# Deriving formulas and determining the area of 2D shapes drawn on a grid of dots

- Algebra, Problem solving

- ISCED 2 = Upper Secondary Education

- English

## Introduction

The Learning Plan provides for activities for pupils that are involved in a real research process with application in services evaluation. Basic issues and stages of the research process are taught, from the formulation of the problem and the goal to the final presentation of the results and conclusions.

## Learning Outcomes

At the end of the lesson, the learner will be able:

- To express a variable through other variables in a given equation
- To performs simple formulas
- To determines the area of 2D shapes drawn on a grid of dots
- To use formulas from math and other subjects
- To present concise, substantiated arguments to explain solutions or generalizations using: symbols, diagrams, or graphs
- To develops a sense of cooperation and empathy with classmates.

## How Does It Work

**Activity 1: **There is a discussion about the preconceptions about the area of regular shapes and how to find the formula for the area of this shape formed by joining a rectangle and two semicircles and shows the model of the figure that has been prepared in advance. The pupils explain their answers that the shape is formed by merging a rectangle and two semicircles, which can be seen by cutting the parts and merging the semicircles and getting a circle. The area is the sum of the areas of the rectangle and the circle and then they write the formula for area of the form

P = w ^ 2 · π / 4 + x · w

**Activity 2: **The teacher gives an activity in which the pupils in pairs look for a connection between the area of shapes drawn on a grid with dots, the points in the form and the points on the perimeter) i.e., examine the formula for area A, on the shapes drawn on a grid with dots (with points in the form and p points on the perimeter) (Peak theorem).

The teacher asks questions:

- How do you write down your findings?
- Do you notice any patterns?
- Can you find a general rule?

It is concluded that the area of the form is one less than the sum of the points in the form and half of the points on the perimeter are and the formula is derived

A = I + p / 2 – 1 known as the Pick’s theorem.

Pupils in pairs draw shapes on dots and then count the dots in the form and the dots on the perimeter of the form and calculate their area using the Pick formula in the same activity using a geometric board.

The obtained solutions are discussed. Couples self-assess.

Pupils in pairs use the GeoGebra app to draw shapes and then count the points in the form and the points on the perimeter of the form and calculate the area with the Peak formula and then check the answer in the area calculation menu.

**Activity 3: **In the school yard, a shape is made with the help of a rope and boats and then the points in the shape and the points on the perimeter of the shape are counted and its area is calculated using the Peak theory.

## Why Is It A Good Practice?

By developing these skills, pupils are trained to learn on their own.

The above activities initiate the development of:

- critical thinking
- creativity
- communication
- cooperation

## Assessment

The first activity allows to diagnose the pupils’ knowledge of the topic and according to the obtained results to plan the future activities. The activity with Peak theorem enables connection with real situations, develops critical thinking, increases communication skills, creativity, and cooperation between pupils. The use of digital tools to validate acquired knowledge provides rapid feedback for both pupils and the teacher.

Formative assessment by monitoring the activity of students during the class.

## Inclusion

The teacher can include elements that the pupil can manipulate such as: elastic shapes on a geometric board. (A 2d shape model made of coloured paper).