gr9_Linear+Equations+and+Inequalities

= Solving Linear Equations and Inequalities =

//It is expected that students will://
 * Prescribed Learning Outcomes


 * B3** model and solve problems using linear equations of the form
 * ax = b
 * x/a = b, a cannot equal 0
 * ax+b=c
 * x/a+b=c, a cannot equal 0
 * ax=b+cx
 * a(x+b)=c
 * ax+b=cx+d
 * a(bx+c)=d(ex+f)
 * a/x=b, a cannot equal 0
 * where a, b, c, d, e, and f are rational numbers [C, CN, PS, V]


 * B4** explain and illustrate strategies to solve single variable linear inequalities with rational coefficients within a problem-solving context [C, CN, PS, R, V] ||
 * **Planning for Assessment** || **Assessment Strategies** ||
 * * ask students to explain the difference between an equation and an expression, giving examples of each.


 * ask students to explain the difference between evaluating and solving, giving examples of each.


 * Have students set up a t-chart labeled left side and right side to determine if a given rational number is a solution to a given linear equation. It is important to stress the fact that the left and right side of the equations must be treated as two different expressions and evaluated separately. Students should be able to do this verification both with and without technology.


 * Provide students with a selection of linear equations to solve. Have students alternate between annotating their work to describe the processes they used and checking their answer using substitution.


 * Provide students with a selection of linear equations. Have students work in partners to solve the equations. One student will hold the pencil and another student will explain to the student what to write. The student holding the pencil is only supposed to write the steps his partner instructs him too. The students should take turns being recorder and instructor after each question. || Verify that students can:


 * model the solution of a given linear equation using concrete or pictorial representations, and record the process

x=3 2x=6 2x+1=7 (2x+1)/3=7/3 4(2x+1)/3=28/3 4(2x+1)/3 - 2x = 28/3 - 2x
 * determine, by substitution, whether a given rational number is a solution to a given linear equation. It is important for students to understand that when verifying a solution to an equation the students must treat the right and left sides of the equation as expressions and evaluate them separately and that they are not allowed to do the same thing to both sides of the equal sign such as multiply through by the common denominator. ||
 * * Have students create linear equations by asking them to start with the answer and work backwards to create the equation. For example:

Then have the students trade questions with a partner and solve each other's questions.


 * Give students a selection of linear equations with solutions some of which are incorrect. Ask students to find which solutions are incorrect, identify the mistakes, explain the errors in thinking that were made and provide correct solutions for each equation. || Verify that students can :


 * solve a given linear equation symbolically

Note: you may assign partners by handing out pairs of equivalent polynomial expressions and requiring students find their match. * Have students translate linear expressions into words and vice versa
 * identify and correct an error in a given incorrect solution of a linear equation ||
 * * Have students work in partners to brainstorm as many English words or phrases to indicate each of the following mathematical symbols +, -, ×, ÷, =.


 * Have students describe orally or draw a picture about a linear relation based on personal anecdotes and have their partner use a linear equation to model the situation and solve the problem. For example:

-My mom will be 25 years younger than my grandma, next year my grandma will be double my mom's age

-I have some change in my pocket consisting of nickels and dimes. If the value of the coins is $1.40 and I have 2 more dimes than nickles, how many of each type of coin do I have?

-I played the ski jump game on Wii Fit last night. The distance of my second jump was 5 feet more than 1.5 times my first jump. Together the distance of the two jumps was 240 feet. How far were each of my jumps? || Verify that students can:


 * write an expression representing a given pictorial, oral, or written pattern


 * write a linear equation to represent a given context


 * describe a context for a given linear equation


 * solve using a linear equation a given problem that involves pictorial, oral, and written linear patterns


 * represent and solve a given problem using a linear equation and record the process ||
 * * Provide students with a selecting of true inequality statements such as 3 or = -1. Ask students to perform the same arithmetic operation to both sides of the inequality and then check to see if the inequality is still true. The operations should include addition, subtract, multiplication and division by both positive and negative rational numbers. Have students work in groups to summarize their results and create a rule for which operations preserve the sense of the inequality. Note: An extension would be to square both sides and see if the sense is always preserved.


 * Display a simple equation on the board or overhead such as 3x+2=8. Ask students to determine a value for x that satisfies the equation. Ask several students for their answers. Then display the same equation as an inequality and ask students to determine a value for x that satisfies the inequality. Plot and compare students' answers. Ask students if rational solutions are possible if they give only examples of integer solutions. Discuss as a class why there are many solutions. Can students display all solutions efficiently?


 * Ask students to explain in their journal the similarities and differences between the process for solving and checking the solution to a given linear equation to the process for solving and checking the solutions to a given linear inequality.


 * Have students create and solve a linear inequalities involving brackets, rational numbers and collecting like terms. The students should then trade questions with a partner and verify they get the same solution. (Note: You may assign partners by handing out English phrases and the corresponding algebraic expression and requiring students to find their match) Then ask students to create a poster with the solution to the equation and a detailed explanation of the steps. The poster should include a graph of the solution on a number line and an explanation of how to verify the solution is correct. Display the posters in your classroom. || Verify that students:
 * understand that when multiply or dividing an inequality by a negative number that the sense of the inequality is changed.


 * understand that when adding or subtracting the same number to both sides of an inequality and when multiplying or dividing an inequality by a positive number that the sense of the inequality does not change


 * can determine if a given rational number is a possible solution of a given linear inequality


 * can solve a given linear inequality algebraically and explain the process orally or in written form.


 * can graph the solution of a given linear inequality on a number line.

Look for evidence that students:
 * understand that when solving an equation, to check to make sure their answer is correct they just need to evaluate each side of the equation and verify the two solutions are equal.
 * understand that when solving an inequality to verify their answer is correct they need to check to see if both the equation and inequality are true. For example when solving 2x -3 <7+x and obtaining a solution of x<10, each side should be evaluated for x=10 to show equality and for a number less than 10 to show that the inequality is true. It is important to check for equality regardless of whether or not the inequality sign is also equal to. ||