Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Plane shapes
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Plane shapes
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 2nd Term
Week: 4
Theme: Mensurtion And Geometry
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State the similarities and differences between the following: square, rectangle, triangle, trapezium parallelogram and circle Find the perimeter of a regular polygon: square, rectangle, triangle, trapezium, parallelogram and circle Find the area of plane shapes such as squares, rectangles, parallelograms Find the area of real life plane objects.
2.1 Definition of Plane Shapes Plane shapes are two-dimensional (2D) figures that have length and width but no thickness. They can be drawn on a flat surface, such as a piece of paper or a whiteboard. Examples include squares, rectangles, triangles, and circles. 2.2 Properties and Comparison of Specific Plane Shapes a.
Square: Definition: A quadrilateral (four-sided polygon) with four equal sides and four right angles (90 degrees).
Properties: All four sides are equal in length. All four interior angles are 90°. Opposite sides are parallel. Diagonals are equal in length and bisect each other at 90°. Real-life
Example: A typical floor tile, chessboard square, a face of a dice. b.
Rectangle: Definition: A quadrilateral with four right angles. Opposite sides are equal and parallel.
Properties: Opposite sides are equal in length. All four interior angles are 90°. Opposite sides are parallel. Diagonals are equal in length and bisect each other. Real-life
Example: A classroom blackboard, a typical door, a naira note. c.
Triangle: Definition: A polygon with three sides and three angles.
Properties: Sum of interior angles is 180°. Can be classified by side lengths (equilateral, isosceles, scalene) or by angles (right-angled, acute-angled, obtuse-angled). Real-life
Example: A slice of bread, a roof truss, the shape of some street signs. d.
Trapezium (or Trapezoid): Definition: A quadrilateral with at least one pair of parallel sides.
Properties: Only one pair of opposite sides is parallel. The non-parallel sides are called legs. Real-life
Example: The cross-section of some drainage channels, a part of a window frame. e.
Parallelogram: Definition: A quadrilateral with two pairs of parallel sides.
Properties: Opposite sides are equal in length. Opposite angles are equal. Consecutive angles are supplementary (add up to 180°). Diagonals bisect each other. Real-life
Example: A pushed-over rectangular box, a pattern in a fabric design, a rhombus (special type of parallelogram). f.
Circle: Definition: A closed curve where all points on the curve are equidistant from a central point.
Properties: Has a radius (distance from center to any point on the curve) and a diameter (distance across the circle through the center, twice the radius). No straight sides or angles. Real-life
Example: A car tyre, a plate, a traditional African drum's top. 2.3 Similarities and Differences | Feature | Square | Rectangle | Parallelogram | Trapezium | Triangle | Circle | | :---------------- | :-------------------------- | :-------------------------- | :-------------------------- | :-------------------------- | :--------------------------- | :--------------------------- | | Sides | 4 equal | 4 (opposite equal) | 4 (opposite equal) | 4 (at least 1 pair parallel) | 3 | 0 (curved) | | Angles | 4 right angles (90°) | 4 right angles (90°) | Opposite angles equal | Sum is 360° | Sum is 180° | 0 | | Parallel Sides| 2 pairs | 2 pairs | 2 pairs | 1 pair | None | None | | Diagonals | Equal, bisect at 90° | Equal, bisect | Bisect each other | May or may not be equal | Not applicable | Not applicable | | Symmetry | High (4 lines, rotational) | Moderate (2 lines, rotational)| Low (rotational) | May have 1 line (isosceles)| 0, 1, or 3 lines | Infinite lines, rotational | Similarities: Squares, rectangles, parallelograms, and trapeziums are all quadrilaterals (4-sided polygons). Squares and rectangles both have four right angles and two pairs of parallel sides. Squares are special types of rectangles, and rectangles are special types of parallelograms. Triangles and quadrilaterals are all polygons (closed figures made of straight line segments).
Differences: Only squares have all four sides equal and all four angles 90°. Rectangles have opposite sides equal, but not necessarily all sides equal (unless it's a square). Parallelograms have opposite sides equal and parallel, but angles are not necessarily 90°. Trapeziums only have one pair of parallel sides, unlike parallelograms, rectangles, and squares. * Triangles have 3 sides, while the others (except circle) have 4. rectangles, and rectangles are special types of parallelograms. Triangles and quadrilaterals are all polygons (closed figures made of straight line segments).
Differences: Only squares have all four sides equal and all four angles 90°. Rectangles have opposite sides equal, but not necessarily all sides equal (unless it's a square). Parallelograms have opposite sides equal and parallel, but angles are not necessarily 90°. Trapeziums only have one pair of parallel sides, unlike parallelograms, rectangles, and squares. Triangles have 3 sides, while the others (except circle) have
4. A circle has no straight sides or angles, distinguishing it from all other polygons. 2.4 Perimeter Perimeter is the total distance around the boundary or outline of a two-dimensional shape. It is measured in units of length (e.g., cm, m, km).
Formulas: Square: P = 4 × side (s) => P = 4s
Example: A square plot of land with side 15m. P = 4 15m = 60m.
Rectangle: P = 2 × (length + width) => P = 2(l + w)
Example: A rectangular football pitch, length 100m, width 60m. P = 2(100m + 60m) = 2(160m) = 320m.
Triangle: P = side1 + side2 + side3 (a + b + c)
Example: A triangular piece of fabric with sides 12cm, 15cm, 10cm. P = 12cm + 15cm + 10cm = 37cm.
Trapezium: P = side1 + side2 + side3 + side4 (a + b + c + d)
Example: A trapezium-shaped flower bed with sides 8m, 5m, 10m, 6m. P = 8m + 5m + 10m + 6m = 29m.
Parallelogram: P = 2 × (side A + side B) => P = 2(a + b) where 'a' and 'b' are adjacent sides.
Example: A parallelogram-shaped signboard with adjacent sides 2m and 1.5m. P = 2(2m + 1.5m) = 2(3.5m) = 7m.
Circle (Circumference): C = 2πr or C = πd, where 'r' is the radius, 'd' is the diameter, and π (pi) ≈ 3.14 or 22/
7. Example: A circular water tank with radius 3.5m. C = 2 (22/7) 3.5m = 2 22 0.5m = 22m. 2.5 Area Area is the amount of surface a two-dimensional shape covers. It is measured in square units (e.g., cm2, m2, km2). Formulas (for shapes specified in objectives): Square: A = side × side => A = s2
Example: A square tile with side 30cm. A = 30cm 30cm = 900cm
2. Rectangle: A = length × width => A = l × w
Example: A rectangular tabletop of length 1.5m and width 0.8m. A = 1.5m 0.8m = 1.2m
2. Parallelogram: A = base × height => A = b × h (The height is the perpendicular distance between the parallel bases).
Example: A parallelogram-shaped pattern on a 'Kente' cloth with base 25cm and perpendicular height 10cm. A = 25cm 10cm = 250cm2. 2.6 Finding Area of Real-life Plane Objects: For real-life objects, the area can be found by:
1. Direct Measurement: Measure the dimensions (length, width, base, height, radius) of the object and apply the appropriate formula.
2. Decomposition: If the object is an irregular shape, it can often be broken down into simpler plane shapes (squares, rectangles, parallelograms, triangles). Calculate the area of each component shape and sum them up.
3. Approximation (using grid paper): For very irregular shapes, place the object on a grid paper and count the number of full squares covered. For partial squares, estimate their contribution or combine them to make full squares. This method gives an approximate area.
Example (Decomposition): A compound wall forming an 'L' shape. This can be split into two rectangles. Calculate the area of each rectangle and add them. 3.1 Introduction (10 minutes)
Teacher Activity: The teacher displays various real-life objects or pictures of objects (e.g., a square mat, a rectangular door, a triangular traffic sign, a circular plate, a parallelogram-shaped brick pattern, a trapezium-shaped garden bed). The teacher asks students to identify the shapes they see.
Student Activity: Students observe the objects/pictures and identify common geometric shapes, sharing their observations with the class. The teacher prompts students to notice if the shapes are flat or solid. 3.2 Exploring Properties and Comparisons (20 minutes)
Teacher Activity: Distributes pre-cut paper models of squares, rectangles, triangles, trapeziums, parallelograms, and circles to small groups. Guides students to examine the properties of each shape: count sides, identify parallel sides, check angles using a square corner (e.g., of a ruler or book), and observe diagonals (by folding or drawing). Facilitates a class discussion on similarities and differences, prompting students to articulate their findings.
Student Activity: In groups, students manipulate the paper shapes, counting sides, identifying parallel lines, and checking for right angles. Students discuss within their groups to identify unique features and commonalities among the shapes. Groups share their observations, contributing to a class-compiled list or table of similarities and differences. 3.3 Understanding Perimeter (25 minutes)
Teacher Activity: Defines perimeter as the distance around a shape. Demonstrates how to calculate perimeter using a ruler and a string on various shapes. Introduces the formulas for perimeter for squares, rectangles, triangles, trapeziums, and parallelograms, and the circumference for circles (P = sum of sides; C = 2πr or πd). Works through a few examples on the board, using real-life contexts (e.g., "A tailor needs to sew a trim around a rectangular tablecloth. If the tablecloth is 2m long and 1.2m wide, how much trim is needed?").
Student Activity: Students define perimeter in their own words. Students use rulers/strings to measure and calculate the perimeter of their paper shapes or classroom objects (e.g., textbook, exercise book). Students copy the formulas and worked examples. Students solve additional practice problems presented by the teacher. 3.4 Understanding Area (25 minutes)
Teacher Activity: Defines area as the amount of surface covered by a shape. Introduces the concept of square units by drawing a 1cm x 1cm square. Demonstrates area calculation for squares and rectangles by counting squares on grid paper, then introduces the formulas (A = s2, A = l × w). Explains the formula for the area of a parallelogram (A = b × h) by showing how a parallelogram can be transformed into a rectangle by cutting and rearranging a triangular part. Works through practical examples for squares, rectangles, and parallelograms (e.g., "A farmer wants to plant maize in a rectangular plot of land 20m by 15m. What is the area of the plot?"). Briefly discusses how to find the area of real-life objects, especially those that can be broken into simpler shapes.
Student Activity: Students understand the concept of square units. Students practice calculating the area of squares, rectangles, and parallelograms using the given formulas. Students engage in group activities to measure and calculate the area of objects in the classroom (e.g., desk top, floor tiles). Students take notes on formulas and examples. 3.5 Consolidation and Application (10 minutes)
Teacher Activity: Recap key definitions, formulas, and concepts. Poses questions that require students to differentiate between perimeter and area and identify situations where each is applied in real life.
Student Activity: Students participate in the recap, answering questions and clarifying any misconceptions. Students offer real-life examples of when perimeter and area calculations would be necessary.
Question 1 (Comparison): State two similarities and two differences between a square and a rectangle.
Solution 1: Similarities: Both a square and a rectangle are quadrilaterals with four sides and four vertices. Both have all four interior angles as right angles (90 degrees). Both have opposite sides that are parallel and equal in length. Both have diagonals that are equal in length and bisect each other.
Differences: In a square, all four sides are equal in length, whereas in a rectangle, only opposite sides are equal (length and width are generally different). The diagonals of a square intersect at a 90° angle, while the diagonals of a rectangle generally do not (unless it is also a square).
Question 2 (Perimeter - Rectangle): A civil engineer is planning to fence a rectangular building site in Abuja. The site measures 35 meters in length and 20 meters in width. Calculate the total length of fencing required.
Solution 2: Concept: The total length of fencing required is the perimeter of the rectangular site.
Formula: Perimeter of a rectangle, P = 2(l + w)
Given: Length (l) = 35 m, Width (w) = 20 m Calculation: P = 2(35 m + 20 m) P = 2(55 m) P = 110 m Answer: The total length of fencing required is 110 meters.
Question 3 (Area - Square): A local government council is renovating a town hall and plans to tile a square section of the floor. Each side of the square section measures 12 meters. What is the area of the section to be tiled?
Solution 3: Concept: The amount of floor surface to be covered by tiles is the area of the square section.
Formula: Area of a square, A = s2 Given: Side (s) = 12 m Calculation: A = 12 m × 12 m A = 144 m2 Answer: The area of the section to be tiled is 144 square meters.
Question 4 (Area - Parallelogram): A farmer in Kano has a farm plot shaped like a parallelogram. The base of the plot measures 50 meters, and its perpendicular height is 25 meters. Calculate the area of the farm plot.
Solution 4: Concept: The amount of land surface available for farming is the area of the parallelogram-shaped plot.
Formula: Area of a parallelogram, A = b × h Given: Base (b) = 50 m, Height (h) = 25 m Calculation: A = 50 m × 25 m A = 1250 m2 Answer: The area of the farm plot is 1250 square meters.
Home Renovation and Construction: Perimeter: Calculating the length of skirting board needed for a room, the amount of decorative trim for a door or window frame, or the length of fencing needed for a compound in Nigerian cities or villages.
Area: Determining the quantity of paint required to cover a wall (e.g., in a house in Lagos), the number of floor tiles needed for a living room (e.g., in an apartment in Enugu), or the amount of carpet to buy for a bedroom. Architects and builders in Nigeria rely heavily on these calculations.
Agriculture and Land Management: Area: Farmers in rural Nigeria (e.g., in Benue State for yam farming or Kano State for groundnut farming) need to calculate the area of their farm plots to estimate crop yield, determine the amount of fertilizer or seeds to purchase, or divide land among family members. Government agricultural extension workers also use area calculations for land surveys and planning.
Local Craft and Industry: Area: Tailors and fashion designers (e.g., making 'Aso-Oke' outfits or 'Ankara' dresses) calculate the area of fabric needed for different parts of a garment to minimize waste. Carpenters measure the area of wood panels for furniture. Local artisans creating patterns on calabashes or weaving mats often subconsciously apply principles of geometric shapes and area when planning their designs.