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Level Five > Geometry and Measurement

Playdough Balls

Keywords:
Achievement Objectives:

Achievement Objective: Deduce and use formulae to find the perimeters and areas of polygons and the volumes of prisms.
Achievement Objective: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders.
Achievement Objective: Form and solve linear and simple quadratic equations.

Specific Learning Outcomes: 

compare the volume of a sphere and a cylinder by either measuring or applying a formula.

devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).

Description of mathematics: 

The Tennis Ball Tubes problem is a useful unit to do prior to this one.

There are number of useful things going on here. First the problem gives students the chance to do some estimation in a situation that is familiar to them but one that they haven’t thought about before. It’s amazing how we can be fooled when estimating empty space.

Second the problem gives a context for measurement tasks. Most students will be familiar with a tube of tennis balls and so the measurement task is in a common setting.

Once again we have a problem that can be tackled on at least two levels. This makes the problem suitable for a range of students. It also means that you can use it to illustrate whichever method you think is appropriate at the time.

Required Resource Materials: 
3 playdough balls the size of tennis balls
Preferably a commercially available tennis ball tube.
Measuring jug
Copymaster of the problem (English).
Activity: 

Problem

Tim's Mum had made 3 balls of playdough the size of tennis balls and gave them to Tim's little brother in a tennis ball tube. Tim was helping his little brother pack the playdough back in the tube. The three balls of playdough fitted neatly in the tube. Tim wondered if he squashed the playdough how many more playdough balls could fit in the tube.

Teaching sequence

  1. Show the students a playdough ball and ask them to think of all the mathematics that could be applied to the ball?
    What mathematics could we apply to this ball?
  2. Illustrate the problem with a tube and the 3 playdough balls.
  3. Ask students to estimate the number of extra balls that could be squeezed into the tube.
    Why did you estimate that?
  4. Allow groups time to invent and carry out a method for solving the problem.
  5. Questions that can help the students get started include:
    What information do you know?
    What mathematical knowledge could you apply to this problem?
    What can you tell me about the amount of space that a ball takes up?     (volume)
  6. As the students work ask questions that focus on their use of volume formulae:
    Can you tell me how you worked out the volume of the tube? Why does that make sense?
  7. Ask the students to write a statement to accompany their solution describing the mathematics that they used to solve the problem.
  8. Share solutions

Solution

Once again we give two methods of solution. The first way is the practical way of getting a tube of tennis balls and calculating the actual dimensions or constructing a tube of their own.

Method 1: Make a (concrete) model.

To find the volume of the tennis tube, fill it with water then measure the amount of water using a measuring jug. You could find the volume of the ball by wrapping it in plastic and immersing it in water and measuring the amount displaced. Or a ball could be squashed to fit tight in the bottom of the tube and then measure how much of the cylinder is full. How many squashed balls will fit?

Method 2: Make a (mathematical) model.

Let the radius of a playdough ball be R. Then the height, h, of the cylinder is 6R and its radius, r, is R. (We assume that the thickness of balls and tube are negligible.) So the volume of the tube (cylinder) is πr2h = π6R3 = 6πR3.

On the other hand the volume of a sphere is 4/3 π>r3. So the volume of a tennis ball is 4/3 πR3, and the volume of three of them is 3(4/3 πR3) = 4πR3.

Hence the volume of empty space is 6πR3 - 4πR3 = 2πR3. So the number of extra tennis balls that could be fitted in is 2πR3/4/3 πR3 = 3/2. So there is an awful lot of space unfilled in a tennis ball tube. If you could fiddle it somehow you could get in another one and a half balls!


 

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PlydoughBalls.pdf37.96 KB

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