SHS1 Additional Mathematics · Semester 2 Week 3: Spatial Sense

Lesson notes

Learning Objectives

Indicator: 1.2.1.LI.8 - Apply knowledge of gradient and the tangent function to find the Acute angle between two intersecting lines.

By the end of the lesson, learners can:

Recall the formula for the gradient of a straight line and state the tangent formula for the acute angle between two intersecting lines, given as tan θ = |(m₁ - m₂)/(1 + m₁m₂)|, where m₁m₂ ≠ -1.
Apply the tangent formula to find the acute angle between two intersecting lines when the gradients of both lines are given.
Rearrange the equation of a straight line into the form y = mx + c to extract the gradient, then use it to find the acute angle between the two lines.
Identify whether two lines are parallel (angle 0°), perpendicular (angle 90°), or neither, by examining the product of their gradients before applying the formula.
Interpret the result of the tangent formula, using the absolute value to guarantee the acute (smaller) angle between the lines, and round answers to two decimal places as required.

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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.8 - Apply knowledge of gradient and the tangent function to find the Acute angle between two intersecting lines.
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.
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. 155

Exemplars (from the NaCCA curriculum)

Learning Experience: Learners in a mixed-ability group are led by a leader to find the acute angle between two intersecting lines.
Activity 1:
- Use collaborative learning to find the acute angle between two intersecting lines.
- The leader of the mixed-ability groups leads by discussing with group members to recollect the slope of lines.
- Learners in groups establish that an angle 𝜃 between two nonparallel lines in the coordinate # $ *# ! plane with slopes 𝑚 ! and 𝑚 ) , such that 𝑚 ! 𝑚 ) ≠ −1, is given by 𝑡 = !$# . # $ !
Activity 2:
- In jigsaw groups, use the formula to find the acute angle between two lines, given their gradient.
- Learners in groups solve examples and share the findings with the whole class.
! Example 1: Determine the acute angle between two straight lines having slopes of 5 and to two , decimal places.
Solution: 𝑚 ! − 𝑚 ) 𝑡 = 1 + 𝑚 ! 𝑚 )
1 5 − i− 4j 𝑡 = 1 1 + (5) i 4j
19 𝑡 = 9
19 𝜃 = 𝑡 *! £ ¤ 9 𝜃 = 62.65°
Activity 3: In collaborative groups, apply and Extension of knowledge. Learners in their groups find the acute angle between two lines given the equation on the lines.
Teaching and Learning Resources:
- SHS Curriculum
- Mathematical set calculators.
Assessment (1.2.1.AS.8). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.