Mathematics — Similarity and congruency

Form 2 · 40 minutes · Term 2 week 8 · 26 June 2026

SchoolKitwe boys secondary school

TeacherClement kasongola

Date26 Jun 2026

ClassForm 2

SubjectMathematics

Duration40 min

TimePeriod 2—08-10--10-10

Learners43

Term/WeekTerm 2 week 8

Topic

Similarity and congruency

Sub-Topic

Similar and congruent figures

General Competences

Analytical Thinking, Problem Solving, Critical Thinking

Specific Competences

Use the concepts of similarity and congruency in real contexts

Learning Activities

Exploring things in the surrounding Exploring the concepts of similarity and congruency

Expected Standard

The concepts and similarity and congruency used in real contexts accordingly

References

Excel and advance in mathematics Mathematics II syllabus form 1-4

Lesson Goal

By the end of this lesson, learners should be able to identify and differentiate between similar and congruent figures in real-life contexts and apply the concepts to solve problems involving scale and shape comparison.

Rationale

This lesson on similarity and congruency builds directly on learners’ prior knowledge of basic 2D shapes, properties of angles and sides, and informal comparison of objects from earlier grades; it is deliberately designed to develop the competences of Analytical Thinking, Problem Solving, and Critical Thinking. Understanding these two geometric relationships is essential in everyday Zambian contexts such as interpreting maps, reading architectural blueprints, matching chitenge patterns, and scaling objects in crafts and construction. Through an active, learner-centred approach that includes exploring objects in the surrounding environment and collaborative group discussions, learners will construct meaning for themselves and then formally articulate definitions and applications of similarity and congruency.

Prior Knowledge / Prerequisite Knowledge

Learners already know the properties of basic 2D shapes (triangles, quadrilaterals, circles) from Grade 7 and Form 1 mathematics. They can measure lengths using a ruler and identify whether two shapes are the same or different in appearance. They also have an intuitive understanding of “same shape” and “same size” from everyday experiences. At the start of the lesson, the teacher will activate this prior knowledge by showing pairs of familiar objects (two identical books, two leaves of different sizes) and asking learners to describe whether they are the same shape, same size, or both, thereby creating a natural bridge to the formal concepts of similarity and congruency.

Learning Environment

Natural Environment: The lesson draws on the immediate surroundings of the school compound. Learners will collect pairs of leaves, stones, or sticks that have the same shape (similar) and pairs that are identical in shape and size (congruent). The teacher references these natural objects throughout the lesson to ground abstract concepts in tangible experience.
Artificial Environment: The classroom is arranged in mixed groups of 4–6 learners with tables pushed together to facilitate collaboration. A display board at the front features a teacher-prepared chart showing pairs of similar and congruent geometric figures, including examples from Zambian patterns (e.g., rectangles in a chitenge design, triangles in a roof truss). The board will be used during the EXPLAIN and ELABORATE phases to anchor discussions.
Technological Environment: A projector (if available) can display real-life photographs of objects that illustrate similarity (different-sized photographs of the same building) and congruency (identical classroom desks). Alternatively, the teacher uses printed images from old magazines or newspapers as a low-tech substitute.

Teaching and Learning Materials / Resources

Chart displaying pairs of similar and congruent figures (teacher-prepared, using coloured card and markers)
Cut-out cardboard shapes in different sizes: equilateral triangles (2 cm, 4 cm, 6 cm sides), squares (2 cm, 4 cm), rectangles (2×3 cm, 4×6 cm)
Natural objects: leaves, stones, sticks (collected by learners during the EXPLORE phase if time permits, or pre-collected by teacher)
Rulers and string for measuring sides and comparing sizes
Worksheet: “Exploring Similarity and Congruency in Our Surroundings” (one per group)
Exercise books and pencils
Homework slip with two examples for learners to find at home

Cross-Cutting Issues

The following cross-cutting issues are integrated into this lesson on “Similar and congruent figures”:

Environmental Sustainability: During the EXPLORE phase, learners collect natural objects (leaves, stones) from the school compound. The teacher explicitly reminds learners to take only fallen objects and not to damage plants, fostering respect for the environment. The discussion of scale and proportion connects to how natural patterns (e.g., leaf venation) exhibit similarity, linking geometry to environmental awareness.
Social and Emotional Learning: The lesson is structured around mixed-group collaboration, requiring learners to share materials, listen to peers’ ideas, and reach consensus when classifying objects as similar or congruent. This develops cooperation, empathy, and positive communication. During the ELABORATE phase, learners have the opportunity to present their findings, building confidence and self-esteem.

Lesson Progression (Model: 5E Model of Instruction)

Phase	Teacher Activities	Learner Activities	Assessment Criteria
INTRODUCTION
ENGAGE 6 min	Hook: Hold up two identical wooden rulers (same brand, same length) and two different-sized leaves. Ask: “Look carefully at these pairs. What do you notice about their shapes? Are the rulers the same shape and the same size? What about the leaves – same shape, different size?” Prior Knowledge Questions: Ask: “How can we tell if two shapes are exactly the same? What does ‘same shape’ mean to you? Have you ever seen two things that looked the same but were different sizes – like a big photograph and a small photograph of the same person?” Lesson Goal: Say: “Today we are going to learn two important words – similar and congruent – that describe when figures have the same shape or the same shape and size. By the end of this lesson, you will be able to tell similar figures from congruent figures and use these ideas in real life.”	Learners respond to the hook by describing what they see – some may say the rulers look identical, the leaves have the same shape but different sizes. (Develops: Analytical Thinking) Learners answer prior knowledge questions, sharing experiences of seeing enlarged photographs or matching coins. (Develops: Critical Thinking) Learners listen to the lesson goal and one or two volunteers restate it in their own words. (Develops: Communication)	Learners accurately describe key similarities and differences between the pairs shown. Learners confidently relate prior experiences to the concepts of same shape and same size. Learners clearly restate the lesson goal in their own words.
DEVELOPMENT
EXPLORE 8 min	Learning Activities Based on Teacher Preferences: This phase focuses on Exploring things in the surrounding and Exploring the concepts of similarity and congruency. Investigation Setup: Say: “Now you will become explorers! In your mixed groups, take a few minutes to walk around the classroom or the area just outside. Find two objects that are the same shape but different sizes, and two objects that are exactly the same size and shape. Use the rulers and string provided to measure sides or lengths if needed. Write down what you find on your group worksheet.” [Distribute worksheet.] Facilitation: Move among groups, asking guiding questions: “Why do you think those two stones are similar? Are they exactly the same size? How can you check? What if we rotate one shape – does it still look the same?” Observation: Note which groups are quickly identifying pairs and which may need a nudge (e.g., suggesting specific objects to compare).	Learners collect objects from the surroundings (leaves, stones, books, tiles) and compare their shapes and sizes. (Develops: Analytical Thinking, Problem Solving) Learners use rulers/string to measure sides and compare purposes of objects to determine if they are same shape only or same shape and size. (Develops: Analytical Thinking) Learners discuss in groups and record findings on the worksheet, classifying under “Similar” or “Congruent” based on their observations. (Develops: Collaboration, Critical Thinking) Learners write a short note summarising what they discovered about the difference between similar and congruent. (Develops: Communication)	Learners correctly identify at least one pair of similar and one pair of congruent objects from the surroundings. Learners accurately measure and compare lengths of sides or objects to support their classification. Learners appropriately justify why they placed each pair under “Similar” or “Congruent”. Learners clearly write a summary of the difference in their own words.
EXPLAIN 8 min	Learner Sharing: Invite 2–3 groups to share one example each of a similar pair and a congruent pair. Ask: “What did you discover? How did you decide whether the pair was similar or congruent?” Formalisation: On the board, write the definitions: “Congruent figures have exactly the same shape and same size. Similar figures have the same shape but may be different sizes – their corresponding sides are in proportion.” Draw two pairs of triangles as examples. Worked Example: Draw two triangles: ΔABC with sides 3 cm, 4 cm, 5 cm; ΔDEF with sides 6 cm, 8 cm, 10 cm. Say: “Check if these triangles are congruent or similar. The sides of ΔDEF are twice the sides of ΔABC, so they are similar because the shape is the same but sizes differ. They are not congruent because the sizes are not the same.” Work through the ratio step by step. Guided Questions: (1) Identify whether the given pairs are similar, congruent, or neither. (2) Fill in the blank: Two figures are ________ if they have the same shape and size. (3) Explain: A rectangle measuring 2 cm by 3 cm and another measuring 4 cm by 6 cm – are they similar? Why?	Learners present their group findings, using terms “same shape” and “same size” to describe similarity and congruency. (Develops: Communication, Critical Thinking) Learners copy the formal definitions and examples into their exercise books. (Develops: Digital Literacy – note-taking) Learners follow the worked example, checking ratios and concluding similarity. (Develops: Analytical Thinking) Learners answer the three guided questions individually and then peer-check answers. (Develops: Problem Solving, Collaboration)	Learners confidently explain their group findings using correct terminology. Learners accurately copy definitions and correctly label examples. Learners correctly apply the ratio method to determine similarity in the worked example. Learners accurately answer guided questions and self-correct through peer discussion.
ELABORATE 12 min	New Task: Project or write on the board: “At the Lusaka market, a tailor has two rectangular pieces of chitenge. One is 40 cm by 60 cm. The other is 80 cm by 120 cm. Are the rectangles similar? Show your working. Extension: A map of Zambia has a scale of 1:5,000,000. The distance between Lusaka and Ndola on the map is 5 cm. What is the actual distance? Is the map similar to the real land area? Explain.” Instructions: “Work in pairs. For the first task, calculate the ratio of corresponding sides and decide if the rectangles are similar. For the extension, if you finish early, attempt the map question and write your reasoning.” Facilitation: Walk around, asking probing questions: “What must be true about the side ratios for two rectangles to be similar? Does the map scale tell you anything about similarity?” Extension: Provide the map scale question for learners who finish quickly; otherwise, save for discussion.	Learners read the tailor problem and identify the need to compare side ratios. (Develops: Problem Solving, Analytical Thinking) Learners work in pairs to calculate 40:80 = 0.5 and 60:120 = 0.5, concluding the rectangles are similar because the ratios are equal. (Develops: Critical Thinking, Collaboration) Learners write the working and conclusion in their exercise books. (Develops: Communication) Learners (fast finishers) attempt the map scale question and present their reasoning to the class during the share-back. (Develops: Analytical Thinking)	Learners accurately identify the need to compare side ratios for similarity. Learners correctly calculate the ratios and justify similarity or congruency in the context. Learners independently write a clear solution with steps and final answer. Learners logically justify their reasoning on the map application.
CONCLUSION
EVALUATE 6 min	Consolidation Questions: Ask: (1) “What is the difference between similar and congruent figures?” (2) “Are all squares similar? Why?” (3) “If two triangles have all sides equal, are they necessarily congruent? Explain.” Learner-Led Summary: Invite two learners to say one thing they learned about similarity and one about congruency. Affirm correct ideas and clarify any remaining confusion. Link Forward: “Tomorrow we will use similarity to solve problems with scale drawings and maps – you’ve already seen how it works with the chitenge fabric!” Homework: “At home, find two pairs of similar figures and two pairs of congruent figures (e.g., tiles, furniture, utensils). Draw or describe them in your exercise book and explain why they are similar or congruent. Due next lesson.” Closure: “Well done, everyone! You have explored the environment and discovered how mathematics describes the world around us. Keep looking for patterns!”	Learners answer consolidation questions individually or as a class, demonstrating understanding of key differences. (Develops: Critical Thinking, Communication) Learners summarise key learning points in their own words when called upon. (Develops: Communication) Learners record the homework task in their exercise books. (Develops: Digital Literacy – self-organisation) Learners reflect briefly by turning to a neighbour and saying one thing they found interesting. (Develops: Social and Emotional Learning)	Learners accurately explain the difference between similarity and congruency using examples. Learners confidently state at least one key point from the lesson. Learners correctly note the homework task and due date. Learners attentively share a reflection with a peer.

Class Exercise

Instructions to learners: Answer all five questions in your exercise book. Show all working where applicable. Time allowed: 10 minutes.

Multiple Choice
Two figures are congruent if they have:
A) The same shape but different sizes
B) The same size but different shapes
C) Exactly the same shape and size
D) Different shapes and sizes
Fill-in-the-Blank
Two squares of different side lengths are ________ because they have the same shape but different sizes.
Short Answer
A rectangle is 5 cm by 8 cm. Another rectangle is 10 cm by 16 cm. Are these two rectangles similar? Explain your reasoning.
Problem-Solving
A photograph is 12 cm by 18 cm. An enlargement is made so that the longer side becomes 36 cm. What will be the length of the shorter side if the photograph and enlargement are similar? Show all working.
Application
In a Zambian village, a traditional hut has a circular base with a diameter of 4 m. A model of the hut for a school project is drawn with a diameter of 20 cm. Is the model similar to the actual hut? If the ratio of the model to the actual hut is 1:20, and the actual height of the hut is 3 m, what is the height of the model? Explain your reasoning.

Answer Key (For Teacher Use Only)

Question 1 — Multiple Choice
Answer: C) Exactly the same shape and size
Marks: 1 mark for correct option letter.
Question 2 — Fill-in-the-Blank
Answer: similar
Marks: 1 mark for correct word.
Question 3 — Short Answer
Answer: Yes, the rectangles are similar because the ratio of corresponding sides is equal: 5:10 = 0.5 and 8:16 = 0.5. They have the same shape but different sizes.
Marks: 1 mark for correct conclusion (yes), 1 mark for showing ratio calculation, 1 mark for explanation of shape same but size different. Total 3 marks.
Question 4 — Problem-Solving
Answer: Scale factor = 36 ÷ 18 = 2. So shorter side = 12 × 2 = 24 cm. Alternatively, set up proportion: 12/18 = x/36 → x = (12×36)/18 = 24 cm.
Marks: 1 mark for identifying scale factor, 1 mark for correct multiplication, 1 mark for final answer 24 cm. Total 3 marks.
Question 5 — Application
Answer: Yes, the model is similar to the actual hut because the shape is the same (a circle) and all dimensions are scaled by the same ratio (1:20). Height of model = 3 m × (20 cm/4 m) but careful with units: Convert 4 m = 400 cm, so scale factor = 20/400 = 1/20. Height of model = 3 m × (1/20) = 0.15 m = 15 cm. OR using ratio: actual height 300 cm, model height = 300 × (20/400) = 15 cm. Also accept direct proportion: model height = (20/400)×300 = 15 cm. Explanation includes mention of same shape and proportional dimensions.
Marks: 1 mark for concluding similarity (yes), 1 mark for correct reasoning (same shape, proportional), 1 mark for correct calculation of model height (15 cm), 1 mark for clear explanation with units. Total 4 marks.

Total Marks: 12 | Suggested Completion Time: 10 minutes

Lesson Evaluation

Instructions to the teacher: Focus on the competences learners were able to demonstrate, use clear evidence from their work and participation, note any difficulties they faced, reflect on what worked or did not work in your teaching, and state what support or next steps are needed.

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