The post Eye-Popping Puzzling Paper-Play appeared first on Bedtime Math.

]]>We’re loving the site Papermatrix, where super-talented people show us how to weave strips of colored paper into wild-patterned boxes, balls, and other shapes. And these cool crafts have a ton of math in them. The tube-shaped box shown here repeats 3 regular diamond shapes lined up to look like cubes. It’s made by weaving red upward to the right, blue upward to the left, and yellow down between them. Because of the angles, it becomes an optical illusion: the blue diamonds look like the tops of cute little cubes with red and yellow sides slanted at you, even though they’re close to flat. Another project on the site – which they say is easy to make but takes lots of time – is a “triacontahedron,” meaning a 30-sided shape where each diamond-shaped side is made of 4 smaller diamonds. It weaves 6 colors such that the 30 sides mix every possible pair of colors. If you have enough hands and eyes to track all those parts, give it a try!

*Wee ones:* How many colors can you count on this cube-patterned box?

*Little kids:* How many rows of “cubes” can you count from top to bottom on the box? *Bonus:* If you count 1 cube in each row and each one’s made of 3 diamond shapes, how many diamonds do they have in total?

*Big kids:* Each blue strip (paired with each red and yellow strip) makes 5 cubes on its way from bottom to top. If the box weaves 14 strips of each color, how many cubes do they make together? *Bonus:* For the triacontahedron, how many different pairs of colors can you choose from 6 colors, ignoring the order?

*The sky’s the limit:* Suppose you’re weaving 1 red strip sideways over 20 blue strips. If you start by making every 2nd diamond (multiple of 2) show blue, then you go back to the start and flip-flop the color on all diamonds that are multiples of 3, then start again and flip-flop the color on every multiple of 4, then every multiple of 5, for all multiples up to 20…which squares from 1 to 20 will end up red?

Answers:

*Wee ones:* 3 colors: yellow, red and blue.

*Little kids:* 5 rows. *Bonus:* 15 diamonds.

*Big kids:* 70 cubes. *Bonus:* 15 pairs. The 1st color has 5 to pair with; the 2nd has 4 new colors to pair with since it already paired with the first…you get 5+4+3+2+1=15. (The box then makes 2 different sides for each of those, with each color in the pointy corners vs. the wide ones.)

*The sky’s the limit: *1, 4, 9, and 16 — and what do you notice about those? They’re all perfect squares! The reason is factors. If all diamonds start off red, any prime number will get switched once to blue — when we do multiples of that number — and never change again. And any composite number (any number that has factors other than itself or 1) also has an odd number of factors other than 1, so for example, 6 will flip-flop to blue for multiples of 2, red for multiples of 3, then back to blue for multiples of 6, and then it’s done. HOWEVER, perfect squares have an*even* number of factors other than 1, because the square root pairs with itself. So for example, 9 changes to blue on 3s, then back to red on 9s. 16 changes to blue on the 2s, to red on the 4s, back to blue on the 8s and to red on the 16s, then it’s done!

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]]>The post Licking Your Own Eyeball appeared first on Bedtime Math.

]]>Of all the animals who blend into what’s around them using shape and color, the leaf-tailed gecko might be one of the coolest. It lives on Madagascar, an island off the coast of Africa, where all kinds of strange animals live, because they’ve been trapped there for thousands of year without mixing with other animals. The leaf-tailed gecko looks the way it sounds like it should: its tail looks like a leaf, and its whole brown-colored body matches the sticks, dirt and tree trunks around it to “camouflage” it, so bigger animals don’t see it and eat it. Even so, the gecko comes out only at night to eat insects with its long, sticky tongue, which it also uses to wipe its eyeballs clean since it has no eyelids. Leaf-tailed geckos grow to be only 6 inches at most, making them the smallest of all geckos and also a popular pet. Hopefully they lick only their own eyeballs and not their owners’.

*Wee ones:* If a gecko has 4 sticky, tree-climbing feet and you have 2 not-so sticky feet, who has more feet?

*Little kids:* If you think you see 15 leaves on the ground, but all but 1 are gecko tails, how many geckos do you see? *Bonus:* The gecko has 5 cute, chubby toes on each foot. How many toes does it have in total?

*Big kids:* If you’re counting leaves on the Madagascar forest floor, and every 3rd leaf is a gecko tail starting with the very 1st, does the 29th leaf belong to a gecko? *Bonus:* If every 9th leaf starting with the 9th is a gecko tail, is the 198th leaf a gecko tail? (Hint if needed: Multiples of 9 have digits that add up to a multiple of 9 themselves…so you can test their total the same way!)

Answers:

*Wee ones:* The gecko has more feet.

*Little kids:* 14 geckos. *Bonus:* 20 toes.

*Big kids:* No, because the 28th does (1 more than a multiple of 3). *Bonus:* Yes, because 1+9+8=18, and that’s a multiple of 9 itself (1+8=9).

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]]>The post Boo to You! appeared first on Bedtime Math.

]]>With the advent of GPS and maps on our phones, our paper maps are gathering dust in the attic. Our kids are so used to an electronic voice telling us when to turn, they might never get the chance to unfold a map or flip the pages of a road atlas. That’s a shame, because old-fashioned map reading teaches many more skills than just finding their way around. This Halloween season, ditch the GPS and give your kids the chance to learn about distance, measurement, estimation and other math skills by developing a *Boo Your Neighbors *attack plan!

Boo’ing neighbors and friends is one of my family’s favorite Halloween traditions. We fill up little bags with candy and treats, attach a *You’ve Been Boo’ed* note (see above), and, without being spotted, leave the Boo Bags on the porches of our neighbors and friends. Each recipient is supposed to “pay it forward” and Boo another neighbor or friend, so that the mystery, excitement, and math skills can spread all over the neighborhood.

Boo Bag delivery requires strategy. You definitely want to leave your friends guessing which friendly ghost stopped by with treats. It can be a challenge to drop the Boo Bags off without being seen. My kids plan this activity with all the intensity of a military operation! It certainly goes a long way towards developing their mapping, measuring, estimating, and timing skills.

1) Print out or draw a map of your neighborhood and locate the houses you plan to Boo.

2) Have your child figure out the best order in which to Boo his friends. Which houses are nearest to yours? What is the shortest route that will take you past all their houses? How many steps or miles will you walk or drive on Boo Night? Make an estimate and use a pedometer or old- fashioned map skills to determine the actual distance.

3) If you’re fast on your feet, ring the doorbell and then rush to the nearest hiding spot (behind a bush or a fence, etc.). How much time does your child think he’ll have to ding-dong ditch? If you’re not in it for the adventure, simply leave the Boo Bag in a spot where it’s likely to be noticed and head off to the next house.

4) Discuss what time would be best to begin the drop-offs; you want it to be fairly dark, but not so dark that your child risks tripping over unseen objects. Go back to your map and ask your child to estimate how long the whole Boo operation will take. Don’t stay out too late; you want to be alert enough for the evening’s Bedtime Math problem!

With all that preparation, Boo Night itself promises all the thrills of a spy mission. You can add to the excitement by bringing a stopwatch and timing your child as he makes each Boo Bag drop-off. The next day, as he listens to his friends wonder who left that bag of treats, your child can silently congratulate himself on a Mission Accomplished and a job well done.

*Images courtesy of Ana Picazo*

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]]>The post Facing the Soccer Ball appeared first on Bedtime Math.

]]>Have you ever wondered how many shapes a soccer ball has on it? It has 12 pentagons and 20 hexagons, and no pentagons touch each other: each one, usually black, has 5 white hexagons around it. It turns out there are only 5 ways to fit together lots of identical shapes with all equal sides – like a cube, which has 6 perfect squares as its faces (it’s also called a “hexahedron” for that reason). And can you picture a pyramid with triangle sides and a triangle bottom? That’s called a tetrahedron since it has 4 faces. You can also make an octahedron out of 8 triangles, a dodecahedron out of 12 pentagons, and an icosahedron out of 20 triangles! These are called platonic solids. Soccer balls “cheat” a bit, since they mix hexagons with the pentagons. Oddly, a soccer ball has 15 lines of symmetry, meaning the ways you can cut it into 2 mirror-image halves. Next time you spin one around, see if you can figure out why!

*Wee ones:* How many sides does a pentagon have? (the black shapes on the ball shown here)

*Little kids:* If you kick the ball and your foot touches a pentagon and all 5 hexagons touching it, how many “faces” or shapes did your foot touch? *Bonus:* How many of the 12 pentagons did you not touch?

*Big kids:* Since the ball has 20 hexagons and 12 pentagons, how many faces does it have in total? *Bonus:* If you can keep the ball in the air by bouncing it 13 times on your knee and then twice as many times off your foot, how many times in a row can you hit the ball to keep it in the air?

*The sky’s the limit:* When someone makes a soccer ball, each shape edge is sewn together with another shape’s edge. How many of those lines does the maker sew in total? (Hint if needed: Every shape edge is shared with 1 other shape…)

Answers:

*Wee ones:* 5 sides.

*Little kids:* 6 shapes. *Bonus:* 11 pentagons.

*Big kids:* 32 faces. *Bonus:* 39 times.

*The sky’s the limit:* 90 lines. The 20 hexagons have 6 edges apiece, giving us 120 edges in total, while the 12 pentagons have 5 edges each, or 60 in total. That gives us 180 edges that need to meet up with each other. Each one takes another one away from the pile, so there are 90 pairs.

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]]>The post Top Pick appeared first on Bedtime Math.

]]>If you’ve ever played with a calculator, you might have seen that some screens make the numbers using all straight lines, and those numbers look like letters when turned upside down: for example 7738 becomes “BELL” when flipped over. But even when they stay right side up, numbers made of sticks give you all kinds of games to play. Here in this picture we see 9 numbers, laid out 3 across and 3 up and down — and if you move just 1 toothpick from one number to another number to change both, you can make every up-and-down set add up to the same number, and every row running across add up to that same number, too. Can you spot it and solve the puzzle? Read on to do more math tricks with picks!

*Wee ones:* What’s the biggest number you see on the board?

*Little kids:* What do the 3 numbers in the top row add up to? *Bonus:* If you could change any number in that row to a 9, what’s the biggest total you’d get, and which one would you have to change? (By the way that isn’t the mystery toothpick.)

*Big kids:* Do you see any other rows or columns that add up to the same as the top row? *Bonus:*And now for our challenge…can you figure out which toothpick you can move to change 2 numbers and make all totals across and down add to the same number?

Answers:

*Wee ones:* 8.

*Little kids:* 10. *Bonus:* If you change the 1 to a 9 you get your biggest jump, to 18.

*Big kids:* Yes: the bottom row, and the rightmost column. *Bonus:* If you take away the toothpick in the middle of the 8 (in the center box) and use it to make the 6 just to its left into an 8, now that row has 8+0+2 = 10, the left column now has 1+8+1=10, and the center column now has 4+0+6=10.

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]]>The post How Many People Have There Ever Been? appeared first on Bedtime Math.

]]>There are about 320 million Americans – right now, at least. But how many have we ever had? This isn’t the easiest number to figure out. It’s not as if all people live exactly 70 years and we always have 320 million of them at a time; then we’d just count out sets of 320 million since the United States was born. But back in 1787 the country had only about 3,400,000 (3 million 4 hundred thousand). We’re now 100 times as big as that. In fact, people who study population guess that there have been about 545 million Americans in total — which means the Americans alive today are more than half of all Americans ever! It’s even harder to figure out the total people the world has seen since the cavemen, since people didn’t count themselves up back then (or even count at all). As of 2011, one group guessed 108 billion, compared to just 7 billion people alive today. But no matter what the math looks like, we all count for something.

*Wee ones:* Which is more, 1 billion or 100 billion?

*Little kids:* Which is more, 320 million people or 545 million? *Bonus:* If PRB thinks we’ve had 108 billion people of whom 7 billion are alive now, how many people have we had in the past?

*Big kids:* If there have been 545 million Americans in total and just 320 million are alive today, how many more citizens have we had in the past? *Bonus:* If by the time today’s people are all gone we have 400 million new Americans, what will the history-long total be then?

Answers:

*Wee ones:* 100 billion.

*Little kids:* 545 million. *Bonus:* 101 billion more people.

*Big kids:* 225 million Americans. *Bonus:* 945 million Americans.

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]]>The post So You Wanna Be A…Professional Clown appeared first on Bedtime Math.

]]>Being a clown might look easy, but making people laugh isn’t always so simple. And it’s one of the most important jobs out there, so we were lucky to speak with Adam Gertascov, who’s been running his own clown company for years, and using plenty of math along the way!

I wouldn’t say I was 100% sure, but I was always interested in theater. I went to “normal” acting school and also attended the Ringling Brothers Clown College, which actually accepts a smaller percentage of its applicants than Harvard, so I was pretty excited about that!

Sure, I joined a circus, but I actually wanted to start my own show, or at least be more center stage. So I kept learning clowning techniques and tricks, and then started performing my own shows. I did my first flea circus in 1992, and a more experienced clown told me I should focus on that, so for the next four years I worked on my flea-training skills!

Well, it’s pretty much what it sounds like! The stars of the show are fleas named Midge and Madge, who perform amazing feats of strength and skill. They pull off stunts like pulling chariots and flying out of a cannon into their dressing rooms.

That’s sort of my secret, but I can tell you that fleas are pretty quick learners and that they do what I ask them to do about 85% of the time. People probably underestimate bugs because they’re so small, especially fleas – for example, you could add 5,000 fleas to your pocket and still not gain an ounce of weight from giving them a ride around town! But fleas can move up to 131,000 times their own weight, which you can see happen during their chariot race.

It takes the fleas about 90 seconds to run 13 inches, or the same length of time as the Kentucky Derby. But those 13 inches are over 200 times the flea’s body length, so it’s pretty impressive. To put it on a human scale, that’s like a 6-foot tall, 150-pound person running 1/4 mile while pulling 245 fully loaded tractor-trailer trucks – which would be impressive at any speed, much less in only 90 seconds. Another human comparison: if a flea was the size of a person, he’d be able to leap over the Great Pyramid.

I do a lot of miniscule math, since my stars are so tiny and the total stage is only 4 feet long. I obviously had to do some careful calculations with the flea cannon, using test dummies to figure out the perfect angle of launch and amount of air pressure to send Midge and Madge flying 2 ½ feet through the 2-inch diameter “Hoop of Flaming Death” and still land softly and safely in their dressing room trailer on the other side. Knowing what my fleas are capable of also allows me to dream up new tricks, create correctly-sized props for them, and estimate how long it will take Midge and Madge to perform each act of the show. Lastly, my audience really enjoys hearing the amazing facts about fleas in numeric terms and making comparisons to their own jumping and lifting abilities.

Timing is very important in any kind of comedy, and clowns follow the Rule of Three: the same punch line can be funny twice, but something needs to change on the third time. For example, if a clown walks down the street and slips on a banana peel two times, then the clown should notice the banana peel the third time, but bump into a light pole instead of simply walking around the banana peel. Timing is also obviously important for juggling with regular and irregular patterns – I know 3 or 4 jugglers who are also mathematicians, which I don’t think is coincidence. And of course, if you want to run your own circus, there’s all the budget and planning math that comes with any other business!

*Does all of this talk of fleas leave you itching for more math? Don’t miss our Ringmaster-ed Math printable activity guide!*

*Image licensed by Ingram Publishing*

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]]>The post 1,000-Pound Pumpkin appeared first on Bedtime Math.

]]>If you live in America, you might want to carve a pumpkin for Halloween, or at least roll a couple onto your front lawn for decoration. Well, you’d better get your muscles in shape if you’re buying your pumpkins from farmer Keith Edwards. Keith has learned how to grow pumpkins that weigh over 1,000 pounds, and not just one of them: this year he’s grown 7 huge pumpkins that together weigh more than 7,000 pounds. Other people have grown giant fruits and vegetables like this, often in places like Alaska where the sun shines almost all day and night during the summer. But Keith lives in regular Indiana farm country. He helps his pumpkins by blocking them from the wind, chasing away bad bugs, and giving the plants plenty of water. They don’t look like they stand up well, but if you do figure out how to carve one, you’ll definitely scare away visitors.

*Wee ones:* Which weighs more, a 500-pound pumpkin or a 700-pound pumpkin?

*Little kids:* If you carve 1 of the 7 pumpkins into a wacky jack-o-lantern and turn 2 more pumpkins into pumpkin pie, how many pumpkins are left? *Bonus:* Even Keith’s smallest pumpkin weighs 798 pounds. If it weighed just 2 pounds more, how much would it weigh?

*Big kids:* If each of Keith’s 7 giant pumpkins can make 50 pumpkin pies, how many pies can they make all together? *Bonus:* The world’s officially heaviest pumpkin ever weighed 2,032 pounds. If Keith’s biggest pumpkin weighs 1,364 pounds, how much more does it need to grow to break the record by 1 pound?

*The sky’s the limit:* Suppose that the 7 pumpkins total 7,600 pounds; the smallest weighs 800 and the largest weighs 1,400; 3 of the remaining pumpkins have identical weight; the last 2 are identical to each other; and the 3 together weigh the same as the 2 together. What weights are the 7 pumpkins?

Answers:

*Wee ones:* The 700-pound pumpkin.

*Little kids:* 4 pumpkins left, since you used 3. *Bonus:* A nice neat 800 pounds.

*Big kids:* 350 pies. *Bonus:* 669 pounds, since you’re growing it to 2,033. (Note that just this week a farmer presented a 2,058-pound pumpkin at a tournament; we’ll see if Guinness crowns it as the new recordholder.)

*The sky’s the limit: *1 pumpkin at 800 pounds, 3 at 900 pounds, 2 at 1,350 pounds and 1 at 1,400 pounds. If they total 7,600 pounds, then the 5 weigh that amount with the biggest and smallest taken out (2,200), leaving us with 5,400 pounds. If then the 3 smaller pumpkins weigh the same as the last 2, then they weigh 2,700 and the last 2 weigh 2,700. That gives us 3 pumpkins at 900 pounds each and 2 at 1,350 each.

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]]>The post Chasing Too Big a Fish appeared first on Bedtime Math.

]]>When animals hunt for food, sometimes they bite off more than they can chew. While fishing off the coast of Vancouver, Canada, Don Dunbar saw a very tired, sick eagle swimming through the water. He rescued it by scooping it up with his net, as you can see in this video, and let it rest in his boat. Don and the animal rescue folks on shore think that the bird tried to grab a really huge salmon, but was pulled down into the water with it. Since the eagle was also sick, he was too weak to fly once his wings were wet and heavy. What’s amazing is that this big bird isn’t even fully grown: eagles can have wingspans of 6 or 7 feet, and some types can weigh up to 15 pounds. They’re also really strong – one time an eagle was spotted carrying a 15-pound baby deer! This eagle is now resting and getting better at a center for birds of prey, where they feed him bits of smaller birds …he’s just as happy to have someone else find his meals for him.

*Wee ones:* If the eagle weighed 3 pounds and went after a 5-pound fish, who weighed more?

*Little kids:* If the eagle weighed 3 pounds and chased a 5-pound fish, how much did they weigh together? *Bonus:* At rescue, the poor eagle weighed only 1/2 of what he’s supposed to weigh at this age. If he should have weighed 8 pounds, then what was his weight?

*Big kids:* If the eagle was picked up on October 8 and stays for 3 weeks, when will they let him fly away? *Bonus:* If you lay down next to an eagle with a 6-foot wingspan, how many inches wider than you would its wings stretch? (Or if you’re taller than 6 feet, how many inches bigger would you be? Reminder: One foot has 12 inches.)

Answers:

*Wee ones:* The fish!

*Little kids:* 8 pounds. *Bonus:* 4 pounds.

*Big kids:* On October 29. *Bonus:* Different for everyone…subtract your height in inches from 72, or subtract 72 from your height.

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]]>The post 7-Foot Shoe appeared first on Bedtime Math.

]]>Yes, you’re seeing that picture right: it shows a man sitting inside a fuzzy, claw-shaped slipper. The slipper was a big accident, made wrong when Tom Boddingham tried to order one slipper in a different size from the other. His right foot is size 13, the left is 14 1/2. But when the factory in China saw the order for size 14.5, they misread it as size 145 and make him this roughly 7-foot-long furry footcover instead (the story reports them as making it size 1,450, but as you’ll see in the math below, that would have been even more enormous). Shoe sizes are a pretty messed-up measurement system to begin with: different countries use different sets of numbers, and none of them says exactly how long your foot is except in Korea. In America, each jump of 1 whole size makes the shoe 1/2 inch longer (so each half-size jump is 1/4 inch longer). Some people thought the company made the big slipper on purpose to get attention; either way, it clearly works well as a warm, fuzzy sleeping bag.

*Wee ones:* Are your two feet exactly the same length? Check them out sole to sole!

*Little kids:* If each fuzzy slipper has 4 claws on it, how many do they have together? *Bonus:* If your foot was 3 inches long when you were born how much does it have to grow to become 3 times as long?

*Big kids:* If one of Tom’s feet is size 13 and the other is size 14 1/2, how many sizes apart are they? *Bonus:* If a full size jump equals 1/2 inch more, how much longer in inches is his longer foot?

*The sky’s the limit:* A size 7 1/2 men’s shoe fits a 10-inch foot. If each full size jump is 1/2 inch and the factory misread the size as 145, exactly how big a foot can this slipper fit?

Answers:

*Wee ones:* Different for everyone…some people’s feet are different lengths, and either one can be longer!

*Little kids:* 8 claws. *Bonus:* 6 inches.

*Big kids:* 1 1/2 sizes. *Bonus:* 3/4 inch longer.

*The sky’s the limit:* 78 3/4 inches. If you’re counting from size 7 to 145, that jump of 138 sizes would match an increase of 69 inches. But we’re counting from 7 1/2, which is 1/4 inch longer to start, giving an increase of just 68 3/4 inches above a size 7 1/2. Added to the 10 inches, that gives us a 78 3/4 inch slipper — a little bit over 6 1/2 feet.

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