Lesson 3: Graphs of simple quadratic functions arising ...



Lesson 3: Graphs of simple quadratic functions arising from real-life problems (Year 9)

|Oral and mental starter |10 minutes |

| |Give a brief reminder of the shape of the graph of a simple linear function y = mx + c and the effect of |

|Objectives |changing the parameters m and c. Introduce the idea of “aerobic graphs” and practice with first y = mx |

|Use vocabulary from previous years and |and then y = x + c |

|extend to: quadratic function |(see ) |

| |Use the teacher graphical calculator and whole class display to show the graph of the quadratic function y|

|Vocabulary |= x2 to the whole class. Ask them to sketch the shape of the graph on their white-boards in black |

|linear, quadratic |together with their guess of the shapes of y = 2x2 in red, y = ½ x2 in blue and y = -x2 in green. |

| |Produce the graphs on the teacher graphical calculator and whole class display and compare with their |

|Resources |results. |

|Pupil white boards and coloured pens, |Finish with “aerobic graphs” for y = ax2 |

|teacher’s graphical calculator and whole | |

|class display. | |

|Main teaching |40 minutes |

| |Display a photograph showing an image which may well be fitted by a parabola such as the cables of a |

|Mathematics Objectives |suspension bridge, the span of a cantilever bridge, a flexed ruler, a jet of water from a hose-pipe… e.g.|

|Construct functions arising from real-life |load the file “golden gate.jpg” |

|problems and plot their corresponding |[pic] |

|graphs; interpret graphs arising from real |Discuss possible shapes of the curve – e.g. circular arc, parabola...also distortions due to perspective, |

|situations, including distance–time graphs. |the angle from which the picture is taken etc. |

|Generate points and plot graphs of linear |Discuss ways in which numerical measurements could be made from a photograph and the need to define axes |

|functions (y given implicitly in terms of x)|and origin – e.g. project image on a grid on the whiteboard, mark some points and measure their |

|e.g. ay + bx = c, y + bx + c = 0, on paper |coordinates – or read off pixel coordinates in imaging software such as MS Photo Editor. An excellent |

|and using ICT; given values for m and c, |tool for reading off coordinates from an image can be found at |

|find the gradient of lines given b equations|The free DigitiseImage program allows you to define your own origin and axis scales. |

|of the form y = mx + c. |Discuss how geometric features of imaging software could be used to make the x-axis horizontal – e.g. by |

|ICT Objectives |rotating by -3 degrees (see the files “brittany bridge.jpg” and “brittany bridge tilt.jpg”. |

|Design and create ICT-based models, testing |[pic] |

|and refining rules or procedures. |Now measure coordinates of important points from the image – commenting on appropriate accuracy and units.|

|Use a wide range of ICT independently and |With DigitiseImage you can define O as origin, say, and units on the x- and y-axis of 300 pixels to export|

|efficiently to combine, refine and present |co-ordinate data for the marked points: |

|information by structuring, refining and | |

|synthesising information from a range of |[pic] |

|sources. | |

|Use a range of ICT tools efficiently to |In MS Photo Editor the points marked A, B, D, C have approximate pixel values: (121,95), (122,138), (486, |

|combine, refine and present information by |83), (485, 135). Transform these into approximate pixel displacements from an origin O: |

|extracting, combining and modifying relevant|(-180, 45), (-180, 0), (0, 0), (180, 0), (180, 45) |

|information for specific purposes. |and rescale into simpler units e.g. 1 unit = 45 pixels |

|Use ICT to draft and refine a presentation, |(-4, 1), (-4, 0), (0, 0), (4, 0), (4, 1) |

|including capturing still and moving images |Enter the x- and y-coordinates of A, O and D into the lists of the graphical calculator, draw a scatter |

|(e.g. using a scanner, digital camera, |graph, choose suitable axes and then fit different models of y = ax2 to the data. Project the graphical |

|microphone), and importing and exporting |calculator display to superimpose over the image (or a sketch taken from it). |

|data and information in appropriate formats.|[pic] [pic] |

| | |

| |(An alternative approach is to use dynamic geometry software to superimpose axes and the graph of a |

|Vocabulary |function directly over an image – see The Geometer’s Sketchpad file “Britanny bridge.gsp”.) |

|parabola, quadratic function, axes, origin, |[pic] |

|perspective, translation, scale, vertex, |Now give pupils the pupil sheet from which to measure and record coordinates, draw a scatter graph and fit|

|symmetry, scatter graph, model |a quadratic function. |

| | |

|Resources | |

|PC with data-projector and white-board, | |

|class set of graphical calculators, | |

|teacher’s graphical calculator and whole | |

|class display, resource sheets M3 and H3, | |

|image files. | |

|Plenary |10 minutes |

| |Discuss principal features of parabolas and quadratic functions – symmetry, vertex, axis….Discuss where |

|By the end of the lesson |parabolas may occur in nature – bridge types, arches, water spouts, lenses … |

|pupils should be able to explain how to |Homework: |

|find, calculate and use: |Find good websites with images of bridges or other parabolic objects (e.g. use a search engine with |

|Be able to generate points and plot the |“bridges pictures parabola”…) |

|graphs of simple quadratic or cubic |Or: |

|functions using ICT. |Use a digital camera to take pictures of a flexed ruler on squared paper and read off coordinates. Fit a |

| |quadratic function. |

| |[pic] |

| |(Pupils without access to Internet or digital camera/PC can draw and measure without need for ICT!) |

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