Clilstore Facebook WA Linkedin Email
Login

This is a Clilstore unit. You can link all words to dictionaries.

SIMULTANEOUS EQUATIONS

INDEX

Justification, context and Objectives
Learning Outcomes
Methodology and Resources
Content/teaching activities. Specific Porcedure for this unit (Lesson 1 to 8)
Scaffolding provived for the Lessons
Hotpotatoes Activities
Self Evaluation Test
Unit Final Test
Assesment
Scoring Rubric

 



Justification

This unit will 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 them.

Students should already know how to solve linear equations, and manage with the algebraic language.

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

 

Context and students

This unit is prepared for 14-15-year-old students,  immersed in a plurilingual program.

Their current level of English is B1, or at least it is the expected one.

 

Time needed

The number of lessons is eight, including the assessment.

 

Aims of the unit

In this unit students will use 3 methods of solving systems of equations and will find that they are seen in everyday life,  helping them to work out real problems.

Those three methods are: estimating solutions graphically, solving using substitution, and solving using elimination methods. Students will develop conceptual skills and will have to change forms of 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 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

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 ; 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.

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 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 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. 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.

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.

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

- Online dictionaries: i.e. wordreference or 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

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

 

 

 


 

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:

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:

 

Task. 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

 


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, offering feedback.


Lesson 7


Lesson 8

Each group of students will have to assess their classmates’ work. Teacher will guide this assessment

 


 

Scaffolding provided for the lessons:

The teacher will prepare some cards with vocabulary and useful expressions depending on the proposed exercise.
For instance these terms or expressions could be:
- Equations, formulae and identities
- Solution: the value of the variable which make the equation true
- Expand brackets
- Add x in both sides
- Every formula has a subject
- ...


Every card will have a short explanation on its reverse.

Here are some examples of these cards:

 

 


 

 ACTIVITIES AND EXERCISES:

 

Activity Unit Project

This project will be due at the end of the unit, and will provide an assessment additional to the final test. Students will have the chance to show their creative side and be rewarded for this as well as the mathematical solutions. This project will also encompass all methods of solving systems of equations and help students realize the importance of these functions in daily life.

Students will create a short story that incorporates systems of equations in it. The story will have one main theme and will either have one main problem. The story must be at least a typed page in length when written in a word processing document.

Then, students must transfer this story onto smaller pages and create drawings in addition to written solutions for each page. The last day of the unit, students will have the chance to show off their work to other students in the class.

  


 

EXERCISES:

Exercise 1.
Solve the equation 2x + 6y = 28 knowing the value of one of the variables.

I) x = 5           IV) y=0

II) x=10          V) y=-2

III) y=2           VI) x=1/2

 

Exercise 2.
Find three solutions for each one of these equations.

I) x + y = 10

II) 2x – y = 14

III) 3x + y = 8

IV) –x – 5y = 0

 

Exercise 3.
Write down an equation following this situation: ‘a group of friends went to the theatre and bought 3 box seat’s tickets and 5 stalls’ tickets. They have paid 80 Euros. How much does it cost each type of ticket?

 

Exercise 4.
Solve this problem by drawing a graph.
Two women are walking on the same long, straight road towards each other. One sets off at 9:00 am at a speed of 4 km/h. The other also sets off at 9:00 am, 15 km away, at a speed of 5 km/h. At 9:10 am, a butterfly leaves the shoulder of the quicker woman and flies to the other woman at 20 km/h. It continues to fly from one woman to the other until they both meet, and take a photograph of it.

a) At what time will the butterfly be photographed?

b) How many times will the butterfly have landed on the woman walking at 4 km/h?

 

Exercise 5.
Solve each pair of simultaneous equations using the elimination method:

 

Exercise 6.
Solve each pair of simultaneous equations using the substitution method:

 

 

Exercise 7.
Solve each pair of simultaneous equations choosing any of the three studied methods.


Exercise 8.
Work out each of the following problems by expressing it as a pair of simultaneous equations, for which you find the solution.


i) Find the value of two numbers if their sum is 12 and their difference is 4.

ii) The difference of two numbers is 3. Their sum is 13. Find the numbers.

iii) Ruth is thinking of two numbers. Adding 4 times the first number and 5 times the second number gives a total of -13. Also, adding 10 times the first number and 10 times the second number gives -20. What are the two numbers?

iv) 1000 tickets were sold. Adult tickets cost 8.50€, children's cost 4.50€, and a total of 7300€ was collected. How many tickets of each kind were sold?

v) Mrs. B. invested 30,000€; part at 5%, and part at 8%. The total interest on the investment was 2,100€. How much did she invest at each rate?
 


 

Hotpotatoes activities.


1. Solving simultaneous equations choosing any of the three methods learnt

 

2. Solving problems using simultaneous equations

 

 

Practicing online exercises:
They can work with this web in pairs or individually. 

For instance here is the link: http://www.livebinders.com/play/play_or_edit?id=330285

If they need some more practice they can also work with these following links, (the answer is shown with an appropriate explanation).

 

- Solving linear systems (solutions with both methods showed):


http://www.regentsprep.org/regents/math/algebra/AE3/PracAlg.htm
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_tutorial_precalc/e/systems_of_equations_word_problems


- Solving simultaneous equations problems with the answer explained:


http://www.regentsprep.org/regents/math/algebra/AE3/PracWord.htm
https://au.ixl.com/math/year-10/solve-simultaneous-equations-using-elimination-word-problems

 

For the lesson before the final test we will have this self-evaluation test in order the student realize where he/she actually is.

 


 

SELF EVALUATION TEST:


At the end of the unit they will pass a 20 minutes self-evaluation test like this:


1. The simultaneous solution of the equations 2x + y = 5 and x – 3y = 13 Is:
a. x = 4, y = 3
b. x = 4, y = -3
c. x = -4, y = 3
d. x = -4, y = -3
(Hint: there is no need to solve the equations simultaneously. Substitute to see what solution satisfies both equations.)

2. Draw a set of axes and graph the lines y = x + 3 and y = 7 – x.
    Write down the simultaneous solution of the two equations shown before.
    (Hint: What are the coordinates of the point of intersection?)

3. Work out the simultaneous equations 7x-2y=5 and y=3x by substitution
   (Hint: Replace y with 3x in the first equation)

4. Solve by elimination 7a+2b=16 and 3a-2b=24
   (Hint: add the two equations together so that b is eliminated)

5. Make y the subject of the equation 2x+y-5=0
   (Hint: the subject of a equation is the value before the equals sign. Rearrange the equation so that it reads y=)
   Solve the equations 2x+y-5=0 and 3x-4y+42=0 by substitution.

6. Solve by elimination 9m+4p=41 and 5m+2p=23
   (Hint: first multiply the second equation by 2. Make sure that you multiply every term in the equation)

 

 


 

FINAL TEST:

1. Look at the equation: (1 p)


x+(x+1)+(x+2)=y


Use it to help you write the missing expression in terms of y.
The first one is done for you.


5 + x + (x+1) + (x+2) = y + 5
(x+5) + (x+6) + (x+7) =
2x + 2(x+1) + 2(x+2) =
(x+a) + (x+1+a) + (x+2+a) =

 

2. Solve these simultaneous equations using a different algebraic method for each one: (3p)

You must show your working.

 

3. Look at the simultaneous equations. (2 p)


x + 2y = a

x + y = b


a) Write an expression for y in terms of a and b.

b) Now write an expression for x in terms of a and b.

Write your expression as simply as possible.

 

4. A chef uses this formula to cook a roast: T = a + bW
    Where T is the time it takes (minutes), W is the weight of the roast (kg) and both a and b are constants.
    The chef says it takes 2 hours 51 minutes to cook a 12 kg roast and 1 hour 59 minutes to cook a 8 kg roast.

   How long will it take to cook a 5 kg roast? (2p)

 

5. At a local tea room I couldn’t help noticing that at one table, where the people had eaten six buns and had three teas, the total cost was 1,65 €.
    At another table, the people had eaten 11 buns and seven teas at a total cost of 3,40 €.
    My guests and I had five buns and had six teas. What would it cost me? (2 p)

 


Assessment


The assessment criteria will be established at the beginning with the students. This will help them to develop what they are supposed to attain and where they are.

The assessment will be formed by:

1. Unit final test. This part will represent the 60% of the unit mark.
2. Unit project. This part will represent the 20% of the unit mark.
3. Student worksheet. This part will represent the 20% of the unit mark.

Student worksheets will be taken up and scored according to ‘Solving Systems of Equations Checklist’. These scores may be placed in the grade book after students have been given sufficient practice to learn the material. This checklist assists the teacher in determining which students need remediation. The sole purpose of this checklist is to aid the teacher in identifying students that need remediation. 

Solving Systems of Equations Checklist:

I.N: improvement needed                                                   F.A: fairly accomplished                                                                                 

                                                                                                                  A: accomplished

Assessment criteria for Unit Project:


Students will be graded based on creativity of the story (problems well tied into the story, interesting problems, different from examples in class), accuracy of mathematical solutions (solutions are correct, students used 3 different methods), and overall presentation (how organized the book is, appearance, drawings, neatness) Each aspect makes up the total grade for the Unit Project. Overall, the Unit Project will be 20% of the entire unit grade.


Here is shown the teacher’s rubric created for the assessment of the project.

Scoring rubric:

  Essential Elements Quality Indicators
  Team has created and solve a system of linear equations that represents the data related to a problem they want to solve. 0= No system of linear equations was created or solved
1= System of linear equations was not a complete representation of the data related to the problem
2= System of linear equations was created but there were three or four errors in the solution
3= System of linear equations was created but there were one or two errors in the solution
4= System of linear equations was created and solved correctly
  Team has used graphing, substitution and elimination methods to solve the systems of equations 0= Team did not solved the system
1= Team did not use any of the listed methods to solve their systems
2= Team used 1 method listed to solve their systems
3= Team used 2 method listed to solve their systems
4=Team used all three methods listed to solve their systems
  Team has created a display that explain their processes and reflects on the answer’s meaning in the context of the original problem 0= Team has not creates a display
1= Team has created a display but has not addressed either their process or reflection of the answer meaning
2= Team has created a display that needs improvement in both describing the processes and reflection
3= Team has created a display that clearly describes their processes, but the reflection needs improvement
4= Team has created a display that clearly describes their processes, and reflections on the meaning of the solution
  Team has given a 3-5 minutes presentation of their book to the class that discussed their project, creative display, appearance, drawings, neatness and explanation of teamwork 0= Presentation does not meet any requirements
1= Presentation meets all but four requirements
2= Presentation meets all but two requirements
3= Presentation meets all but one requirements
4= Presentation meets all requirements

 

Clilstore

Short url:   https://clilstore.eu/cs/3564