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1、Solving Linear Equations in Algebra I,Jim Rahn J www.J,What must all Algebra I students know about solving and working with Linear Equations?,Students should Understand the big ideas of equivalence and linearity Modeling real situations with variables Use appropriate tools such as algebra tiles and
2、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),Goals from the
3、New Jersey Algebra I Core Content Standards,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 numbe
4、rs 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.,Lineari
5、ty 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 propor
6、tionately. 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. Descr
7、ibe, 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 contex
8、t of the problem.,Benchmarks Related to Linear Relationships,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. S
9、olve 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.,Building Understanding for Writi
10、ng and Evaluating Expressions,Algebra should become a language through which we can describe various situations,What is three plus five times two?,Try entering this problem on the homescreen of the graphing calculator,How many ways can you enter it on the homescreen?,Is there an order for the operat
11、ions when the problem is written horizontally? 4+3*2,Order of Operations,Evaluate expressions within parentheses or other grouping symbols. Evaluate all powers. Multiply and divide from left to right. Add and subtract from left to right.,How would you have to write 4 + 32 so the answer is 14?,Learni
12、ng to build mathematical expressions,Lets 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, d
13、ivide 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 perform
14、ing 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
15、the four steps equivalent to this one relationship?,After we have written the expression we can test it:,On the graphing calculator homescreen type: 6 = x:10-(2x+6)/2,= means STO,General Format: Your Number = x:10-(2x+6)/2,Confirm 20 and -4,Using the Description/Expression Template Pick any number D
16、ivide the number by 4 Add 7 Multiply the result by 2 Subtract 8 Find the value of your expression when x=2, -5, 8,Set up the expression for this problem:,Number Tricks,Each 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 ans
17、wer 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.,This is a pretty complex expression. Can we put these in an equation and solve for x?,If you
18、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?,You create a trick,Create your own trick that ha
19、s 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
20、same result no matter what number you choose. Can you explain why this number trick work?,Analyzing a Number Trick,Given the expression on the left, you might want to think of subtraction as adding the opposite and re-write the expression Write, in words, the number trick that is described above. Te
21、st 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?,Daxun, Lacy, Claudia, and Al are working on a number trick. Here are the number sequences their number trick generates:,a. Describe the
22、 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 measurement Les
23、son 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
24、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 number Lesson 2.8: write linear equations to represent sequences of steps and solve those equatio
25、ns by undoing.,In Chapter 2,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
26、final number Record 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 2,Ans+2,x,X+2,Multiply by 5,A
27、nsx5,Subtract 4,Ans-4,Divide by 8,Ans 8,2,5,4,8,Add 2, Multiply by 5, Subtract 4, And divide by 8,Reveal the results,What was my starting number?,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 th
28、at 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.,Solve Equations is Just Undoing Operations,Use the Build and Undo Expression or Equations Chart to complete the following number trick. Complete
29、the first three columns only. Pick a number Divide the number by 4 Add 7 Multiply the result by 2 Subtract 8,Solve Equations is Just Undoing Operations,36,18,11,44,Instead of building an equation, lets start with an equation and solve it,Here is an equation. What is it saying? First build the equati
30、on, then well solve it.,Pick a number,x,Try solving these equations,Place 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 th
31、e bottom of the table to solve the equation. Write a few sentences explaining why this method works to solve an equation,Simplifying the Technique of Solving an Equation,-3,x,4,+7,42,-7,35,x4,140,+3,143,An equation is a statement that says the value of one expression is equal to the value of another
32、 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
33、chapter builds toward the concept of function, which is formalized in Chapter 8.,Chapter 3,Lesson 3.1: development of linear growth with recursive sequences. Lesson 3.2: linear plots. Lesson 3.3: walking instructions to study motion Lesson 3.4: intercept form of a line with starting value and rate o
34、f change Lesson 3.5: rates of change Lesson 3.6: balancing technique for solving equations Lesson 3.7: Model real-world data with linear equations,Solving Equations by Balancing Equations,The figure illustrates a balanced scale. This is because 4 yellow square tiles balances with 4 square yellow squ
35、are 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 bot
36、h 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 lef
37、t 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 t
38、he 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. Build this on your scal
39、e. 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 is equal to 2 yellow squares. Show at least two ways this can be accomplished.,x + -3 = -4,Make a sketch of the balance scale that matches with this equati
40、on. 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 +-2,3x + -3 =2(x 1),Making the Transition to solving an Equation Algebraically with symbols,If the equation was 1 + 2x + 3 = 7 you would have built the balance scale in the figure
41、. 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
42、sides into two equal groups so the green rectangle equals 2 yellow squares or x = 2.,This 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.,Chapter 4,Lesson 4.1: formula for determining slope L
43、esson 4.2: use the intercept form to fit lines to data Lessons 4.3 and 4.4: point-slope form through application Lesson 4.5:Use the point-slope form to fit lines to data Lessons 4.6 and 4.7: method for determining lines of fit Lesson 4.8: activity day for reviewing lines of fit.,In Chapter 5, studen
44、ts 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 operat
45、ions on matrices. Lesson 5.5: Inequalities in one variable are introduced Lesson 5.6: graph inequalities in two variables Lesson 5.7: graph and solve systems,Students, through using Discovering Algebra are going to discover and learn much useful algebra along the way. Learning algebra is more than l
46、earning 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 opportunities,Through the investigatio
47、ns, 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 situation
48、s, 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 Ka
49、ppan. February, 1999,Michael T. Battista of Kent State University writes,technology, 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 representationsnumerical, graphical, symbolic, and verbalto deepen understanding for all students and to serve a variety of learning styles. Exploratio