Lesson ACTIVITY: Tree Growth

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Lesson 3.1 - ACTIVITY: Tree Growth Obj.: use arrow diagrams to represent expressions. evaluate expressions. write expressions to model realworld situations.

Coefficient - The factor or factors multiplying a specific variable in a given term. Evaluate an expression - Find the value of an expression.

These guidelines tell you the order in which the operations must be performed.

$Perform the operations within any grouping symbols, which include parentheses, brackets, and fraction bars. 2. Evaluate powers. 3.$

2 Lesson ACTIVITY: Tree Growth Obj.: use arrow diagrams to represent expressions. evaluate expressions. write expressions to model realworld situations. Algebraic expression - A symbol or combination of symbols that contains one or more numbers, variables, and arithmetic operation symbols. Coefficient - The factor or factors multiplying a specific variable in a given term. Evaluate an expression - Find the value of an expression. Order of operations - When you evaluate expressions that contain many mathematical operations, guidelines must be followed. These guidelines tell you the order in which the operations must be performed. ***Order of Operations Guidelines*** To evaluate an expression that contains more than one operation, the following order is used: 1. Perform the operations within any grouping symbols, which include parentheses, brackets, and fraction bars. 2. Evaluate powers. 3. Perform multiplications and divisions in order from left to right. 4. Perform additions and subtractions in order from left to right. In this Activity, you will write expressions to describe the growth of trees. Go to pages 57 & 58 of the textbook. This sequence of drawings shows successive years in the tree s growth. Answer questions 1-8 in the space below Complete the table by measuring each of the drawings and using a proportion to find the actual 3. height of each tree. work a. b EXAMPLE 1 Evaluate the expression 4t + 8 2t 2 + t for t = 5 **Practice for Lesson 3.1 pages 59 & all**

3 Lesson INVESTIGATION: From Expression to Equation Obj.: write an equation that models a real-world situation. Equation - A statement that two expressions are equal. ***STRATEGIES FOR CONSTRUCTING MATHEMATICAL MODELS*** Write a simple word equation. Make sure your words state the big idea of the problem. Then gradually add smaller pieces of the problem until you have a symbolic equation. Make a table. Your table should contain specific values for the quantities that you are investigating. Once your table is complete, look for a pattern. Draw a sketch or diagram. This strategy is often used in situations that include aspects of geometry. Check your model. When you have found a model you think is a good one, substitute values for any variables in the model to see if they predict reasonable results. EXAMPLE 1 A giant sequoia tree has an initial diameter of meters and it grows meters in diameter each year thereafter. a. Find an equation that models the diameter d of the tree after t years. D = t b. Find the diameter of the tree after 100 years m **Practice for Lesson 3.2 pages 63-66, 1-25 all**

Lesson 3.3 - Using Arrow Diagrams and Tables to Solve Equations Obj.: use arrow diagrams to solve two step equations. use tables to solve equations.

Properties of Equality - The properties that, when applied to an equation, result in an equivalent equation.

4 Lesson Using Arrow Diagrams and Tables to Solve Equations Obj.: use arrow diagrams to solve two step equations. use tables to solve equations. Equivalent equations - Equations that have the same solution (or solutions). Inverse operations - Two operations that undo each other. For example, addition and subtraction are inverse operations. Properties of Equality - The properties that, when applied to an equation, result in an equivalent equation. The Properties of Equality state that an equation can be changed into an equivalent equation by; exchanging the expressions on the two sides of the equation; adding the same number to each side of the equation; subtracting the same number from each side of the equation; multiplying each side of the equation by the same non-zero number; dividing each side of the equation by the same non-zero number. Example 1 a. Draw an arrow diagram that models the solution of the equation. a = 1.8 b. Solve the equation for a. a = 8 a = 1.8 Example 2 a 2.6 Use a calculator table to solve 3 = 1.8 You must use parentheses!!!! Y= (a-2.6)/3 **Practice for Lesson 3.3 pages 71-74, 2-32 even**

5 Lesson Review And Practice Obj.: solve problems that require previously learned concepts and skills. EXAMPLE 1 Fill in the blank. a. The word means parts per hundred. b. Subtraction and are called inverse operations. EXAMPLE 2 Choose the correct answer. Which percent is equivalent to 0.025? A. 25% B. 2.5% C % D % EXAMPLE 3 Multiply or divide. Write your answer in simplest form. a b EXAMPLE 4 Add or subtract. a b EXAMPLE 5 Write the phrase as a variable expression. a. a number multiplied by 8 b. twice a number divided by 3 EXAMPLE 6 Evaluate the expression. a. 6x x 2 + y 8 for x = 3 and y = 4 b. 5+ m2 m 2 5 for m = 2 Do Lesson 3.4 R. A. P. pages 73 & 74, 2-32 even.

6 Lesson INVESTIGATION: Solving Multi-Step Equations Obj.: combine like terms in an expression. solve equations. use equations to solve real-world problems. Like terms - Terms in which the variable parts are exactly the same. Term - The parts (variables, numbers, or products or quotients of variables or numbers) of an expression that are added. EXAMPLE 1 Combine like terms in each expression. a. 12d - 7d 5d b. 5m - 2n - 4m m - 2n c. 14x + 2y - 3z + x - 6y 15x - 4y - 3z EXAMPLE 2 Solve 7b b = 4 for b. -4/5 EXAMPLE 3 Solve 12t - 2(3t - 1) = 44 for t. 7 EXAMPLE 4 Solve 8q - 10 = q + 4 for q. 2 **Practice for Lesson 3.5 pages 79-81, 2-30 even**

7 Lesson INVESTIGATION: Solving Inequalities Obj.: solve inequalities. graph inequalities. Properties of Inequality - The properties that, when applied to an inequality, result in an equivalent inequality. Properties of Inequality Any number can be added to or subtracted from both sides of an inequality without changing the direction of the inequality. When both sides of an inequality are multiplied or divided by a positive number, the direction of the inequality is unchanged. When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed. EXAMPLE 1 Solve the inequality 5-4x > 1 for x. x < 1 EXAMPLE 2 a. Solve the inequality 6z < 1-4(3-2z) for z. z > 11/2 b. Graph the solution set on a number line. **Practice for Lesson 3.6 pages 85 & 86, 1-13 all**

8 Lesson Compound Inequalities Obj.: write compound inequalities. graph compound inequalities on a number line. solve compound inequalities. Compound inequality - Two inequalities that are joined by the word and or the word or form a compound inequality. ***Compound Inequalities*** ***Compound Inequalities*** Joined by and Joined by or x >1 and x < 3 x < 0 or x > 2 A solution to a compound inequality joined by the word and is any number that makes both inequalities true. Its graph is the intersection of the graphs of the two inequalities. To find the intersection of the two graphs, graph each inequality and find where the two graphs overlap. So, the graph of x >1 and x < 3 is A solution to a compound inequality joined by the word or is any number that makes at least one of the inequalities true. Its graph is the union of the graphs of the two inequalities. To find the union of the two graphs, graph each inequality. So, the graph of x < 0 or x > 2 is These inequalities can be combined and written without using and. 1 < x < 3 These inequalities cannot be combined. They must be written using the word or x < 0 or x > 2 Example 1 Write an inequality for each statement. a. t is less than 4 and greater than 0. b. p is greater than 1 and no more than 5. 0 < t < 4 1 < p < 5 c. q is less than -2 or greater than 1. d. b is between 0 and 6 inclusive. q < -2 or q > 1 o < b < 6 Example 2 Graph each compound inequality on a number line. a. -4 < t < 1 b. w < 1 or w > 2

9 Example 3 Solve -6 < 2(d - 3) < 4 for d. 0 < d < 5 (3.7 cont.) Example 4 a. Solve 3-2b > 1 or b - 6 > 4 for x. b < 1 or b > 10 b. Graph the solution. **Practice for Lesson 3.7 pages 89 & 90, 1-13 all**

Exponents. Reteach. Write each expression in exponential form (0.4)

Exponents. Reteach. Write each expression in exponential form (0.4) 9-1 Exponents You can write a number in exponential form to show repeated multiplication. A number written in exponential form has a base and an exponent. The exponent tells you how many times a number,