Introduction to Differential Calculus Presentation Secondary School Listowel (1)
Transcript of Introduction to Differential Calculus Presentation Secondary School Listowel (1)
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Maths CountsInsights into LessonStudy
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Presentation SecondarySchool: Listowel:
• Jacqueline Normile, Norma Dowlingand Elaine Hickey
• Si!th "ear
• #ssociating deri$ati$es with slo%es o&tangent lines
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Insights into Lesson Study
• (ntroduction: )ocus o& Lesson
• Student Learning : *hat we learned a+outstudents understanding +ased on data
collected• -eaching Strategies: *hat we noticed
a+out our own teaching
•
Strengths . *eaknesses o& ado%ting theLesson Study %rocess
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Introduction• Topic investigated:
Associate derivatives with slopes of tangent lines.• How we planned the lesson:
Maths Team got together and we looked at anddiscussed!lanning lesson !rior knowledge"hat we wanted to teach"hat we wanted the students to know at the
end.• Interested in:
Strategies to improve learning and understanding.•
#esources used:The Teaching and Learning !lan the sylla$uswhite$oards rulers pencils coloured markersactivity sheet and !ower!oint%s.
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• !rior &nowledge:oStudents understand:
• #ate of change and slope are
e'uivalent.• The di(erence $etween the average
rate of change and the instantaneousrate of change.
• Transformation of )uadratic *unctions.
Introduction
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Introduction
• "hy did we choose to focus onthis mathematical area+ –
The class had started the calculussection of the course and thisTeaching and Learning planprovided a solid foundation from
which to continue.
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Introduction
• ,nduring understandings: – Students will $e a$le to analyse this
information and present their -ndingsin mathematical form.
– The aim of this lesson study was forstudents to understand why weassociate derivatives with tangentlines.
– Students should $e a$le to apply theirknowledge and skills to solve pro$lems.
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Learning 4utcomes:
• Students will understand: – The slope function of a 'uadratic
function allows us to predict the slope
of the function at any point. – Slope functions may $e determined $y
eamining the pattern of the slopes of
tangents to the function. – Slope functions may $e determined $y
a simple rule.
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#e/ections on theLesson
• Student Learning : *hat welearned a+out studentsunderstanding +ased on data
collected
• Teaching Strategies: *hat wenoticed a+out our own teaching
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Student Learning
• Data 7ollected &rom the Lesson:
1 #cademic eg sam%les o& students
work
' 8oti$ation
/ Social 9eha$iour
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#cti$ity &rom -.L Plan
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We handed this activity to the class and told them they were to fill in the
table, individually, and see if there was a discernible pattern.
Then they collaborated in pairs and in small groups.
One person then reported from each group on their findings.
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7ommon misconce%tions: 1
• Students &ound the slo%e o& the
secant o& two ad;acent %oints, ratherthan the tangent
See ne!t ' slides
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7ommon misconce%tions: '
• Students &ound the slo%e o& the
secant &rom the origin to the %oint o&interest, rather than the tangent
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7ommon misconce%tions: /
• *hen students used the &ormula,
they took run as a negative distance, rather than a negative mo$ement
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See ne!t slide
Slope =rise
run
Slope = riserun
± ÷
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Student 0nderstanding:1
"hat we learned a$out the waydi(erent students understand thecontent of this topic+
• Students saw two points andfound the slope using these twopoints rather than the tangents.
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Addressing
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AddressingMisconceptions•
#ecommendationsIn future when we teaching rates ofchange we should put more emphasis on
1 -he idea o& a constant rate o& change
' -he secant line is used to
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*urther #ecommendation
• 3efore teaching this lessonagain the students should havegood prior knowledge of
– !atterns – Slopes of linear functions
– #ates of 4hange
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Teaching Strategies
"hat did I notice a$out my ownteaching+
– To $e /uent in the correct mathematical
language. – To $e organised 5 timing resources.
– Increased level of sound.
– "ait time 5 give the students time tocomplete activity and then discussactivity and -ndings.
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Teaching Strategies
"as it di6cult to ask 'uestions toprovoke students% deep thinking+• "e circulated the classroom and listened to
the progress.
• "e encouraged students to ask each other'uestions.
• "e noticed di(erent mem$ers of a group haddi(erent results.
•"e asked them to discuss this with eachother.
• Students had to tease out which piece of workthey thought was correct.
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Teaching Strategies
How did I assess what students knewand understood during the lesson+• A mem$er from each group reported $ack
with their conclusions using the visualiser.
Some )uestioning used during thelesson –
"hy did the group decide that this wasthe $est answer+
– "hat was wrong with the other answers+
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Teaching Strategies
How did I put closure to the lesson+• Having seen eamples of work we asked the
class which piece of work they thought wascorrect.
• "e also asked what were the commonmisconceptions.
• "e then showed a fully correct eample ofthe activity on the visualiser including thecorrect pattern for the slope function.
• "e then graphed the f%89 2 function andasked the class what type of function thiswas.
• The class recognised it was linear.
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Teaching Strategies• Homework:
• ,ach student was asked to -ll in theta$le formulate a pattern i.e. slopefunction and graph the slope
function for di(erent 'uadraticfunctions such as:
• x2 + 1
• x2 – 1
• 2 x2
• 2 x2 + 1
•
2 x2
– 2 /2
u en omewor :
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u en omewor :,ample 1
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f ( x ) = x 2 + 1
u en omewor :
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f(x) = x 2 – 1
u en omewor :,ample 2
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f(x) = 2x 2 + 1
,ample ;
u en omewor :
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f(x) = 2x 2 – 2
u en omewor :,ample <
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– Students used the visualiser to
display their homework – They noticed that all 'uadratic
functions gave rise to linear slopefunctions
– "e asked students could they comeup with a general formula forderiving the slope function $ased
on the homework – "e then introduced the derivative
formula as a simpler method for-nding the slope function.
– =erivative function Slope function01
Ne!t Lesson
> ? ti L St d
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>e?ections on Lesson StudyProcess• >reatly improved
communication colla$orationand collegiality $etween
mem$ers of the Maths Team.• Increased my con-dence.
• Increased my knowledge of more
varied teaching methodologies.
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>e?ections on Lesson Study
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>e?ections on Lesson StudyProcess
#ecommendations as to how LessonStudy could $e integrated into aschool contet. –
"ritten lesson studies are very timeconsuming.
– "e found it to $e a great learningeperience.
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It has thought us the importance ofsitting down and talking a$out thecontent of a topic and how to teach it 5even for -ve minutes.
– Sharing of la$our.
– The im ortance of a common a roach.0/
P t ti S d S h l
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Presentation Secondary School:Listowel:
Thank ?ou
@ac'ueline 7ormile 7orma =owling and
,laine Hickey.