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Linear Algebra

Linear Algebra. 1.4. This is about . writing equations, Drawing straight line graphs Solving simultaneously Substituting in values into equations Writing about why and what you are doing. Keep this in mind for achieved.

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Linear Algebra

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  1. Linear Algebra 1.4

  2. This is about • writing equations, • Drawing straight line graphs • Solving simultaneously • Substituting in values into equations • Writing about whyand whatyou are doing

  3. Keep this in mind for achieved • Students will demonstrate competency in at least three different methods. • This means • Writing equations • Drawing graphs • Simultaneous equations • Substitution

  4. Examples of Achieved answers: • (graphing linear models) • students have correctly drawn a distance/time graph for one competitor • (using formulae) • students have found the average speed for at least two different events • (simultaneous equations) • students have read from their graphs at least two distances where the competitors cross

  5. Merit level • This will mean at least one of the following: • selecting and carrying out a logical sequence of steps; • forming and using a model; • connecting different concepts and representations, • demonstrating understanding of concepts.

  6. Student has connected their knowledge of graphs, rates of change (speed) and simultaneous equations (passing times) to write a commentary for the race, including key information such as: average speed over different events, times when competitors swapped positions and concluding with who finished first; and has communicated their thinking using appropriate mathematical statements. • Distance/time graph for both competitors (allow for minor error) has been drawn • The graph allows passing times to be read within 5 minutes • Average speeds for at least two events for both competitors have been calculated.

  7. Excellence • This will mean at least one of • demonstrating understanding of abstract concepts, • developing a chain of logical reasoning, forming a generalisation. • This will also mean using correct mathematical statements • or communicating mathematical insight.

  8. student has demonstrated understanding of abstract concepts to write an accurate commentary for the race, including key information such as: average speed for the race, exact times when competitors swapped positions, who came first and has suggested a valid strategy for the loser of the race to improve their time; and has used correct mathematical statements. • Distance/time graph for both competitors (allow for minor error) has been drawn. • Event sections converted into equations • Simultaneous equations used to find passing times

  9. LOOK AT THE INFORMATION GIVEN

  10. First nominate times from the table to be your times for the three events. You will use this information in completing the task.

  11. Swimming

  12. Equations

  13. Choose to be ‘somewhat athletic’

  14. Your friend starts two minutes after you in the swim event. They swim as fast as they can to make up for lost time and finish the swim event two minutes ahead of you.

  15. Add this to your graph • They have the points (2, 0) and (28,750) on their graph. So their gradient is

  16. Add this to your graph • They have the points (2, 0) and (28,750) on their graph. So their gradient is • And the general equation of their graph is

  17. Add this to your graph • Substitute (2, 0) to find the ‘c’ value

  18. Add this to your graph • Equation is

  19. Note where your friend passes you.

  20. Can now do a simultaneous equation Our graph agrees with this

  21. Biking

  22. Biking Choose to be the non-athletic one as 60 divides into 18000 well

  23. Biking Choose to be the non-athletic one as 60 divides into 18000 well

  24. Your graph

  25. Unfortunately for them, the swim tires them out and they bike on average 10 metres per minute slower than you.

  26. Unfortunately for them, the swim tires them out and they bike on average 10 metres per minute slower than you. • You are travelling at 300m/min so they are travelling at 290m/min but they have 2 mins start on you so they start at 580 m. • Their equation is

  27. It is hard to tell where they pass

  28. Simultaneous

  29. Check with the graph

  30. Choose ‘somewhat athletic’

  31. Run • You both complete the run in the same time. • This means they take 25 mins and win the race by 2 mins. • Now comment on the total times for both competitors and look for a strategy for the loser.

  32. You then need to suggest suitable new target times for the loser, and a strategy, so that they can work towards improving their time and placing in relation to the person they have just lost against.

  33. They need to make up 2 mins so suggest they make them up in their ‘somewhat athletic’ categories improving by 2 minutes in each category. • The new times would be 28mins for the swim and 23 mins for the run or they could improve their bike riding by 4 mins. • You could give equations for this.

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