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1 Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic function; etermine the nature of these stationar points; an to iscuss an unerstan these processes. Learners shoul have some knowlege of: ifferentiation of polnomials; fining stationar points of a quaratic function; using or f (x) to etermine their nature. For each learner ou will nee: mini-whiteboar; Sheet 1 Matching. For each small group of learners ou will nee: Car set A Stationar points; large sheet of paper for making a poster; glue stick; felt tip pens. At least 1 hour. Level of challenge: D C5 Fining stationar points of cubic functions C5 1

Learners shoul have some knowlege of: ifferentiation of polnomials; fining stationar points of a quaratic function; using or f (x) to etermine their nature.

2 Level of challenge: D C5 Fining stationar points of cubic functions Suggeste approach Beginning the session Use mini-whiteboars an whole group questioning to revise ifferentiation of polnomials an stationar points of a quaratic function b asking a range of questions such as: If = 4x + 5x 1 what is? If = x 3 8x + 6x 1 what is? Fin the x coorinate of the stationar point of = x 8x + 3. How o ou know whether it is a maximum or a minimum? Explain that, to fin the stationar points on a cubic graph, exactl the same metho is use. You ma like to encourage learners to think about the likel shape of the graph. Working in groups (1) Ask learners to work in pairs. Give out Car set A Stationar points an ask learners to sort the cars into functions. Explain that the cars inclue three cubic functions an all the steps require to fin their stationar points an to etermine the nature of those stationar points. Learners have to sort the cars/steps into an appropriate orer an stick them onto the poster paper in three columns, one for each function. Learners who struggle can be given just one or two functions to sort. Learners who fin the task eas coul be aske to fin examples of cubic functions that have onl one stationar point or no stationar points. You shoul encourage them to relate their answers to the erive quaratic functions. Whole group iscussion When all learners have complete at least one function, iscuss the orer in which the have stuck own the steps. Ask learners to compare their orers with pairs sitting near them an to report an ifferences. Discuss how the orers can be checke an which orers are logicall correct. Working in groups () Ask learners to write a comment besie each step of one of their functions, explaining its purpose. Finall, ask them to sketch a graph of the function, marking the stationar points. C5

How o ou know whether it is a maximum or a minimum? Explain that, to fin the stationar points on a cubic graph, exactl the same metho is use.

3 What learners might o next Further ieas Reviewing an extening the learning Give iniviual learners Sheet 1 Matching an ask them to match the equations to the graphs, justifing their choice as full as the can. You ma wish to ask them to match onl two equations an graphs. Learners coul be aske to explore an then sketch a given cubic function that factorises. This woul connect work using the factor theorem with this session on stationar points. This approach coul be use to fin stationar points of other tpes of functions (e.g. functions with fractional or negative inices). This wa of sorting the orer of a solution can be use for solving multi-stage problems on an topic. Level of challenge: D C5 Fining stationar points of cubic functions C5 3

$This woul connect work using the factor theorem with this session on stationar points. This approach coul be use to fin stationar points of other tpes of functions (e.g. functions with fractional or negative inices).$

4 Level of challenge: D C5 Fining stationar points of cubic functions BLANK PAGE FOR NOTES C5 4

5 C5 Car set A Stationar points (page 1) 3 3 = x 4x + 5x + 11 = x x x + 3 = x 7x 5x + 9 3x 8x + 5 = 0 6x 8 = 5 = 3x 14x 5 C5 Fining stationar points of cubic functions x = 1 3, = K x = 5 3, x = 1 = 3x x 1 6x = x = 1, = K x = 1 3, x = 5 ( x )( x ) = 0 x = 1, = K 3x 14x 5 = 0 3x x 1 = 0 C5 5

of cubic functions x = 1 3, = K x = 5 3, x = 1 = 3x x 1 6x = x = 1, = K x

6 C5 Car set A Stationar points (page ) ( 3x 5)( x 1) = 0 ( x )( x ) x = 5, x = 1 3, = K = 0 = 3x 8x + 5 = K x = 1 3, x = 1 C5 Fining stationar points of cubic functions = 6x 14 x = 5 3, = K Maximum is at Minimum is at Minimum is at Maximum is at Maximum is at Minimum is at C5 6

functions = 6x 14 x = 5 3, = K Maximum is at.......... Minimum is at.......... Minimum is at.......... Maximum is at.......... Maximum is at.......... Minimum is at.......... C5 6

7 C5 Sheet 1 Matching Name: Match these equations to their graph. Give as much evience as ou can to justif our matchings. The graphs are not rawn to scale. Equation 1 Equation 3 3 = x + x 5x = x + 7x + 15x + 1 Equation Equation 4 3 = x 9x + x 10 3 = x x 13x 10 C5 Fining stationar points of cubic functions Graph A Graph B Graph C Graph D C5 7

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level C1 of challenge: D C1 Linking the properties and forms of quadratic of quadratic functions functions Mathematical goals Starting points Materials required Time needed To enable learners to: identif