Posted in Geometry

Regular Polygons Problems

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One  of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.

Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.

Problem 1

The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.

equal perimeter

Problem 2

What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?

perimeter problem

Problem 3

What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.

Problem 4

What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.

Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.

You may want to view another similar lesson on quadrilaterals.

 

Posted in Algebra, Math videos

Teaching Equations of Sequence with Mr Khan

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In the following video Mr. Khan’s gave an excellent task and solution on finding the equation of a sequence of blocks. I suggest you stop the video after the presentation of the problem. Let the students solve it first before you let Mr. Khan do the talking.

 

Mr Khan did give an excellent explanation especially the one  about x – 1.  The last solution involving the slope and equation of lines was not as clear. This is the part where your students need you. So I suggest that after viewing the video ask the students what part of the video made sense to them and which part was not very clear.

I think it would be best to ask students first about the rate at which the number of blocks is increasing rather than use the term slope. If you want to relate this to slope ask the students to plot the values in the table on a grid. You make then ask what the slope is of the line containing the points.

Additional solutions

Here are two more solutions to the problem. The first solution involve dividing then adding. This leads to a a different expression but will still simplify to 4x-3.

divide then add
Dividing and putting together

The second solution involve completing the figure into rectangles for easy counting then taking away what was added. This leads directly to the simplified equation. Don’t you love it:-) I do. So please share this post to FB and Google. Thanks.

algebraic expressions
Adding and taking away

This post is the second in my series of post on Teaching Math with Mr Salman Khan. The first is about Teaching Direct Variation with Mr Khan.

If students find Khan Academy’s math videos helpful and cool then by all means let’s use them in teaching mathematics. Just don’t let Mr Khan do all the teaching. Remember you are still the didactician.

Posted in Algebra, Geogebra, Geometry

The Pythagorean Theorem Puzzle

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Math puzzles are great activities for enjoying and learning mathematics. The following is an example of Tachiawase. Tachiawase is a popular puzzle in Japan which involve dissecting a geometrical figure into several parts and then recombining them to form another geometrical figure. The puzzle below is credited to Hikodate Nakane (1743). This was one of the puzzles distributed at the booth of Japan Society of Mathematical Education during the ICME 12 in Seoul this year.

Make a shape that is made from two different sizes of squares by dividing them into three parts  then recombine them into one square. [Reformulated version: Make two cuts in the shapes below to make shapes that can be recombined into a bigger square.]

two squares puzzle

Here’s how I figured out the puzzle: I know that it must have something to do with Pythagorean Theorem because it asks to make a bigger square from two smaller ones. But where should I make the cut? I was only able to figure it out after changing the condition of the puzzle to two squares with equal sizes. It reduced the difficulty significantly. This gave me the idea where I could make the cut for the side of the square I will form. The solution to this puzzle also gave me an idea on how to teach the Pythagorean theorem.

I made the following GeoGebra mathlet (a dynamic math applet) based on the solution of the puzzle. I think the two-square math puzzle is a little bit tough to start the lesson so my suggestion is to start the lesson with this mathlet and then give the puzzle later.  As always, the key to any lesson are the questions you ask. For the applet below, here’s my proposed sequence of questions:

  1. What are the areas of each of the square in the figure? Show at least two ways of finding the area.
  2. How are their areas related? Drag F to find out if your conjecture works for any size of the squares.
  3. Can you think of other ways of proving the relationships between the three squares without using the measures of the sides?
  4. If the two smaller squares BEDN and GFNH have sides p and q, how will you express the area of the biggest square LEJG in terms of the area of the smaller ones?
  5. Express the length of the sides LEJG in terms of the sides of BEDN and GFNH.

[iframe https://math4teaching.com/wp-content/uploads/2012/08/Pythagorean.html 550 450]

After this lesson on Pythagorean relation you can give the puzzle. Once they have the correct pieces, ask the students to move the pieces using transformation in the least possible moves. They should be able to do this in three moves using rotation. Click here to download the applet. Note: If you don’t see the applet, enable java in your browser

Use the comment sections to share your ideas for teaching the Pythagorean relation. If you like this post, share it to your network. Thank you.

Posted in Elementary School Math, Geometry

A triangle is a fish

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Why is it that students find it easier to calculate the area of triangle ABC but will have difficulty calculating the area of triangle DEF? Middle school students even believe that it’s impossible to find the area of DEF because the triangle has no base and height!right triangles

That knowing the invariant properties that makes a triangle a triangle (or any geometrical shape for that matter), is not an easy concept to learn is illustrated by this conversation I had with my 4-year old niece who proudly announced she can name any shape. The teacher in me has to assess.

Thinking about how a four-year old could possibly think of these meaning of the shapes made me ask: If four-year olds are capable of thinking this way then why do we think that there are students who can’t do math or doubt the idea that algebra is for all?