Math is everywhere, even inside a watermelon, as four children discovered at a vacation cabin in the mountains. It all began when a family friend arrived with a plump watermelon from a roadside stand.

Placing it on the dining room table, he wondered aloud, “How many seeds do you think it has?” That simple question led to an unplanned math project more complex and more enjoyable than anyone could have predicted. To start,everyone ventured a guess.

“Ninety-five,” Grace Linderholm, 10, said confidently. “More,” said her sister, Amelia Gurley, 12. “Way more, like 250.”

Daria and Adam Fixler, 9 and 12 respectively, proffered their own guesses: 65 and 82. Once the adults had piped in, there were nine guesses, ranging from 65 to 280.

“That’s quite a wide range,” observed Grace and Amelia’s dad Owen. “What should we do next?”

“Eat it!” exclaimed Amelia.

Adam and Daria’s mom, Rae, sliced through the melon, revealing its glistening red flesh speckled with seeds. The children were again intrigued by the original question.

“Don’t eat it yet,” Grace implored. “Let’s estimate the seeds.”

Grace understood the distinction between estimating and guessing. When the children had merely looked at the melon and called out numbers, they were guessing.To estimate, they will need more information about the seeds and they will have to apply math.

Estimation is an important math skill used by everyone from young children to advanced students and scientists. Parents can encourage estimation with questions like “How many candies are in the jar?” or “How many people are in the stadium?” or “What time will we get to Grandma’s?” It’s a good idea to guess first, then discuss possible strategies that would refine the guesses into estimates.

At the cabin, the watermelon project becomes a brainstorming session with each child offering a different strategy for estimating the seeds. All agree on one thing: the process has to include consumption of the melon! Finally, with some parental persuasion, it is decided that half of the watermelon will be saved in the fridge for tomorrow’s picnic, and the other half will be divided into four wedges, each to be cut into four pieces.

Everyone will count the seeds in one piece as it is eaten. From this data and a little multiplication, the group can estimate the total for the entire fruit. But some of the kids want to know how close their estimate will be to an actual count.After all, the seeds are not evenly distributed through the melon. Will counting only nine sections be enough? To find out, they also decide to count every seed.

Minutes later, only rind and seeds are left on the plates. The counts for nine sections are averaged, then multiplied by 32 (the number of sections in the whole watermelon). The group arrives at an estimate of 580 seeds for the full watermelon. Everyone is surprised at how much higher it is than the initial guesses. Tomorrow, once the other half of the watermelon has been eaten and its seeds counted, they will do a reality check on their estimate.

At the picnic, the final count is 640. Some are disappointed that it is 60 seeds more than their estimate, but Daria and Adam’s father, Craig, puts it in perspective: “It’s really not bad. You were off by only 10 percent. Congratulations!” The two older children do some mental math and agree with him before heading off to go swimming.

Craig’s congratulations are certainly deserved and not only because the estimate was close. These kids attacked a simple question – “How many seeds?” – and turned it into a math project that involved high-order thinking, problem- solving and lots of computation. Best of all, they had fun.

Parents on the lookout for Math Moments will find no shortage if they ask questions – realistic or ridiculous – and encourage kids to find answers by estimating. The process can help sharpen math skills – and appetites.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series.

Share Your Math Moments
David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address. David will award a signed copy of one of his books to those whose submissions he uses in this column.

 

A family road trip offers many possibilities for side trips and sightseeing. For the Hart children – Josh, 12, Kallie, 9, and Marissa, 6 – a journey to Yellowstone National Park also includes opportunities for excursions into a landscape of mathematical learning. Math Moments abound on the American roadside.

In Minnesota, the Harts stop to ponder the proportions of a famous steel and concrete figure of Paul Bunyan and his blue ox Babe, which towers over the mortals who stop to admire. How large are the outsized figures? Kids love guessing games, and this can be a quick estimation activity: Josh is almost 5 feet tall, and Big Paul looks to be as high as four of Josh … so the lumberjack must be a little less than 20 feet. (Checking the sign after estimating reveals that the technique works: the statue is 18 feet high.) Now spread the fun (and the math) around: Kallie, at 52 inches, can be a benchmark to gauge Babe’s oxenly dimensions. And how many Marissas, end to end, would it take to span the big bovine’s horns? Now that’s a longhorn!

A few hundred miles down the road, the Harts are in Gillette, Wyo., touring a working coal mine. Never mind the coal – what fascinates the kids are the behemoth mining trucks. A tire, available for inspection, can bring up words like “diameter,” “radius” and “circumference.”

Josh’s height is just about equal to the tire’s radius, so what is its diameter? How far does the truck move in a single tire revolution? To figure it out, compute the circumference (multiply the diameter by 3.14, or pi, possibly with a calculator; or mentally multiply by 3 to approximate). Compare that with the family minivan!

In the Black Hills of South Dakota, everyone gets a chance to pan for riches at Big Thunder Gold Mine. The yellow flakes may be small, but there is a lode of math to be mined. Pans come in diameters of 8, 10, 12 and 14 inches, and in the gold miner’s lexicon, there are 12 Troy ounces to the pound (instead of the cus- tomary 16). A parent could wonder aloud, “So which is heavier,a Troy ounce or a U.S. ounce?”

A similar math principle comes from examining the panning sieves that miners use to screen material. The holes range in size from 1/4-inch (“4 mesh”) to 1/100inch (“100 mesh”). Which size has smaller holes? A simple Math Moment spent considering that question will lead to a key principle of fractions: the larger the denominator,the smaller the amount.The 1/4 holes are large compared to the tiny 1/100-inch holes, which thus screen out the most.

An impressive numerical comparison comes up when the family learns that in 35 years of operation, Big Thunder mine produced a mere 10 ounces of gold from 250,000 pounds of excavated ore! (The Holy Terror Mine across the stream averaged 26 ounces of gold per ton of ore. Which mine did better?) But today, Big Thunder is producing: the Hart kids get to take home a few gleaming yellow specs they find in their pan bottoms.

With its geysers and grizzlies (seen at a safe distance), Yellowstone is the highlight of the trip, but Old Faithful is less faithful than expected. It doesn’t erupt at perfectly-timed intervals. Instead, say the rangers, the time between blasts ranges from 35 to 120 minutes with an average of 94 minutes – more math vocabulary! Still, geologists can predict the next eruption with accuracy. So, to see it again in 95 minutes, what time should we return?

In visiting America’s natural and human-made wonders, children encounter fascinating facts and figures that add to their appreciation of the sights. Don’t let those numbers pass unnoticed. By asking questions that lead to mathematical thinking, parents and children can enjoy Math Moments that add another dimension to the family vacation.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

In his camouflage trousers and brown sweatshirt,Ty Evenich, 6, looks like he’s getting ready for a hunting trip. Which, in a way, is what he’s doing. He and his great-aunt Debbie are standing over a sewing table, pinning the hem of an apron.The last time Ty went hunting with his dad, he decided he wanted to help cook breakfast at the camp. But there was no apron even close to his size.

Aunt Debbie places a pattern, folded in half, atop the fabric. At first puzzled by its size and shape, Ty wonders aloud why it looks the way it does. Finally, his confusion turns to understanding.

“I get it. When you open it up … it will be bigger,” he says.

“How much bigger?” Debbie asks.

“Twice as big.”

She tells him that they need a 1-inch hem, and folds some fabric to demonstrate. She sets her “hem gauge” (a 6-inch ruler with sliding pointer) to 1 inch and shows Ty how to use it.

“So how much smaller will the apron be after we sew this hem all the way around?” Debbie runs a finger 1 inch from the outside edge of the pattern.

Ty almost says “1 inch,” but seems to know it isn’t right.Then he realizes the finished apron will be 1 inch shorter on each side.“Two inches!”

“You got it!” Debbie says, tousling his hair. She loves to see children think through her queries.

Almost any home craft project – sewing, knitting, beading, macramé and myriad others – can also be math projects. If adults ask questions and point out math connections, they will provide valuable lessons in measuring, computation, estimation, geometry and other skills.

While Ty pins the hem, his sister Shelby, 8, works on a quilt square, part of a 4-H project. Quilting can be a splendid introduction to geometry. The more complex the quilt pattern, the more sophisticated the math. Even children who don’t get to work on actual quilts can read delightful picture books that tie quilting to math.

Shelby is holding up two fabric pieces that she is about to sew together.

“What shape are they?” Debbie asks.

“Rectangles,” Shelby says with confidence.

“What shape will you have after you sew them together?”

This is harder.After a minute at the sewing machine, Shelby has the answer, fascinated to discover that two oblong rectangles can make a square. (An older child might be asked how to predict when two joined rectangles will make a square.)

Back at the sewing table,Ty has finished pinning the hem and, after folding the inner edge under,Aunt Debbie runs the apron through the machine.

The next step is planning the pocket. The pattern calls for one long pocket across the front, seamed down the middle.

“Where should we put the seam?” Aunt Debbie asks.

“In the middle,” says Ty.

“But where is the middle?”

He thinks, but not for long: “I don’t know.” Debbie won’t take that answer: “Think harder,Ty!”

Shelby comes to the rescue. Without a moment’s hesitation, she folds the fabric in half.

“Right there!” she pronounces with big-sister superiority, pointing to the folded edge.

Ty is still unsure. He looks to his great-aunt, who doesn’t resolve the issue but instead takes a piece of chalk and marks the folded edge. She opens the pocket.

“Is the chalk down the middle?” Revelation dawns across Ty’s face: “Yessssss!”

In minutes, he is donning his new apron and savoring the thought of flipping flapjacks. But his great-aunt is just as happy to know she has facilitated many Math Moments that will serve Ty well long after he outgrows the custom-made apron.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

Elizabeth Todd and her mother walk through their neighborhood almost every day. At almost 4 years old, Elizabeth loves to look for special things along the way: the carved figure of an Indian on a nearby roof, purple flowers in a particular garden, a bench tucked under the trees, a bright yellow antique car parked in a driveway. But most of all, Elizabeth loves to look for numbers. She finds them on mailboxes, fences, garage doors, license plates and street signs. There is no limit to the number of places to find numbers, and no limit to the pleasure Elizabeth gets from finding and naming them. Her mother, Cathy Sharp, has been nurturing Elizabeth’s number nature for a long time.

“When we find them, we read them and talk about them,” says Cathy. Elizabeth doesn’t need any prompting. She is pointing and reading the house numbers posted on a wooden retaining wall outside a neighbor’s home. “One–eight–four–eight,” she says with obvious satisfaction.

Elizabeth does more than just read numbers. She and her mother count all kinds of things, including their steps. Sometimes, it’s just “One, two, three, four, five …” as they walk along. Other times, they play a game.“One,” says Cathy. “Two,” says Elizabeth.“Three,” says Cathy. “Four,” replies Elizabeth.And so forth.A little later, they may reverse roles. Elizabeth can usually count to 20 without a problem, but when Cathy stops by a mailbox bearing the number 12 and asks her daughter to name the number, the girl says,“One-two.”

“And what do we call it if it’s written ‘one-two’?”

“One-two,” says Elizabeth.

Cathy tells her it’s also called “twelve.” Then she asks, “What’s bigger than 12?” To her mother’s surprise, Elizabeth points to another 12, in larger numerals, on the side of the house! Her mother laughs. “Now that’s confusing,” she admits. But, because Elizabeth has a strong foundation in basic counting and because she encounters numbers daily in an enjoyable way, Cathy knows that her daughter will soon be reading two-digit numbers and understanding concepts like “bigger than.”

Variations on simple counting games abound and parents can adapt them to their child’s level.“What number comes after (or before) this one?” … “What number is two more than this?” … Looking at a three-digit number – 268, for example – you can play with place value:“If we changed this 6 to a 7, what would the number be called? What if we put a 1 in front of this 268?” (To make it more visual, carry number cards.) Even multiplication (and division) can come into play:“What’s twice (or half) as much as this 12?” Like reading, appreciating the joys of numbers should start at a very early age. Charles Richards and his mom count his feet and hands when it’s boots and mittens time. “We count fingers and toes when it’s fingernail and toenail clipping time,” says the two-year-old’s mother.

Counting goes in two directions for Charles. “We count down with the microwave oven as it finishes heating the bottle: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, BEEEEPP! Charles always looks forward to the beep part.”

When Charles’ older brother was 18 months and in the hospital recuperating from surgery, his parents discovered a way to keep him happily occupied.

“We rode in the elevators counting up and down along with the flashing numbers. It occupied hours,” his mother recalls.

The simple act of naming numbers can occupy many pleasurable hours in a young child’s life. Children introduced to number concepts in this way will probably be eager to take on more advanced math challenges as they grow. Parents need merely to ensure that, in their family, everybody counts.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

When Phyllis Licht decided it was time to replace her lawn with a lush garden, she knew where to turn for expert advice: her 8 year-old grandson. “I love plants,” says Sam Licht who, with his father Mark, had already created a butterfly garden that is the envy of their neighborhood, “and I knew I could give Grandma a nice garden instead of her un-nice garden.”

Sam’s grandmother didn’t have a large space to work with – all the more reason that Sam had to plan carefully. First, he measured the area that was to be planted, including landscape features that could not be altered – walkways, fences and a concrete pillar. Scaling down the dimensions to notebook size, he sketched the area, then drew in the plants.

“I had to think about the spacing of each plant,” Sam recalls.“So if a border was 56 inches and I put in a butterfly ginger that needs 8 inches, then I was down to 48 inches.” If he then had four variegated Scheffleras that took 3 inches apiece, he had to subtract another foot, and so on. Sam marked the position of each plant on his sketch.

He didn’t know it but, as a third-grader, Sam was practicing not only basic computation, but also real-life mathematical thinking – exactly the kind of math that bedevils many American students. In a recent international study,American teens ranked 24th among those from 29 industrialized nations in practical math applications. Ironically, Sam does not enjoy math in school, but in the context of a garden, he says,“It’s not like math. It’s just like fun. I don’t mind that kind of math.”

Natalie Milnikoff enjoys math at school, at home, in the car – anywhere she finds it. Her love of the subject is not surprising, since her parents have been working it into her everyday routines since Natalie was very young. Now, at 11, she’s eager to help with projects around the house, and they often involve math.

When Natalie’s father, Vladimir, decided to make a patio of terra-cotta tiles, Natalie knew that a math problem was on its way. Each tile is a 16-inch square (16” x 16”) and the finished patio will be an 18-foot square (18’ x 18’). Natalie figured out how many tiles to buy.

“First, I multiplied 18 feet by 12 to find out how many inches long each side is.Then, I divided by 16 to see how many tiles would fit on that side,” she explains. Natalie had to square that number to get the total number of tiles needed, allowing for space between the tiles.

With a long measuring tape, she helped her dad mark the boundaries of the patio, and with a builder’s square, she checked to be sure they were angled right, which is to say at right angles.The project took a few weeks to complete, but the results were beautiful, useable and, for Natalie, educational.

Nearly every home-improvement project involves solving mathematical problems through measurement, estimation, computation, geometry, proportional thinking – or all of the above:

• How many cans of paint will we need to cover the deck? Figure the area of the deck and read the label to see the coverage of each can.
• Where should we nail the hanger to center this picture on the wall? Measure the width of the wall and divide it in half.
• How many yards of soil should we order for the flower bed? Multiply the bed’s length by width by height in feet and convert to cubic yards, with 27 cubic feet equaling one cubic yard (for “extra credit” your child can figure out why!).

Many children enjoy helping their parents around the house, and “doing the math” can be one of the enjoyable ways they can be helpful. As Sam Licht would say, “It’s not like math. It’s just like fun.”

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

On a frosty January morning, 5year-old Miya found icicles hanging from the garage and decided to collect them. She noticed that the biggest icicles were much larger than the “baby” icicle.

“How much bigger?” asked her mother, Joanne. It was an opportunity for Miya and her mom to compare sizes:Was the biggest icicle twice as large as the smallest one, or more? By holding the icicles next to each other and “walking” the little one up the side of the big one five times, Miya could easily see how many times bigger the “papa” icicle really was.

Miya’s icicle encounter lasted only a few minutes, but in this Math Moment she learned an important principle that will help in her understanding of multiplication. For a phrase like “five times as big” to be meaningful, children must learn it in a way that has meaning to them, and usually that involves using actual objects (often called “manipulatives”).When parents use a phrase like “five times as big” in everyday conversation – not just in the context of homework – it becomes a natural way of thinking for the children.

Miya decided to arrange the icicles like stair steps, from smallest to largest. With the bottoms aligned horizontally, the tops made a sloped line. By ordering the icicles, Miya expressed an appreciation for mathematical order that is innate to young children.

Parents can develop that sense further – and develop mathematical vocabulary – by asking their children to make comparisons:Which is the smallest flower? The biggest toy? The heaviest dog? The shortest path through the park (while looking at a map)? Which holds more volume, this truck or that one? Can we arrange these flowerpots in order so they will nest inside each other? Ordering by size leads to discussions of numerical order, a pre-requisite for numerical computations and higher math.

All in Order

Many family Math Moments will sharpen children’s understanding of order and help them see its relevance in their lives. Here are a few ideas:

• Play simple games of dice or cards (like the game of “War”) where players compare numbers. Make up variations – the player with the smallest card could be declared the winner of each round.
• Put coins in order of their value. (Counting money is a more advanced skill, but this comes first.)
• Read route numbers (or distances) on highway signs and ask which is larger or smaller?
• Play numerical guessing games where one player thinks of a number and the other uses “greater than” and “less than” to narrow the field:

Parent: I’m thinking of a number between 1 and 20.

Child: Is it more than 10?

Parent:Yes.

Child: I’m thinking of a number between 1 and 100.

Parent: Is it a multiple of five?

During a game, compare scores to see who is ahead and by how much. Effortlessly, your child will slide into the world of computation.

These Math Moments take little time but they go a long way in helping children to think mathematically. Just as important, they make math a pleasurable aspect of daily life.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

It takes less than a second for 12-year-old Jennifer Sirrine to look at the 12 cards on the table and call “set!” as she points at three cards.At nearly the same time,her brother Alex, 14, picks out different cards and cries out the same word.Their mother, Sumiko, stares at the cards and then her children. She says nothing. Unlike the two younger players, she has not yet found a “set.” She is baffled, amused and only slightly chagrined.

In the game of set – invented by a mathematician – players peruse an array of cards with various attributes to find “sets” – three cards in which each of the characteristics is the same on each card or different on each card.The rules sound easy, but try it against experienced players like Jennifer and Alex!

Set is a game of perception and logic, and good players develop a mental agility that leaves inexperienced opponents shaking their heads.

“They beat me every time,” says Sumiko with apparent satisfaction, despite her poor performance.“They’ve gotten much better at logical thinking than I am.”

Age means little in an analytical game like set. In a recent tournament in the Sirrines’ community, a 9-year-old emerged victorious from a field of competitors ranging in age from 6 to 80! Sumiko believes that her children’s love of games like set is part of the reason why both are “A” students in math.“Games help them to think quickly and recognize complex patterns,” she says.

At the age of 6, David Green discovered the card game rummy and became the official family scorekeeper. His passion was adding everyone’s score as quickly as possible.A run like 5-6-7 would give 18 points and an 8-8-8 triple would yield 24. As with any math skill, proficiency comes from practice, and David so loved beating his parents that he got plenty of practice!

In another favorite game,“Crazy 8’s,” the scoring includes subtraction. Now 9, David still plays card games and he can mentally add strings of three-digit numbers in seconds.

the game of cribbage caught Oren Stoelting’s fancy when he was 10, and he even made his own wooden cribbage board – a math project in itself! Scoring in cribbage requires flexible thinking and the ability to see subtle number combinations.A pair is worth two points and any run of three cards is worth three, so a hand with 4, 5 and two 6’s would yield eight points: two for the double 6’s and six for the two runs (each 6 combined with the 4 and 5 makes a separate run).Try scoring 5, 5, 6, 7, 7 to get 16 points and you’ll see how agile your mind must be.

A few additional rules make the scoring even more complex.A player who fails to calculate how many points he has earned will suffer because his uncredited points go to an astute opponent who sees them.That’s Oren’s favorite part of the game.

“I definitely like taking the points that my opponent didn’t realize he had. It’s quite fun!” he says.

Not all games are created equal.The best math games require players to think analytically, develop a winning strategy, use basic math skills – or all of the above. Games that rely mostly on chance will do less to hone a child’s mathematical thinking. But even a simple game of chance like “War” (where each player turns over a card and the one with the higher card takes both) can be improved.Why not have each player draw two or more cards and add them together? Or multiply them. Once mastered, any game can be changed. Coming up with variations is part of the fun, and with each variation, players must adapt their strategy.

Research shows that children learn best when they are happy and relaxed, not anxious or stressed.What better way to encourage relaxed and happy mathematical learning than through games? There’s only one catch: you must be ready to be trounced by your kids!

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.