RECREATIONAL MATHEMATICS IN THE PARK

PART 2 EXAMPLES OF MATH TRAILSWe give four examples of trails in particular situations: A park playground,a city, a zoo, and a shopping mall. The playground and the shopping mallare each a single actual location, the city and the zoo trails are compositesdrawn from several locations. For each trail stop you will find one or morepictures, and then ideas and questions suggested by the pictures. The trailin the park playground is more specific than the others, in order for you tosee examples of the development from what you see to questions and ideasto actual items for a trail guide.RECREATIONAL MATHEMATICS IN THE PARKParks are popular places for many people, including family groups. In thischapter we will take a tour of a local park to blaze a math trail in the parkplayground. We’ll discuss our thinking as we spot opportunities tohighlight some math, identify questions to ask, and note choices to make insettling on a final guide design. We’ll also include a draft of the trail guidefor each stop.THE FIRE ENGINEOur park has an attractive redfire engine. Children of all ageshave fun climbing aboard the fireengine, pretending to be thedriver or one of the firefightersrushing to a fire across town. Thefire engine certainly is large, buthow large is it? Here’s a chanceFigure 1.for walkers to estimate some16PART 2 - MATH TRAILS

measurements using their hands and feet, although a tape measure wouldhelp move things along more quickly.Think about the size of the tires, the length of the truck or various pieces,or the height of the seat from the ground. Walkers could recordmeasurements on the trail guide in order to compare their results. Theymight also try estimating some of the lengths before measuring.As an extension of this exploration, trail walkers can compare theirnumerical results when they divide the height of the tire into itscircumference. While we would not expect everyone to get exactly thesame quotient, we would expect the trailers to obtain similar results. Thisis a neat way to let everyone discover that π did not just “pop” out of thesky! It (π) really does have a true and valid meaning in life andmathematics. There will be other opportunities during our visit at the parkto explore π.GUIDE TO STOP 1—The Fire TruckThis truck is bigger than most cars on the road, but how big isit? Even the tires are large. Each of you estimate how longyour hand is. Now use your hand to measure the height ofa tire and then measure the distance around the outside ofthe tire. Compare your answers. Are they different? Talkabout why.Name:Divide the height of the tire into its circumference. Is thisnumber familiar? Compare your results.Bumper length:Tire height:Circumference:Quotient:Truck length:Seat height:How long is the truck? How long is the front bumper? Thedriver’s seat is high off the road, how high is it? Why is the seat so high? Try estimating the height firstand then measuring. Is it easier to estimate longer or shorter lengths?MATH TRAILS - PART 217

TILING BLOCKSThis gate has a grid of square tiles, light on one side and dark on the other.The tiles are mounted so that they rotate on a vertical wire. Rotating themgenerates a great variety of patterns and that affords a variety of countingproblems. The most directproblem is to estimate or countthe number of square tiles. Amore complex problem is tocount the number of differentpatterns. You might want toask exactly what you meanby different.Figure 2.GUIDE TO STOP 2—Tile GridSee how each small square tile rotates independently so that youcan turn either its light face or its dark face to the outside. Countthe number of rows of tiles. Count the number of columns. Howmany individual tiles are there in the grid?Each time you turn a tile, you change the pattern. One patternhas all the light faces showing. The opposite pattern has all thedark faces showing. There are many, many more patterns mixinglight and dark tiles.Look at the four tiles in the corner that make a 2 x 2 square.How many different patterns can you make with them?RowsColumnsTiles2 x 2 patterns3 x 3 patterns4 x 4 patterns5 x 5 patternsTotal patternsHow many patterns can you make with a 3 x 3 set of tiles?How many patterns can you make on the whole board? The number is very large! Think about 4 x 4 and5 x 5 arrays first and see if you can spot a pattern.18PART 2 - MATH TRAILS

SHAPES AND NUMBERS BOARDThis panel on the carousel is notonly decorative, but also aninstructive opportunity for verysmall children to recognizenumbers and simple shapes. Thiscan be a prompt for the youngestof the trail walkers to look fornumbers and shapes along the trail.Figure 3.GUIDE TO STOP 3—Shapes and NumbersLook at all of the shapes and numbers on the board.Name them and trace them with your finger. As youwalk through the playground today, see if you canfind all of these shapes and numbers. Keep a tally foreach of the shapes and numbers. At the end of yourwalk, check the tallies to see which shape you foundmost often and which number you found most often.How many?How Chimes have always fascinatedchildren and adults alike.Windchimes catch the breeze andplay beautiful musical notes. Otherchimes, such as the brightly coloredpanel containing eight chimespictured here, need to be struck withsomething like a stick in order for usFigure 4.to hear musical notes.MATH TRAILS - PART 219

GUIDE TO STOP 4—ChimesWhat a terrific place to stop and have some fun with musical chimes! Areall the eight chimes the same length?Hit each of the chimes starting from the shortest to the longest, left-toright. Do all of the chimes make the same sound? Which chime has thelowest sound? Which has the highest sound? How do you think the lengthof the chime and the pitch of the tone are related?GAME BOARDGame boards can be foundthroughout many playgrounds,recreational picnic areas, andzoos. This game board in ourplayground is made up of Xs andOs. Each of the nine faces cancome up either as an X or an Oor a blank. The game board is inan arrangement of a 3 x 3 grid.Figure 5.SLIDESSlides are loads of fun for everyone! We never seem to outgrow the thrill ofcoming down a slide—the steeper and faster, the more fun! One of thethings you learn in mathematics has to do with the slopes of lines. We cancombine the fun and thrill of a slide with the concept of the slope of theslide. Once the walkers are comfortable finding the slope of the first slide,it is a good idea to have them find other slides or ramps in the playgroundand compare their slopes.20PART 2 - MATH TRAILS

GUIDE TO STOP 5—The Game BoardSee how each square face of the game board rotates independentlyso that you can turn an X or an O or a blank to the outside.Work with a partner to see how many different ways you canarrange the Xs and Os on the game board grid with no blanks. Howis this like Stop 2?How many different ways can you arrange the Xs and Os if the ninesquares were lined up in a row instead of in a grid?Team 1 Team 2Game 1Game 2Game 3Game 4What if you decided that the Xs and Os must alternate? How many different ways can this be done?Choose teams and play some games of tic-tac-toe.GUIDE TO STOP 6—Straight SlidesMeasure and record the height of the slide at its tallest point and at its lowest point. Subtract the lowestfrom the highest and record your results in the numerator of the fraction below.height of tallest point – height of lowest pointdistance from tallest point to end of slide Now measure and record how far it is along the ground from the tallest point of the slide to the end of theslide. Record this answer in the denominator of the fraction above.This fraction is called the average slope of the slide.Divide the first result by the second result.MATH TRAILS - PART 221

Some slides are not perfectlystraight, but rather have a slightbend toward the end of the slide.This bend acts as a brake forchildren, slowing them downbefore they come to the end ofthe slide. Once trail walkers arecomfortable finding the slope ofa straight slide, have them tackleFigure 6.finding the slope of a slide with abend in it.GUIDE TO STOP 6—Slides With BendsDiscuss ways of defining the slope of this slide. There might be more than onesuggestion for defining the slope. How close are the different slopes? Why arethere different answers? If you see other slides or ramps, find their slopesand compare them.GUIDE TO STOP 6—SlidesHow long does it take to get down eachof the different slides you find on theplayground? Suppose your friend weighsmore than you. Would she/he get downthe slide faster than you? Record yourfindings in the following table.22Time to getWalker 1 Walker 2down each slideWalker 3Slide 1Slide 2Slide 3Slide 4PART 2 - MATH TRAILS

GUIDE TO STOP 6—SlidesIf you roll a ball down the slide, how far does it land fromthe end of the slide?What do you think affects this distance?Distance from the end of the slideSlide 1Slide 2Slide 3Slide 4SWINGSHave you ever noticed howchildren run over eagerly to aswing set once they spot it?Many times they will also screechwith delight as they run towardthe swings. Adults also enjoyswinging! There just seems to besomething relaxing and carefreeFigure 7.about swinging.GUIDE TO STOP 7—SwingsNow walk over and watch the children as they swing. Do some of the childrenneed to be pushed by someone else in order to go higher? Why? Take turnsswinging with your friends. How would you describe the motion of theswings? How do you make yourself swing higher? Why does the swingeventually stop?Some trail walkers could work the with the following situation using themotion of the swing or “damping.”MATH TRAILS - PART 223

GUIDE TO STOP 7—SwingsAt this stop along the math trail you are to decide if the motion of the swing or “damping” is constant. Youwill need two friends to help you out with this activity. Have one of your friends sit in the swing. Standbehind your friend and mark the position in the sand from where you will let the swing go. Another friendshould stand to the side of the swing to mark the distance the swing will travel. Now bring the swing backto your marked position in the sand and let go. The friend on the side should place marks in the sand toindicate the distance that the swing travels on each successive swinging motion. After the swing comes to astop (or near-stop), repeat the swinging and distance measuring activity again using the same person inthe swing and letting the swing go from the same position. Repeat once more and see if you can make aconjecture about your observations.GUIDE TO STOP 7—SwingsNow explore another activity at the swings. You will need a stopwatch or a watch with a second hand, andtwo friends for this exploration. Do you remember what a period is? It is the time it takes for one backand-forth motion of the swing. Have a friend sit in one of the swings. Stand behind your friend and bringthe swing back and start your friend going in the swing. Have another friend time 10 back-and-forthswings and divide that time by 10. Do thisTime for 10 back - and - forth swingsseveral times giving your friend a different10amount of push on each trial. Does this affectTrial 1the period?Trial 2Repeat the whole process using differentdistances from which to start the swing. Does thisaffect the period?Trial 3Trial 4GUIDE TO STOP 7—SwingsFind some other swings in the playground withdifferent length chains. Try the same experimentusing these swings and see if the length of thechain affects the period of the swing.Time for 10 back - and - forth swings10Swing 1Swing 2Swing 3Swing 424PART 2 - MATH TRAILS

engine, pretending to be the driver or one of the firefighters rushing to a fire across town. The . Tiles 2 x 2 patterns 3 x 3 patterns 4 x 4 patterns 5 x 5 patterns Total patterns. . children ru