Perimeter and Area

Today’s problem (Jan 9):

In order to activate our brains we brainstormed the words “perimeter” and “area”.  Here is what the students came up with:

We also looked at a simple shape and determined the perimeter and area:

Today’s question:

Students found many shapes that had a perimeter of 12 units:

One misconception was noticed.  Many students were counting the diagonal of a triangle as one unit.  So we looked at a triangle to see if the length and width were equal to the diagonal.  It was not.

Consolidation:  We focused on quadrilaterals, and how you go about getting the perimeter.  Some students noticed that you simply add all the sides together.  Other students noticed that you could simply double one side and double the other (2l+ 2w), and others noticed that it was the same as adding the length and width and doubling it.  And so we explored the formula for perimeter 🙂

Today’s problem (Jan 10):

To activate our brains, we reviewed the formulas for perimeters of quadrilaterals:

Today’s problem:

Some students used the formula P=s+s+s+s:

Some students used the formula P=(lx2)+(wx2):

Some students used the formula P=(l+w)x2

Consolidation:  We focused our discussion on what a formula is (a general statement that can be used to solve a problem regardless of what the numbers are) and the importance of writing out the formula being used.  The students connected the idea that writing down the formula demonstrates or helps explain your thinking!  Well done!

Today’s problem (Jan 11):

We focused on the area when discussing our solutions.

Some students calculated the area of the carpet by counting the squares:

Some students calculated the number of squares using multiplication:

Some students used the formula for area:

Consolidation:  Our discussion focused on two things. 

First, that showing the formula is a very efficient way of demonstrating your thinking (in this case A = l x w).

Second, we talked about why there is a “2” used to indicate “square units”.  The students had some interesting ideas:

Today’s problem (Jan 12):

Some students doubled the shape (visually) but did not double both the length and the width:

Some students doubled the length and width and discovered the statement to be false:

Some students found a pattern, that the area does not double it quadruples:

Consolidation:  We focused our discussion on the question, “Do you need to show all the squares on the inside of a shape in order to calculate (or show) the area?  The students talked about how you could represent any size rectangle by simply changing the numbers of the length and width.  The rectangle is a representation.  The area is simply a calculation.

Today’s Problem (Jan 16th):

To activate our brains, we reviewed the formulas for perimeter and area:

Today’s question:

Some students found only 1 shape (which did not allow for any proof or comparison):

Some students compared 2 shapes to find out which one has the biggest area:

Some students compared 3 shapes to find out which one has the biggest area:

Some students compared 4 shapes to find out which one has the biggest area:

Some students compared 5 shapes to find out which one has the biggest area:

Consolidation:  When the students looked at the “shortest” side of all the shapes they saw a pattern.  We discussed what would be the best way to show this pattern, and it was decided a chart or table would be good.  So we created a table to show the information:

We then had a discussion around why using a chart or table is a good idea when answering questions like the one we answered today:

Today’s problem (Jan 17th):

Some students concluded (based on yesterday’s work) that the answer would be a square (and therefore did not prove their work).  It was not, but it was a good idea:

Some students correctly fenced three sides, and calculated the area of more than one shape:

Some students drew all the possible shapes, and correctly identified the shape with the biggest area (6m x 3m):

Some students included a table, to make sure they had all the possible shapes by following the pattern:

Some students were able to efficiently figure out the answer without drawing any pictures (only using a table):

Consolidation: We had a discussion about the effectiveness of using tables and charts to display our thinking (and also to discover patterns).

Today’s problem (Jan 18th):

To activate our brains we discussed how we would go about figuring out the perimeter and area of the top of our desks.  Afterwards we read today’s problem:

Some students solved the problem by measuring the entire length of three desk and then the width:

Other groups explored the idea that by figuring out the area of one desk, they could calculate the area of three simply by tripling the answer (of one):

 

Consolidation:  We had a discussion around the idea that regardless of the shape of the desks, the area would have been the same.  We explored the idea of breaking irregular shapes into smaller rectangles in order to figure out the area.

Today’s problem (Jan 19):

Some students focused on finding the missing 2 measurements (but then ended up calculating the perimeter instead of the area… oops):

Some students broke the shape up into 2 rectangles and then calculated the area of each rectangle (A=lxw) and added the two areas together:

Some students solved the problem by calculating the area of the entire rectangle and then subtracting the area of the missing piece:

Consolidation:  We focused our discussion on the importance of breaking up irregular shapes into rectangles in order to use our formula for area (A=lxw)

Today’s problem (Jan 23):

Some students drew their walls to scale, and counted the squares to check their work:

Some students found the area of the wall and then subtracted the area of the door:

Some students broke the wall up into smaller rectangles, calculated the area of each smaller rectangle and then added the areas together:

Some students didn’t even need to make a diagram:

Consolidation:  We had a discussion about the variety of ways this question could be solved (we focused our discussion on the area of the wall)

 

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