Spatial Sense · SHS2 Mathematics

Lesson notes

Learning Objectives

Indicator: 2.3.1.LI.4 - Carry out an enlargement of a plane shape given a scale factor.

By the end of the lesson, learners can:

Define enlargement and reduction in their own words, distinguishing between the two based on the scale factor.
State the properties of enlargement, including the relationship between the object and image (similarity) and the meaning of scale factor k for k > 1, 0 < k < 1, k = 1, and k < 0.
Construct the image of a given plane shape (triangle or quadrilateral) under an enlargement, given a centre of enlargement and a positive scale factor, using a ruler and a pair of compasses.
Determine the coordinates of the image of a shape on a Cartesian plane when the centre of enlargement is the origin (0, 0) and the scale factor is a positive integer.
Draw both the object and its image on a graph sheet, verifying that corresponding sides are in the ratio of the scale factor.

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Curriculum details

Strand: Geometry Around Us (Strand 3)
Sub-strand: Spatial Sense (3.1)
Content standard: 2.3.1.CS.1 - Demonstrate a conceptual understanding of spatial sense regarding changes and invariance achieved by performing a combination of successive transformations (reflection, translation, rotation) in a 2D shape. 2.3.1.LO.1 Carry out a variety of transformations through translation, reflection, rotation and enlargement of plane shapes and identify scale drawing as an enlargement/reduction of a plane shape.
Indicator: 2.3.1.LI.4 - Carry out an enlargement of a plane shape given a scale factor.
Suggested placement: Semester 2, Week 3 (Week 23 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Curriculum reference: NaCCA curriculum document, p. 225

Exemplars (from the NaCCA curriculum)

Think-pair share activities: Learners in pairs research and make presentations on the concept of enlargement/reduction in transformations.
Example: In transformation, enlargement/reduction is when the size of an object is changed without changing its original shape. When the size of the object is increased, it is called an enlargement, and when the size of an object is decreased, it is called a reduction. Look at the images here.

An apple and a triangle each reduced along rays from a centre O, object and image labelled. — A reduction drawn twice. On the left a large apple labelled "Image" and a small apple labelled "Object" lie on rays from a point marked "O (Centre)"; on the right the same idea is drawn with a large triangle ABC and its smaller image A prime B prime C prime on rays from O.

Enlargement Reduction
The enlargement is made with the help of a fixed point called the centre of enlargement and by the fixed ratio called scale factor, i.e., the ratio of the corresponding sides of the image and object.
Think-pair-share activities: Learners in pairs discuss the properties of enlargement/reduction. And, as learners talk about properties and rules of enlargement/reduction, encourage discussions on the need to obey rules and the consequences of doing otherwise in everyday life.
Example 1: Properties of enlargement/reduction 1. The object and the image under the enlargement are similar. 2. Scale factor (k) =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑚𝑎𝑔𝑒 𝑓𝑖𝑔𝑢𝑟𝑒 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡 𝑓𝑖𝑔𝑢𝑟𝑒
3. If the scale factor k>1, then the transformation is called enlargement. 4. If the scale factor 0<k<1, then the transformation is called reduction. 5. If the scale factor k=1, then the transformation is identity. 6. If the scale factor k<0, then the image will be on the opposite side of the object from the centre of enlargement.
Example 2: Find the image of ΔABC under the enlargement with the centre of enlargement O and scale factor 2. [It is denoted by E(O, 2)]

Blue right-angled triangle ABC with a separate point O marked below it as the centre of enlargement. — A blue right-angled triangle ABC, with A at the top left, B at the foot left and C at the foot right, and a separate point marked O below and to the left as the centre of enlargement.

Solution: For the image ΔABC under the given enlargement, perform the following steps: Step 1: Join OA, OB and OC. Step 2: Produce OA up to A' such that OA' = 2OA. Step 3: Produce OB up to B' such that OB' = 2OB. Step 4: Produce OC up to C' such that OC' = 2OC. Step 5: Join A'B', B'C' and C'A'.

Worked enlargement with a table mapping A(1,2) B(1,1) C(2,1) D(2,2) to A'(4,8) B'(4,4) C'(8,4) D'(8,8). — A worked enlargement. A faint triangle diagram sits above the text "Hence, triangle A prime B prime C prime is the required image of triangle ABC under the enlargement E(O, 2)", and a coloured table of the next example maps the preimage A(1, 2), B(1, 1), C(2, 1) and D(2, 2) to the image A prime (4, 8), B prime (4, 4), C prime (8, 4) and D prime (8, 8).

Hence, ΔA'B'C' is the required image of ΔABC under the enlargement E(O, 2).
Example 3: Find the coordinates of the vertices of the image square ABCD and draw ABCD and its image on the same graph paper if a shape ABCD with vertices A(1, 2), B(1, 1), C(2, 1) and D(2, 2) is enlarged with the centre of enlargement O(0, 0) and scale factor 4.
Solution: First, obtain the coordinates for the image as:
Preimage Image
A(1,2) A 1 (4, 8) B(1,1) B 1 (4, 4) C(2,1) C 1 (8, 4) D(2,2) D 1 (8, 8)

A small square on a grid enlarged from the origin to a square with vertices (4,8), (8,8), (4,4) and (8,4). — A coordinate grid showing a small pink square with vertices near (1, 1) and (2, 2) enlarged from the origin to a large blue square with vertices A(4, 8), D(8, 8), B(4, 4) and C(8, 4), the corresponding vertices joined by dashed rays. A Teaching and Learning Resources row listing mathematical sets, graph sheets, computers, GeoGebra, a compass and a clock face runs along the foot.

Teaching and Learning Resources:
- Mathematical sets. Graph sheet.
- Technology tools such as computers, mobile phones, etc.
- Computer software applications like
- GeoGebra.
- Compass
- clock face, etc.
Assessment (2.3.1.AS.4). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.