Fall Math

Fall presents special opportunities for bringing math to life in meaningful ways, as our students observe and quantify changes in the world around them.

LEAF MATH

Leaf weight-loss plan:

1. Weigh a small pile of freshly-gathered leaves which can flex without breaking.
2. Subtract out the weight of the container in which the leaves have been placed (if any).
3. Weigh the same pile of leaves each day for a week.
4. Graph the daily weight of the leaves.

Suggestions:

• If you can't wait for nature to dry the leaves, you can crisp them by microwaving them on low heat until they break when folded.
• Another fast-drying method is to iron the leaves on low heat, covered by a thin piece of cloth or several layers of newspaper.
• Use a kitchen scale or a balance scale. Variations:
  1. Separate the leaves by type. Weigh each type separately over a period of several days. Which type of leaf has the fastest/slowest weight loss? Make a line graph of your findings.
  2. Calculate the weight loss as a percentage of the original weight.

  Seeking a symmetrical leaf:
  
  Collect a variety of leaves and compare them: Which one is the most symmetrical? Is any leaf perfectly symmetrical?
  
  Suggestions:

  1. Record the leaf's shape using one of these methods:
     • Lay each leaf on a sheet of paper and trace around it with a pencil.
     • Make a crayon rubbing of each leaf by placing a piece of paper over it and rubbing gently with the side of a crayon.
     • Flatten a piece of clay with a rolling pin or bottle, then make a leaf imprint.
  2. After you have preserved the shape of the leaf with one of the above methods, flip the leaf over and set it down onto its recorded shape. How well does it fit? The better the fit, the more symmetrical the leaf.
  3. Another method of checking symmetry after transferring the leaf's shape to paper is to cut out the paper outline and fold it in half. How well do the edges match up?
  Variation:
  
  On graph paper, trace around one-half of a leaf, making sure that the stem points straight down. Remove the leaf. Try to draw the mirror image of this half-leaf to form a completely symmetrical leaf. Can you do it? If not, why not? If it's not symmetrical, make minor adjustments to both halves of the drawing until you can complete a symmetrical leaf.

  What's my size?

  1. Estimate the surface area of a leaf based on graph paper units. If you have several leaves, or each student has one leaf, try to guess which leaf is the largest.
  2. Lay each leaf on graph paper and trace around it or make a crayon rubbing.
  3. Count the number of complete squares within the leaf outline. Work out a system for counting the partial squares, such as pairing up two partial squares whose combined areas are close to one square.
  4. Compare the total number of squares to your estimate.
  5. Rank/order the leaves by size.
  Variation:
  
  Measure the length and width of each leaf and use those measurements to compare the leaves: Which is longest/shortest; widest/narrowest? Which has the largest/smallest ratio of length to width?

  What's my line?
  
  Place a leaf on graph paper and trace around it. Try to determine a method for estimating the length of the leaf's "perimeter." (If the outline were stretched out to become a straight line, how long would it be?)
  
  Suggestions:

  1. Shape a length of thread or thin wire so that it roughly follows the shape of the leaf, then stretch it out and measure it.
  2. Use the grid lines to estimate the length of each segment of the leaf's outline.

  FALL WEATHER MATH

  Daily highs and lows

  1. Record the daily high and low temperature on a line graph for several weeks. (Use data from the local newspaper or record each day's high and low temperature during school hours.)
  2. Analyze the data:
     • During the period for which the class is tracking temperatures, what is the range of the daily high? the daily low?
     • Which shows more variability, the daily high or daily low?
     • What is the mean difference between the daily high and low?

  Rain, rain, go away!

  1. Set up a rain gauge on the playground, or set out a can or jar for this purpose. Each morning, measure and record any rainfall in the past 24 hours, and empty any collected water. (Is there any difference in water depth in a wide-mouthed vs. a narrow can or jar? Why or why not?)
  2. Compare your class's informal rain measurements with data from the local newspaper.
  3. Collect rainfall data for a month. How does this data compare to the typical rainfall in your area for that month?

  Stormy Weather

  • Track a hurricane (track one for an entire season) or learn about some of the deadliest hurricanes of all time!
  • We use math to describe a hurricane's position, size, direction, maximum wind speed, and barometric pressure. We also use math to describe damage and casualties. Your students can track the daily changes in a developing tropical storm or hurricane, or can compare data for several or all of this fall's tropical storms and hurricanes.
  • One easy-to-use online resource is Weather Underground's Tropical Weather. Be sure to visit the Hurricane Archive, where you can select any year from 1851 to the present. A tracking map displays the year's tropical storms and hurricanes; clicking on any of the paths pulls up detailed data on that storm.

  SUNNY MATH

  Sunrise, sunset...
  
  From the spring equinox (around March 21) until the fall equinox (around September 22), day lasts longer than night. But from the fall equinox until next spring's equinox, night lasts longer than day. Your students can

  • record and graph the time of sunrise and sunset each day, from the local newspaper or their own observations.
    • How much change is there in the time of sunrise, from day to day or week to week?
    • How much change is there in the time of sunset?
    • How much change is there in the length of each day?
    • Do these changes progress at a steady rate?
  • Compare your region's sunrise and sunset times with data in another part of the world (closer or farther from the equator, or in the Southern Hemisphere).
  • Think about it: Does any location on Earth get the same number of sunshine hours each year?

  Angle of the sun
  
  Have your students noticed that the sun isn't right overhead at noon on a fall day? Your class can track changes in the sun's noon position over the course of a month or a season.

  1. Place a posterboard on the ground and mark a spot on it where you will position a ruler vertically.
  2. At noon each day, mark the end of the ruler's shadow. Label that position with the date, and use a second ruler to draw a straight line from the vertical ruler to the end of its shadow.
  3. Measure the shadow's length.
  4. Measure the shadow's changing angle using a protractor.
  5. Record each day's data on a line graph.
     • How much does the shadow's length grow over the course of a month? (Is the rate of change steady?)
     • How much does the shadow's angle change?
  For more information, see
  • To Every Season There Is a Reason
  • Seasons of the Year
  • The Angle of the Sun's Rays