An idea for the classroom - weighty problems
Think of a balance situation!
You can ask a variety of questions and pose problems about weights on balances and so, with encouragement, can your learners. By thinking about what can happen when weights are put on the pans of a balance it is possible to create problems that are weighty also in the sense that they are rich – they may be approached in different ways, and modified or extended.
Here is the starting point for an investigation in which students can set their own rules, and then try to find out as much as they can under the rules that they have set:
Using a set of four weights, for example 1kg, 2kg, 3kg and 4kg, what different amounts of flour can you weigh for customers?
Students are likely to ask: “Can we put the weights on both sides?”
You might suggest – “Why not try to see what happens with a ‘weights on one side only’ rule, and/or with a ‘weights on both sides’ rule – you decide what your rules are going to be!”
Let students decide what weights they want to use.
Encourage them to ask their own questions.
What happens if I change my set of four weights?
Can I now weigh more or less amounts?
With what set of four weights can I weigh the greatest number of different amounts of flour?
Does my strategy for finding the four weights that allow me to weigh the greatest number of different amounts still work if I am using five, six, … weights?
Starting points for other explorations involving weight can be found in problems and activities such as the Swings and roundabouts interactive activity, this Weights problem, and this Number Balance problem, all from NRICH.
Valuing methods more than answers
A fascinating aspect of students solving problems is discovering how they started the problem and then how they found the solution. If students share their ideas with the whole class, and you value methods more than answers, they will get the idea that doing mathematics is partly about finding different ways of getting unstuck.
To build confidence while encouraging creativity, you might present a variety of problems, of different types and degrees of difficulty, and let your students choose just one to try to solve. For example, they might choose one of the following six problems, which are ideal for students to work on in pairs, discussing possibilities and bouncing ideas off each other.
We have nine weights that all look identical.
Eight weights weigh exactly the same.
One weight is slightly less than the other eight weights.
Using a balance scale how can you find out which is the lighter weight?
One brick is one kilogram and half a brick heavy. What is the weight of one brick?
A grocer had just received a 20kg bag of rice.
She had 10 customers that morning and each customer wanted to buy 2kg of rice.
Unfortunately she only had a 5kg and a 9kg weight to go with her balance scale.
How did she weigh out 2kg of rice?
A particular fish’s tail weighs 9lbs. The head of the fish weighs as much as the tail + 1/3 of the weight of the body. The body of the fish is the same as the weight of the tail and the head. How much does the fish weigh?
Imagine that you find 10 sacks of gold coins arranged in a row. Only one of the sacks contains true gold coins; the coins in the other nine sacks look like pure gold coins, but actually they are gold plated, and almost worthless. You must identify the one sack of pure gold coins.
A pure gold coin weighs 2 ounces, while a gold-plated copper coin weighs only 1 ounce. You have a weighing scale, but you are allowed only one weighing. You might take a coin from bag number 7 and weigh it. If the scales show 2 ounces you've found the gold, but if it shows just 1 ounce, the gold coins are in one of the other nine sacks. How can you find the sack of gold coins with just one weighing?
A pumpkin grower has five pumpkins. When he weighs them two at a time, he gets the following results 110, 112, 113, 114, 115, 116, 117, 118, 120, and 121 pounds. What is the weight of each pumpkin?
You might find or devise a problem, present it at the end of a lesson, and then let the students think about it before talking about it during the next lesson.
This is Lewis Carroll’s ‘Monkey Puzzle’, which is in Mathematical Puzzles of Sam Loyd, Volume 2:
“If to a rope, passed over a frictionless pulley, is suspended a 10-pound weight that exactly balances a monkey at the other end, what happens to the weight if the monkey attempts to climb the rope?”
Inviting students to contribute to a collection of objects can help them develop their understanding of weight.
For example, invite them to bring in, for a classroom display:
- items that weigh exactly 1g
- photographs of items that weigh 1 ton.
Or challenge them to find out about the history of weights, and weights in different cultures – posters showing their findings would also make a good display for the mathematics department.