One of the large ideas involved in algebraic reasoning is the notion of balance and the role of the equal sign. For many elementary school children, the equal sign indicates that there is some sort of operation "to do" in order to find an answer. In high school algebra, students must develop the understanding that the equal sign indicates that the expressions on either side have the same value. This shift in the concept and meaning of the equal sign can be difficult for children if they are never exposed to the second interpretation before they get to formal high school algebra.
This link will allow you do download the GSP file balance.gsp which was designed by Daniel Scher as a part of the Sketchpad for Young Learners project. The set up of this file provides several possibilities for activities to be used with preservice teachers as well as children in grades 3 to 5.
My particular style of teaching in working with my PST is to model for them / with them, activities using approaches similar to those I might make while working with children. The largest differences between my work with PST and children being the extension activities I use, the types of numbers that I choose, and the levels of questioning involved in the activity.
Introducing the sketch
I begin with a whole group activity to explore some of the facets of the sketch and to talk explicitly about some of the underlying ideas involved in the task at hand. As we move shapes over to the balance scale, the scale will react to the "weight". Two important things to mention, or more importantly, to have students describe, are the differences between our representation of a balance scale and a true balance scale.
1. The balance always "tips" the same distance regardless of the difference between the values of each pan. That is to say, it only displays equal and not equal, and cannot be used to determine how much lighter or heavier things need to be in order to balance.
2. The weights do not remain sitting on the pans when the balance tips up or down.
As the PST interact with the scale, it is important to bring out some of the tacit understandings that they have about this environment that may not be as obvious to children. For instance...
All of the same shapes weigh the same. This is not surprising to us, but the idea that every star represents the same amount is an important idea in developing the concept of a variable, and is something that children will want to / need to explore. This is a nice opportunity to think about making conjectures and what is needed to justify these ideas.
Activities and Challenges Here are some of the activities that could be used with PST in this environment. In doing these activities it a good idea to have students keep track of their methods and ideas so there can be some sharing of problem solving approaches during a wrap up discussion of the activity. These beginning activities can be used with the default value settings for each shape.
1. Rank the five shapes from lightest to heaviest.
2. Find as many different combinations of singular shapes that will balance as you can. For example, triangle + star = square + circle. This activity can be used to lead into a discussion of
3. How much does each shape weigh? When looking at actual values of the shapes, I suggest having the PST describe the values in terms of other shapes instead of letting on that the star is equal to one. If they should say that the star is the unit, that is fine, but we would also want to explore the values of the other shapes in terms of the star as an unknown as well. when working with children, it might be helpful to give them the information that the star is equal to one unit, but then again, you might not want to either.
4. Rather than rank all the shapes, find with as few weighings as possible, the shape with the value of three.
5. Using just a single, star, square, circle, triangle, and diamond, it is possible to balance the scale using all five pieces? How many different ways can it be done?
6. How many ways can you balance the scale using just four shapes, all different? What is similar about your solutions? What is different?
7. How many different ways can you balance the shape with a value of 5?
One of the great features of this sketch is the ability to change the assigned values of the different shapes. By employing this function, each of these activities can be changed and modified. It is possible to set them up as a two player game and keep track of the fewest moves. I have found that my PST like to be able to set up their own values and pose questions to their classmates. Like I'm sure some students will do, my PST try to pick some "tricky" numbers for their peers. Some of them like to choose large numbers, some try to use fractions and decimals (which sadly are not supported by this sketch), and some try to use negative numbers (which thankfully are supported in this environment).
I have found that this sketch is engaging for students and simple for them to use, even by those who haven't had much experience with GSP. It is a fruitful environment for exploring the notion of balance and is ripe with potential for developing reasoning skills involving the relationship between different variables.