We started talking about 3 dimensional shapes today. We looked at geometric shapes in the real world:
And we talked about attributes. First we reviewed the attributes used to define 2D shapes, then we discussed what kind of things might define one 3D shape from another. The students came up with:
- corners (vertices)
- edges (sides)
Today’s problem (April 20):
To activate their brains the students were asked what the difference was between a prism and a pyramid. This is what they came up with:
For today’s problem they were asked to organize the attributes of prisms and pyramids onto tables:
Here are some of the solutions…
And for pyramids:
Consolidation will take place next class.
Today’s problem (April 23):
Today we looked at our tables from last class. We filled in a class table and from that tried to find some patterns for prisms and pyramids. The students found some really good patterns (we wrote them at the top of each chart).
Today’s problem: The students were put into groups and asked to come up with a song about a three dimensional shape.
Today’s problem (April 27th):
Today we looked at ornithographic representations of shapes (front, side, and top views). The students were asked to build shapes using connecting cubes, and then draw the shape from different views. They also added a “hint”, being the number of cubes used to make the shape.
The rules were:
(“garages” being hidden missing cubes)
Here were some of the results:
Next class we will begin solving the puzzles.
Today’s problem (April 30):
Today we solved the problems we created yesterday. We had a discussion at the end about some of the puzzles that had mistakes and explored the importance of seeing and uderstanding our mistakes as AHA moments. We also discussed the connections between the “hints” in our puzzles (the number of cubes used) and measurement.
Students worked on isometric drawings using dot paper. We looked at faces of shapes creating ornitographic representations of shapes. With this activity students move from 2 dimensions to 3 dimensions. They looked like this:
Today’s problem (May 2):
Today we looked at nets. Nets are designs that, when folded, will make 3D shapes. You need to have all the faces of the shape attached together. We decided to work with a cube, because all 6 faces are square.
And some of the solutions:
Consolidation will take place next class.
We had a discussion today about the nets used to make cubes. The question was asked, “What happens in your head when you try to imagine a net as a shape?” The students talked about how they imagine themselves folding the net into a shape. Some students imagined the cube first and tried to unfold it into various nets. But in general, the consensus was that the students choose one face to act as a base, and try to fold the rest of the shape up around the base. It was a good discussion. For today’s work the students were asked to make nets of shapes and try to attach the shapes to make a sculpture of some kind. Their work can be found on the Nets to Shapes page.
Today’s problem (May 7th):
Students created a variety of shapes and hence tables:
Consolidation: We put a number of solutions on the board and asked the students to see if they could find a pattern. first they noticed that the total was always a multiple of the L (length). Then it was noticed that if you multiplied LxWxH you it always equaled the TOTAL.
We then connected this to measurement, 1 cm being a measurement of distance, 1 cm squared being a measurement of area, and today, 1 cm cubed being a measurement of capacity, or VOLUME:
We will explore volume some more this week.
Today’s problem (May 8):
To activate our brains, we looked how to multipy 3 numbers. When we asked for the answer, and how people got it (4 x 2 x 5), we discovered some people did 4×2=8 x5=40, while others did 2×5=10×4=40, to make friendly numers. We then realized we could also do 4×5=20×2=40.
Consolidation: As opposed to looking at how they did their calculations, we focused our discussion on how they solved a problem that didn’t have any numbers. Here were their ideas:
…and just as a reminder, the students were told to include the formula in any question (when they know it).
Today’s problem (May 10th):
To activate our brains we reviewed the formula for volume and talked about possible units of measurement:
(Grade 4’s had to figure out the volume of the first box only, but were given the option of doing the entire question)
Some students solved the problem using a calculator:
Some students used a grouping and adding strategy:
Some students were able to do a significant amount of the work using mental math:
Some students used traditional multiplication strategies:
And some of the groups quickly realized that the third block is a cube because all the measurements are 10 cm:
Consolidation: We reviewed the different strategies students used to solve the problem.
Today’s problem (May 11th):
We activated our brains by reviewing what we me mean when we talk about the dimensions of a 2D or 3D shape. The students responded that for 2D shapes we have length and width, and for 3D shapes we have length, width, and height. We also reviewed the formulas for perimeter, area, and volume (focusing on volume being lxwxh)
Prove or disprove:
If you double the dimensions of a cube, you double the volume.
Some students started with a single cube and worked from there:
Some students stared with a cube (2x2x2) and doubled that (4x4x4):
Some students started with 5x5x5 and compared it with 10x10x10:
Some students started with 10x10x10 and compared it with 20x20x20:
Some students began setting up a table wich was a good organizational idea:
Consolidation: When we put all the results onto a table the students discovered there was a pattern. The rule they found was that when you double the dimensions of a cube, the volume increases eight fold:
Today’s problem (May 14th):
To activate our brains we reviewed the formula for volume, and the students were given a quick question to solve (turn and talk):
Since… volume = l x w x h , the students looked at the question algebraically as:
8 = 1 x 2 x ?
And the missing digit is 4
Some students solved the problem using (20) unifix cubes, and then showed their answer (some using orthographic representations):
Some used unifix cubes, and came up with more than one answer:
Some students were able to solve the problem mathematically:
Some students were able to solve the problem using mental math, and found more than one solution:
Consolidation: We focused on the solutions than did not use math manipulatives. The students were asked how they solved the problem. Some students explained that they worked the problem backwards (20 = ? x ? –> 20 = 4 x 5 –> 5 = ? x ? –> 5 = 5 x 1 –> therefore 20 = 4 x 5 x 1). And some students explained that they were simply able to to the work in their heads:
Today’s problem (May 15th):
To activate our brains we answered this question:
Some of the grade 4 students answered the question using unifix cubes (each cube representing a sugar cube):
Some of the grade 4 students were able to solve the problem using the equation:
There were two ways the grade 5’s answered the question. The first strategy was to find the volume of the box (288 cm cubed), find the volume of the sugar cube (8 cm cubed), and then divided the volume of the box by the volume of the cube (288 divided by 8 = 36). Several different strategies were used for the division part of the question:
The most efficient strategy used to solve the problem divided the dimensions of the box from cm to sugar cubes first (12 cm = 6 cubes; 4 cm = 2 cubes; 6 cm = 3 cubes), and they used those numbers in the formula:
Consolidation: Our conversation focused on the importance of trying to solve problems in different ways, the goal always to find the most efficient way because (the students said) it ends up being less work 🙂