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

• Students will demonstrate competency in at least three different methods.

• This means • Writing equations• Drawing graphs• Simultaneous equations• Substitution

• 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

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.

• 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.

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.

• 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

LOOK AT THE INFORMATION GIVEN

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

task.

Swimming

Equations

Choose to be ‘somewhat athletic’

• 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.

Add this to your graph

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

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

Add this to your graph

• Substitute (2, 0) to find the ‘c’ value

Add this to your graph

• Equation is

Note where your friend passes you.

Can now do a simultaneous equation

Our graph agrees with this

Biking

Biking

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

Biking

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

Your graph

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

• 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

It is hard to tell where they pass

Simultaneous

Check with the graph

Choose ‘somewhat athletic’

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.

• 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.

• 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.