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CHAPTER TWO Variables Expressions and Properties 2-1 USING VARIABLES TO WRITE EXPRESSIONS Objective: write numerical expressions with variables to represent relations How: take notes, think pair share, read word phrases to convert to algebraic expressions VARIABLE a quantity (or amount) that can change, usually represented with a letter Variable like the word vary meaning it changes or varies. Example: 5a where a is a variable Non-example: 4 2 there is no variable COEFFICIENT A number multiplied by a variable Example: 5a (5xA) Think of Co- like sharing so coefficients have to share multiplication with a variable Think of Efficient- like achieving maximum productivity with minimum wasted effort or expense Non-example: 5+a no multiplication ALGEBRAIC EXPRESSION Mathematical phrase having at least one variable and one operation Example: 5+T or w/7 Non-example: w=8 or 6x8= 48 MEGAN BOUGHT SOCKS ON EBAY FOR $10 A PAIR. How can you represent the total cost of the socks bought? Pairs of Socks Cost 1 $10 x 1 2 $10 x 2 3 $10 x 3 4 $10 x 4 A $10 x A When we don’t know how many pairs of socks Megan bought we can use the variable A to represent potential socks bought. That amount can change which is why we use a VARIable. Word Phrase five minutes more than time t ten erasers decreased by a number n six times a width w n nectarines divided by three eight more than four times an amount x Operation Algebraic Expression Word Phrase five minutes more than time t ten erasers decreased by a number n six times a width w n nectarines divided by three Operation addition subtraction multiplication division eight more than four times multiplication and addition an amount x Algebraic Expression Word Phrase five minutes more than time t ten erasers decreased by a number n six times a width w n nectarines divided by three Operation Algebraic Expression addition t+5 subtraction 10 - n multiplication 6 x W or 6w division n ÷ 3 or n 3 eight more than four times multiplication and addition an amount x 4x + 8 NOW LET’S TRY SOME TOGETHER 12 times a number g The difference of a number m and 18 P pennies added to 22 pennies Yuri walk p poodles and b bulldogs. Write an algebraic expression to represent how many dogs were walked. EXIT TICKET Keeshon bought packages of pens. There are 4 pens in each package. Keeshon gave 6 pens to his friends. Write an expression that show this situation. (Hint- there are 2 operations that take place) STARTER FOR 2-2 Juanita sells homemade jam at the farmers’ market. She sold 35 jars during the first hour and 85 jars during the second hour. Write an algebraic expression to show the number of jars Juanita has left to sell. Explain how the expression relates to the problem. 2-2 PROPERTIES OF OPERATIONS Objective: give missing addends and factors in equations and state the property used How: discuss properties of addition and multiplication then use these properties to label equations and determine missing information COMMUTATIVE PROPERT Y OF ADDITION The order numbers are added does not change the sum. Think about when you commute you can go different ways and still get to work. a+b=b+a 8 + 18 = 18 + 8 6 + c = c + 6 COMMUTATIVE PROPERT Y OF MULTIPLICATION The order numbers are multiplied does not change the product. Think about when you commute you can go different ways and still get to work. axb=bxa 8 x 18 = 18 x 8 6 x c = c x 6 ASSOCIATIVE PROPERT Y OF ADDITION The way numbers are GROUPED does not affect the sum Think my associates/friends: sometimes I hangout with one group sometimes another and they are all my friends a+(b+c)=(a+b)+c 2+(3+4)=(2+3)+4 3+(a+4)=(3+a)+4 ASSOCIATIVE PROPERT Y OF MULTIPLICATION The way numbers are GROUPED does not affect the product Think my associates/friends: sometimes I hangout with one group sometimes another and they are all my friends a(bxc)=(axb)c 2(3x4)=(2x3)4 3x(ax4)=(3xa)x4 IDENTIT Y PROPERT Y OF ADDITION The sum of any number and zero is that number Think about what you can do to a number that won’t change it’s value. Their “name tag” a+0=a 24 + 0 = 24 IDENTIT Y PROPERT Y OF MULTIPLICATION The product of any number and one is that number Think about what you can do to a number that won’t change it’s value. Their “name tag” ax1=a 24 x 1 = 24 NOW LET’S PRACTICE __ x (14x32) = (5x14) x 32 5 + 23 + 4 = 23 + 4 + __ 25 + 0 + 3 = 25 + __ (7 + 12) + 4 = 7 + (12+__) (5 x 7) x (3 x 8) = (5 x 3) x (8 x __) (43 x 1) x 4 = ___ x 4 CHALLENGE (41 x 43) x (3 x 19) = (41 x __) x (19 x 43) (5 + 3) + __ = 5 + (8 + 3) EXIT TICKET 328 x 1 8 + __ = 4 + 8 STARTER 2-3 Can you use Associative, Commutative, or Identity Properties with subtraction or division? Explain. 2-3 ORDER OF OPERATIONS Objective: Use the order of operations to evaluate expressions How: Watch a video, learn a song, and evaluate numeric and algebraic expressions. THERE IS AN AGREED UPON ORDER IN WHICH OPERATIONS ARE CARRIED OUT IN A NUMERICAL EXPRESSION. ORDER OF OPERATIONS ORDER OF OPERATIONS The order to perform operations in calculations Compute inside parentheses. Evaluate terms with exponents. Multiply and Divide from left to right. Add and Subtract from left to right. PEMDAS NOW LET’S PRACTICE! 9 2 - 8 x 3 24 / 4 + 8 + 2 18 – 3 x 5 + 2 49 – 4 x (49 /7) 5 2 – 6 x 0 24 / (4+8) + 2 EXIT TICKET Use parentheses to make each number sentence true. 8 x 9 – 2 – 3 = 53 6 2 + 7 + 9 x 10 = 133 2 2 + 4 x 6 = 48 2-4 STARTER Mrs. Nerren is decorating her rectangular bulletin board by placing stars along the edges. It is 5 feet wide and 3 feet tall. She places stars every 6 inches. How many stars does she need? Explain your reasoning. 2-4 THE DISTRIBUTIVE PROPERT Y Objective: use the distributive property to evaluate expressions and to compute mentally. How: take notes, video clip, work with a partner DISTRIBUTIVE PROPERT Y Multiplying a sum (or difference) by a number gives the same result as multiplying each number in the sum (or difference) by the number and adding (or subtracting) products WHITE BOARD 2-6 STARTER Provide the missing information, then solve. Aki must take turns with his sisters mowing the lawn. One of them must mow the lawn every week. How many times in 12 weeks will Aki mow the Lawn? 2-6 EVALUATING EXPRESSIONS Objective: Evaluate algebraic expressions using substitution. How: take notes on how to replace variables with given numbers and solving the expression. EVALUATE Find the value of an expression EXAMPLES: 5 3 = 125 2+8 = 10 SOLVE! NON-EXAMPLES: 5 3 = 5 x 5 x 5 2 + 8 = W Get a number for an answer!!! SUBSTITUTION Replace the variable with a number EXAMPLE: y+9 Y = 10 10+9= 19 NON-EXAMPLE: y+9 Y = 10 W+9 =9 LET’S PRACTICE Evaluate each expression for 2, 5, and 8. 9x 3x+6 48 ÷ x x(0) 1x x(4) ÷ 2 X 2 + 1 EXIT TICKET Evaluate the expression for the values of n. N 2+3n 3 5 8 12 25 2-7/2-8 STARTER Max’s farm has 480 acres. His farm is divided into fields of n acres each. Write an expression that shows the number of fields on Max’s farm. EXPLAIN YOUR THINKING. 2-7/2-8 USING EXPRESSIONS TO DESCRIBE PATTERNS AND MAKING A TABLE TO SOLVE PROBLEMS Objective: identify missing numbers in a pattern and write algebraic expressions to describe the pattern, make and use tables to solve word problems. How: take notes, read and create tables, discuss patterns with a partner, write expressions. LET’S CHECK OUT OUR ONLINE ACCESS! Go to PEARSON SUCCESS NET on the internet Student Login: Username: student #* Password: [email protected] *Because the username needs to be unique not only within our district but across all districts using Pearson products there are a handful of students that we had to add a “x” to the end of their username to make the username unique. If a student is unable to login with their student # as the username then to try adding a “x” to the end of the username.