What is the Effect of Changing the Cup Style?

Posing questions about the effect of changing parts of the problem give students the opportunity to look for patterns, look for additional relationships and make generalizations. It also gives them the opportunity to think about causes and effects, and converse relationships.

What does this have to do with our stacking cups problem? By changing the size of the cup, students have the opportunity to explore the effects of that change, and how the new context relates to the old. In this case, the students will have the opportunity to discover the principle that if two lines are parallel, then they have equal slopes. A subtlety here is that the converse is also a valid statement--that is, if two lines do not have equal slopes, then they are not parallel.

Let's return to our cup problem. Students will have the opportunity to explore relationship between the attributes of our cups. If we are looking at two different styles of cups, what must they have in common to achieve identical slopes? That's right, the lip size, because that is the variable responsible for the growth of the stack.

Now, if the lip size of the cup corresponds to the slope, what does the height of the cup correspond to? Observe what the students have discovered about the height of the cup in this situation.

Classroom Clip Reflection:

• What do you notice about the students? understanding of slope, y-intercept, and parallel lines?
• What do you think they might not yet understand?
• How would you find out what they know and do not know?
• What are the implications for your work with students?