SHS1 Additional Mathematics · Semester 2, Week 2
Spatial Sense
Full lesson notes coming
Notes for this lesson are being prepared. The curriculum details below are complete and ready to use for your planning.
Curriculum details
- Strand
- Geometric Reasoning and Measurement (Strand 2)
- Sub-strand
- Spatial Sense (2.1)
- Content standard
- 1.2.1.CS.1 - Demonstrate knowledge and understanding of spatial sense in relation to lines and angles between intersecting lines. 1.2.1.LO.1 Describe the properties of lines, including parallel, perpendicular and midpoints. 1.2.1.LO.2 Derive the equation of a line in various forms, find the shortest distance between a point and a line and the perpendicular distance from an external point to a line. 1.2.1.LO.3 Solve problems on acute angles between two intersecting lines. 1.2.1.LO.4 Perform algebraic manipulations of Vectors and resolve vectors using the triangle, parallelogram and polygon laws of addition.
- Indicator
- 1.2.1.LI.6 - Deduce the shortest distance between a point and a line and use the knowledge of intercepts and right-angled triangles to find the perpendicular distance from an external point to a line.
- Suggested placement
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Semester 2, Week 2
(Week 22 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
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The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:
- exemplars - p.152: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
- Curriculum reference
- NaCCA curriculum document, p. 152
Exemplars (from the NaCCA curriculum)
Learning Experience: Investigate the shortest distance between a point and a line as well as the shortest distance between two lines. Real-life activities to measure these distances in the classroom and around the school Activity 1: Use collaborative groups to investigate the shortest distance between a point and a line. - Learners in convenient groups investigate the shortest distance between a point and a line as well as the shortest distance between two lines. - Learners conduct real life activities to measure these distances in the classroom and around the school. - Learners establish that the shortest distance between a line and a point is the perpendicular distance. Activity 2: Using a formula for finding the perpendicular distance. Theorem: The shortest distance (or the perpendicular distance), 𝐷, between the point 𝑃(𝑥 ! , 𝑦 ! ) and |&0 $+ 'A $ $;| the line 𝐿: 𝑎 + 𝑏 + 𝑐 = 0 is given by: 𝐷 = √& ! $' ! - Learners in convenient groups identify that the numerator of the formula is the equation of a line and the denominator is the Pythagoras theorem for finding the length of the hypotenuse. - Learners in groups use the formula to solve examples. Example 1: Find the length of the perpendicular drawn from the point 𝐴(1,9) to the straight line −5𝑥 + 12𝑦 + 13 = 0. Solution: |&0 'A $;| - Recollect the formula as 𝐷 = $+ ! $ ! √& $' - Identify the components of the question. From the question 𝑥 ! = 1, 𝑦 ! = 9, 𝑎 = −5, 𝑏 = 12 and 𝑐 = 13. - Learners substitute these values into the formula and find the solution |*.(!)$!)(2)$!"| !!/ , i.e. 𝐷 = = √. $!) ! ! !" Activity 3: - Learners in groups apply knowledge in the equation of a line and perpendicular distance. - Learners in groups extend the knowledge gained to solve application questions. Example 1: Find the length of the perpendicular drawn from the point 𝐴(−1, −7) to the straight line passing through the points 𝐵(6, −4) and 𝐶(9, −5). - Learners in their convenient groups discuss and discover that they have to first find the equation of the line using the points 𝐵(6, −4) and 𝐶(9, −5). The equation of the line is 𝑥 + 3𝑦 + 6 = 0 - Learners in groups use the equation of the line and the point 𝐴(−1, −7) to find the perpendicular distance. Expected answer 1√!( . Activity 4: - Learners in groups find the distance between Two Parallel Lines. - Learners in groups investigate and extend their knowledge to find the distance between two parallel lines. Note: The distance between two parallel lines can be found as the perpendicular distance between any point on one line and the other line. - Learners in groups discuss and build on what others say to realise that given two parallel lines: - Find one point on one of the lines - Find the distance from this point to the other line Teaching and Learning Resources: - SHS Curriculum - Mathematical set calculators. Assessment (1.2.1.AS.6). The document marks these depth-of-knowledge levels for this indicator: Level 4 Extended critical thinking and reasoning.