Tag: trapezoids

Posted in Geometry

Pentagon to Quadrilateral Puzzle

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Puzzles involving cutting shapes and forming them into different shapes helps reinforce the idea that area do not necessarily change with change in shape. It is also a good activity for developing visualisation skill and spatial ability.

The puzzle below is from one of the leaflets at the booth of Japan Society of Mathematical Education last ICME 12 in Seoul, Korea. The original puzzle is suited for Grade 4. The instruction was to cut the pentagon along the dotted lines and then form them into the shapes shown. The shapes shown in the leaflet is a parallelogram, a rectangle, an isosceles trapezoid, and a general trapezoid. I modified the puzzle for students in the higher level. I have indicated the measure of the two angles just in case you want your students to justify that the pieces really form into quadrilaterals. This is one way to assess your students knowledge of the properties of these parallelograms, trapezoid and trapezoids as they justify each shape formed.

pentagon puzzle

Here are two solutions – rectangle and isosceles trapezoid. Form the other two shapes.

trapezium and rectangle

Posted in Algebra

Visual representations of the difference of two squares

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Students’ understanding of mathematics is a function of the quality and quantity of the connections of a concept with other concepts. As I always say in this blog, ‘To understand is to make connections’.

There are many ways  of helping students make connections. One of these is through activities involving multiple representations. Here is a lesson you can use for teaching the difference of two squares, x^2-y^2.

Activity: Ask the class to cut off a square from the corner of a square piece of paper. If this is given in the elementary grades, you can use papers with grid. If you give it to Grade 7 or 8 students you can use x for the side of the big square and y for the side of the smaller square. Challenge the class to find different ways of calculating the area of the remaining piece. Below are two possible solutions

Solution 1 – Dissect into two rectangles

 

Solution 2 – Dissect into two congruent trapezoids to form a rectangle

 

Extend the problem by giving them a square paper with a square hole in the middle and ask them to represent the area of the remaining piece, in symbols and geometrically.

Solution 1 – Dissect into four congruent trapezoids to form a parallelogram

 

Solution 2 – Dissect into 4 congruent rectangles to form a bigger rectangle

These two problems about the difference of two squares will not only help students connect algebra and geometry concepts. It also develop their visualization skills.

This is a problem solving activity. It’s important to give your students time to think. Simply using this to illustrate the factors of the difference of two squares will be depriving students to engage in thinking. They may find it a little difficult to represent the dimensions of the shapes but I’m sure they can dissect the shapes. Trust me.