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SIMULTANEOUS EQUATIONS

INDEX

Justification, context and Objectives
Learning Outcomes
Methodology and Resources
Content/teaching activities. Specific Porcedure for this unit (Lesson 1 to 8)

 



Justification

This unit is going to show the students how to construct and solve different types of linear simultaneous equations, how to use trial and improvement and how to solve problems involving solving simultaneous equations.

Students should already know how to solve linear equations (equations, formulae and identities) and they must be familiarized with the algebraic language.

This unit is required in the official syllabus, inside ‘Algebra’.

 

Context and students

This unit is prepared for 14-year-old students, studying 2nd class of ESO and immersed in a plurilingual program.

Their current level of English is B1 and they were taught using English as the vehicular language last course in three different subjects, including maths.

 

Time needed

The number of lessons is eight, including the assessment.

The procedure syllabus for each lesson is explained later on, at the beginning of the Content/teaching activities section. 

 

Aims of the unit

In this unit students will be able to use methods of solving systems of equations in applications and will find that systems of equations are seen in everyday life. Learning to solve systems of equations using any of the three methods contained in this unit will help students work out real life problems such as the examples showed later.

Solving systems in this level class includes estimating solutions graphically, solving using substitution, and solving using elimination methods. Students gain experience by developing conceptual skills, using models that develop into abstract skills of formal solving of equations. Students also have to change forms of equations (from a given form to slope-intercept form) in order to compare equations.

 

CONTENT OBJECTIVES

At the end of this unit, students should be able to:

 

LANGUAGE OBJECTIVES

At the end of this unit, students should be able to:

 

Learning outcomes

COMPETENCY BASED EDUCATION-SKILL-BASED LEARNIG:

Students and teachers will collaborate on students learning to reach the following competencies:

Mathematical competency

Students will understand that one of the keys to solving problems lies in the understanding of basic skills such as simplifying algebraic expressions and solving equations and systems of equations. 

Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. Mathematics is, among other things, a language, and students should be comfortable using the language of mathematics.

Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. It is valuable for students to learn with a teacher and others who get excited about mathematics, who work as a team, who experiment and form conjectures.

Linguistic and communication competency

Students can construct viable arguments to support their own reasoning and to critique the reasoning of others. Students need extensive experiences in oral and written communication regarding mathematics, and they need constructive, detailed feedback in order to develop these skills. The goal is not for students to memorize an extensive mathematical vocabulary, but rather for students to develop an ease in carefully and precisely discussing the mathematics they are learning. Memorizing terms that students don't use does not contribute to their English or their mathematical understanding. However, using appropriate terminology so as to be precise in communicating mathematical meaning is part and parcel of mathematical reasoning.

Knowledge and interaction with physical and social environment competency

Students can work out a range of real life problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Problem solving is not a collection of specific techniques to be learned; it cannot be reduced to a set of procedures. Problem solving is taught by giving students appropriate experience in solving unfamiliar problems, by then engaging them in a discussion of their various attempts at solutions, and by reflecting on these processes.

Students will understand that linear functions can be used to model relationships between many real-world quantities.

Technological competency

Students will be able to manage web pages and online tools in order to get information or do their homework. They should use technology respectfully and apply it effectively, manage projects, produce results, and create media products. 

Social and civic competency

Experience in solving real problems gives students the confidence and skills to approach new situations creatively, by modifying, adapting, and combining their mathematical tools; it gives students the determination to refuse or accept an answer until they can explain it.

Students will understand that systems of linear equations can be used to model real-world situations in which many different conditions must be met. This will help them to develop a critical attitude against `social´ reality, demonstrate respect, collaboration, and leadership in working with others.

Artistic and cultural competency

Students will develop this competency through their Unit Project. They have to be creative not only with ideas, also by the use of varying unusual materials, different designs…

Learning to learn competency

Students are facing different challenges, using intuition and deduction methods, learning how to use trial and improvement and how to work out problems involving solving simultaneous equations. They are going to develop good habits as discipline, study or individual and team working.

Autonomy and personal initiative competency

Students will be aware of their improvement, leading them to a positive attitude towards problems’ resolution. They should feel confidence in their own ability to cope with problems successfully and acquire an adequate level of self-esteem that allows them to enjoy creative, manipulative, aesthetic and utilitarian aspects of mathematics.

 

Methodology

We are looking for an active, intuitive and motivating methodology, which wakes up interest and promotes the learning by the discovery of the concepts from knowledge and personal experiences. We have a clear instructional strategy: Individual and group learning through lecture, discussions, demonstrations, research, and investigation.

It is well known that if the pupil discovers the concepts itself, these will last further in its logical structure. We are trying to design and to elaborate activities in order that students discover the concepts and not only store them.

Students should be familiar (prior knowledge) with basic algebraic operation skills such as addition, subtraction, multiplication, division, exponents, fractions, decimals, solving equations, and point-slope form of a linear equation.

There will be promoted active classes, in which the pupils develop their skills. The activities have to make them asking, thinking and expressing their thoughts verbally. While they are working in the activities they need to establish a relation between the major numbers of possible concepts. The activities proposed have several levels in order to allow different rhythm according to each student.

Solving problems should be one of the main matters. Each problem has to be planned, carefully read in order to be correctly understood, it could be approached using a diagram to show the information. It is important how they choose the unknown variables and how they lay out the system of equations they need to reach the answer.

We will work with diverse groups and different resources will be used (text book, audio-visual, web pages …). Algebra has to be used in different contexts: games, personal and familiar situations, science… Usually at the end of each task the teacher will show the final conclusions that each student must have gained. Students have to realize how they are reaching main goals or perhaps to think about the reasons for those who are not achieving the goals.

Proposed activities will pursue the following sequence: Make – discuss – discover – explain and expressing, sometimes orally and, sometimes writing them down. 

 

Usually activities are shown with a theoretical support and enough exercises with a gradual difficulty degree.

Lessons will last 55 minutes and this will be the general structure:

 

Assessment will be continuously done by the teacher, writing down notes on his/her grade book.

The teacher will provide scaffolding before explaining any task. He/she will prepare different cards with vocabulary, expressions and questions, to support students learning in a second language.

The specific procedure for this unit is showed later, inside ‘Content/teaching activities’ section. Here you can find the development of each lesson widely explained.

Materials

Resources to be used to achieve the aims of the unit:

- Introductory session: http://www.shmoop.com/video/cahsee-math-11-algebra-and-functions

- Introduction to systems: https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/introduction_to_systems_of_linear_equations/v/trolls-tolls-and-systems-of-equations

- Solving simultaneous equations: http://www.videojug.com/film/how-to-solve-simultaneous-equations

- Solving systems of equations by graphing: http://www.shmoop.com/video/solving-systems-of-equations-by-graphing

 

- Text book, e.g. Maths Frameworking. Ed. Collins

- Moodle platform. Here the teacher will provide new online activities developed, worksheets and other useful material.

- Online dictionaries: i.e. Worldreference, Macmillan dictionary, Linguee,…

 If you are looking for different mathematics concepts and how to pronounce them correctly ‘MATH SPOKEN HERE!’ is a richly illustrated 460-word arithmetic and algebra dictionary: http://www.mathnstuff.com/math/spoken/here/

- Online activities: i.e. 

http://nrich.maths.org/5674

http://www.helpingwithmath.com/printables/worksheets/equations_expressions/8ee8-simultaneous-equation-generator01.htm

 

Solving graphically simultaneous equations we will work with this link, where examples, worksheet with the key and checklist are provided:

http://www.beaconlearningcenter.com/documents/1753_01.pdf

A link with interesting free stuff:

http://justmaths.co.uk/

A great web with video links and explanations and also activities related to systems of equations:

http://troup612resources.troup.k12.ga.us/curriculum1/mathematics/8_math/systems/4_systems.html

 

Examples prepared for different moments during the lessons:

 

At the very beginning we can start solving problems in pairs. The teacher forms these ‘two people’ groups.

Work out each of the following problems by expressing it as a pair of simultaneous equations, for which you find the solution

  1. The two people in front of me in the post office bought stamps. One bought 10 second-class and five first-class stamps at a total cost of 3,35€. The other bought 10 second-class and 10 first-class stamps at a total cost of 4,80 €.

 

How much would I pay for:

  1. a) One first-class stamp?
  2. b) One second-class stamp?
  3. In a fruit shop, one customer bought five bananas and eight oranges at a total cost of 2,85€. Another customer bought five bananas and 12 oranges at a total cost of 3,65€.

 

If I bought two bananas and three oranges, how much would it cost me? EXPERTO UNIVERSITARIO EN COMPETENCIA PROFEIONAL PARA LA ENSEÑANZA EN INGLÉS (CAPACITACIÓN) Blanco Gimenez C.Raquel 9

 

 


 

Content/teaching activities

SPECIFIC PROCEDURE FOR THIS UNIT:

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8

 

Lesson 1: Introduction procedure

This lesson will introduce students the concept of systems of equations and the methods of solving two equations with two unknowns. Through examples, students will learn that systems of equations are present in everyday life and are useful in problem solving.

The scheme followed for this initial lesson should be:

o Why would we need more than one equation?

o What types of real world situations require more than one variable?

o Have students theorize what the answer will be and why.

Trolls, tolls, and systems of equations: A troll forces us to use algebra to figure out the make-up of his currency. We end up setting up a system of equations.

 

 


 

Lesson 2:

How to solve a system of linear equations graphically (example): Sal graphs the following system of equations and solves it by looking for the intersection point: y=7/5x-5 and y=3/5x-1.

Task 1.The students have to come up with one of their own problems and create solutions for it graphically. The students will write their solutions on a separate sheet of paper and this will be their answer key. Then the next lesson students will explore their classmates’ ideas and further practice with solving with this method.

Task 2.Practice solving systems graphically using this web:

http://www.beaconlearningcenter.com/documents/1753_01.pdf


Lesson 3:

 

Solving linear systems by substitution: Solving Linear Systems by Substitution

 

Continue solving problems with the substitution method by using previous examples used with the graphing method and also with some new problems. Have students work individually on problems then discuss as a class.


Lesson 4

 

Solving systems of equations by elimination: Solving Systems of Equations by Elimination

http://www.helpingwithmath.com/printables/worksheets/equations_expressions/8ee8-simultaneous-equation-generator01.htm


Lesson 5

 They will also do online activities from the web page:

http://www.regentsprep.org/regents/math/algebra/AE3/PracWord.htm


Lesson 6

 

The teacher will move from student to student observing their work and lending assistance, offer feedback that is specific in praise and instruction.


Lesson 7


Lesson 8

 

 

 

Short url:   https://multidict.net/cs/3523