you explain yourself than just whether you got it right.

You may talk about this with your parents, but I want you to do the work of figuring this out. Please don’t talk about the problem with other students, no matter what.I will grade this according to the points listed above, and I will also occasionally grade these on the 6 Traits of Writing. Take your time, think, and write well.

The Bridge of Doom

Four people have to cross an old, rickety bridge over deep cold water infested with sharks, alligators, crocodiles and shrieking eels. The bridge is so old that it can hold no more

than two people at any one time.It’s the middle of the night, so on every trip across the bridge, the person crossing needs to use the group’s only flashlight to cross safely.The four people who need to cross the bridge all walk at different speeds. One person takes 1 minute to cross, one person takes 2 minutes, one takes 5 minutes, and one takes 10 minutes. If two people cross together, they need to stay together to

share the flashlight, and so they cross at the speed of the slower person.For example, the 1-minute person can cross with the 10-minute person. This trip will take 10 minutes. How quickly can you get all four people across the bridge?

Got Game?There are 6 teams in a

basketball tournament. Each team will play each other

team once. How many total games will be played?

What if there were 12 teams?What about 20 teams?

Can you find a rule that will let you predict the number of

games for any number of teams?

Alls I sees is vertices…

Take 2 straight lines. No matter how you arrange them, they can have at most 1 vertex.

Now add another line, and you can have up to 3 vertices.

How many vertices can you create from 4 lines?What about from 5 lines?From 6?

Can you find a pattern? An equation? A function?Can you use your pattern to predict the number of vertices for 10 lines, without drawing it out?Can you extend your pattern to predict the number of vertices for n lines?

vertex

Have You Lost Your Marbles?

Andy was on his way to a marble tournament when he met Bob. Andy’s bag of marbles was heavy so he gave Bob 1/2 of his marbles plus 2. Then he met Cathy and gave her 1/4 of his remaining marbles plus 5. Just before arriving at the tournament he met Dave and gave him 1/2 of the marbles he had left. When he got to the tournament, he had 2 marbles left. How many marbles did he start with?

Do You Play Well With Others?

A college sold tickets to a play at $5 per ticket if you bought them in advance. Anyone who attended and

purchased a ticket at the door had to pay $6 per ticket.

A total of 680 people attended the play, and the income from ticket sales was $3660.

How many people bought tickets in advance, and how many people bought tickets at the door?

Keep On Truckin’A person wants to rent a moving truck for one day. He can rent from AAA Movers or Bob’s Rentals, and each company has different prices. AAA Movers charges $30 per day plus $0.60 per mile. Bob’s Rentals charges $70 per day plus $0.50 per mile, and the first 60 miles are free. Which company should he hire?

Using your knowledge of algebraic modeling, write functions to represent the cost C for each company, in terms of miles driven m. Use CA for AAA Movers and CB for Bob’s Rentals.Use these equations to decide when, if ever, it makes economic sense to rent from Bob’s Rentals for the trip described above. You are encouraged to use charts, graphs or other means to represent the situations.Now, let’s say this person needs the truck for two days. How might this change your answer? Is a different company now more cost-effective?Finally, can you come up with an equation, chart or graph to help you decide which company is the better value if I give you a distance m and a number of days d? As an example, which company would be cheaper for a 300 mile, 3 day rental? What about a 900 mile, 2 day trip? A cross-country move of 3000 miles and 5 days? Can you predict which is cheaper without actually solving each equation each time?

AAA Movers

Trucks are $30 per day.

Mileage charge: $0.60 per mile.

Bob’s Rentals

Trucks rent for $70 per day.

Mileage charge is $0.50 per mile.(No charge for trips less than 60 miles.)

This Problem is CornyOne hundred bushels of corn are to be divided

among 100 men, women and children. Men get 3 bushels each. Women get two bushels each. Children get ½ bushel each. How can the 100

bushels be distributed among 100 people with no leftovers? Is there more than one solution?

That’s a Cool Pool, Fool!

The Gillis family has a swimming pool with the shape as shown. They want to make a pool cover to protect it during the oh-so-harsh Tucson winter. How many dollars, to the nearest cent, will they need to spend on material for the cover if the fabric costs $5.00 per square meter?Next they want to build a fence completely around the pool, exactly on the edge. If the fence costs $3.00 per meter, how much will the fence cost? If you need to round, round to the nearest penny.

Locker Rocker!There is a school hallway with 1000 closed lockers in a row. One thousand students go down the hall one at a time. The first student opens every locker. The second student closes every 2nd locker. The third student changes the state of every 3rd locker – if the locker is closed he opens it, and if the locker is open he closes it.

This pattern continues with every student going down the hallway, changing the state of the lockers he or she arrives at.

The 1000th student goes down the hallway, and changes the state of the 1000th locker. When she is done, which lockers are open?

Why are these particular lockers open? What is different about them, compared to all the lockers that are closed?

4m

6m

9m

The Art of the MatterFranklin’s Art Supplies wants to make a new sign for their front window, using the design below. They will paint the shaded section. How many square meters will they need to paint?

O 1 m 2 m 3 m

Tied Up in Knots

One day I realized I was out of ties to wear to school – the horror, the horror! I immediately set out walking to my favorite store, Ties R Us, to get a new supply. I saw Mr. Carney there – he was shopping for a nice Sponge Bob tie, while I was looking for a Cosmo.Like any good math teacher, I computed my average speed for each section of my trip. I realized that I walked 2 miles per hour while walking uphill, 6 miles per hour while walking downhill, and 3 miles per hour on level ground.After I arrived at the store and purchased some lovely new ties, I walked back home along the same route. When I got home, I realized I had walked for 6 hours total.What is the total distance I walked?

Mislead Me!This POTW is a little different from others we have done. Your assignment is to create a Misleading Graph using what you have learned in this class, and any propaganda techniques you have learned in Writing class.This is your chance to be creative! Make a graph for an invention, for a political party, for a company’s product, or for anything else you can dream up. Be artistic! Use construction paper, posterboard, glitter, paints and colored ink – you name it. Well-done graphs will be displayed in the room.For the written part of the assignment, I need you to do the following on a separate sheet of paper:

Describe what you did to make the graph misleading. What did you change, compared to an accurate graph of that information?

Why did you choose that type of graph? What facts are you hiding or exaggerating to make

your graph more persuasive? Make a smaller sketch of your

graph, but make this one accurate. This graph does not need to full-size or colorful – just show me what the accurate and correct version should look like.

Finally, talk about who you think would be persuaded by your misleading graph. Who would your “audience” be? Can you think of a real-life ad

or commercial that uses this type of graph to persuade customers?

Care to Share?Three generous friends have some cash. They decide to redistribute their money as follows:

Ami gives enough money to Jan and Troy to double the amount that each has.

Jan then gives enough money to Ami and Troy to double their amounts.

Finally, Troy gives Ami and Jan enough to double their amounts.

If Troy has $36 when they begin and $36 when they end, what is the total amount that all three friends have?

Don’t Be a Square, Daddy-O!

A B

D C

Circles A, B, C and D are tangent to each other as shown, and each has a radius of 8 cm. Find the area of the central diamond-shaped region. Express your answer both

in terms of π and to the nearest hundredth.

The Room of DoomThree friends are traveling together, and they decide to share a hotel room for a night. The clerk tells them that the room costs $30, so the three friends each pay $10.A few minutes later, the clerk realizes that the room actually only costs $25 for the night, so he takes five $1 bills out of the cash register. He gives each friend $1 back, but keeps the other $2 as a tip for himself.The three men each paid $9 for the room. That’s $27. The clerk kept $2. That’s a total of $29. Where did the missing dollar go?

Football Fever?

In this diagram ABCD is a square, and points A and C are the centers of two congruent circles. AB is 5 units long. What is the area of the football-shaped region in the center of the diagram?Please express your answer to the nearest hundredth or in terms of π.

D

C

B

A

Overlap Overload

Let two 8 x 12 rectangles overlap as shown, where AB is 7. What is the area of the shaded region of overlap?

Milli-Fraction?How many proper, simplified fractions are there with a denominator of 1000?

is too small, and is too big.

You may wish to use the concepts of relatively prime and factors in your answer and explanation.

Heavy, Man, Heavy

The world weighs approximately six sextillion tons. In standard form, that is 6,000,000,000,000,000,000,000 tons, and of course 1 ton contains 2000 pounds.If all the nations of the world united to build four-lane automobile bridges to connect New York City to London, Los Angeles to Honolulu, and Honolulu to Tokyo, how much more would the earth weigh?Assume each bridge will be 50 feet wide and an average of 100 feet deep with towers, supports and all the other structures needed. Also assume that the building materials weigh an average of exactly 100 pounds per cubic foot. (The bridge will be thicker in some places and thinner in others, and some materials weigh more and others weigh less. These are reasonable averages, though.)If you estimate any of the distances between cities, please explain how you arrived at that estimate. You may consult a map, a globe, an atlas or any other reference materials.

Triangulation Nation

In the diagram below, ∆RTS ∆UTV, m °, and mT42°. Find the number of degrees in mQV.Please answer using a 2-column proof. This will cover both the explanation and the justification. If you wish to add a written paragraph describing the process you used to solve, that would be helpful.If you need to look up the official name for a theorem you already know, like the fact that quadrilaterals contain 360°, you are more than welcome to.

Reunion ConfusionThe Diaz family went to Disneyland for their vacation. The tickets cost $73 for 3 adults and 5 children. The Anderson family paid $93 for 7 adults and 2 children. The Owens-Faulkner family reunion will need tickets for 55 adults and 53 children. What will be the total cost for the Owens-Faulkner family reunion to enter the amusement park? (No group discounts were given.)

These Ladders Are Making Me Madder!

Two ladders, one of which is twice as long as the other, each have one end resting on the floor and the other end leaning against a wall, reaching the same

vertical height. The shorter ladder makes a 60° angle with the floor. What angle (to the nearest degree)

does the longer ladder make with the floor?

The Two-Digit GameA carnival game has two boxes with an

undetermined number of chips in each box. All the chips are numbered 1, 2, or 3.

A chip is randomly selected from each box to form a two-digit number. The first chip forms the tens’ place

digit, and the second chip forms the ones’ place digit. For example, drawing a 1 and then a 3 forms

the number 13, but not 31.The following probabilities (P) are possible:

P(drawing 11) = 16

P(drawing 22) = 124

P(drawing 33) = 18

What is the probability of drawing a number greater than 22?

Tens Place Box

Ones Place Box

Birthday Bash!Tommy invited some friends from school and several cousins to see a movie for his ninth birthday party. Each school friend is 9 years old, and each cousin is 11 years old. None of his cousins attend his school. If the sum of the ages of Tommy, his friends, and his

cousins is 122, how many of Tommy’s school friends and how many cousins attended the movie?

Kung Pao Kunfusion!Jeannete, Raj and Henry go to a Chinese restaurant

for lunch and order three different luncheon combination platters. Jeanette orders 2 portions of fried rice and 1 portion of chicken chow mein. Raj orders 1 portion of fried rice, 1 portion of chicken

chow mein and 1 portion of sautéed broccoli. Henry orders 1 portion of sautéed broccoli and 2 portions of chicken chow mein. Jeannete’s platter costs $5.00,

Raj’s costs $5.25 and Henry’s costs $5.75. How much does each menu item cost for 1 portion?

Answer KeyBRIDGE OF DOOM

Please note that I tell the kids that there are no “tricks” in solving this one: You cannot throw the flashlight across to your buddies, leave it in the middle of the bridge for the next crossing, carry someone piggyback, swing across on vines, or anything else. With that said, it is logical and reasonable to have the 1-minute person make all the trips back and forth across the bridge, shuttling the flashlight back and forth. This will result in a total time of 19 minutes.

This is reasonable, and it is wrong. If you think creatively, you can get 17 minutes.

If kids are confused as you give them this problem, act it out. Get three kids to join you up front, grab a “flashlight”, and act it out. Assign yourself the role of the 10-minute person. Walk across with someone while you hold the flashlight (10 minutes), then walk back (10 more minutes). Escort the next person across (10) then go back (10). Finally escort the final person across (10, for a total of 50 minutes). Tell the kids that this is a legal solution, but that they may be able to do better. They will all see that having the slowest person be the escort is silly, and will all jump in.GOT GAME?

This is a great problem, because it has several plausible wrong answers, and because the answer can lead into some pretty heavy-duty math.

The first common wrong answer is to say that with 6 teams, each team will play 6 games, go there will be 6*6 = 36 games played. This makes sense until you realize that a team can’t play itself. This leads to the second common wrong answer of 6*5 = 30 games played. This is even more logical, but is wrong because this has team A playing team B, then later team B playing team A. The problem says that each team plays each other team exactly once, not twice.

So – the correct answer is that 6 teams will play 5 games, for 30 games, but half of these games are duplicates. The correct answer is (6*5)/2 or 15 games.

Following that, 12 teams will play (12*11)/2 for 66 games, and 20 teams will play (20*19)/2 for 195 games.

The formula for n teams becomes [n(n –1)]/2, or (if you distribute) (n2 – n)/2. This is commonly called the Triangular Number Formula, and it relates to all kinds of real-world problems, Pascal’s Triangle, and much more. Tell the kids they’ll see it again…

ALL I SEES IS VERTICESThis is a visual version of GOT GAME. Don’t tell the kids, but this is a

different application of triangular numbers. I suggest you use this POTW in third quarter during all the geometry PO’s. It will be interesting to see how many kids recognize and remember the triangular number pattern.HAVE YOU LOST YOUR MARBLES?

This is a difficult problem, but it requires no advanced math to solve – just the ability to think carefully and logically, and the ability to work backwards. The correct starting amount is 28 marbles, but there are several other common answers depending on which mistake you make in solving it. DO YOU PLAY WELL WITH OTHERS?

Algebra teachers should give this POTW before you teach solving systems. Other classes can give it at any time. The solution is 260 people bought the $6 tickets, and 420 people bought the $5 tickets. There are at least 4 different ways of solving this – see how many your kids can find.KEEP ON TRUCKIN’

This is a visual version of GOT GAME. Don’t tell the kids, but this is a different THIS PROBLEM IS CORNY

Algebra students can solve this graphically or possibly by a system. More often, it becomes a giant guess-and-check to find the first solution. Usually once the first combination is found, the students can work logically to find the rest. They are:

2 men 30 women 68 kids5 men 25 women 70 kids8 men 20 women 72 kids11 men 15 women 74 kids14 men 10 women 76 kids17 men 5 women 78 kids 20 men 0 women 80 kids

THAT’S A COOL POOL, FOOL!Really two problems in one. There is no simple way to find the area

of the pool in one step, since it is an irregular shape. However, you can pretty easily divide it into a semicircle with a radius of 2m, a rectangle of 4m by 6m, and a right triangle with a base of 3m and a height of 4m.

Some students don’t realize that they know how to find the area of a semicircle, but most will understand that they just find the area of a whole circle then divide by 2. You need to make a decision here: If students use 3.14 for π, they will get the cost for the area of the semicircle as $31.40. If they use the more-accurate π key on their calculator, or manually solve using 3.14159, they will get a cost of $31.42. You need to decide how precise you want them to be, depending on how you have taught π. The rectangle costs $120 to cover, and the triangle costs $30 to cover, for a total cost for the pool cover of $181.40 or $181.42, depending on how you round.

The fence adds another element. They need to know how to find the circumference of a semicircle. The same issue with π shows up here, giving you a cost for the fence around the semicircle of either $18.84 or $18.85. The third side of the triangle is not given, but if students have been taught the Pythagorean Theorem it is easy to find that the missing side is 5m. The total cost for the pool fence is then $78.84 or $78.85.

This is a good opportunity to talk about rounding error, and the fact that π does not equal 3.14, and how precise do you need to be, and so on. It’s not fatal to be off by a penny or two in real life, but think about NASA sending a spaceship to Mars – they can’t use 3.14 or the ship will crash. It’s not accurate enough.LOCKER ROCKER

This is a cool number theory problem. Locker 1 remains open because only Person 1 ever touches it. Locker 2 gets closed, because it is opened by 1, closed by 2, and then skipped by everyone else. Locker 3 is also closed: Opened by 1, skipped by 2, closed by 3, skipped by the rest. 4 remains open: Opened by 1, closed by 2, skipped by 3, opened by 4, skipped by the rest. 5, 6, 7 and 8 are all opened and closed, finishing closed.

Most kids will realize that each locker is touched by its factors. For example, Locker 30 is touched by 1, 2, 3, 5, 6, 10, 15 and 30. Almost every number has an even number of factors, so it gets opened then closed, and finishes closed. Only a rare set of numbers has an odd number of factors and so remains open: the perfect squares. 12 = Locker 1, 22 = Locker 4, 32 = Locker 9, 42 = Locker 16, and so on.

So, the lockers that remain open are lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961 (or numbers 1 through 31, squared).THE ART OF THE MATTER

A bunch of different solution methods, but all boil down to this: If you “flip” the lower half of the diagram, you will have a shaded circle with a circular hole in it. The shaded circle has a diameter of 2m, and the “hole” has a diameter of 1m . The shaded circle (if it was complete) would have an area of 1π m2, or about 3.14159 m2. (Let them know

how you want them round…) The “hole” has an area of m2, or about .7854 m2. The shaded area is the entire shaded circle minus the

“hole”, or 1π - , which gives you m2, or about 2.36 square meters. Make the kids draw diagrams of how they solve this one – you’ll enjoy them.

TIED UP IN KNOTSA diagram is helpful on this problem. Many kids are confused by the

fact that the problem doesn’t tell you how far I walk uphill, how far downhill, etc. If you draw it out, you’ll see that the sections that are uphill on the way to the store are downhill on the way back, and vice versa. With that in mind, you can start guessing distances. The simplest version is to say, what if the entire trip was level? It took me 6 hours total, so it must be three hours there and three hours back. I walk at 3 mph on level ground, so 3 hours times 3 mph means 9 miles each way, for 18 miles total round trip.

No matter what kind of trip you set up, it will work. Say I walk entirely uphill on the way there. I will guess that it took me 6 hours of walking uphill. If I walk at 2mph uphill, then that means I walked three miles uphill. Those same three miles downhill will only take half an hour, for 3.5 hours total – but the problem says I walked 6 hours. Not far enough. I’ll guess and check my way up to 9 miles uphill. At 2 miles per hours, that will take me 4.5 hours. That same 9 miles back will take me only 1.5 hours, for 6 hours total – there’s my answer.

No matter how they divide the 9 mile trip into uphill, downhill and level, it will solve.MISLEAD ME

This is the most fun POTW of them all. Do this while you are teaching graphs, and all the components that must be in a proper, complete graph. Hopefully, this will be about the same time that the Language Arts teachers are teaching bias, propaganda and persuasive techniques. Tell the kids that is their one and only chance to lie to you using a graph. The green Passport book has a good section on misleading graphs… show them how misleading a pictograph can be, like so:

I drink one soda per day. Nunes drinks 4. On the left it’s correct, and on the right it’s misleading. They can see how distorted it is, even though both graphs reach to 4. It looks like Nunes is drinking gallons! (Ha ha )

Edwards Nunes Edwards Nunes

Let them have fun with this. You will, too.

CARE TO SHARE?The three friends have a total of $252. It is easy to make this

problem too complicated, so we’ll just focus on Troy. We are told that Troy has $36 when the problem starts. In the first step, his money is doubled, so he now has $72. In the second step, his money is doubled again, so he now has $144. In the third step, Troy gives away an unknown amount of money, but we know he finishes with $36, so he must give away $108 to the two girls. We are told that this amount is enough to double the amount they have, so they must have $108 between them before Troy adds his $108. So, they girls have $108, they are given $108, and Troy finishes with $36, for a total of $252.DON’T BE A SQUARE, DADDY-O!

If students focus on finding the area of the curved diamond-shaped region in the middle, they might think it’s impossible. However, if you connect the centers of the circles, you can form a 16 by 16 square. This square has four quarter-circles in it, and it’s pretty easy to see that these 4 quarters can rearrange to form a complete circle. So, the area of the region is the area of the square minus the area of the circle, or 256 – 64π cm2. To the nearest hundredth, that is 54.94 cm2. (Or 54.04 if you use 3.14 for π.)THE ROOM OF DOOM

This is just a trickily-worded problem. There is no missing dollar, but it will take some careful reading and logical thought to prove it. Tell the kids that their explanations will be by far the most important part of this one.FOOTBALL FEVER

This is not as hard as it looks. Picture the quarter-circle centered at point A, with vertices at B and D. The radius is 5, so the area of the quarter-circle is will be 25π, divided by 4, or 6.25π. Then, if you subtract the area of right triangle ABD, you will be left with the area of the curved region between B, D and C. (It’s easier to follow if you look at the diagram.) If you double that area, you’ll have the area of the whole “football”. So: (6.25π – 12.5)(2), or about 14.27 u2.OVERLAP OVERLOAD

Takes some thought, and knowledge of the Pythagorean Theorem. Draw a line from point A to the upper-left-hand vertex of the shaded region. This divides the shaded region into two right triangles.

Pythagoras tells us that this line is units long. You can now use this measurement and Pythagoras to find the length of the segment between the lower-left-hand vertex and point A., which works out to be

exactly 9. (Convenient, isn’t it?) The area of the lower triangle is (9*8)/2, or 36, and the upper triangle is (1*12)/2, or 6. Add to get 42 u2.

MILLI-FRACTION?This problem is especially good for 6th graders learning factors and

how to really manipulate fractions. The brute-force way to do it would be to list out all 1000 fractions, from 1/1000, 2/1000, 3/1000 and all the way up to 999/1000 and 1000/1000. This would work, and it would be silly.

Instead, look at the factors of 1000, which are 2*2*2*5*5*5. Since the only factors of 1000 are 2’s and 5’s, the only non-reducible numerators will be those that don’t have any 2’s or 5’s as factors. There are 500 even numerators (divisible by 2), so that leaves you 500 odd numerators. Some of these are multiples of 5, though, so we’re not done yet. 1000 divided by 5 is 200, so this tells us that there are 200 numerators that are multiples of 5 that we need to discard. BUT - - all the even multiples of 5 have already been discarded. This leaves us 100 multiples of 5 that we need to discard. So, there are 1000 possible numerators. Throw out 500 of them for being multiples of 2. Throw out another 100 for being multiples of 5. That leaves 400 correct, proper, non-reducible fractions over 1000.HEAVY, MAN, HEAVY

Decide how evil you want to be. The trick in this one is that no matter how large you make this bridge, the net weight of the earth won’t change, since the materials come from the earth – we’re just moving them around. Every time I give this, a few kids go nuts with the research and determine that the bridge will weigh 27,000,000,000 tons or something, and fail to realize that even if this were possible, it wouldn’t change a thing.TRIANGULATION NATION and THESE LADDERS ARE MAKING ME MADDER

These are pretty hard-core Geometry and Trig, respectively. I’ll leave the solutions to any teacher that want to use them. REUNION CONFUSION

Once algebra kids know how to solve systems, this isn’t too hard. Otherwise, it’s a long guess-and-check. However you solve it, you find that adult tickets cost $11 each and kids’ tickets cost $8 each, so the 55 adult tickets and 53 kid tickets needed would cost $11(55) + $8(53) = $1029.

THE TWO-DIGIT GAMENotice that each probability can be converted to something out of

24. That suggests that the number of chips in each box is a factor of 24. 1 and 24 won’t work, since there need to be at least one of each chip in each box to get 11, 22, or 33. For the same reason, 2 and 12 won’t work. So the boxes contain either 3 chips and 8 chips , or 4 and 6 (and can be solved either way).

3 and 8 is easier. If the first box has three chips in it, then there must be one each of 1, 2, and 3. Then the second box must have the correct number of each to make the listed probabilities, and that means four 1’s, one 2, and three 3’s.

With that set up, the numbers greater than 22 are 23, 31, 32 and 33. P(23) is 3/24. P(31) is 4/24. P(32) is 1/24, and P(33) is 3/24, for a total P of 11/24, or about 46%.BIRTHDAY BASH!

This has a tricky part. The question tells you that the sum of the friends, the cousins, and Tommy is 122. Since this is Tommy’s 9th birthday, that means the sum of the friends and the cousins is 113. Some guess and check leads to 4 friends and 7 cousins at the party.KUNG PAO KUNFUSION

This is a three-variable system. I recommend you teach it after students know how to solve 2-variable systems, but before you introduce 3-variable systems. Let them see if they can substitute or combine to make the 3-variable equations into 2-variable equations. Algebra II students can apply Cramer’s Rule.OFF THE CHEEZIE???

This is pretty simple. Draw a diagram of a clock and cross off the numbers as you go. You will find that 8 is the last piece eaten. If you want to make an extension, tell the kids the following story: There are 100 prisoners in a faraway prison. The guards tell them to form a large circle, and tell them that they will count around the circle, starting at prisoner number 1. They will count one, skip one, count one, skip one, just like the mouse did with the cheese. If a prisoner gets counted, they are locked back up. However, the guards will release the last prisoner counted. If you wanted to be released, in which spot on the circle should you stand?