Today’s question (May 3):
We introduced 3D geometry today. We began by looking at architecture that used geometrical shapes:
Then we named a number of 3D shapes, and reviewed math vocabulary, including: base, faces, vertices (corners), and edges.
Today’s question was to create charts (one for prisms and one for pyramids) that included the properties of each shape. Here are examples of what the students came up with:
Consolidation: the students were asked to try to find a pattern in the charts that listed prisms. After some discussion, here is what they came up with:
Using this pattern the students could easily tell how many faces, vertices, and edges a prism had, as long as they know how many sides the base has (e.g.: the base of an octogonal prism has 8 sides, so it has (8 + 2) 10 faces, (8 x 2) 16 vertices, and (8 x 3) 24 edges).
We then, together, put up a chart of pyramids. For homework the students were asked to find patterns for the faces, vertices, and edges of pyramids.
Today’s (and yesterday’s) question:
Yesterday the students were asked create a building using interlocking cubes. Then they were asked to draw this 3D shape using 2D drawings of different views. They were also asked to give a “hint”, being the number of cubes used to create the building:
There were some rules given:
And from this the students created orthographic representations of their buildings (2D drawings that represent a 3D shape). Here are some of the results:
Answer cards were then made (they used the top view, and in each square they put the number of cubes high it was).
Today we spread the orthographic puzzles about the class and hallway, and the students went around and tried to build the buildings based on the ‘views’ provided.
Consolidation: We discussed if there were any shapes that did not work. We also talked about why some shapes were harder to re-create than others. We discussed why people would represent shapes this way.
Today’s Question (May 6):
Using isometric paper (dot paper), we discussed and then demonstrated how to draw our ‘buildings’ so that they look three-dimensional:
Today’s Question (May 9):
To activate our brains, we discussed how our orthographic puzzles used 2D images to describe a 3D shape. We then looked at a net of a cube and the students were asked if they could make any connections between the net and our orthographic puzzles. The students noticed that the net is similar because it is like each face of the net is like a different view (front, side, side, back, top, and bottom) of the shape. Then we asked the question:
And here are some of the results:
Consolidation: Out of all the work we put up, we found that there were 7 different nets that made a cube.
For homework they were asked to try to find out at least one more (I believe there are 11 possibilities).
We then had a discussion about what is happening in their brains when they look at a net of a shape (Can they picture it as a shape? How?). A few interesting points were made. Students often noticed that when they were picturing the net being folded in their heads, they often chose one face as the base, and it was often the face that had the most other ‘faces’ attached to it. They said it made it less work for their brains if they didn’t have to flip the shape in their heads. It was an interesting discussion.
Yesterday’s math question:
We had a discussion about volume today. They were shown 4 buildings and asked what the volume (the number of cubes used to build the building) of each building was.
From left to right, the volume is: 1, 10, 100, 1000 (cm cubed). We then had a discussion about area and the formula we used (l x w). We applied this idea, but the students noticed there was a 3rd dimension (height… or depth). So we made a graph.
The students were asked if they could find a pattern within the graph. What they noticed was that if you multiplied one column, by the next, by the next, the answer was the volume. So they were asked if they could come up with a formula, and they did:
Today’s Question (May 17):
Today we activated our thinking by talking about the meaning of volume.
Our question was this:
Then they went to work.
Some students used an orthographic representation to try to solve the problem:
Some students used a table and traditional long, or repetitive addition strategies:
Some students used traditional strategies for multiplication:
Some students used Napier’s Bones:
Consolidation: The students were asked (and discussed) how they solve the problem if there were no dimensions given. There were many excellent solutions including: finding actual shoe boxes, finding boxes that were similar in size, estimating, etc…
The students were then asked if it mattered that they all had different dimensions. They discussed the importance of the thinking, as opposed to the numbers or the answer. They agreed that it was more important to know how to solve the problem, than it was to have the same answer.
Homework: Find the volume of a box at home.
Today’s question (May18):
A quick poll at the beginning of the class revealed that most students thought this statement to be true. They were then asked to prove it.
Some students had some of the right ideas, but a misconception was discovered in their understanding of how to do the question. The students had an opportunity to talk about it and discover the misconception (they simply doubled the volume of their first cube, instead of doubling the dimensions):
The majority of the groups used cubes 10 x 10 x 10, and then doubled the dimensions to 20 x 20 x 20 (disproving the original statement):
Some groups were able to visualize the problem. If they had one block (cube) and then added another block (doubling the volume), the shape they ended up with (2 cubes stacked) was not a cube. In order to make it a cube they needed 8 blocks. Therefore if you double the dimensions, the volume increases by 8 times!
Consolidation: We had a discussion around how math sometimes seems obvious, but upon further evaluation is not always as evident as we think.
We also discussed the idea of patterns, and how they apply to this question (doubling dimensions = 8 x volume).