Equations (AICLE unit)

Course: 4 ESO  Matemáticas académicas

Introduction:
This lesson is about equations, system of equations and inequalities.A project called "Three squirrels and a pile of nuts" is included as a final task. You also can find the objectives, the activities and several rubrics for assessment.

Learning outcomes:

1. ü  To factorize polynomials in order to solve polynomial equations whose degree is greater than two
2. ü  To express in algebraic language a real situation, to study this situation and to solve the problem with the appropriate algebraic tool: equation, inequality, system of equations… 
3. To give an interpretation of the results in terms of the considered problem.
4. To apply these concepts in a class project.
5. To use the appropiate technological tool such as spreadsheet, text procesor.
6. To write a report using mathematical lenguage.
7. To know the corresponding vocabulary in English

Evaluation criteria:

1. ü  To use correctly algebraic language
2. ü  To compute algebraic calculations
3. ü  To know and apply algebraic properties.
4. ü  To represent and analyze situations involving mathematics using inequalities, equations and system of equations.
5. ü   To apply equations, system of equations and inequalities in order to solve mathematical and real problems.

Subject content:

1. ü  Polynomial equations, equations with the unknown in denominator (algebraic  fractions) and equations with the unknown inside a radical (irrational equations).
2. ü  Simultaneous linear equations (systems of linear equations)
3. ü  Simultaneous non linear equations (systems of non linear equations).
4. ü  Linear and quadratic inequalities. Numerical and graphical solutions.
5. ü  System of inequalities (one variable)
6. ü  The spreadsheet

Vocabulary

BICS CALP Nouns Squirrel, pile, nut, report, summary, height, weight, length, width Equation, unknown, variable, solution, interval, substitution, elimination, root, line, intersection, algebra, algebraic fraction, LCM, common denominator, union, intersection, sign, coefficient, term, degree, spreadsheet. Verbs To check,  to belong, to flip, to calculate, to isolate, to remain, to leave, to weigh To solve, to square, to reduce(common denominator), to extract (common factor), to pass(through) Adjetives Open, closed, high, long, wide Linear, quadratic, biquadratic, radical irrational, algebraic, polynomial, cubic, leading(coefficient), independent(term), squared

Comparative  Greater than, less than, equal to, greater than or equal to,…Students will be these structures when they are solving inequalities . Equal Student have to know how to use this word as a verb and as a noun. For example  the expression x=3 can be read in two different way.         X is equal to 3 (here equal is a noun)         X equals 3 (and now is a verb) Derivative words High/height vs weigh/weight It is necessary to pay attention to these pairs of words High is an adjective and height is a noun but to weigh is a verb and weight is a noun. The students are usually confused about the pronunciation and tend to pronounce height and weight in the same form. These words are very frequent in words problems. Long/length Wide/width The pronunciation is different again, and student usually are mistaken

Activities:

In each type of equation there are different activities, active watching, watching, vocabulary exercises, application exercises and word problems

Quadratic equation:

1. Interactive watching http://learningapps.org/display?v=pfkwrq95c16
2. Video http://www.youtube.com/watch?v=LSANfj3ozSk
3. Vocabulary  activity. Complete the following sentences 
   
4. Application exercises:
   
5. Word problems

• Translate each English phrase into an algebraic expression. Let x stand for the number:

a)      twice the number
b)      nine more than the number
c)      the number multiplied by seven
d)      nine divided into the number
e)       the number divided by nine
f)       the number increased by fourteen
g)       six decreased by the number
h)      a number less 9
i)       nine less than a number
j)       nine less a number
k)      nine is less than a number

•  A ladder is resting against a wall. The top of the ladder touches the wall at a height of 15 feet. Find the distance from the wall to the bottom of the ladder if the length of the ladder is one foot more than twice its distance from the wall
•  The longest leg of a right triangle is one meter longer than the shortest leg. The hypotenuse is one meter shorter than twice the shortest leg. Find the length of the shortest leg of the triangle.
•  The length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle.
•  The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers
• The length of a rectangle is 2 times its width. The area of the rectangle is 72 square inches. Find the dimensions of the rectangle.
• The length of a rectangle exceeds its width by 3 inches. The area of the  rectangle is 70 square inches, find its dimensions.
• The product of two consecutive integers is 56. Find the integers.
•  The product of two consecutive odd integers is 99. Find the integers.
• Find two consecutive even integers such that the square of the smallest is 10 more than the largest.
• The product of two consecutive odd integers is 1 less than twice their sum. Find the integers.
• The medium side of a right triangle is 7 more than the shortest side. The longest side is 7 less than 3 times the shortest side. Find the length of the shortest side of the triangle.
• 
  

Radical equations

1. Interactive watching http://learningapps.org/display?v=p1powy2pn16
2. Vocabulary. 
   
   
   
3. Application exercises

Rational equations:

1. Interactive watching http://learningapps.org/display?v=p9my5mvpj16
2. Vocabulary 
   
3. Application exercises 
   

Polynomial equations

1. To watch the following video http://www.youtube.com/watch?v=g7_llQnLepA&feature=channel
2. Vocabulary . Complete the sentences: To solve a polynomial equation first _____________ the polynomial and then ____________ for x each _______________
3. To solve the exercises
   

Vocabulary exercise

Match each equation with its name
Systems of linear and non-linear equations

1. Watching activities 

• http://www.youtube.com/watch?v=KNwwu5wjcaA
• http://www.youtube.com/watch?v=JDmmRVUKQww&feature=relmfu
• http://www.youtube.com/watch?v=cLQtqANYg6U&feature=relmfu

1. Word problems

• Find the ages of two people knowing that ten years ago, the age of the first person was four times the age of the second person, and in twenty years, the age of the first person will be double the age of the second person.
•   A group of students have paid € 144 for 3 tickets in the shade and 6 tickets in the sun to a bull fight. Another group paid € 66 for 2 tickets in the shade and 2 tickets in the sun. Calculate the price of each ticket.
• Michael buys two bags of chips and three boxes of pretzels for $5.13. He then buys another bag of chips and two more boxes of pretzels for $3.09. Find the cost of each bag of chips and each box of pretzels.
•  There are 420 students in a high school. Forty-two percent of “ESO” and 52% of “bachillerato” are girls, which means 196 in total. Calculate the amount of students in ESO and bachillerato.
• A total of 925 tickets were sold for $5,925. If adult tickets cost $7.50, and children's tickets cost $3.00, how many tickets of each kind were sold? 
• The perimeter of a rectangle is 26 cm and its surface is 40 cm². Calculate its dimensions.
•  The sum of two numbers is 10 and their product is 24. Calculate both numbers. 
• Find two consecutive odd numbers whose product is 49 times the square of the smaller.
• Calculate the surface of a square knowing that the diagonal length exceeds 5 cm its side length.

Inequalities

1. Watching activities http://www.youtube.com/watch?v=1wPJfzStBqE&feature=related
2. Word problems     Solve the following word problems by :
   a.     Clearly identifying your variables
   b.     Setting up an inequality
   c.      Solving the inequality
   d.     Stating your answer.
   
   
   a) Tom earned 400€ and 550€ in interest the last 2 years. How much interest must he earn this year so that his average earnings over the three year period are more than 600€?
   
   
   c) Robert makes $3.50 per hour working at a convenience store. If he gets a bonus of $25 this week, how many hours must he work to make at least $165?
   
   
   d) The low temperatures for the last two days were 28° and 15° . What must the low temperature for the next day be in order for the average temperature for the 3-day period to be less than 19°?
   
   
     Project

THREE SQUIRRELS AND A PILE OF NUTS.

THE PROBLEM.

Three squirrels, Steve, Keith and Benjamin, have spent all day collecting nuts. At the end of the day they are very tired and go to bed.

During the night, Steve wakes up and eats a nut. He then decides to take one third of the remaining pile and hides it before going back to sleep. Keith then wakes up. He too is hungry, so eats a nut. He also takes one third of the remaining pile and hides it before going back to sleep. Finally, Benjamin wakes up, eats a nut and hides a third of the remaining pile.

In the morning, all of the squirrels share the remaining nuts equally between them and each goes off with two more nuts. Each squirrel then eats all the nuts he now has.

How many nuts has Steve eaten in total?

HOW TO SOLVE IT.

You have three different strategies to solve the problem. You have to work in a team and you must develop completely the three strategies and write a document explaining the solution step by step.

STRATEGY 1.

WHAT TO DO: With this strategy, you'll be solving the problem the way a squirrel would - one nut at a time!

1. Count out the number of nuts your group would like to start with (not necessarily all you have). Put these in your starter pile.

2. Make one member of your team Squirrel 1, and have him or her tackle the pile of nuts the way the first squirrel in the story did.

Warning: No fractions of nuts allowed! Let Squirrel 1 take several tries until she or he ends up with whole amounts after each step. Record these numbers.

3. Now the teammate who's playing Squirrel 2 should work through the steps. Again, no fractions - so both Squirrel 1 and 2 may need to go back and adjust their piles. Record the numbers that work.

4. Finally, Squirrel 3 should take his or her turn. Adjust as needed, and record the results in your document.

5. When Squirrel 3 is through, divide the nuts that are left. If the pile doesn't divide by three, adjust all the way back through the problem. Record your tries and your final answers.

6. From all your notes, write several sentences (with pictures if you'd like) describing the steps and numbers that finally gave you the right answer.

STRATEGY 2.

WHAT TO DO: In this step, you'll be setting up a spreadsheet that will let you try out various combinations of numbers to solve the problem.

1. Label your rows for Squirrel 1, Squirrel 2, and Squirrel 3. Label your columns "No. of nuts in pile," "No. of nuts after s/he eats 1," and "No. of nuts s/he leaves behind."

	A	B	C	D	E
		No. of nuts in pile	No.of nuts after s/he eats 1	Of nuts s/he leaves behind
Squirrel 1
Squirrel  2
Squirrel 3

2. Now, figure out formulas for all the cells that will let you plug in and test out numbers. Remember to make each cell relative to the others, so that a number you plug into one cell's formula will affect the rest. Here are a few hints:

The formula in cell C2 should be =B2-1, because the squirrel eats one nut from the pile in B2.

The formula for D2 is =(C2/3)*2, because Squirrel 1 leaves two-thirds of the pile behind.

The formula for B3 is =D2, because Squirrel 2 finds the pile the way Squirrel 1 leaves it.

3. Once all your formulas are in place, test out your spreadsheet by plugging a number into B2. You may end up with a lot of weird-looking fractions - and you know these squirrels aren't going to settle for anything less than whole nuts! Keep trying numbers until you find one that gives you whole numbers throughout. What patterns do you see?

4.- Include your spreadsheet in the document twice, once showing the formulas and once with your whole-number solution in place.

STRATEGY 3(algebra power).

WHAT TO DO: With this problem-solving strategy, you'll be setting up an algebra equation to solve.

1. Open the Excel worksheet used for Strategy 2, leave the column and row headings. Now you have plenty of room to create a table for defining your variables.

2. The variables you most need to solve for are:

x= nuts in original pile

y= number of nuts each squirrel gets in the end.

Record these at the bottom of your worksheet.

	A	B	C	D
		No. of nuts in pile	No.of nuts after s/he eats 1	Of nuts s/he leaves behind
Squirrel 1
Squirrel  2
Squirrel 3
	x= nuts in original pile
	y= number of nuts each squirrel gets in the end

3. Here are some hints: Put in x as Squirrel 1's "No. of nuts in pile."

When it comes to solving for y, remember that it is one-third of however many nuts are left in the end, so y=1/3 (last entry in table).

4. After you fill in your table, put together an equation. You can type this in your worksheet below your table. Show steps for simplifying the equation. Then test it out with several numbers.

5. Make sure your worksheet shows all the steps you followed to write and solve the equation.

THE FINAL REPORT

Your final report must include

1.      A cover (make a beautiful one, including photos or drawing a picture)

·        Title

·        Names of the members of the group and their degree of involvement in the different tasks.

·        Date

2.      Index

3.      Objectives

·        The main objective of this project is to ___________________________

·        Other objectives are___________________________________________ and ____________________________________________________________

4.      Description and explanation of the three strategies

·        Strategy 1:

ü Description: In this strategy, each member of the group behaves as a ______________, then____________________________

ü Results: We think that the number of nuts is______

·        Strategy2:

ü Description: In this strategy, we have used a ______________, and then____________________________

ü Results: We think that the number of nuts is______

·        Strategy2:

ü Description: In this strategy, we have used ______________, and then____________________________

ü Results: We think that the number of nuts is______

5.      Resources

6.      Conclusion and personal opinion:

·        Combining the ______________ strategies, we think that the number of nuts can be ___________________ because ______________________________

·        I like or dislike this project because __________________________________

·        The easiest strategy was ___________ because_________________________

·        The most difficult strategy was______because_________________________

Assessment

·        Interactive activities  and quizzes

                  http://www.math.com/school/subject2/S2U3Quiz.html

http://www.shmoop.com/equations-inequalities/quiz-3.html

http://quiz.econ.usyd.edu.au/mathquiz/inequalities/quiz1.php

            http://quiz.econ.usyd.edu.au/mathquiz/sim-eqns/quiz1.php

Rubrics

1.      Self-assessment:

Students will use the following table:

SELF ASSESSMENT

Tick the following items when you are sure that you know them

	Well known	Medium	Don’t know
Basic vocabulary
To solve complete quadratic equations
To solve incomplete quadratic equations
To solve biquadratic equation
To solve radical equations
To solve rational equations
To solve polynomials equations
To translate English into algebraic language
To solve linear system of equations by substitution
To solve linear system of equations by elimination
To solve linear system of equations by “igualación”
To solve linear system of equations graphically
To solve non linear system of equations
To solve linear inequalities
To solve quadratic inequalities
To use scientific calculator
To use geogebra (CAS)

1.      Project ‘three squirrel and a pile of nuts’  (adapted from eduteka.org)

The project will be assess  using the following rubric

	Category	3	2	1	0
1	Personal work			Trabaja de forma de adecuada	No trabaja nada
2	deadlines			Cumple los plazos	Incumple los plazos
3	REPORT
3.1	Format			Incluye portada, índice, objetivos, trabajo desarrollado y conclusiones	Falta algún apartado
3.2	Vocabulary		Escrito en inglés sin grandes errores	Escrito en inglés con errores	Escrito en castellano
4	COMPREHENSION			Lee el texto y entiende la tarea	Necesita ayuda para comprender la tarea
5	STRATEGY 1
5.1	Finding a solution		Más de una	Una	No encuentra
5.2	Making their own estrategy			Sí	No
6	STRATEGY 2
6.1	Format of the spreadsheet		Claro y buena presentación	Normal	No hace o incomprensible
6.2	Spreadsheet		Correcta y sin ayuda	Correcta con ayuda	No hace o incorrecta
6.3	Finding solutions with the spreadsheet		Encuentra varias	Comprueba las que tenía	No encuentra
7	STRATEGY 3
7.1	Equation		Correcta sin ayuda	Correcta sin ayuda	Incorrecta o no encuentra
	Finding the set of solutions		Correcta sin ayuda	Correcta sin ayuda	Incorrecta o no encuentra
8	CONCLUSSION
8.1	Objetives			Describe	No describe
8.2	Describe estrategies	Describe 3	Describe 2	Describe 1	No describe
8.3	Explain strategies	Explica 3	Explica 2	Explica 1	No explica
8.4	Final conclussion			Hace	No hace