SHS3 Additional Mathematics · Semester 1, Week 12
Spatial Reasoning
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 Reasoning (2.1)
- Content standard
- 3.2.1.CS.1 - Demonstrate an understanding of Parabola and its properties. 3.2.1.LO.1 Construct a parabola of a given quadratic equation (𝑦 = 𝑎𝑥 ) + 𝑏 + 𝑐) and explain its key features 3.2.1.LO.2 Sketch a parabola and use it to deduce the relation 𝑦 ) = 4𝑎 3.2.1.LO.3 Sketch a parabola given the directrix Communication: Provide learners the opportunity to engage and and focus. 3.2.1.LO.4 Deduce the equation of the tangent and normal to a parabola
- Indicator
- 3.2.1.LI.5 - Apply the formula of distance between two points to find the general equation of a parabola.
- Suggested placement
-
Semester 1, Week 12
(Week 12 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:
- exemplars - p.485: 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. 485
Exemplars (from the NaCCA curriculum)
Experiential Learning, Talk for Learning, Group work, Building on what others say, and Project-based Learning. Learning Experience: Learners in mixed-ability groups derive the standard equations of a parabola with the vertex at the origin. Activity 1: Drawing Parabola Learners in mixed-ability draw a parabola and label the coordinates of the focus and directrix.
Activity 2 Algebraic expression of a parabola with vertex at the origin. Learners in mixed-ability groups use the well-labelled diagram and apply their knowledge of the distance of a line to express a parabola algebraically. Distance formula is given as w(𝑥 ) − 𝑥 ! ) ) + (𝑦 ) − 𝑦 ! ) ) Using the diagram, the relationship between the focus and the directrix implies that |𝑃| = |𝑃| ∴ w(𝑥 − 𝑎) ) + (𝑦 − 0) ) = w(𝑥 + 𝑎) ) + (𝑦 − 𝑦) ) Activity 3 Equation of a parabola (𝑦 ) = 4𝑎) In mixed-ability groups, learners work to resolve the expression of the relationship between directrix and focus (𝑎, 0) to obtain the equation of a parabola that is 𝑦 ) = 4𝑎. Activity 4 Deriving standard equations of a parabola with vertex at origin. Learners in their groups brainstorm and share ideas on how to derive the other three standard equations of a parabola and state their observations. Learners observe that: - If the equation has the term with 𝑦 ) , then the axis of symmetry is along the 𝑥 − 𝑎, and if the equation has the term with 𝑥 ) , then the axis of symmetry is along the y-axis. - When the axis of symmetry is along the 𝑥 − 𝑎, the parabola opens to the right if the coefficient of the 𝑥 is positive and opens to the left if the coefficient of 𝑥 is negative. - When the axis of symmetry is along the 𝑦 − 𝑎, the parabola opens upwards if the coefficient of 𝑦 is positive and opens downwards if the coefficient of 𝑦 is negative.
Activity 5 Deriving standard equations of a parabola with vertex not at the origin. Learners in their mixed-ability groups brainstorm and explore the equation of a parabola with the vertex not at the origin. Learners establish that: if the vertex of the parabola is not at the origin, then it is translated. if a parabola is translated ℎ units horizontally and 𝑘 units vertically, the vertex will be (ℎ, 𝑘). this translation will cause 𝑥 to be replaced with (𝑥 − ℎ) and 𝑦 replaced with (𝑦 − 𝑘) in the standard form of the equation.
Activity 6: Learners in groups brainstorm and write the equation of a parabola given the vertex and focus. Example: Find the equation of a parabola with a focus at (0, 4) and vertex at (0, 0). Solution - Learners observe that the given vertex at (0, 0) and focus at (0, 4) implies that the parabola opens up on the y - axis. Therefore, the equation of the parabola is given by 𝑥 ) = 4𝑎 Distance between the vertex and focus |𝑎| = w(0 − 0) ) + (0 − 4) ) |𝑎| = 4 ) If the equation of the parabola is 𝑥 = 4𝑎 Then 𝑥 ) = 4(4)𝑦 𝑥 ) = 16𝑦 Activity 7 Modelling real life problems on parabolas Learners in groups create and pose problems on the derivation of equations of parabolas given the vertex and focus. Teaching and Learning Resources: - SHS Curriculum, Graph boards, mathematical set, ICT tools Assessment (3.2.1.AS.5). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.