Translations

Code	Language		Translator	Run

Credits

weelookang@gmail.com; Francisco Esquembre; Felix J. Garcia Clemente

Source 1: "Area of Trapezium (Proof using Parallelogram) JavaScript HTML5 Applet Simulation Model - Open Educational Resources / Open Source Physics @ Singapore"

This source outlines a lesson plan designed for students to discover the formula for the area of a trapezium by relating it to the area of a parallelogram and, indirectly, to the area of a rectangle and triangles. The core theme is active learning through exploration and manipulation of physical models (paper cut-outs).

Key Ideas and Facts:

• Relational Understanding: The lesson prioritizes students forming "meaningful links between newly-acquired knowledge and existing schema." The goal is for students to understand how the formulas are derived and why they are important, rather than just memorizing them. The source states, "In this way, students will not just memorize the formula given but know how the formulas of perimeter and area come about, why learning the concepts of perimeter and area are important and how they can apply the concepts."
• Connecting Prior Knowledge: The lesson explicitly builds upon students' prior knowledge of "Properties of triangles and rectangles" and "Areas of triangles and rectangles."
• Properties of Parallelograms as a Foundation: Before tackling trapeziums, the lesson focuses on understanding the properties of parallelograms and deriving their area from the area of a rectangle. This involves a hands-on activity where students "make a straight cut across opposite parallel sides on the rectangle and rearrange the pieces to form a parallelogram," leading to the understanding that "Area of parallelogram = Area of rectangle = base x height" or "A = b x h."
• Deriving the Area of a Trapezium: The lesson plan for trapeziums details a "hands-on approach where students investigate and explore properties and relationships using a guided worksheet." The source mentions multiple ways to derive the formula using paper models:
• Using two trapeziums to form a parallelogram: Students arrange two congruent trapeziums to create a parallelogram. The area of this parallelogram is the sum of the parallel sides of the trapezium multiplied by the height ((a+b)h). Since the parallelogram is made of two trapeziums, the area of one trapezium is half of that: "( \frac{1}{2}(a+b)h )."
• Other methods involving relationships with parallelograms and triangles: The lesson encourages students to explore alternative ways to find the area, potentially involving dividing the trapezium into simpler shapes.
• Emphasis on Justification and Verbalization: Teachers are instructed to ask students to "justify their answers and verbalized their thinking processes," promoting deeper understanding and the development of "Critical and Inventive Thinking (focus – sound reasoning)."
• Use of Resources: The lesson plans specify the use of "Coloured paper cut-outs of parallelograms & trapeziums, Scissors, [and] Glues" as essential resources for the hands-on activities.
• 21st Century Competencies: The lesson aims to foster "Critical and Inventive Thinking (focus – sound reasoning)" and "Information and Communication skills (Focus – Communicating and collaborating effectively)" through pair work and presentations.
• Formula Reinforcement: The lesson explicitly states the goal for students to be able to "find the area of a trapezium" and reiterates the formula: "Area of trapezium: ( \frac{1}{2}(a+b)h ),where (a + b) is the sum of parallel sides." It also provides the formula for the area of a parallelogram: "area of parallelogram = ( (a+b)h )." (Note: This formula for the parallelogram seems to imply that 'a' and 'b' might represent something different in this specific context, perhaps related to how the parallelogram was formed from two trapeziums. In the earlier derivation, it was simply base x height).

Source 2: "area of trapezium ( one piece) is 1/2 ( a + b ) h"

This source, likely related to a specific interactive simulation or resource, directly states the formula for the area of a trapezium.

Key Ideas and Facts:

• Direct Statement of the Formula: The title itself, "area of trapezium ( one piece) is 1/2 ( a + b ) h," clearly presents the standard formula for the area of a trapezium.
• Attribution and Licensing: The source provides information about the authors (weelookang@gmail.com; Francisco Esquembre; Felix J. Garcia Clemente), the copyright year (© 2021), the compilation tool (EJS 6.1 BETA), and the licensing terms.

Connecting the Sources:

Both sources share a common goal of teaching the area of a trapezium. The first source provides a detailed pedagogical approach emphasizing discovery and relational understanding through hands-on activities and connecting to the area of a parallelogram. It aims to help students understand why the formula for the area of a trapezium is what it is. The second source directly presents the formula, likely as part of a resource (perhaps a simulation) that would allow students to explore or apply this formula, potentially building upon the conceptual understanding fostered by the methods described in the first source. The mention of Francisco Esquembre and Felix J. Garcia Clemente as contributors to both suggests a connected effort in developing these educational materials.

In conclusion, the primary theme across these sources is promoting a deeper, conceptual understanding of geometric formulas, specifically the area of a trapezium, through active learning, manipulation of models, and linking new knowledge to existing mathematical foundations. The use of interactive simulations and hands-on activities are highlighted as effective pedagogical strategies.

Sample Learning Goals Mathematics / Perimeters and Areas of Plane Figures / Area of parallelogram

The students will discover the relationship between the area of parallelogram from the area of rectangle using the intuitive-experimental approach. It is a hands-on approach where students investigate and explore properties and relationships of rectangles and parallelograms

For Teachers

It is important to let students learn through relational understanding that is by forming meaningful links between newly-acquired knowledge and existing schema. In this way, students will not just memorize the formula given but know how the formulas of perimeter and area come about, why learning the concepts of perimeter and area are important and how they can apply the concepts. Students will also be asked to justify their answers and verbalized their thinking processes.

Prior Knowledge:

• Properties of triangles and rectangles
• Areas of triangles and rectangles

 

Concepts and Skills

By the end of the lesson, students will be able to:

• explain the properties of a parallelogram
• find the area of a parallelogram

 

21st Century Competencies:

• Critical and Inventive Thinking (focus – sound reasoning)
• Information and Communication skills (Focus – Communicating and collaborating effectively)

Lesson Plan Parallelograms

Focus: The intent of this phase of the lesson is to introduce the properties of parallelograms to the students as a trigger to pique students’ interest. (5 mins)

Teacher will recap the areas of rectangle, triangle and circle.

What is a parallelogram?

Teacher will show students a slide on examples and non-examples of parallelograms. 

Teacher will ask students to describe a parallelogram

• What can you say about the pairs of equal sides?

Opposite sides of a parallelogram are equal and parallel.

• What can you say about the pairs of equal angles?

Opposite angles of a parallelogram are equal.

• Is square a rectangle? Yes.
• Is rectangle a parallelogram? Yes.
• Is square a parallelogram? Yes.
• Is rhombus a parallelogram? Yes.

Properties of parallelogram

It is a quadrilateral with two pairs of parallel lines.

Teacher will show examples of parallelogram in real life.

Focus: Finding the area of parallelogram, Pair-work & Think-Pair-Share( 35 mins)

Pair work activity to find the formula for the area of parallelogram

Areas of triangles, parallelograms and trapeziums are related to the area of a rectangle. The use of paper model can help visualization and to establish relations. Teacher will distribute the worksheets and coloured paper with rectangle cut-outs for each pair of student. Each person to get 2 rectangle cut-outs. 

The pair will discuss how to obtain a parallelogram from the rectangle.

Students will make a straight cut across opposite parallel sides on the rectangle and rearrange the pieces to form a parallelogram.

                Area of parallelogram = Area of rectangle = base x height

                      A = b x h

where                                              h = perpendicular height

Teacher to get a pair to present the solution on visualiser. The pair will be asked to explain the steps to obtain the area of parallelogram from rectangle.

Teacher will get the class to work out questions in their textbook.

Recap on the characteristics of parallelogram. Emphasize on the key concepts.

Teacher reiterates that the area of a parallelogram = base x height 

Issue homework questions. Set deadline to hand in.

Resources: 

1. Coloured paper cut-outs of parallelograms & trapeziums
2. Scissors
3. Glues

Lesson Plan Trapezium Perimeters and Areas of Plane Figures / Area of trapezium

The students will discover the relationship between the area of trapezium from the area of parallelogram and triangle using the intuitive-experimental approach. It is a hands-on approach where students investigate and explore properties and relationships using a guided worksheet.

It is important to let students learn through relational understanding that is by forming meaningful links between newly-acquired knowledge and existing schema. In this way, students will not just memorize the formula given but know how the formulas of perimeter and area come about, why learning the concepts of perimeter and area are important and how they can apply the concepts. Students will also be asked to justify their answers and verbalized their thinking processes.

Knowledge

After this lesson, students will be able to: 

• find the area of a trapezium

21st Century Competencies:

• Critical and Inventive Thinking (focus – sound reasoning)

Focus: The intent of this phase of the lesson is to introduce the properties of trapezium to the students as a trigger to pique students’ interest. (5 mins)

Recap on previous lesson

Teacher will recap with students the area of a parallelogram and a triangle. 

 

Teacher writes the formula for area of  parallelogram: b x h and formula for area of triangle: ½ x b x h on white board.

 

What is a trapezium?

Teacher will show students a slide on examples and non-examples of parallelograms. 

It is a quadrilateral with one pair of parallel lines

 

Properties of trapezium.

The parallel sides of a trapezium are called the bases, here symbolized by a and b.
The height of the trapezium is the perpendicular distance between the bases, here symbolized by h.

 

If the two sides which are not parallel have equal lengths, then the trapezium is called an isosceles trapezium. The base angles are equal in measurement.

 

Where can we find trapeziums in our daily life?

• In landmarks and buildings
• Aesthetic beauty in homes and offices

Focus: Finding the area of trapezium, Pair-work & Think-Pair-Share (25mins)

Pair work activity to derive the formula for the area of trapezium

 

Area of trapezium is related to the areas of rectangle, triangle and parallelogram. The use of paper model can help visualization and to establish relations. Teacher will distribute coloured paper with trapezium cut-outs for each pair of students. Each person to get 4 trapezium cut-outs.

 

Each pair to make use of the trapezium shapes cut out from the coloured paper to find formula of area of trapezium, knowing the formulae of areas of rectangles, triangles and parallelograms.

1st way: half of area of parallelogram :

Teacher informs students that two trapeziums are for Method 1 of the activities on the worksheet.

Students are given 5 mins complete method 1 by using two trapeziums to find area of trapezium. Teacher walks around to supervise them.

Teacher asks a pair to come up to the visualiser to arrange the two trapeziums to form a parallelogram. 

Possible arrangements:

2nd way, half height x total length:

Teacher then asks the students whether there is another way of finding the area of trapezium by using the area of parallelogram besides the one described in Method 1.

Teacher then asks students to work on Method 2 of the activity for 5 minutes. She walks around the class to supervise the students.

3rd way: area of 2 triangles (By cutting into 2 triangles across opp. vertices) 

Formula for area of trapezium.

Teacher reiterates Area of trapezium:   a +b = sum of parallel sides, h = height

Teacher will get the class to work out questions in their textbook.

Recap on the characteristics of trapeziums. Emphasize on the key concepts.

Teacher reiterates that the area of a trapezium is 

½ x h x (a + b),where (a + b) is the sum of parallel sides. 

Issue homework questions. Set deadline to hand in.

Resources: 

1. Coloured paper cut-outs of parallelograms & trapeziums
2. Scissors
3. Glues

What is the area of the trapezium, in terms of a,b and h?

area of trapezium = $\frac{1}{2}(a+b)h$

What is the area of the parallelogram, in terms of a,b and h?

area of parallelogram = $(a+b)h$

 

Video

[text]

 Version:

1. https://weelookang.blogspot.com/2019/09/area-of-trapezium-proof-using.html

Other Resources

1. https://www.geogebra.org/m/d2hrHaGt by Lew WS
2. https://www.geogebra.org/m/dyawu2us by Lew WS
3. https://www.geogebra.org/m/VTrYdaZt by David T
4. https://www.geogebra.org/m/jRbCAFVx by Mark Dabbs

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