Flavian Brian Fernandez (06)25 October 2005
QCM 520 (TG11)
BrightsideSecondary School
Lesson Plan
Subject:Elementary Mathematics
Level:Secondary 2 (Express)
Duration:70 minutes
Topic:Solving Simultaneous Equations
Subtopic(s):i)Using the Elimination Method
ii)Using the Substitution Method
Location:Computer Laboratory equipped with whiteboard and computer terminals with Internet access for the teacher and students.
Pre-requisite Knowledge:
Prior to this lesson, students should be able to:
i)distinguish the difference between an algebraic expression and an algebraic equation
ii)differentiate between variables and constant
iii)manipulate an equation so as to make an unknown variable the subject of the equation
iv)multiply an algebraic expression by a constant to obtain another algebraic equation
Specific Instructional Objectives:
By the end of this lesson, students should be able to:
i)solve simultaneous equations involving two variables, by eliminating one of the variables
ii)express one of the variables in terms of the other, and to solve the simultaneous equations using the substitution method
iii)identify simultaneous equations as being solvable only when the number of equations is equal to the number of variables
Key concepts:
Variables, constants, expressions, equations, multiplication of an equation with a constant, addition and subtraction of two equations, elimination and substitution
Significance of topic to curriculum:
The ability to solve simultaneous equations is a fundamental skill that students should learn to master. Many scientific problems can be reduced to a system of simultaneous equations. Within the domain of Mathematics itself, simultaneous equations can be found in many other topics such as trigonometry and area and perimeter.
Solving simultaneous equations is a basic and highly useful skill that can be applied extensively in mathematics and science subjects. Furthermore, students should be made aware of the use of simultaneous equations in real-life scenarios and problems.
Learning Theories:
a)Assimilation, Accommodation & Equilibration (Piaget)
Piaget identified four concepts to explain cognitive development; schema, assimilation, accommodation and equilibration. Schema refers to cognitive structures by which individuals organize information. Assimilation is the integration of new concepts into existing schemata while accommodation is the process of creating schema. Equilibration is finding the balance between assimilation and accommodation. With regards to this lesson, the skills involved (manipulating algebraic equations) should already be part of the students’ existing schema. The lesson involves applying these skills to a new situation.
b)Socio-cultural View of Knowledge Construction (Vygotsky)
Vygotsky believed that the social and cultural backgrounds of students’ influence their learning styles. He felt that children learn, not as solitary actors, but by using ways of acting and thinking provided by their culture. In a developed country like ours, ICT is being used more and more widely in almost all sectors of our lives. Students’ are far more ICT savvy than students of previous generations. With our students becoming so comfortable with the ICT culture, the use of ICT in the classroom, wherever appropriate, will aid their learning. The interactive online applet is used in this lesson to achieve this purpose. When students are comfortable with their medium of learning, they are bound to learn better.
c)Co-operative Learning Groups (Johnson & Johnson)
A co-operative learning environment is one in which students are encouraged to work together in groups to achieve an objective. Johnson and Johnson believed that achievement is increased when students work in a collaborative, rather than competitive, environment. Most of the class work in this lesson is done in pairs or groups of four students. The team scoring system also encourages peer teaching/learning.
d)Guided Discovery Approach
The worksheets are designed to lead students towards discovering two different methods of solving simultaneous equations. Another approach advocated by Bruner is the ‘spiral approach’. This approach builds on the students’ prior knowledge to develop new concepts. Students who discover these methods of solving simultaneous equations, are more motivated to understand, remember and apply them correctly.
e)Value and Expaectancy (Feather)
Feather postulated that the effort that students are willing to put into completing a task is a product of: (a) the degree to which they value the rewards of successfully completing the task, and (b) the degree to which they expect to be able to perform the task. In this lesson, the inter-disciplinary nature of simultaneous equations is emphasized. Students will see the value in learning to complete this task well once they are aware that they will come across simultaneous equations in other chapters of Mathematics, in their science subjects and even in their daily lives. In addition, the worksheets are designed such that they are not too challenging. While the worksheets are slightly progressive in terms of difficulty level, they are within the capabilities of even the weaker students.
Teaching approaches:
a)Builds upon students’ prior knowledge of being able to manipulate algebraic equations.
b)While Simultaneous Equations is a rather new topic which most students would not have encountered before, the concepts and skills involved in the calculations are actually part of the students’ prior knowledge. It is the manner in which these skills are applied that is new.
Common Learning difficulties / Misconceptions:
a)As mentioned earlier, the knowledge or skills required for this topic is not really new. As such, the learning difficulties often arise from earlier lessons when the fundamentals were being taught.
b)Elimination Method:
-Students often forget to multiply both sides of the equation by the appropriate constant
Eg. 3x + 2y = 8 ------(1)
(1) × 2:6x + 4y = 8 ------(1a)
c)Substitution Method
- Students make the mistake of substituting the expression for the unknown variable back into the same equation from which the expression was derived. This, of course, will yield the trivial solution i.e. 0 = 0.
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Flavian Brian Fernandez (06)25 October 2005
QCM 520 (TG11)
BrightsideSecondary School
Secondary 3 (Express)
Elementary Mathematics
Time(min) / Teaching/Learning Activity / Resources / Rationale / Remarks
10 / Recap of Prior knowledge/ Introduction
-Recap: Revision Worksheet/Quiz
-Divide students into Maths Groups and give each group a worksheet to do.
-Assess students’ prior knowledge by randomly picking one student from each group to give examples of constants, variables, algebraic expressions and algebraic equations. Award group points for correct answers.
-Write a few algebraic equations on the board and ask some students to make the unknown variable the subject of the equation.
-Motivational Activity: Give students Worksheet 1 to show them the inter-disciplinary nature of simultaneous equations. Students will not be expected to work through the worksheet at this stage. The purpose of giving it out is simply to make them aware of the various domains in which simultaneous equations are used / Whiteboard,
Revision Worksheet
Worksheet 1 / -These are fundamental concepts/terms that must be understood by the student.If the students are confused or unclear about these terms, it must be addressed immediately so that this confusion does not propagate throughout the later chapters.
-Besides giving me a rough gauge of the students’ prior knowledge, getting students to do these questions on the board also acts as a revision or reinforcement for the rest of the class.
-Students will be interested in the topic once they learn how applicable it is to their studies and their lives in general. The worksheet will show them that simultaneous equations are used in science, mathematics and even in their daily lives.
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10 / Development of Lesson
-Demonstrate to the students that in order to obtain a unique solution for two unknowns, we need to have a pair of simultaneous equations involving only these two unknowns and constants. This will be done by showing them counter-examples as well as worked examples.
-Explain to the students briefly, that there are 3 methods of solving simultaneous equations:
- the elimination method
- the substitution method
- the graphical method (not taught in this lesson)
Worksheet 2 / -Students must understand clearly that in order to obtain a unique solution, the number of unknowns must be equal to the number of equations involving these unknowns.
-The worksheet guides students along in solving simultaneous equations. In following the steps, students will be able to see that they are actually eliminating one of the variables in the process.
10 / Evaluation/Practice:
-Ask the students to complete the second part of worksheet 2 using the interactive online applet. / Java Applet
(see reference for website) / -This allows the students to practice using the method they have just learnt to solve simultaneous equations.
-The website is very useful because it allows the students to proceed at his/her own pace and it provides immediate feedback on students’ answers
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10 / -Give students the following question and ask them to solve it, as a group, using the elimination method:
11x + y = 37
13x─8y = 7
Solution: x = 3, y = 4
-Show the students the solution to the same question using the substitution method
Evaluation/Practice:
-Allow students to work through in pairs, or groups of 4, the solutions to the questions in Worksheet 3, using the substitution method / PowerPoint
PowerPoint
Worksheet 3 / -The rationale is to show the students that for some questions, the elimination method can be quite tedious. Hence they will be able to better appreciate the substitution method when it is shown to them.
5 / Consolidation
-Refer students back to Worksheet 1 and get them to solve questions 2 – 4 using either of the 2 methods they have just learnt. Go through answers in the next lesson. (Question 1 is omitted because they have not learnt trigonometry yet) / Worksheet 1 / -This to consolidate their learning. They will be able to discover which method they are more comfortable with. It will also reinforce the inter-disciplinary nature of simultaneous equations
5 / Lesson Closure / Summary
-Give students a concept map that summarizes the key ideas involved in using both methods / Handout 1 / -To allow students to view the chapter as a whole. It also acts as a good summary of the chapter which the students can use when revising for exams.
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