I was constructing towers as you see below. I noticed that each time I made the tower higher, I had to add more blocks on the sides to stabilize the structure. I would like to know how many cubes I will need to build a 5-block high tower and a 10-block high tower. Generalize, if you can, on how many blocks I will need for any size tower?
My sixth grade class just finished a unit on integers and graphing simple equations. The class was able to predict where a certain equation would be located on a coordinate grid. We worked with equations like x + 4 = y, (-x) + 4 = y, x + (-4) = y and (-x) + (-4) = y. The graphing unit came out of my desire to work with integers - to show them that you could see equations on graphs and that graphs help to find other solutions to equations. I wanted to continue with the idea that an equation is really describing a pattern. Some patterns from equations can be graphed and some equations can come from building blocks. Sixth grade is a little early for this, but I knew that all of my students would be able to access this problem and those that were ready would reach a generalization. This task makes the connection for some students between a physical pattern and an equation or numerical pattern. Some students asked for blocks to construct the buildings. Others asked for graph paper to construct the next pattern. Some students made a chart of cubes in each tower and cubes in the whole building to look for a pattern. Other students could visualize how the structure was growing and could almost come up with a generalization from that information. 45 minutes Most of the students investigated and collected their data in a 45-minute period. Some needed a few more minutes to pull it all together. Some links could be the study of patterns in nature or in man-made objects. Many of my students asked about the block that you cannot see in the towers. They wanted to be sure there was a cube there. None of my students asked for blocks, but I had them available in case a student felt s/he wanted to build the structures.
- Blocks
- Paper
- Pencil
- Graph paper
A five-block high tower takes 21 blocks and a 10-block high tower takes 46 blocks (five blocks are added each time - one to the tower and one to each of the four supports). Some observations that lead to a generalization are that there is one more block in the tower than the four supports. So if h = height of tower then...5h - 4 = total blocks. You need to subtract four because each support has one less block than the height of the tower. Another way to think of this is...4(h -1) + h = total blocks.
A Novice student will not be able to become engaged in the problem. Other Novices, as in the benchmark, may seem to understand that they need to find the total blocks, but their numbers make little sense and there is no explanation as to how they came up with their numbers.
Apprentices will have a good approach. They may have a chart of the number of blocks in the tower and the total blocks in the structure, but will make a mathematical error in their calculations. They may also make a drawing that is not complete.
Practitioners will be able to show the towers growing by one block and the total number of blocks growing by five. They will be able to find the number of blocks for a five and 10-block tower. They should be able to see the pattern that the tower increases by one block each time and that the total blocks increases by five. They should have a chart or a diagram of the towers.
Experts will be able to find the solution to the five and 10-block high towers, but will also be able to generalize the solution. Experts will solve the problem two ways to verify their solutions. A chart will let them see a pattern that the total blocks increase by five each time. They should also be able to see the relationship between the tower and the supports. They should be able to explain their equation using the diagram of the tower.
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