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6th Grade Big Idea 3 Teacher Quality Grant Big Idea 3: Write, interpret, and use mathematical expressions and equations. MA.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. MA.6.A.3.2: Write, solve, and graph one- and two- step linear equations and inequalities. MA.6.A.3.5 Apply the Commutative, Associative, and Distributive Properties to show that two expressions are equivalent. MA.6.A.3.6 Construct and analyze tables, graphs, and equations to describe linear functions and other simple relations using both common language and algebraic notation. Big idea 3: assessed with Benchmarks Assessed with means the benchmark is present on the FCAT, but it will not be assessed in isolation and will follow the content limits of the benchmark it is assessed with. MA.6.A.3.3 Work backward with two-step function rules to undo expressions. (Assessed with MA.6.A.3.1.) MA.6.A.3.4 Solve problems given a formula. (Assessed with MA.6.A.3.2, MA.6.G.4.1, MA.6.G.4.2, and MA.6.G.4.3.) Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big idea 3: Benchmark Item Specifications Big Idea 3: Prerequisite knowledge Order of Operations Fractions and ratios Decimals Percent Big idea 3: Variable video Writing Algebraic Expressions Be able to write an algebraic expression for a word phrase or write a word phrase for an expression. Although they are closely related, a Great Dane weighs about 40 times as much as a Chihuahua. When solving real-world problems, you will need to translate words, or verbal expressions, into algebraic expressions. Since we do not know the can weight of the Chihuahua we can represent it with the variable c =c So then we can write the Great Dane as 40c or 40(c). Notes •In order to translate a word phrase into an algebraic expression, we must first know some key word phrases for the basic operations. On the back of your notes: Addition Subtraction Multiplication Division Addition Phrases: • More than • Increase by • Greater than • Add • Total • Plus • Sum Subtraction Phrases: • Decreased by • Difference between • Take Away • Less • Subtract • Less than* • Subtract from* Multiplication Phrases: •Product •Times •Multiply •Of •Twice or double •Triple Division Phrases: •Quotient •Divide •Divided by •Split equally Notes •Multiplication expressions should be written in side-by-side form, with the number always in front of the variable. •3a 2t 1.5c 0.4f Notes Division expressions should be written using the fraction bar instead of the traditional division sign. c x 15 , , 4 24 y Modeling a verbal expression • First identify the unknown value (the variable) • Represent it with an algebra tile • Identify the operation or operations • Identify the known values and represent with more tiles Modeling a verbal expression Example: Lula read 10 books. Kelly read 4 more books then Lula. • There is no unknown value • More means addition • The known values are: Lula 10 books and Kelly 4 more 10 books 4 books 10 +4 Singapore math introduction • Level 1 • Level 2 • Level 3 • Enrichment • Links for other strategies Modeling a verbal expression Modeling a verbal expression Examples • Addition phrases: • 3 more than x • the sum of 10 and a number c • a number n increased by 4.5 Examples • Subtraction phrases: • a number t decreased by 4 • the difference between 10 and a number y • 6 less than a number z Examples • Multiplication phrases: • the product of 3 and a number t • twice the number x • 4.2 times a number e Examples • Division phrases: • the quotient of 25 and a number b • the number y divided by 2 • 2.5 divide g Example games • Snow man game • Millionaire game Examples • converting f feet into inches 12f • a car travels at 75 mph for h hours 75h • the area of a rectangle with a length of 10 and a width of w 10w Examples i • converting i inches into feet 12 • the cost for tickets if you purchase 5 adult tickets at x dollars each 5x • the cost for tickets if you purchase 3 children’s tickets at y dollars each 3y Examples • the total cost for 5 adult tickets and 3 children’s tickets using the dollar amounts from the previous two problems 5x + 3y = Total Cost Example Great challenge problems are located on the website bellow: Challenges PROBLEM SOLVING What is the role of the teacher? “Through problem solving, students can experience the power and utility of mathematics. Problem solving is central to inquiry and application and should be interwoven throughout the mathematics curriculum to provide a context for learning and applying mathematical ideas.” NCTM 2000, p. 256 Instructional programs from prekindergarten through grade 12 should enable all students to- •build new mathematical knowledge through problem solving; •solve problems that arise in mathematics and in other contexts; •apply and adapt a variety of appropriate strategies to solve problems; •monitor and reflect on the process of mathematical problem solving. Teachers play an important role in developing students' problem-solving dispositions. 1. They must choose problems that engage students. 2. They need to create an environment that encourages students to explore, take risks, share failures and successes, and question one another. In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies. • Three Question Types – Procedural – Conceptual – Application • Procedural questions require students to: – Select and apply correct operations or procedures – Modify procedures when needed – Read and interpret graphs, charts, and tables – Round, estimate, and order numbers – Use formulas • Sample Procedural Test Question A company’s shipping department is receiving a shipment of 3,144 printers that were packed in boxes of 12 printers each. How many boxes should the department receive? • Conceptual questions require students to: – Recognize basic mathematical concepts – Identify and apply concepts and principles of mathematics – Compare, contrast, and integrate concepts and principles – Interpret and apply signs, symbols, and mathematical terms – Demonstrate understanding of relationships among numbers, concepts, and principles • Sample Conceptual Test Question A salesperson earns a weekly salary of $225 plus $3 for every pair of shoes she sells. If she earns a total of $336 in one week, in which of the following equations does n represent the number of shoes she sold that week? (1) 3n + 225 = 336 (2) 3n + 225 + 3 = 336 (3) n + 225 = 336 (4) 3n = 336 (5) 3n + 3 = 336 • Application/Modeling/Problem Solving questions require students to: – Identify the type of problem represented – Decide whether there is sufficient information – Select only pertinent information – Apply the appropriate problem-solving strategy – Adapt strategies or procedures – Determine whether an answer is reasonable • Sample Application/Modeling/Problem Solving Test Question Jane, who works at Marine Engineering, can make electronic widgets at the rate of 27 per hour. She begins her day at 9:30 a.m. and takes a 45 minute lunch break at 12:00 noon. At what time will Jane have made 135 electronic widgets? (1) 1:45 p.m. (2) 2:15 p.m. (3) 2:30 p.m. (4) 3:15 p.m. (5) 5:15 p.m. According to Michael E. Martinez There is no formula for problem solving How people solve problems varies Mistakes are inevitable Problem solvers need to be aware of the total process Flexibility is essential Error and uncertainty should be expected Uncertainty should be embraced at least temporarily What steps should we take when solving a word problem? 1. Understand the problem 2. Devise a plan 3. Carry out the plan. 4. Look back Reads the problem carefully Defines the type of answer that is required Identifies key words Accesses background knowledge regarding a similar situation Eliminates extraneous information Uses a graphic organizer Sets up the problem correctly Uses mental math and estimation Checks the answer for reasonableness K W E S What do you What does the Is there an What steps did KNOW from the question WANT EQUATION or you use the word problem? you to find? model to solve SOLVE the the problem? problem? UNDERSTAND THE PROBLEM Ask yourself…. •What am I asked to find or show? •What type of answer do I expect? •What units will be used in the answer? •Can I give an estimate? •What information is given? •If there enough or too little information given? •Can I restate the problem in your own words? K W E S What do you What does the Is there an What steps did KNOW from the question WANT EQUATION or you use the word problem? you to find? model to solve SOLVE the the problem? problem? Pattern: What are the next 1 The amount 1, 3, 6, 10, 15, … 4 numbers? 1+2=3 being added 3+3=6 increases by 1 6+4=10 each time so: 10+5=15 15+6=21 21+7=28 28+8=36 36+9=45 K W E S What do you What does the Is there an What steps did KNOW from the question WANT EQUATION or you use the word problem? you to find? model to solve SOLVE the the problem? problem? Number of What fraction of What you want Green Chips chips: the total chips is 3 green green? Total Total Chips 4 blue 1 red 8 total chips 3 8 Solve problems out loud Explain your thinking process Allow students to explain their thinking process Use the language of math and require students to do so as well Model strategy selection Make time for discussion of strategies Build time for communication Ask open-ended questions Create lessons that actively engage learners Jennifer Cromley, Learning to Think, Learning to Learn LOOK BACK This is simply checking all steps and calculations. Do not assume the problem is complete once a solution has been found. Instead, examine the problem to ensure that the solution makes sense. Hierarchical diagramming Sequence charts Compare and contrast charts Algebra Geometry MATH Calculus Trigonometry Compare and Contrast Category Illustration/Example What is it? Properties/Attributes Subcategory Irregular set What are some What is it like? examples? Compare and Contrast - example Numbers Illustration/Example What is it? Properties/Attributes 6, 17, 25, 100 Positive Integers Whole -3, -8, -4000 Numbers Negative Integers 0 Zero Fractions What are some What is it like? examples? Prime Numbers 5 7 11 13 2 3 Even Numbers Multiples of 3 4 6 8 10 6 9 15 21 Right Equiangular 3 sides 3 sides 3 angles 3 angles 1 angle = 90° 3 angles = 60° TRI- ANGLES Acute Obtuse 3 sides 3 sides 3 angles 3 angles 3 angles < 90° 1 angle > 90° Word = Category + Attribute = + Definitions: ______________________ ________________________________ ________________________________ Word = Category + Attribute = + 4 equal sides & Square Quadrilateral 4 equal angles (90°) Definition: A four-sided figure with four equal sides and four right angles. 1. Word: 2. Example: 4. Definition 3. Non-example: 1. Word: semicircle 2. Example: 4. Definition 3. Non-example: A semicircle is half of a circle. Divide into groups Match the problem sets with the appropriate graphic organizer Which graphic organizer would be most suitable for showing these relationships? Why did you choose this type? Are there alternative choices? Parallelogram Rhombus Square Quadrilateral Polygon Kite Irregular polygon Trapezoid Isosceles Trapezoid Rectangle Counting Numbers: 1, 2, 3, 4, 5, 6, . . . Whole Numbers: 0, 1, 2, 3, 4, . . . Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . . Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1 Reals: all numbers Irrationals: π, non-repeating decimal Addition Multiplication a+b a times b a plus b axb sum of a and b a(b) ab Subtraction Division a–b a/b a minus b a divided by b a less b a÷b Use the following words to organize into categories and subcategories of Mathematics: NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division. POLYGON Square, rectangle, Parallelogram: rhombus has 2 pairs of parallel sides Trapezoid: has 1 Trapezoid, Quadrilateral isosceles trapezoid set of parallel sides Kite: has 0 sets of Kite Kite parallel sides Irregular: 4 sides w/irregular shape REAL NUMBERS Addition Subtraction ____a + b____ ____a - b_____ ___a plus b___ __a minus b___ Sum of a and b ___a less b____ Operations Multiplication Division ___a times b___ ____a / b_____ ____a x b_____ _____a(b)_____ _a divided by b_ _____ab______ _____a b_____ Mathematics Geometric Numbers Figures Operations Rules Symbols Rational Addition Postulate m║n Triangle Prime Subtraction √4 Corollary Integer Hexagon Multiplication Irrational Division Quadrilateral Whole Composite {1,2,3…} Mike, Juliana, Diane, and Dakota are entered in a 4- person relay race. In how many orders can they run the relay, if Mike must run list? List them. Mrs. Stevens earns $18.00 an hour at her job. She had $171.00 after paying $9.00 for subway fare. Find how many hours Mrs. Stevens worked. Try solving this problem by working backwards. Use the work backwards strategy to solve this problem. A number is multiplied by -3. Then 6 is subtracted from the product. After adding -7, the result is -25. What is the number? Big Idea 3: Patterns and Equations Analyzing patterns and sequences (lesson ENLVM) Properties of Addition & Multiplication Why do we need rules or properties in math? Lets see what can happen if we didn’t have rules. Before We Begin… • What is a VARIABLE? A variable is an unknown amount in a number sentence represented as a letter: 5+n 8x 6(g) t+d=s Before We Begin… • What do these symbols mean? ( ) = multiply: 6(a) or group: (6 + a) * = multiply · = multiply ÷ = divide / = divide Algebra tiles and counters • Represent the following expressions with algebra tiles or counters: 1.3 + 4 and 4 + 3 2.3 - 4 and 4 – 3 3. 3 4 and 3 4 Algebra tiles and counters • Represent the following expressions with algebra tiles or counters: 1.9x+ 2 and 2 + 9x 1.9x - 2 and 2 – 9x Commutative Property • To COMMUTE something is to change it • The COMMUTATIVE property says that the order of numbers in a number sentence can be changed • Addition & multiplication have COMMUTATIVE properties Commutative Property • One way you can remember this is when you commute you don’t move out of you community. Commutative Property Examples: (a + b = b + a) 7+5=5+7 9x3=3x9 Note: subtraction & division DO NOT have commutative properties! a b As you can see, when you have two lengths a and b, you get the same length b a whether you put a first or b first. b a a b The commutative property of multiplication says that you may multiply quantities in any order and you will get the same result. When computing the area of a rectangle it doesn’t matter which side you consider the width, you will get the same area either way. Commutative Property Practice: Show the commutative property of each number sentence. 1. 13 + 18 = 2. 42 x 77 = 3. 5 + y = 4. 7(b) = Commutative Property Practice: Show the commutative property of each number sentence. 1. 13 + 18 = 18 + 13 2. 42 x 77 = 77 x 42 3. 5 + y = y + 5 4. 7(b) = b(7) or (b)7 You can change + to + You can change to And the result will not change Keep in mind the and do not have to be numbers. They can be expressions that evaluate to a number. Lets see why subtraction and division are NOT commutative. The commutative property: a + b = b + a and a*b=b*a 7 + 3 = 3 + 7 and 7*3 = 3*7 10 = 10 21= 21 Try this subtraction: 8–4 = 4–8 8÷4 = 4÷8 and division 4 ≠ -4 2 ≠ 0.5 Associative Property Practice: Show the associative property of each number sentence. 1. (7 + 2) + 5 = 7 + (2 + 5) 2. 4 x (8 x 3) = (4 x 8) x 3 3. 5 + (y + 2) = (5 + y) + 2 4. 7(b x 4) = (7b) x 4 or (7 x b)4 Identity property Multiplication: 1. 4 x 1 = 4 2 6 2. why is 3 9 Division: 1. 10 1 10 Distributive Property • To DISTRIBUTE something is give it out or share it. • The DISTRIBUTIVE property says that we can distribute a multiplier out to each number in a group to make it easier to solve • The DISTRIBUTIVE property uses MULTIPLICATION and ADDITION! Distributive Property Examples: a(b + c) = a(b) + a(c) 2 x (3 + 4) = (2 x 3) + (2 x 4) 5(3 + 7) = 5(3) + 5(7) Note: Do you see that the 2 and the 5 were shared (distributed) with the other numbers in the group? Distributive Property Practice: Show the distributive property of each number sentence. 1. 8 x (5 + 6) = (8 x 5) + (8 x 6) 2. 4(8 + 3) = 4(8) + 4(3) 3. 5 x (y + 2) = (5y) + (5 x 2) 4. 7(4 + b) = 7(4) + 7b Ella sold 37 necklaces for $20.00 each at the craft fair. She is going to donate half the money she earned to charity. Use the Commutative Property to mentally find how much money she will donate. Explain the steps you used. Use the Associative 1 1 Property to write two 4 5 2 2 equivalent expressions for the perimeter of the triangle 6 Six Friends are going to the state fair. The cost of one admission is $9.50, and the cost for one ride on the Ferris wheel is $1.50. Write two equivalent expressions and then find the total cost. Identity and Inverse Properties Identity Property of Addition The Identity Property of Addition states that for any number x, x + 0 = x 5+0=5 27 + 0 = 27 4.68 + 0 = ¾+0=¾ Identity Property of Multiplication The Identity Property of Multiplication states that for any number x, x (1) = x Remember the number 1 can be in ANY form. The number 1 can be in ANY form. In this case 3/3 is the same as 1. 2 3 6 2 33 9 3 same Inverse Property of Addition The inverse property of addition states that for every number x, x + (-x) = 0 4 and -4 are considered opposites. 4 + -4 = 0 -4 +4 What number can be added to 15 so that the result will be zero? -15 What number can be added to -22 so that the result will be zero? 22 Inverse Property of Multiplication The Inverse Property of Multiplication states for every non-zero number n, n (1/n) = 1 The non-zero part is important or else we would be dividing by zero and we CANNOT do that. Properties of Equality In all of the following properties Let a, b, and c be real numbers Properties of Equality Addition property: If a = b, then a + c = b + c Subtraction property: If a = b, then a - c = b – c Multiplication property: If a = b, then ca = cb Division property: a b for c ≠ 0 If a = b, then c c Addition Property This is the property that allows you to add the same number to both sides of an equation. STATEMENT REASON x=y given x+3=y+3 Addition property of equality Subtraction Property This is the property that allows you to subtract the same number to both sides of an equation. STATEMENT REASON a=b given a-2=b-2 Subtraction property of equality Multiplication Property This is the property that allows you to multiply the same number to both sides of an equation. STATEMENT REASON x=y given 3x = 3y Multiplication property of equality Division Property This is the property that allows you to divide the same number to both sides of an equation. STATEMENT REASON x=y given x/3 = y/3 Division property of equality More Properties of Equality Reflexive Property: a=a Symmetric Property: If a = b, then b = a Transitive Property: If a = b, and b = c, then a = c Substitution Property of Equality If a = b, then a may be substituted for b in any equation or expression. You have used this many times in algebra. STATEMENT REASON x=5 given 3+x=y given 3+5=y substitution property of equality Solving One-Step Equations Definitions Term: a number, variable or the product or quotient of a number and a variable. examples: 12 z 2w c 6 Terms are separated by addition (+) or subtraction (-) signs. 3a – ¾b + 7x – 4z + 52 How many Terms do you see? 5 Definitions Constant: a term that is a number. Coefficient: the number value in front of a variable in a term. 3x – 6y + 18 = 0 What are the coefficients? 3 , -6 What is the constant? 18 Solving One-Step Equations A one-step equation means you only have to perform 1 mathematical operation to solve it. You can add, subtract, multiply or divide to solve a one-step equation. The object is to have the variable by itself on one side of the equation. Example 1: Solving an addition equation t + 7 = 21 To eliminate the 7 add its opposite to both sides of the equation. t + 7 = 21 t + 7 -7 = 21 - 7 t + 0 = 21 - 7 t = 14 Example 2: Solving a subtraction equation x – 6 = 40 To eliminate the 6 add its opposite to both sides of the equation. x – 6 = 40 x – 6 + 6 = 40 + 6 x = 46 Example 3: Solving a multiplication equation 8n = 32 To eliminate the 8 divide both sides of the equation by 8. Here we “undo” multiplication by doing the opposite – division. 8n = 32 8 8 n=4 Example 4: Solving a division equation x 11 9 To eliminate the 9 multiply both sides of the equation by 9. Here we “undo” division by doing the opposite – multiplication. x 11 9 x 9 (11)(9) 9 x 99 Identify operations Undo operations Balance equation Repeat steps Solve for variable Check solution Identify Operations Minus sign means subtraction x 38 2 Fraction bar means division Use Opposite Operations or “undo” Operations Addition is opposite of subtraction (addition undoes subtraction) Subtraction is opposite of addition (subtraction undoes addition) Multiplication is opposite of division (multiplication undoes division) Division is opposite of multiplication (division undoes multiplication) Keep Equation Balanced What ever you do to one side of the equation you do to the other side of the equation. Repeat these steps until the equation is solved. 1-step equations 2-step equations Example: 7x + 15 = 85 7x +15 – 15 = 85 - 15 7x = 70 7 7 x = 10 Example: 2 x 6 28 3 2 x 6 6 28 6 3 2 x 28 3 3 2 3 x 28 2 3 2 x 42 Graphing a Linear Equation When graphing the solution to a linear equation with one- variable on a number line you would put a dot (point) on the answer. x – 3 = -7 x – 3 + 3 = -7 + 3 x = -4

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