gr8_patterns+and+relations

= Patterns and Relations =

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

B1 Graph and analyze two-variable linear relations. [C, ME, PS, R, T, V]

B2 Model and solve problems using linear equations of the form:
 * ax = b
 * x/a = b, a not equal to 0
 * ax + b = c
 * x/a + b = c, a not equal to 0
 * a(x + b) = c [C, CN, PS, V] ||
 * **Planning for Assessment** || **Assessment Strategies** ||
 * Provide students with tables of values for a variety of linear and non-linear relationships (some examples are skydiving free-fall speed vs time, cab fare vs distance traveled, commission earnings vs sales, and area vs side length of a square). Have students graph the ordered pairs in their table of values and compare their graph with another students graph, looking for similarities and differences. Discuss as a class what a linear system is, in light of student observations.

Have students tie knots in a string, measuring the length of the string after each successive knot. Students record the information in a table of values. Have students graph the ordered pairs. Students discuss why they do or do not think the relationship between the number of knots and the length of the string is a linear relationship. Challenge more advanced students to come up with an equation that gives approximately the same results.

Provide students with equations for a variety of linear relationships. Have students generate a table of values by substituting arbitrary values for one variable in the equations. Give students the context or meaning of each equation, and ask them to explain how the real-life meaning of the equation affects the values they choose to record in their table. Finally, have students construct a graph from the equation. As an example, the profit earned at a school dance might be described by the equation P = 5n - 500 where P is the profit in dollars and n is the number of tickets sold. In this situation, students should identify that the number of tickets must be a positive integer and must have a reasonable upper limit. It is also notable that the profit can be negative (loss) if too few tickets are sold. Additionally, students could discuss how the elements of organizing a dance (ticket price, DJ service, decorations) are represented in the equation.

Using one of the linear situations students have already worked with, provide students with the equation and one value in an ordered pair. Ask students to determine the missing value in the ordered pair. Using the school dance example, students would be provided with the equation and asked how many tickets must be sold to earn a profit of $600. || Verify that students are able to identify the characteristics of a linear relationship, and to differentiate between linear and non-linear graphs.

Assess students' abilities to: create a table of values by substituting values for a variable in the equation of a linear relation by substitution construct a graph from a linear relation determine the missing value in an ordered pair for a given linear relation

Through discussion and students' writing, assess the degree to which they are able to describe relationships between the variables in a linear relationship. ||
 * Have students model and solve linear equations. They will need to have a way to represent both integers and variables. For instance, students may use round chips to represent integers (red chips for positive numbers, blue chips for negative integers) and squares to represent variables (red for positive variables and blue for negative variables).
 * Have students model and solve linear equations. They will need to have a way to represent both integers and variables. For instance, students may use round chips to represent integers (red chips for positive numbers, blue chips for negative integers) and squares to represent variables (red for positive variables and blue for negative variables).

Provide a number of equations for students to solve using these manipulatives. See example 1 2(x + 3) = 14 and example 2 3x + 1 = 5 on the example page [|algebra examples.pdf].

Have students draw a visual representation of the steps they have used to solve a given linear equation (such as in the example) and record each step symbolically. || Verify that students can
 * model a given linear equation and solve it using concrete models.
 * drawn a visual representation of the steps used to solve a linear equation and record each step symbolically.
 * correctly apply the distributive property in solving a particular linear equation
 * verify the solution to a given linear equation using a variety of methods (i.e., concrete materials, diagrams, and substitution). ||
 * Solving simple linear equations can be thought of conceptual in two different ways:
 * Solving simple linear equations can be thought of conceptual in two different ways:

1) Stress the idea that an equation is like a balanced scale, When solving the equation what ever is done to one side of the equation the same thing must be done to the other side in order to keep the scale balanced.

or

2) Solving an equation can be analogous to unwrapping a present. To wrap a present you put something in a box, put paper around it, then tape it. To unwrap you take the tape off, take the paper off, then discover what is in the box. Similarly, to solve an equation the opposite operations must be done in reverse order to determine the value of x. For example. If a number was doubled then 3 was added and the result is 9. Then to determine x, 3 must first be subtracted from 9, and then the result should be halved.

You also might want to note, that in terms of solving linear equations the variable in this case represents an unknown, not a quantity that varies. || * determine the missing value in an ordered pair for a given equation
 * create a table of values by substituting values for a variable in the equation of a given linear relation
 * construct a graph from the equation of a given linear relation (limited to discrete data)
 * describe the relationship between the variables of a given graph ||
 * || * model a given problem with a linear equation and solve the equation using concrete models (e.g., counters, integer tiles)
 * verify the solution to a given linear equation using a variety of methods, including concrete materials, diagrams, and substitution
 * draw a visual representation of the steps used to solve a given linear equation and record each step symbolically
 * solve a given linear equation symbolically
 * identify and correct an error in a given incorrect solution of a linear equation
 * apply the distributive property to solve a given linear equation (e.g., 2(//x// + 3) = 5; 2//x// + 6 = 5; …)
 * solve a given problem using a linear equation and record the process ||