SHS2 Mathematics · Semester 1 Week 9: Proportional Reasoning

Lesson notes

Learning Objectives

Indicator: 2.1.2.LI.2 - Establish the relationships among ratio, rates and proportions.

By the end of the lesson, learners can:

Define ratio, rate and proportion in their own words and state at least two similarities and two differences among them.
Verify the cross-products property of proportions (if a : b = c : d, then ad = bc) using at least three numerical examples.
Solve for an unknown term in a proportion when any three of the four numbers are known, using the cross-multiplication method.
Use the proportion property of corresponding sides of similar triangles to find missing side lengths.
Justify their solution steps orally or in writing, explaining why the cross-products property works for any proportion.

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Curriculum details

Strand: Numbers for Everyday Life (Strand 1)
Sub-strand: Proportional Reasoning (1.2)
Content standard: 2.1.2.CS.1 - Demonstrate knowledge and understanding ratios, rates and proportions and use it to solve real-world problems. 2.1.2.LO.1 Establish the similarities among ratios, rates and proportions and use these to solve problems.
Indicator: 2.1.2.LI.2 - Establish the relationships among ratio, rates and proportions.
Suggested placement: Semester 1, Week 9 (Week 9 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.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

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Curriculum reference: NaCCA curriculum document, p. 181

Exemplars (from the NaCCA curriculum)

- Using Collaborative Learning, initiate small group discourse on additional fractional quantities. Ensure participation by all and the correct language usage. Champion the course for respecting each other's views and contributions even if they differ from yours.
Example: Cross Products Investigate the cross-products of ratios and make generalisations. If a : c then the cross products 𝒂𝒂𝒂𝒂 and 𝒃𝒃𝒃𝒃 are equal. Also a : b b d c d
- In mixed groupings, learners investigate four numbers in a proportion, to demonstrate that if any three of these numbers are known, the fourth can be found.
Example: i. Solve for x in the proportion 36 : 7 x 3 ii. Solve the equation a − 2 : a + 1 5 3 iii. Similar triangles are said to be similar if they have the same shape (but not necessarily the same size). Similar triangles have sides that are proportional.
For example, the figure shows two simi
Figure: Similar triangles.
3 3 6 3 4.5 3 Notice that the ratios of the corresponding sides are all equal = , = , = 2 2 4 2 3 2
Extend the application of ratios to angles, Pythagoras's theorem and some algebra application of rates to speed: distance-time graphs.
Teaching and Learning Resources:
- * Geodot * Rubber bands * Computer
- * GeoGebra * Google search * Mobile phone
- * Calculator * YouTube videos, etc. * Geodot,
- * Rubber bands * Cardboards * Measuring instruments
Assessment (2.1.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 4 Extended critical thinking and reasoning.