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Jim RahnJ www.J Students should Understand the big ideas of equivalence and linearity Modeling real situations with variables Use appropriate tools such as algebra tiles and graphing calculators,and spreadsheets regularly Understand that geometric objects can be represented algebraically(lines can be described using coordinates),and algebraic expressions can be interpreted geometrically(systems of equations and inequalities can be solved graphically)EQUIVALENCE:EQUIVALENCE:Numbers,expressions,functions,or equations have many different but equivalent forms.These forms differ in their efficacy and efficiency in interpreting or solving a problem,depending on the context.Algebra extends the properties of numbers to rules involving symbols to transform an expression,function,or equation into an equivalent form and substitute equivalent forms for each other.Solving problems algebraically typically involves transforming one equation to another equivalent equation until the solution becomes clear.Linearity The relationship between two quantities can often be represented graphically by a linear function.Linear functions can be used to show a relationship between two variables with a constant rate of change Linear functions can be used to show the relationship between two quantities that vary proportionately.Linear functions can also be used to model,describe,analyze,and compare sets of data.Understanding linear functions should be prominent in the Algebra I content.Recognize,describe and represent linear relationships using words,tables,numerical patterns,graphs and equations.Describe,analyze and use key characteristics of linear functions and their graphs.Graph the absolute value of a linear function and determine and analyze its key characteristics.Recognize,express and solve problems that can be modeled using linear functions.Interpret solutions in terms of the context of the problem.Solve single-variable linear equations and inequalities with rational coefficients.Solve equations involving the absolute value of a linear expression.Graph and analyze the graph of the solution set of a two-variable linear inequality.Solve systems of linear equations in two variables using algebraic and graphic procedures.Recognize,express and solve problems that can be modeled using single-variable linear equations;one-or two-variable inequalities;or two-variable systems of linear equations.Algebra should become a language through Algebra should become a language through which we can describe various situationswhich we can describe various situationsWhat is three plus five times two?Try entering this problem on the homescreen of the graphing calculatorHow many ways can you enter it on the homescreen?Is there an order for the operations when the problem is written horizontally?4+3*21.Evaluate expressions within parentheses or other grouping symbols.2.Evaluate all powers.3.Multiply and divide from left to right.4.Add and subtract from left to right.43 24610How would you have to write 4+32 so the answer is 14?Learning to build mathematical expressionsLets try performing a string of operations to see what we get.On paper:Start with 6.Multiply 2 times a starting number,then add 6,divide this result by 2,and then subtract your answer from 10.Start with 20.Multiply 2 times a starting number,then add 6,divide this result by 2,and then subtract your answer from 10.Start with-4 Multiply 2 times a starting number,then add 6,divide this result by 2,and then subtract your answer from 10.These problems appear pretty simple because we are giving all the directions in short steps and you are performing them in the order in which they are described.Lets see if we can learn to write expressions through a similar activity.Start with a chart and complete each line based on the directions given.Is there any relationship between the starting number and the resulting answer?How is what we did in the four steps equivalent to this one relationship?DescriptionDescriptionExpressionExpressionDescriptionDescriptionExpressionExpressionStart with a numberMultiply the number by 2Add 6Divide by 2Subtract the result from 10262x 26102x 26x 2xxOn the graphing calculator homescreen type:6=x:10-(2x+6)/2=means STOGeneral Format:Your Number=x:10-(2x+6)/2Confirm 20 and-4 Using the Description/Expression TemplatePick any numberDivide the number by 4Add 7Multiply the result by 2Subtract 8Find the value of your expression when x=2,-5,8Each person pick any number from 1 to 25.Add 9 to it.Multiply the result by 3.Subtract 6 from the current answer.Divide this answer by 3.Now subtract your original number.Compare your results.Will the answer be the same regardless on the number you begin with?Why is this?Write out the algebraic expression for this number trick.DescriptionDescriptionExpressionExpressionStart with a numberAdd 9Multiply the result by 3Subtract 6 from the resultDivide by 3Subtract the original number3(9)6x 3(9)63x 3(9)x 9x x3(9)63xxThis is a pretty complex expression.Can we put these in an equation and solve for x?3(9)673xxIf you were told the expression on the left describes several operations that were performed to a given number and that the result equals to 7,describe all the operations that were performed on x and what order they were performed to arrive at the answer 7?Create your own trick that has at least 5 stages.Test it on your calculator with at least four different numbers to make sure all the answers are the same.When you think your trick works,test it on your other group members.Write in words the number trick that is described above.Test the number trick to be sure you get the same result no matter what number you choose.Can you explain why this number trick work?32582xxAnalyzing a Number TrickAnalyzing a Number TrickGiven the expression on the left,you might want to think of subtraction as adding the opposite and re-write the expressionWrite,in words,the number trick that is described above.Test the number trick to be sure you get the same result no matter what number you choose.Which operations that undo previous operations make this number trick work?42(4)2xx42(4)2xx Daxun,Lacy,Claudia,and Al are working on a number trick.Here are the number sequences their number trick generates:a.Describe the stages of this number trick in the first column.b.Complete Claudias sequence.c.Write a sequence of expressions for Al in the last column.Lessons 2.1 and 2.2:review proportions and introduce the idea of undoing to solve a proportion.Lesson 2.3:deriving linear expressions from measurementLesson 2.4:introduces direct variation equations as an alternative to solving proportions(a special linear function),create a scatter plot of a real data set,model with a line through the points,and write an equation in the form y=kx to describe that line.Lesson 2.5:introduces the related topic of inverse variation(not a linear function)Lesson 2.7:rules for order of operations by analyzing how the steps in linear expressions that describe“number tricks”undo each other to end with the same numberLesson 2.8:write linear equations to represent sequences of steps and solve those equations by undoing.What does it mean to solve an equation?Is it any more than just undoing the procedure of building an equation?Choose a secret number.Now choose four more non-zero numbers and in any random order add one of them,multiply by another,subtract another,and divide by the final numberRecord in words what you did and your final result on the communicator with a blank Building and Evaluating an Expression or Equation template.(Do not record your secret number.)Switch communicators and have another students find your secret number.?Add 2Ans+2xX+2Multiply by 5Ansx552x Subtract 4 Ans-4524x Divide by 8Ans 85248x 2,5,4,8Add 2,Multiply by 5,Subtract 4,And divide by 8Reveal the results2138What was mystarting number?52421388x To some number,add 3,multiply by 2,add 18,and finally divide by 6.a.Convert the description into an expression,and write an equation that states that this expression is equal to 15.b.Find the starting number if the final result is 15.c.Test your solution to part b using your equation from part a.Use the Build and Undo Expression or Equations Chart to complete the following number trick.Complete the first three columns only.Pick a numberDivide the number by 4Add 7Multiply the result by 2Subtract 8Description/SequenceExpressionPick a number?XDivide by 4 /4x4Add 7 +7-7Multiply by 2 x2/2Subtract 8 -8+8 284x74x274x2784x36181144Here is an equation.What is it saying?First build the equation,then well solve it.32(4)6115xPick a numberx32(4)6115xPlace the Building and Undoing an Expression Template in your Communicator.Record an equation in the cell at the top.Complete the description column using the order of operations.Complete the undo column.Finally,work up from the bottom of the table to solve the equation.Write a few sentences explaining why this method works to solve an equation 32(4)6115xSimplifying the Technique of Solving an Equation(3)7424x-3x4+742-735x4140+3143An equation is a statement that says the value of one expression is equal to the value of another expression.Solving equations is the process you used to determine the value of the unknown that makes the equation true.This is called the solution.In Chapter 3,students use equations to model linear growth and graphs of straight lines and learn the balancing method for solving equations.This chapter builds toward the concept of function,which is formalized in Chapter 8.Lesson 3.1:development of linear growth with recursive sequences.Lesson 3.2:linear plots.Lesson 3.3:walking instructions to study motionLesson 3.4:intercept form of a line with starting value and rate of changeLesson 3.5:rates of change Lesson 3.6:balancing technique for solving equationsLesson 3.7:Model real-world data with linear equationsSolving Equations by Balancing EquationsThe figure illustrates a balanced scale.balanced scale.This is because 4 yellow square tiles balances with 4 square yellow square tiles.Build this scale in front of you.Lets discover some things we can do to balanced scale that will keep it that keep the scale in figure 1 balanced._What would happen if you added 2 yellow squares tiles to both sides of the figure?_What would happen if you added 1 red square tile to both sides of the figure?_What would happen if you added 1 red square to the left side and one yellow square tile to the right side of the figure?_What would happen if you added double the number of tiles on both sides of the figure?_What would happen if you removed one yellow square from the left side and added one red square to the right side of the figure?_What would happen if you cut the number of tiles in half on each side of the figure?_What would happen if you doubled the left side and divided the right side by 2 in the figure?_What would happen if you added one red square to the left side only in the figure?_What would happen if you added one yellow square to the right side only in the figure?_What would happen if you added red square to the left and removed one yellow square from the right in the figure?The figure illustrates a balanced scale.balanced scale.Build this on your scale.How many red or yellow squares would the green rectangle be equal to?Using one of the ideas from above,we can show that the green rectangle isequal to 2 yellow squares.Show at least two ways this can be accomplished.x+-3=-4Make a sketch of the balance scale that matches with this equation.Solve the equation by using the algebra tiles.2x+-3=5-2x+-3=-4+-1x-3+x=2x+1-4=2(x+2)x+4=-2x+-23x+-3=2(x 1)Making the Transition to solving an Equation Algebraically with symbolsIf the equation was 1+2x+3=7 you would have built the balance scale in the figure.One step you might do first is combine the like terms.This would result in the next figure.This figure says that 2x+4=8.Now you might think about remove 4 yellow squares from both sides.This would leave you with the next figure.This figure says that 2x=4.Then you would have divided both sides into two equal groups sothe green rectangle equals 2 yellow squares or x=2.1238combine like terms248remove 4from both sides24divide both sides into 2 equal groupsx=2xxxThis time our steps will be more algebraic,but based upon what we did with the balance scale.Chapter 4 emphasizes slope in the context of finding lines of fit.Lesson 4.1:formula for determining slope Lesson 4.2:use the intercept form to fit lines to data Lessons 4.3 and 4.4:point-slope form through applicationLesson 4.5:Use the point-slope form to fit lines to dataLessons 4.6 and 4.7:method for determining lines of fitLesson 4.8:activity day for reviewing lines of fit.In Chapter 5,students look at systems of linear equations and consider linear inequalities.Then they put these two ideas together to think about systems of linear inequalities.Lessons 5.1 to 5.4:five ways to solve a system of equations:tables,graphs,the substitution method,the elimination method,and row operations on matrices.Lesson 5.5:Inequalities in one variable are introducedLesson 5.6:graph inequalities in two variablesLesson 5.7:graph and solve systemsStudents,through using Discovering Algebra are going to discover and learn much useful algebra along the way.Learning algebra is more than learning facts and theories and memorizing procedures and then trying to apply them through applications sections.Through the text students be involved in mathematics and in learning“how to do mathematics.”Success in algebra is a gateway to many varied career opportunitiesThrough the investigations,students will make sense of important algebraic concepts,learn essential algebraic skills,and discover how to use algebra.that algebra teaching should focus on the basic skills of today,not those of 40 years ago.Problem solving,reasoning,justifying ideas,making sense of complex situations,and learning new ideas independentlynot paper-and-pencil computationare now critical skills for all Americans.In the Information Age and the Web era,obtaining the facts is not the problem;analyzing and making sense of them is.“The Mathematical Miseducation of Americas Youth,”The Phi Delta Kappan.February,1999technology,along with applications,is used to foster a deeper understanding of algebraic ideas.The explorations emphasize symbol sense,algebraic manipulations,and conceptual understandings.The investigative process encourages the use of multiple representationsrepresentationsnumerical,graphical,numerical,graphical,symbolic,and verbalsymbolic,and verbalto deepen understanding for all students and to serve a variety of learning styles.Explorations from multiple perspectives help students simplify and understand what formerly were difficult algebraic abstractions.Investigations actively engage students as they make personal and meaningful connections to the mathematics they discover.Traditional algebra teaches skills and ideas before examples and applications.The investigative approach works the other way.Interesting questions and simple hands-on investigations precede the introduction of formulas and symbolic representations.By providing meaningful contexts for students,the investigations motivate relevant algebraic concepts and processes.The investigations are accessible.They use inexpensive and readily available materials,require little prerequisite technical knowledge,and follow simple procedures.Students can conduct them with a minimum of direction and intervention from you.Teaching with Discovering Algebra decreases the time students spend on rote memorization,teacher exposition,and extended periods of paper-and-pencil drill.It changes the rules for what is expected of students and what they should expect of their teacher.Teaching from Discovering Algebra requires nontraditional thinking and behavior and a nontraditional classroom.Success depends on your sensitivity,patience,enthusiasm,and determination.
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