Math 7 Notes Unit One: Algebraic Reasoning Numbers and Patterns Objective: The student will learn to identify and etend patterns. Note: This is not one of the 7 th grade benchmarks. However, this material will support Ch 4-5 and CCSD Benchmark 3.: The student will create tables, charts, and graphs to etend a pattern in order to describe a linear rule, including integer values. Sequence a set of numbers in a particular order. Each number is called a term of the sequence. Arithmetic Sequence a sequence in which every term after the first term is obtained by adding a fied number. Eamples: 5, 0, 5, 0, 5,, 5, 8,, 4, 3,, 9, 7, 35, General Strategy To find the net term in a sequence, subtract the terms. Notice 3 = 8, 9 = 8, suggesting you are adding 8 to each term to find the net term. Some patterns are not arithmetic sequences. The number you add to each term is not constant, but there is a pattern to the number that is added to find the net term. Eamples:, 3, 5, 8,, (add, then add, add 3 and so on), 3, 7, 3,, (add, then add 4, add 6 and so on) Students will also be responsible for other patterns like the following: Eamples: 4,, 6, 3, (divide the preceding term by ),,,,,, (,, keeping the same symbol pattern),,,, Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page of 4 Revised 8/00
Eponents Syllabus Objective: (.) The student will translate written and oral epressions including ratios, proportions, eponents, radicals, scientific notation, and positive and negative numbers to numerical form. Eponent the superscript which tells how many times the base is used as a factor. eponent 3 base In the number 3, read to the third power or cubed, the is called the base and the 3 is called the eponent. Eamples: 3 5 6 4 = = 5 5 = 6666 To write an eponential in standard form, compute the products. i.e. 5 = 5 5= 5 Special Case Eample: 0 = 0 = 0 00 = 3 0 000 What pattern allows you to find the value of an eponential with base 0 quickly? Answer: The number of zeroes is equal to the eponent! Caution: If a number does not have an eponent, it is understood to have an eponent of ONE! Writing Numbers in Eponential Form Eamples: Write 8 with a base of 3. 8 = 3, 8 = 3 3 3 3, therefore 8 = 3? 4 Write 5 with a base of 5.? 3 5 = 5, 5 = 5 5 5, therefore 5 = 5 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page of 4 Revised 8/00
Let us look at a pattern that will allow you to determine the values of eponential epressions with eponents of or 0. 4 4 4 = iii= 6 3 = 3i3i3i3= 8 4 = 4i4i4i4= 56 3 3 3 = i i = 8 3 = 3i3i3= 7 4 = 4 i4 i4 = 64 = i = 4 3 = 3i3= 9 4 = 4 i4 = 6 =? = 3 =? 3 = 3 4 =? 4 = 4 0 0 0 0 0 0 =? = 3 =? 3 = 4 =? 4 = Any number to the power of is equal to the number. That is, n = n. Any number to the power of 0 is equal to one. That is, n 0 =. Again remind students that if there is no visible eponent, the eponent is always. Metric Measurement Syllabus Objective: (6.) The student will compare units of measure within the same system. The customary system is the measurement system we use in the United States. The metric system is used almost everywhere else in the world. We will concentrate on the metric system in this chapter (and address the customary system in Chapter 5). Measurement the metric system in particular is embedded in the science program at the middle school level. Measurement is an objective is to be addressed repeatedly throughout the science course. Be sure to collaborate with your science department! If you do not have a measurement tool, like a ruler, measuring cup or a scale, it is good to have a benchmark or estimate that you can use. It will also help you to choose the appropriate measurement when asked to measure an object, as well as make a comparison between the two systems of measurement. You may want to make an eercise having students determine their own benchmarks. Metric Units: LENGTH Unit Abbr. Benchmark millimeter mm thickness of a CD (or dime) centimeter cm width of your pinky finger meter m about the distance from the floor to your belly button (for an average 6 th grader) kilometer km about half a mile Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 3 of 4 Revised 8/00
Metric Units: WEIGHT/MASS Unit Abbr. Benchmark milligram mg grain of sand gram g large paper clip kilogram kg hardcover tetbook Metric Units: CAPACITY Unit Abbr. Benchmark milliliter ml or ml drop from eyedropper liter L or l large quart; liter bottles now common for soda Converting Units: Metric System kilo- hecto- dekaor deca- 3 0 or 000 0 or 00 0 or 0 base or unit 0 0 or deci- centi- milli- 0 or 0. 0 or 0.0 3 0 or 0.00 Each number increases by a power of 0 as it moves one column to the right. (Multiply by 0.) Each number decreases by a power of 0 as it moves one column to the left in the table. (Divide by 0.) You can use both conversion factors and proportions to convert metric units. However, since the system is based on factors of 0, you are multiplying or dividing by powers of 0. All we need to know is how to move the decimal point what direction and how many places. Eample:.005.05.5 5. 5 50 5,00 We need to start with the meaning of the metric prefies. Again, note that to move from one unit to another is simply multiplying or dividing by 0. kilo- 000 hecto- 00 deka- or deca- 0 base or unit (meter, liter, gram) deci-. centi-.0 milli-.00 Note: If the prefi ends in an a or o, the number is greater than one. If the prefi ends in an i, the number is less than one. Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 4 of 4 Revised 8/00
A way to remember the order of these units is to think: King Henry Doesn't Usually Drink Chocolate Milk or King Henry Died By Drinking Chocolate Milk or King Henry Doesn t Bother Drinking Chocolate Milk where the first letter matches the first letter of the prefi, and the U refers to the unit or the B refers to the base (meter, liter or gram). If you memorize them in this order, you will know the answer to our questions of how to move the decimal point. For eample, let s convert 4 hectometers to centimeters. We would list: km, hm, dkm, m, dm, cm and mm. Then we need to determine, how many jumps I would make from hectometers to get to centimeters. km hm dkm m dm cm mm I would jump four places, to the right. I will move my decimal that way, filling in zeros as place holders: 4 4.0 4.0 40000 Therefore, 4 hm = 4,000 cm. Eample: Convert 54,653 m to km. Let s determine the number of jumps and the direction to move. km hm dkm m dm cm mm We need to move the decimal 3 places to the left. 54,653 54653.0 5 4 6 5 3. 0 54.653 Therefore, 54,653 m = 54.653 km. Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 5 of 4 Revised 8/00
Applying Eponents Syllabus Objective: (.) The student will translate written and oral epressions including ratios, proportions, eponents, radicals, scientific notation, and positive and negative numbers to numerical form. Scientific Notation: A number is written in scientific notation when it is written as a product of a number between one and ten and some power of ten. Eample: 4. 3 0 Converting Scientific Notation to Standard Form 5 To convert a number written in scientific notation to standard form, the eponent tells you how many times to move the decimal point to the right or left. 5 Eample: Write 4.3 0 in standard form. The eponent 5 indicates you move the decimal point 5 places to the right. 5 4.3 0 = 43,000 7 Eample: Write 5.4 0 in standard form. The eponent 7 indicates you move the decimal point 7 places to the right. 7 5.4 0 = 54,000,000 Converting a Number in Standard Form to Scientific Notation To convert a number in standard form to scientific notation, rewrite the number as a product of the number between one and ten and some power of ten. Eample: Write 630 in scientific notation.? Rewrite 630 as a product: 6.3 0. How many places do you need to move the decimal point to get 630? places, so 630 = 6.3 0 Eample: Write 735,000 in scientific notation.? Rewrite 735,000 as a product: 7.35 0. How many places do you need to move the decimal point to get 735,000? 5 5 places, so 735,000 = 7.35 0 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 6 of 4 Revised 8/00
Order of Operations Syllabus Objective: (.) The student will evaluate variable and numerical epressions, including powers, using the order of operations. The Order of Operations is just an agreement to compute problems the same way so everyone gets the same result, like wearing a wedding ring on the left ring finger or driving on the right side of the road. Order of Operations (PEMDAS or Please ecuse my dear Aunt Sally s loud radio)*. Do all work inside the Parentheses and/or grouping symbols.. Evaluate Eponents. 3. Multiply/Divide from left to right.* 4. Add/Subtract from left to right.* *Emphasize that it is NOT always multiply-then divide, but rather which ever operation occurs first (going from left to right). Likewise, it is NOT always add-then subtract, but which of the two operations occurs first when looking from left to right. Eample: Simplify the following epression. 3+ 5 3+ 5 3+ 0 3 Underline the first step. Simplify underlined step, & underline net step. Repeat simplify and underline process until finished. Note: each line is simpler than the line above it. Eample: Simplify the following epressions. (a) 3+ 5 3+ 5 Work: 3+ 0 3 (b) 4+ 4 6 + 4+ 4 6 + 4+ 4 + Work: 4+ 8 + + 3 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 7 of 4 Revised 8/00
(c) 8 (+ 3) 5 8 8 (+ 3) 5 8 8 4 5 8 Work: 5 8 0 8 Properties of Real Numbers Syllabus Objective: (.0) The student will identify the Distributive, Commutative, and Associative Properties of Rational Numbers to solve problems. Syllabus Objective: (.) The student will apply the Distributive, Commutative, and Associative Properties of Rational Numbers to solve problems. The properties of real numbers are rules used to simplify epressions and compute numbers more easily. Property Operation Algebra Numbers Key Words COmmutative + a+ b= b+ a 3+ 4= 4+ 3 Change Order COmmutative a b = b a 5 = 5 Change Order Associative + ( a+ b) + c = a+ ( b+ c) (4 + 9) + 6 = 4 + (9 + 6) Change Grouping Associative ( a b) c= a ( b c) 9(5) = (95) Change Grouping Identity + a+ 0 = a 7+ 0= 7 Add 0 Identity a = a 6 = 6 Multiply by Distributive ab ( + c) = a b+ a c 5(3) = 5(0) + 5(3) 00 + 5 5 Distribute Over ( ) Eamples: Name the property shown below.. 4+ 0= 4 Identity Property of Addition. (6 + 3) + = 6 + (3+ ) Associative Property of Addition 3. 7 5 = 5 7 Commutative Property of Multiplication 4. 4(7 + 3) = 47 + 43 Distributive Property 5. 4(73) = (47)3 Associative Property of Multiplication 6. 4(73) = 4(37) Commutative Property of Multiplication Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 8 of 4 Revised 8/00
Nevada CRT eample: Zachary puts some cherry and some lemon jellybeans into each of 6 bags. He puts 50 cherry jellybeans and 4 lemon jellybeans into each bag. Which epression could be used to determine the total number of jellybeans of each flavor that Zachary puts in the bags? A 6 54 B 6 50+ 4 C 6 + (50 4) D 6 (50+ 4) Variables and Algebraic Epressions Syllabus Objective: (.) The student will evaluate variable and numerical epressions, including powers, using the order of operations. Syllabus Objective: (3.3) The student will evaluate formulas and algebraic epressions for given values of a variable. Variable a letter or symbol that represents a number. Eamples: a, b, n,, _,, Constant a value that does not change; it is represented by a number 3 Eamples: 5, 0,, ½,, 5 n, 4 y + 3 y, 8, Algebraic Epression an epression that consists of numbers, variables, and operations. Eamples:, 5n, 4y + 3y, 8, 3 Evaluating Algebraic Epressions. Substitute the assigned value to each variable. Simplify the epression, using Order of Operations if there is more than one operation. Order of Operations From left to right, in this order:. Parentheses or Grouping. Eponents 3. Multiply / Divide 4. Add / Subtract Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 9 of 4 Revised 8/00
Eamples: Evaluate 3 5 for = 4. 3(4) 5 Substitute 4 for 5 Simplify 7 Evaluate y + 5 for = 40 and y = 3. 40 (3) + 5 Substitute 40 for and 3 for y 0 6 + 5 Simplify: divide and multiply from left to right 4 + 5 Simplify: subtract and add from left to right Translate Words into Math Symbols Syllabus Objective: (.) The student will translate written and oral epressions including ratios, proportions, eponents, radicals, scientific notation, and positive and negative numbers to numerical form. Operation Verbal Epression Algebraic Epression Addition + a number plus 7 n + 7 Addition + 8 added to a number n + 8 Addition + a number increased by 4 n + 4 Addition + 5 more than a number n + 5 Addition + the sum of a number and 6 n + 6 Addition + Tom s age 3 years from now n + 3 Subtraction a number minus 7 7 Subtraction 8 subtracted from a number 8 Subtraction a number decreased by 4 4 Subtraction 5 less than a number 5 Subtraction the difference of a number and 6 6 Subtraction Tom s age 3 years ago 3 Multiplication ( ) multiplied by a number n Multiplication ( ) 9 times a number 9n Multiplication ( ) the product of a number and 5 5n Division a number divided by Division the quotient of a number and 5 Division 8 divided into a number 5 8 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 0 of 4 Revised 8/00
More Advanced Translations Operation Verbal Epression Algebraic Epression Multiplication twice a number Addition + two consecutive integers, + Addition + consecutive odd integers Let = st odd, + = nd odd Addition + consecutive even integers Let = st even, + = nd even Subtraction separate 5 into two parts, 5 Multiplication number of cents in quarters 5 Eamples: Write an algebraic epression for each of the following epressions.. 5 subtracted from twice a number n n 5. the product of 8 and y 8y 3. 4 less than 4 Simplifying Algebraic Epressions Syllabus Objective: (.4) The student will simplify variable epressions. Syllabus Objective: (.5) The student will identify and use properties of real numbers to simplify variable epressions, including absolute value of integers. Syllabus Objective: (3.4) The student will simplify variable epressions by combining like terms. Term: A number, a variable, or a product or a quotient of a number and variable. At term is not a sum or difference. Eamples: 5,, 5, 5 are all terms. Coefficient: A numerical factor in a term. Eample: In 8 3 the coefficient is 8. Like Terms: Terms that have the same variables with the same corresponding eponents. For eample, in 9 + 4y + 6y, 4 y and 6y are the like terms. Student may incorrectly identify 5 4, 4, and 4 as being like terms. While each term has the same coefficient of 4, these are NOT like terms because each term has a different variable part. Remember is not the same as ( ), which is different from 5 ( ). Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page of 4 Revised 8/00
Combining Like Terms To combine like terms, group the terms with the same variable raised to the same power. Then add or subtract the coefficients as indicated. Eample: 5 + = (5 + ) = 7 Distributive Property Eample: ( c + 4) = Distributive Property ( c) + ( 4) = c + 8 Eample: 8 + 9 + 7+ + 4+ 3 Mark like terms. (8 + ) + (9+ 4 ) + (7 + 3) (8 + ) + (9 + 4) + (7 + 3) 0 + 3+ 0 Simplify. Group like terms. Group coefficients of like terms. An important link can be made to previous learning for students. Review place value and epanded notation. Eample: 67 = 6(00) + 7(0) + () = 6(0 ) + 7(0) + () This parallels polynomials = 6 + 7+ Now show that you combine like terms in algebra the same way you combine terms in arithmetic. To add horizontally, from left to right, group the hundreds, the tens and the ones. Eample: 4+ 35 = (00) + 4(0) + () + 3(00) + 5(0) + () = ( + 3)00 + (4 + 5)0 + ( + )() = 5(00) + 9(0) + 3() = 593 To add polynomials, you add from left to right, grouping terms. Using the same numerals, an eample: ( + 4+ ) + (3 + 5+ ) = + 4+ + 3 + 5+ = ( + 3) + (4 + 5) + ( + ) = 5 + 9 + 3 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page of 4 Revised 8/00
Equations and Their Solutions Syllabus Objective: (.3) The student will solve one-step equations using mental math. Equation A mathematical statement that shows two epressions are equivalent. Solution The value(s) of a variable in an equation that make the statement true. Determining if a Value is a Solution of an Equation Substitute the given value in place of the variable and simplify to see if this value makes a true statement. If it is true, it is a solution of the equation. Eample: Given 5 = 7, determine whether or not = 8 is a solution. 5 = 8 7 Substitute the 8 (value of ) for 5 = Simplify and determine if statement is true or false. 5 This is false, so 8 is NOT a solution to this equation. Solving Equations Syllabus Objective: (.6) The student will solve one-step equations using addition and subtraction. Syllabus Objective: (.7) The student will solve one-step equations using multiplication and division. Solving Equations finding the value(s) of which make the equation a true statement. Strategy for Solving Equations: To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use OPPOSITE (or INVERSE) OPERATIONS. Let s look at a gift wrapping analogy to better understand this strategy. When a present is wrapped, it is placed in a bo, the cover is put on, the bo is wrapped in paper, and finally a ribbon is added to complete the project. To get the present out of the bo, everything would be done in reverse order, performing the OPPOSITE (INVERSE) OPERATION. First we take off the ribbon, then take off the paper, net take the cover off, and finally take the present out of the bo. To solve equations in the form of + b = c, we will undo this algebraic epression to isolate the variable. To accomplish this, we will use the opposite operation to isolate the variable. Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 3 of 4 Revised 8/00
Eample: Solve for, 5= 8. 5 = 8 + 5= + 5 = 3 3 5 = 8 Eample: Solve: + 7= 6. + 7 = 6 7= 7 = 9 9+ 7= 6 Eample: Solve: 3 = 7. 3 = 7 3 7 = 3 3 = 9 3(9) = 7 To isolate the term, undo subtracting 5 by adding 5 to both sides. Check to see that the answer is a solution. To isolate the term, undo adding 7 by subtracting 7 from both sides. Check to see that the answer is a solution. To isolate the term, undo multiplying by 3 by dividing both sides by 3. Check to see that the answer is a solution Eample: Solve: =. 4 = 4 To isolate the term, undo dividing by 4 by multiplying both sides by 4. (4) = (4) 4 = 48 48 Check to see that the answer is a solution = 4 Math 7, Unit 0: Algebraic Reasoning Holt: Chapter Page 4 of 4 Revised 8/00