6 th Grade Mathematics Curriculum

Transcription

1 6 th Grade Mathematics Curriculum Units 1, 2, 3, 4, 5, 7, 8, 10 Course Description: The 6 th Grade Mathematics course serves as a culminating year for a lot of work with fractions, decimals and percents. Students convert among fractions, decimals and percents as well as perform operations with fractions. Students knowledge in the area of algebra continues to expand to include solving more complicated equations and evaluating more complicated expressions. Circles, triangles, and quadrilaterals are all addressed in geometry units while the measurement focus is on volume and capacity. Finally, students continue their work with the collection, display and analysis of data and build upon the probability knowledge they developed in 5 th grade. Problem solving and communication are emphasized across all these areas. Course Essential Questions and Big Ideas: Any number can be represented in multiple ways (fractions, decimals, percents, powers). You can make the math easier by choosing the best representation. Ratios, fractions, percents, and division are just different ways of representing the same process. 6th Grade State Assessment Information: Approximate Percentage of Questions Assessing Each Strand Strand: Percent: Time allotted: Units: Number Sense and Operations: 37% (10 weeks) (Unit 1, 3, 4, 5, 8) Algebra: 19% (3 weeks) (Unit 2) Geometry: 17% (4 weeks) (Unit 6) Measurement: 11% (2 weeks) (Unit 7) Probability and Statistics: 16% (2 weeks) (Unit 9) AECSD Grade_6_Math_Curr Rev 7-08.doc 1

2 Table of Contents Section Page # 6 th Grade Mathematical Language...3 Post-March 5 th Grade Performance Indicators th Grade Local Math Standards...8 Math 6 Unit Sequence and Timeline:...10 Unit 1 Whole Numbers, Place Value and Properties...12 Unit 2 Algebra...14 Unit 3 Adding and Subtracting Fractions...17 Unit 4 Multiplying and Dividing Fractions...20 Unit 5 Ratio, Proportion and Percents (N.1, N.5, N.12, G.3)...23 Unit 6 Geometry (G.1, G.5)...26 Unit 7 Measurement (M.5)...30 Unit 8 Rational Numbers (N.3, N.7)...32 Unit 9 Data and Statistics (D.2)...34 Review Unit for State Assessment (Test dates: March 15 and 16, 2006)...35 Unit 10 Algebra...36 Unit 11 Coordinate Geometry (G.7)...38 Unit 12 Collection and Display of Data (D.1)...39 Unit 13 Probability (D.3)...40 AECSD Grade_6_Math_Curr Rev 7-08.doc 2

3 6 th Grade Mathematical Language The language below is language that all students should be familiar with and should be used throughout instruction. Definitions for most words and expressions can be found in the PK-8 Glossary. Problem Solving analyze apply collaboration counterexample differentiate discuss draw a graph draw a picture explain formulate identity interpret Reasoning and Proof algebraically appropriate mathematical terms argument conjecture (noun) counterexample develop formulas Communication accurately label work analyze clarifying questions comprehend consolidate Connections apply coherent conjecture (verb) connections invalid approach irrelevant information language of logic (and, or, not) logic logical reasoning model using manipulatives monitor observe patterns organized chart organized list process of elimination explain graphically interpret investigate justify manipulative(s) mathematical relationships decode distinguish explain extend mathematical relationships draw conclusions explore investigate irrelevant information reasonableness of a solution reflect relevant information solution solve a simpler problem strategies trial and error valid approach verify results work backwards write an equation methods of proof models numerically special case(s) verbally verify claims of others organize work rationale solution verbal symbols written symbols mathematical relationships model (noun) model problems relevant information AECSD Grade_6_Math_Curr Rev 7-08.doc 3

4 Representation apply describe explain explore extend Number Sense and Operations absolute value additive inverse associative property of addition associative property of multiplication base (of percent) base ten number system commutative property of addition commutative property of multiplication counting (natural) numbers distributive property equivalent fractions equivalent numerical expressions equivalent ratios estimate exponent exponential form extremes (of a proportion) fraction Algebra algebra algebraic expression algebraic solution equation evaluate exponents interpret investigate mathematical phenomena model(s) identity element identity property of addition identity property of multiplication inequality integer inverse element inverse operation like (common) denominators lowest terms mathematical statement means (of a proportion) mixed number multiplicative inverse (reciprocal) multiply (multiplication) negative number line number system numerical problem formula input values interest inverse operations proportion solve nonstandard representations physical phenomena social phenomena standard representations numeric (arithmetic) expression operation order (verb) order of operations percent positive power properties of real numbers proportion rate rate of interest ratio rational number repeating decimal round (verb) terminating decimal unlike denominators verbal expression whole number zero property of multiplication substitute translate variable verbal expression AECSD Grade_6_Math_Curr Rev 7-08.doc 4

5 Geometry arc area central angle chord circle circumference coordinate geometry coordinate plane corresponding sides develop formulas diameter geometric shape Measurement calculate volume cup customary units of capacity distance equivalent customary units of capacity Statistics and Probability circle graph compound events data dependent events favorable outcomes frequency frequency table fundamental counting principle geometry height irregular polygon length perimeter pi (!) plot point proportional reasoning quadrant quadrilateral radius estimate area circumference volume gallon liter measure capacity histogram impossible outcomes interpret graphs justify line graph mean median mode population rectangle rectangular prism regular polygon rhombus sector similar triangles square trapezoid triangle vertex volume width metric system metric units of capacity milliliter personal references for capacity pint quart possible outcomes predict probability range record data sampling statistics Venn diagram AECSD Grade_6_Math_Curr Rev 7-08.doc 5

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7 Post-March 5 th Grade Performance Indicators The 5 th Grade state performance indicators below are denoted by the state as post-test. Therefore, students will be responsible for this knowledge of the 6 th Grade assessment. Attention should be given to them during the normal course of instruction or during review. 5.A.2 Translate simple verbal expressions into algebraic expressions 5.A.3 Substitute assigned values into variable expressions and evaluate using order of operations 5.A.4 Solve simple one-step equations using basic whole-number facts 5.A.5 Solve and explain simple one-step equations using inverse operations involving whole numbers 5.G.12 Identify and plot points in the first quadrant 5.G.13 Plot points to form basic geometric shapes (identify and classify) 5.G.14 Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes) 5.S.5 List the possible outcomes for a single-event experiment 5.S.6 Record experiment results using fractions/ratios 5.S.7 Create a sample space and determine the probability of a single event, given a simple experiment (e.g., rolling a number cube) AECSD Grade_6_Math_Curr Rev 7-08.doc 7

8 6 th Grade Local Math Standards Numbering Key: Local.Grade level.mathematics strand.standard # e.g. L.6.N.5 (L = local; 6 = 6th Grade; N = Number Sense and Operations; 5 = 5th standard) Number and Operations: L.6.N.1 Representation and Notation Read and write whole numbers to trillions; convert among fractions, decimals (repeating and non-repeating), and percents (0 to 100). L.6.N.3 Number Order Order rational numbers (positive and negative) including locating them on a number line L.6.N.5 Estimation Estimate a percent (0 to 100%) of a quantity; justify the reasonableness of answers using estimation including rounding L.6.N.7 Properties and Laws Identify, use, and explain: the commutative and associative properties of addition and multiplication, the distributive property of multiplication over addition, the identity and inverse properties of addition and multiplication, and the zero property of multiplication; define absolute value and determine the absolute value of rational numbers Evaluate numerical expressions using order of operations (may include exponents of two and three) L.6.N.8 Computation and Facts L.6.N.10 Fractions Add, subtract, multiply, and divide fractions and mixed numbers with unlike L.6.N.12 Percent, Ratio, Proportion denominators Understand the concepts of rate and ratio and distinguish between them; express equivalent ratios as proportions, solve proportions (see L.6.A.2), and verify proportionality (e.g. verify two triangles are similar, see L.6.G.3)); read, write, and identify percents of a whole (0% to 100%); solve percent application problems L.6.N.13 Power and Roots Convert between repeated multiplication and power notation Algebra: L.6.A.1 Patterns and Translate between two-step verbal and algebraic expressions and equations. Representations L.6.A.2 Solving Equations and Inequalities Solve and explain simple (whole number) two-step equations using inverse operations; solve proportions (see L.6.N.12) L.6.A.3 Expressions Substitute values (for 1 or 2 variables) into an expression (including exponents up to 3) and evaluate it (e.g. circumference, area, volume, distance, temperature, interest, etc.) (See L.6.N.8). Geometry: L.6.G.1 L.6.G.3 L.6.G.5 Shapes and Figures Similarity and Congruence Perimeter, Area, and Volume L.6.G.7 Coordinate Geometry Measurement: L.6.M.5 Volume (Capacity) Identify the radius and diameter (and the relationship between them), chords, and central angles of a circle; understand the relationship between the circumference and the diameter of a circle. Calculate the length of corresponding sides of similar triangles using proportional reasoning (see L.6.N.12 and L.6.A.2). Estimate and determine the area of triangles and quadrilaterals (square, rectangle, rhombus, trapezoid) and develop area formulas; estimate and find the area of regular and irregular polygons; estimate the volume of rectangular prisms by counting cubes and calculate the volume after developing a formula; determine the circumference and area of a circle using a formula, and calculate the area of a sector of a circle, given the central angle and radius Identify and plot points in all four quadrants of the coordinate plane; calculate the area of shapes with sides parallel to the axes (see L.6.G.5). Measure volume (capacity) using the appropriate tool and technique; identify and convert between units of volume (capacity) within a given system (metric or customary); determine personal references for customary and metric units of AECSD Grade_6_Math_Curr Rev 7-08.doc 8

9 Statistics and Probability: L.6.S.1 Collect and Display Data customary); determine personal references for customary and metric units of volume (capacity); justify the reasonableness of an estimate of the volume of an object. Explore data collection through sampling; record (e.g. with a frequency table) and choose an appropriate display (e.g. Venn diagram, pictograph, bar graph, line graph, histogram, or circle graph) for real-world data; construct Venn diagrams. L.6.S.2 Analyze Data Calculate and use the mean, mode, median, and range of a set of data; read, interpret, and predict from (with justification) graphs L.6.S.3 Probability For a compound-event: determine the sample space using the fundamental counting principle, list all possible outcomes, and determine the probability of a single outcome; determine the probability of dependent events. Problem Solving: L.6.PS.1 Organization Analyze situations (identify the problem, identify needed and relevant information, find relationships, observe patterns, and generate possible strategies) and organize work to solve problems (e.g. use Auburn Problem Solving Process). L.6.PS.2 Strategies Solve problems using a variety of strategies and representations (e.g. modeling with manipulatives, acting out, drawing pictures or diagrams, guess and check (trial and error), making a list or chart, and process of elimination). L.6.PS.3 Reflection Estimate possible solutions; examine solution to ensure it is reasonable in context of problem; compare solution to original estimate; verify results. Reasoning and Proof: L.6.RP.1 Communication: L.6.CM.1 Connections: L.6.CN.1 Representations: L.6.Rep.1 Make, investigate, and evaluate conjectures; support (or refute) mathematical statements or conjectures with valid arguments including the use of models, facts, relationships, and (counter)examples to help explain their reasoning; express arguments verbally, numerically, algebraically, pictorally, and in writing. Decode and comprehend mathematics expressed verbally and in writing; clearly and coherently communicate mathematical thinking verbally, pictorally, numerically, algebraically and in writing using appropriate mathematical vocabulary and symbols; organize and accurately label work. Recognize and use connections among branches of mathematics and real life (e.g. determine the perimeter of a bulletin board, construct tables to organize data showing book sales, find the missing value: (3 + 4) + 5 = 3 + (4 + )) Represent mathematical ideas in a variety of ways (verbally, in writing, pictorally, numerically, algebraically, or with physical objects); switch among different representations; investigate how different representations can express the same relationship but may differ in efficiency. AECSD Grade_6_Math_Curr Rev 7-08.doc 9

10 Math 6 Unit Sequence and Timeline: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Whole Numbers, Place Value and Properties (N.1, N.7) ~ 2 weeks Early September to Mid-September Algebra (N.5, N.8, N.13, A.1, A.3) ~ 3 weeks Mid-September to early October Adding and Subtracting Fractions (N.1, N.10) ~2 weeks Mid-October to Late October Multiplying and Dividing Fractions (N.1, N.10) ~2 weeks Early November Ratio, Proportion and Percents (N.1, N.5, N.12, G.3) ~ 3 weeks Mid-November to Mid-December Geometry (G.1, G.5) ~4 weeks Mid-December to Mid-January Measurement (M.5, G.5) ~2 weeks Late January Rational Numbers (N.3, N.7) ~1 week First week of February Data and Statistics (D.2) ~ 2 weeks February State Assessment Review and Administration ~ 2 weeks Early to mid-march (State Assessment: 3/15 and 3/16) Unit 10 Algebra (A.2) ~ 1 week End of March (after assessment) AECSD Grade_6_Math_Curr Rev 7-08.doc 10

11 Unit 11 Unit 12 Unit 13 Coordinate Geometry (G.7) ~ 2 weeks Early April Collection and Display of Data (D.1) ~ 3 weeks Late April to Early May Probability (D.3) ~ 2 weeks Mid-May AECSD Grade_6_Math_Curr Rev 7-08.doc 11

12 Unit 1 Whole Numbers, Place Value and Properties ~ 2 weeks Early September to Mid-September State Standards (Shaded statements are identified as Post-March Indicators): 6.N.1 Read and write whole numbers to trillions 6.N.2 Define and identify the commutative and associative properties of addition and multiplication 6.N.3 Define and identify the distributive property of multiplication over addition 6.N.4 Define and identify the identity and inverse properties of addition and multiplication 6.N.5 Define and identify the zero property of multiplication 6.N.19 Identify the multiplicative inverse (reciprocal) of a number Local Standards (Stricken text is covered in a different unit): L.6.N.1 Read and write whole numbers to trillions; convert among fractions, decimals (repeating and non-repeating), and percents (0 to 100). L.6.N.7 Identify, use, and explain: the commutative and associative properties of addition and multiplication, the distributive property of multiplication over addition, the identity and inverse properties of addition and multiplication, and the zero property of multiplication; define absolute value and determine the absolute value of rational numbers Big Ideas: Whole numbers are an infinite set. Using the properties of addition and multiplication will simplify the mental math process. Properties are the ground rules for the game of math. Essential Questions: How is each place value related to the one on the right? How can the associative property help add a large group of numbers in a series? Prior knowledge: to know basic addition and multiplication math facts to know place value of digits in a number to read and write whole numbers to millions to use parenthesis to show the grouping of numbers (Holt 1-4) to represent, compare and order decimals (Holt 3-1) to add and subtract decimals (Holt 3-3) Unit Objectives: to read whole numbers to trillions to write whole numbers to trillions to identify the commutative, associative, distributive, identity, and zero properties of addition and multiplication to use the commutative, associative, distributive, identity, and zero properties of addition and multiplication AECSD Grade_6_Math_Curr Rev 7-08.doc 12

13 to explain the commutative, associative, distributive, identity, and zero properties of addition and multiplication Resources: Holt: 1-1 (read and write large numbers) 1-5 (distributive, commutative, associative) Holt Text: Skills Bank page 757 (identity, and zero properties) Holt WB: Are You Ready- Skill 48 (identity, and zero properties) Review Template (No Calculators): All operations with whole numbers 1. Find the sum of 1665 and 18, Find the difference of 4,835 and 1, The product of 3,655 and 61 is: 4. What is the quotient of 4,810 and 10? Rounding whole numbers and place value to millions 1. Round 81,294,537 to the given place: a. thousands b. millions 2. Give the place value of the digit 4 in 2,549,013. Use of parenthesis 1. Simplify: a. 7 ( 8 + 3) b. ( 11-5 ) 3 Problem Solving: 1. At every hour of the day, a clock beeps the same number of times as the hour. It beeps once at 1:00, twice at 2:00, three times at 3:00, and so on. How many times does it beep in one day? (Answer: 156 times) 2. Oops! The 5 key on your calculator just broke. Using this calculator, how could you find the answer to 597 x 84. [One answer: Using the distributive property: (600-3) x 84 = (600 x 84) (3 x 84) = 50,148] 3. From: SFAW: DT 2-7: On her birthday, Jennifer spent half of her savings at the mall and then donated $5 to charity. She received $25 as a birthday gift. Now she has $128. How much money did Jennifer have before she went to the mall? (Answer: $216) AECSD Grade_6_Math_Curr Rev 7-08.doc 13

14 Unit 2 Algebra ~ 3 weeks Mid-September to Mid-October State Standards (Shaded statements are identified as Post-March Indicators): 6.N.22 Evaluate numerical expressions using order of operations (may include exponents of two and three) 6.N.23 Represent repeated multiplication in exponential form 6.N.24 Represent exponential form as repeated multiplication 6.N.25 Evaluate expressions having exponents where the power is an exponent of one, two, or three 6.N.26 Estimate a percent of quantity (0% to 100%) 6.N.27 Justify the reasonableness of answers using estimation (including rounding) 6.A.1 Translate two-step verbal expressions into algebraic expressions 6.A.2 Use substitution to evaluate algebraic expressions (may include exponents of one, two and three) 6.A.6 Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest, etc.) Local Standards (Stricken text is covered in a different unit): L.6.N.8 Evaluate numerical expressions using order of operations (may include exponents of two and three) L.6.N.13 Convert between repeated multiplication and power notation L.6.N.5 Estimate a percent (0 to 100%) of a quantity; justify the reasonableness of answers using estimation including rounding L.6.A.1 Translate between two-step verbal and algebraic expressions and equations. L.6.A.3 Substitute values(for 1 or 2 variables) into an expression (including exponents up to 3) and evaluate it (e.g. circumference, area, volume, distance, temperature, interest, etc.) (See L.6.N.8). Big Ideas: There are alternate solutions to math problems. The set of rules known as the order of operations, guarantees that everyone gets the same solution for a problem. Phrases and situations can be translated into mathematical expressions. Essential Questions: What is the difference between a constant and a variable? What is the advantage of using exponents to write numbers? Is n 7 the same the same as n * 7? What happens when you raise a base number to the zero power? First power? Prior knowledge: to know the difference between a constant and a variable to know sum, difference, product, and quotient to know the words that suggest each operation AECSD Grade_6_Math_Curr Rev 7-08.doc 14

15 to know multiplication is repeated addition to know the difference between an expression and an equation to solve 1-step equations using inverse operation Unit Objectives: to translate 1 or 2 step verbal and algebraic expressions to evaluate numerical and algebraic expressions using substitution and order of operations to determine if an equation is true or false to find the value of a variable to make an equation true to identify the base number and exponent in a power to use exponents to express repeated multiplication to write numbers in standard form, exponential notation, and as a product of factors 6A: Enriched Objectives and Resources to solve two step equations (POST MARCH) (Holt pages 90-91) (Are You Ready Workbook: Skill 60) to use grouping symbols /brackets and braces (supplement materials) to use powers of ten to represent place value (supplement materials) Resources: Holt: 1-3 (exponents) 1-4 (order of operations) 2-1 (variables and expressions) 2-2 (translate expressions) Holt pages (translate 2 step verbal and algebraic expressions) 2-4 (equations and their solutions) 2-5 through 2-8 (solving 1-step equations) Supplement: evaluating 2 variables in an expression using substitution Hands-On Equations Review Template (No Calculators): Constant and variables Identify each as a variable or a constant. 1. the price of a pair of sneakers 2. the number of days in October 3. the number of people in Auburn Expressions, equations 1. Write each phrase as an expression. a. a number, n, decreased by 7 b. half of k AECSD Grade_6_Math_Curr Rev 7-08.doc 15

16 c. r cubed d. 18 more than x 3. Which one is not an equation? a. 6 x 9 b n = 9 c = 8 Problem Solving: Rufus has $3.45 in quarters and dimes. He has four more quarters than dimes. How many of each coin does he have? (Answer: 11 quarters, 6 dimes) AECSD Grade_6_Math_Curr Rev 7-08.doc 16

17 Unit 3 Adding and Subtracting Fractions ~2 weeks Mid-October to Late October State Standards (Shaded statements are identified as Post-March Indicators): 6.N.16 Add and subtract fractions with unlike denominators 6.N.18 Add, subtract, multiply and divide mixed numbers with unlike denominators 6.N.20 Represent fractions as terminating or repeating decimals 6.N.21 Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100) Local Standards (Stricken text is covered in a different unit): L.6.N.1 Read and write whole numbers to trillions; convert among fractions, decimals (repeating and non-repeating), and percents (0 to 100). L.6.N.10 Add, subtract, multiply, and divide fractions and mixed numbers with unlike denominators Big Ideas: Fractions are used in real-life situations. Fractions represent parts of a whole, a group, or a set. Divisibility rules make working with fractions easier. Essential Questions: Why do you need like denominators to add or subtract fractions? Why is it easier to add fractions using the least common denominator instead of any common denominator? Could you add mixed numbers by first changing them to decimals? Explain. In a subtraction problem, when might you need to convert a whole number into a mixed number? Prior knowledge: to know the rules of divisibility (Holt 4-1) to identify the numerator and denominator of a fraction to know the difference between a proper and improper fraction (Holt 4-6) to compare and order fractions (Holt 4-7) to know prime and composite numbers (Holt 4-1) to find the LCM of two or more numbers (Holt 5-1) to find the GCF of two or more numbers (Holt 4-3) to find equivalent fractions (Holt 4-5) to reduce fractions to lowest terms (Holt 4-5) to convert between mixed numbers and improper fractions (Holt 4-6) to add and subtract like fractions and mixed numbers (Holt 4-8) Unit Objectives: to find equivalent fractions AECSD Grade_6_Math_Curr Rev 7-08.doc 17

18 to find a lowest term fraction to find the LCD to add and subtract fractions with unlike denominators to add and subtract mixed numbers with unlike denominators 6A: Enriched Objectives and Resources to find the LCM of two or more numbers using prime factorization (Holt: 5-1) (see Holt: Know It Notebook) to find the GCF of two or more numbers using prime factorization (Holt: 4-3) (see Holt: Know It Notebook) to solve addition and subtraction fraction equations (Holt: 5-5) (see Holt: Chapter 5 Resource Book) Resources: Holt: 4-5 (Understanding equivalent fractions) 4-5 (Fractions in lowest terms) 5-2 (Adding and subtracting fractions with unlike denominators) 5-3 (Adding and subtracting mixed numbers) 5-4 (Regrouping to subtract mixed numbers) Review Template (No Calculators): Divisibility rules, LCM, GCF 1. Test each number for divisibility by 2, 3, 5,6,9 and 10. a. 104 b. 660 c. 450 d. 1, Find the LCM for each number pair a. 14 and 24 b. 21 and Find the GCF of the given pair of numbers. a. 15 and 20 b. 18 and 45 c. 17 and 21 Lowest terms, compare and order fractions 1. Write in lowest terms. a. 8/10 b. 21/28 c. 36/54 d. 12/15 2. Compare these fractions: a. 2/3 8/12 b. 3/6 6/9 c. 7/11 2/3 3. Order from least to greatest using repeated inequalities a. 2/3, 7/8, 3/4 b. 2/5, 1/4, 1/2 c. 3/4, 3/5, 5/6 Add and subtract like fractions/mixed numbers /4 + 3/ / /8 AECSD Grade_6_Math_Curr Rev 7-08.doc 18

19 / / /7 Prime and composite numbers 1. Given the number and its factors, tell whether it is prime or composite. a. 29: 1,29 b. 57: 1,3,19,57 c. 92: 1,2,4,23,46,92 d. 83: 1,83 2. Which number is a prime number? a. 63 b. 78 c. 115 d. 29 Problem Solving: A string is cut in half, and on e of the halves is used ot bundle newspapers. Then, one fifth of the remaining string is cut off. The piece left is 8 feet long. How long was the string originally? Answer: 20 feet AECSD Grade_6_Math_Curr Rev 7-08.doc 19

20 Unit 4 Multiplying and Dividing Fractions ~2 weeks Early November State Standards (Shaded statements are identified as Post-March Indicators): 6.N.17 Multiply and divide fractions with unlike denominators 6.N.18 Add, subtract, multiply and divide mixed numbers with unlike denominators 6.N.20 Represent fractions as terminating or repeating decimals 6.N.21 Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100) Local Standards (Stricken text is covered in a different unit): L.6.N.1 Read and write whole numbers to trillions; convert among fractions, decimals (repeating and non-repeating), and percents (0 to 100). L.6.N.10 Add, subtract, multiply, and divide fractions and mixed numbers with unlike denominators Big Ideas: The same value can be expressed as both a fraction and a decimal. Multiplying by a number less than 1, results in a smaller product. Dividing by a whole number is the same as multiplying by its reciprocal. (e.g. dividing by 3 is the same as multiplying by 1/3) Essential Questions: Can every fraction be changed into a decimal? Why or why not? How is multiplying two fractions different from adding two fractions? What is really happening when you divide a whole number by a proper fraction? What does it mean? Will the quotient be larger or smaller than the whole number? Why? Prior knowledge: to know the rules of divisibility to know the difference between a proper and improper fraction to find the GCF of two or more numbers to reduce fractions to lowest terms to convert between mixed numbers and improper fractions note: to multiply and divide decimals (Holt 3-5, 3-6, 3-7) Unit Objectives: to know that the fractional bar means to divide the numerator by the denominator to convert between fractions and decimals to identify a repeating and non-repeating (terminating) decimals to multiply fractions with unlike denominators by a whole number, a fraction, and a mixed number to find the reciprocal of a whole number, mixed number, and fraction to simplify common factors before multiplying AECSD Grade_6_Math_Curr Rev 7-08.doc 20

21 to divide fractions with unlike denominators by a whole number, a fraction, and a mixed number 6A: Enriched Objectives and Resources to solve multiplication and division fraction equations (Holt: 5-10) (see Holt: Chapter 5 Resource Book) to interpret the quotient in word problems (Holt: 3-8) (see Holt: Ready to Go On workbook page 57) Resources: Holt: 4-4 (Converting fractions to decimals) 5-6 (Multiplying by a whole number) 5-7 (Multiplying fractions) 5-8 (Multiplying mixed numbers) 5-9 (Dividing fractions and mixed numbers) Review Template (No Calculators): GCF 1.Find the GCF of the given pair of numbers. a. 15 and 20 b. 10 and 12 c. 18 and 45 d. 21 and 28 Proper and improper fractions 1. Identify each fraction as proper or improper. a. 9/12 b. 12/3 c. 3/2 Problem Solving: 2. Write as a improper fraction or mixed number. a. 3 1/3 b. 99/10 c. 14/5 d. 2 9/12 e. 1 1/4 f. 5 4/5 A box of laundry detergent contains 35 cups. If you use 1 and 1/4 cups per load of laundry, how many loads can you wash with 1 box? Answer: 28 loads AECSD Grade_6_Math_Curr Rev 7-08.doc 21

22 Placido, Dexter, and Scott play guard, forward, and center on a team, but not necessarily in that order. Placido and the center dove Scott to practice on Saturday. Placido does not play guard. Who is the guard? Answer: Scott AECSD Grade_6_Math_Curr Rev 7-08.doc 22

23 Unit 5 Ratio, Proportion and Percents (N.1, N.5, N.12, G.3) ~ 3 weeks Mid-November to Mid-December State Standards (Shaded statements are identified as Post-March Indicators): 6.N.6 Understand the concept of rate 6.N.7 Express equivalent ratios as a proportion 6.N.8 Distinguish the difference between rate and ratio 6.N.9 Solve proportions using equivalent fractions 6.N.10 Verify the proportionality using the product of the means equals the product of the extremes 6.N.11 Read, write, and identify percents of a whole (0% to 100%) 6.N.12 Solve percent problems involving percent, rate, and base 6.N.20 Represent fractions as terminating or repeating decimals 6.N.21 Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100) 6.N.26 Estimate a percent of quantity (0% to 100%) 6.N.27 Justify the reasonableness of answers using estimation (including rounding) 6.A.5 Solve simple proportions within context 6.G.1 Calculate the length of corresponding sides of similar triangles, using proportional reasoning Local Standards (Stricken text is covered in a different unit): L.6.N.1 Read and write whole numbers to trillions; convert among fractions, decimals (repeating and non-repeating), and percents (0 to 100). L.6.N.5 Estimate a percent (0 to 100%) of a quantity; justify the reasonableness of answers using estimation including rounding L.6.N.12 Understand the concepts of rate and ratio and distinguish between them; express equivalent ratios as proportions, solve proportions (see L.6.A.2), and verify proportionality (e.g. verify two triangles are similar, see L.6.G.3)); read, write, and identify percents of a whole (0% to 100%); solve percent application problems L.6.A.2 Solve and explain simple (whole number) two-step equations using inverse operations; solve proportions (see L.6.N.12) L.6.G.3 Calculate the length of corresponding sides of similar triangles using proportional reasoning (see L.6.N.12 and L.6.A.2). Big Ideas: Ratios and proportions are found in real-life situations. Proportions are the key to solving percent problems. Working with ratios is the same as working with fractions. Essential Questions: What does it mean to say quantities are proportional? Can you solve a proportion without using cross products? Explain. Is it possible to have more than 100% of a quantity or less than 1% of a quantity? What do you know about the sides of similar triangles? AECSD Grade_6_Math_Curr Rev 7-08.doc 23

24 Why is it necessary to change a percent to its fraction or decimal equivalent? Prior knowledge: to know the concept of ratio to know the concept of percent to identify similar triangles and their corresponding sides and angles to use multiplication to find cross products to be able to estimate and round to evaluate algebraic expressions Unit Objectives: to write a ratio three ways to express two quantities as a ratio to find ratios equal to a given ratio to express two quantities with different units as a rate to know a proportion is two equal ratios to decide if two ratios form a proportion to solve proportions using cross products to use proportions to calculate the lengths of the corresponding sides of similar triangles to read, write, and identify a quantity as a percent of a whole to estimate percents of a quantity to express percents as fractions and decimals to express fractions and decimals as percents to use estimation (including rounding) of percents to justify the reasonableness of answers to solve percent application problems using proportions (only % of a number) to find interest by substituting values into the formula and evaluating (I = PRT) 6A: Enriched Objectives and Resources to read and use maps and scale drawings (Holt: 7-6) (see Holt: Chapter 7 Resource Book) to solve all percent application problems using proportions (supplement: Holt Grade 7 Text: 6-5) to solve percent problems using discounts, tips and sales tax (Holt 7-10) AECSD Grade_6_Math_Curr Rev 7-08.doc 24

25 Resources: Holt: 7-1, 7-2 (Ratios and Rates) 7-3 (Proportions; solving proportions using cross products) 7-4 (Similar figures) 7-7 (Percent) 7-8 (Percents, decimals and fractions) 7-9 (Percent problems) Holt Text: (Finding Interest) Extension pages Supplement: (Estimating a percent) (see Holt Grade 7 Text: 6-3) Review Template (No Calculators): Define ratio and percent, similar triangles and corresponding parts 1. Complete each statement with one of the following terms: ratio, percent, corresponding parts, similar figures a. Figures that have the same shape, but not necessarily the same size are called. b. A comparison of two quantities, often written as a fraction is a. c. If two figures are similar, their are proportional. d. A ratio that compares a part to a whole using the number 100 is a. Cross products 1. Compare ( >, <, = ) using cross products. a. 5/6 7/8 b. 8/11 5/7 c. 3/4 5/6 Evaluate algebraic expressions 1. Evaluate: a. 4b + 6 if b = 3 b. 24 t if t = 8 AECSD Grade_6_Math_Curr Rev 7-08.doc 25

26 Unit 6 Geometry (G.1, G.5) ~4 weeks Mid-December to Mid-January State Standards (Shaded statements are identified as Post-March Indicators): 6.G.2 Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas 6.G.3 Use a variety of strategies to find the area of regular and irregular polygons 6.G.4 Determine the volume of rectangular prisms by counting cubes and develop the formula 6.G.5 Identify radius, diameter, chords and central angles of a circle 6.G.6 Understand the relationship between the diameter and radius of a circle 6.G.7 Determine the area and circumference of a circle, using the appropriate formula 6.G.8 Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle 6.G.9 Understand the relationship between the circumference and the diameter of a circle 6.M.1 Measure capacity and calculate volume of a rectangular prism 6.M.7 Estimate volume, area, and circumference (see figures identified in geometry strand) Local Standards (Stricken text is covered in a different unit): L.6.G.1 Identify the radius and diameter (and the relationship between them), chords, and central angles of a circle; understand the relationship between the circumference and the diameter of a circle. L.6.G.5 Estimate and determine the area of triangles and quadrilaterals (square, rectangle, rhombus, trapezoid) and develop area formulas; estimate and find the area of regular and irregular polygons; estimate the volume of rectangular prisms by counting cubes and calculate the volume after developing a formula; determine the circumference and area of a circle using a formula, and calculate the area of a sector of a circle, given the central angle and radius Big Ideas: Geometric patterns and designs are evident everywhere, such as in art, history, and science. Perimeter measures the length or distance around a shape or an object. Circumference is the distance around or the perimeter of a circle. Area measures square units inside the figure. Volume is the number of cubic units needed to fill a solid figure. Essential Questions: How is finding the area of a triangle different than finding the area of a rectangle or square? For a given rectangle, if you switch the numbers for the base and height, do you get a different area? Explain. What is difference between perimeter, area and volume? Why do we use 3 as an estimate for pi? If two circles have the same circumference, must they have the same area? AECSD Grade_6_Math_Curr Rev 7-08.doc 26

27 Prior knowledge: to identify triangles and quadrilaterals (Holt Text: 8-5; 8-6) to evaluate algebraic expressions to identify and plot points in the first quadrant (Holt Text 6-6) to plot points to form basic geometric shapes (identify and classify) to calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with whole number lengths and parallel to the axes) (supplement: Coach Workbook) Unit Objectives: to know the difference between perimeter, area, and volume of polygons to estimate the area of quadrilaterals and triangles to identify the base and height of triangles and quadrilaterals to develop the area formulas for triangles and quadrilaterals to find the area of triangles and quadrilaterals using formulas to find the area of regular and irregular polygons to estimate the volume of rectangular prisms by counting cubes to calculate the volume of rectangular prisms by developing a formula to identify the radius, diameter, chords, and central angles of a circle to know the relationship between radius and diameter to know the relationship between the circumference and the diameter of a circle (value of pi is circumference divided by diameter) to know pi is approximately = to (Note: Do not use this approximation in calculations) (to estimate circumference and area, use 3 as an approximation for!) to find the circumference of a circle using a formula (Note: students should leave answer in terms of!, the only approximation of! that is acceptable is the! key on a calculator) to find the area of a circle using a formula (Note: students should leave answer in terms of!, the only approximation of! that is acceptable is the! key on a calculator) to calculate the area of a sector of a circle given the central angle and radius 6A: Enriched Objectives and Resources to calculate the radius or diameter when given the circumference or area of a circle (Holt: 7-6) to identify pairs of angle relationships (Holt 8-3) (Holt Text: Hands on Lab: pages ) to find the area of composite figures when comprised of quadrilaterals, triangles, and/or circles (Holt 10-3) (Supplement: Holt Grade 7 Text 9-6) AECSD Grade_6_Math_Curr Rev 7-08.doc 27

28 Resources: Holt: 9-7 (Perimeter) 10-1 (Estimating and finding area) 10-2 (Area of triangles-only) 10-3 (Area of composite figures) 9-8 (Circles and circumference) 10-4 (Comparing Perimeter and Area) 10-5 (Area of a circle) note: answer must be in terms of! when estimating use 3 instead of 3.14 to find the answer Supplement: chord central angle area of sector of a circle (Holt text page 764; Coach Workbook) 10-6 (Three-dimensional figures) 10-7 (Volume of prisms) Problem solving: A rectangular prism has a volume of 36 cubic inches. Its length is 2 inches, its depth 9 inches. How wide is it? Explain how you found your answer. (Answer: 2 inches) Review Template (No Calculators): Geometry vocabulary 1, Which quadrilaterals have 2 pair of parallel sides? 2. Classify a triangle that has two sides that measure 4 cm. each, and one side with a length of 5 cm. 3. Explain why all squares are rectangles, but not all rectangles are squares? 4. True or false? a. A square is also a parallelogram. b. Every four-sided figure can be classified as more than one type of quadrilateral. c. All triangles are quadrilaterals. d. A rectangle is a parallelogram. e. A square is a quadrilateral. f. Triangles have 4 sides. AECSD Grade_6_Math_Curr Rev 7-08.doc 28

29 Plot points in first quadrant Plot points to form, identify, and classify basic geometric shapes Calculate perimeter of basic geometric shapes on the coordinate plane AECSD Grade_6_Math_Curr Rev 7-08.doc 29

30 Unit 7 Measurement (M.5) ~2 weeks Late January State Standards (Shaded statements are identified as Post-March Indicators): 6.M.1 Measure capacity and calculate volume of a rectangular prism 6.M.2 Identify customary units of capacity (cups, pints, quarts, and gallons) 6.M.3 Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons) 6.M.4 Identify metric units of capacity (liter and milliliter) 6.M.5 Identify equivalent metric units of capacity (milliliter to liter and liter to milliliter) 6.M.6 Determine the tool and technique to measure with an appropriate level of precision: capacity 6.M.8 Justify the reasonableness of estimates 6.M.9 Determine personal references for capacity Local Standards (Stricken text is covered in a different unit): L.6.M.5 Measure volume (capacity) using the appropriate tool and technique; identify and convert between units of volume (capacity) within a given system (metric or customary); determine personal references for customary and metric units of volume (capacity); justify the reasonableness of an estimate of the volume of an object. Big Ideas: The metric system is used in almost every country in the world while customary units are used in the United States. Conversions are simple in the metric system because every unit is based on the number 10. Essential Questions: Is it easier to convert in the metric or the customary system? Why? Prior knowledge: to determine personal references and approximate comparisons for metric units to determine personal references and approximate comparisons for customary units to identify the basic metric units and prefixes for measuring length, mass, capacity to convert equivalent metric units of length and mass to identify the customary equivalent units of length and mass to convert equivalent customary units of length and mass Unit Objectives: to identify metric units of capacity (liter and milliliter) to convert equivalent metric units of capacity(liter to milliliter and milliliter to liter) to identify customary units of capacity(cups, pints, quarts, and gallons) to convert equivalent customary units of capacity(cups to pints, pints to quarts, quarts to gallons) to determine the tool and technique to measure an appropriate level of precision: capacit AECSD Grade_6_Math_Curr Rev 7-08.doc 30

31 Resources: Holt: 9-1 (Understanding Customary Units of Measure) 9-2 (Understanding Metric Units of Measure) 9-3 (Converting Customary Units) 9-4 (Converting Metric Units) Supplement: tools and techniques for measuring capacity Review Template (No Calculators): Calculate elapsed time in hours and minutes 1. Reuben is going out, but has promised he would be home by 5:30 PM. It takes him 20 minutes to skate over to his friend s house. He will stay there for two hours. On the way home, he always stops for a snack, so the return trip takes 30 minutes. If Reuben is to keep his promise, by what time must he leave home? 2. Bryan got to school at 8:05 AM. It took him 1/4 hour to walk from home, 1/4 hour to look at some magazines along the way, and 1/2 hour to eat breakfast. What time did he sit down to eat? Metric basic units and prefixes 1. Choose the most appropriate unit of measure to estimate the length of each object. Write centimeter or meter. a. book b. school bus c, postcard d. computer desk Process for converting within metric system Complete: mm = cm cm = mm cm = m mm = m 5. 5 m = mm ml = L L = ml L = ml kg = g g = kg Customary units for length and mass Complete: in. = ft yd. = ft ft. = in yd. = in. Personal references/approximate comparisons for both metric and customary units AECSD Grade_6_Math_Curr Rev 7-08.doc 31

32 Unit 8 Rational Numbers (N.3, N.7) ~1 week First week of February State Standards (Shaded statements are identified as Post-March Indicators): 6.N.13 Define absolute value and determine the absolute value of rational numbers (including positive and negative) 6.N.14 Locate rational numbers on a number line (including positive and negative) 6.N.15 Order rational numbers (including positive and negative) Local Standards (Stricken text is covered in a different unit): L.6.N.3 Order rational numbers (positive and negative) including locating them on a number line L.6.N.7 Identify, use, and explain: the commutative and associative properties of addition and multiplication, the distributive property of multiplication over addition, the identity and inverse properties of addition and multiplication, and the zero property of multiplication; define absolute value and determine the absolute value of rational numbers Big Ideas: Numbers can be either positive or negative. Negative numbers are necessary in real-life situations, such as to express temperatures below zero. On a number line, any number to the left of another number is smaller. Positive and negative numbers are used to express opposites. Essential Questions: Is a negative number always less than a positive number? Explain. Is the absolute value of a 3 the same as the absolute value of a + 3? Explain. Prior knowledge: to order and compare whole numbers to order and compare decimals and fractions Unit Objectives: to define rational numbers to locate rational numbers on a number line (including positive and negative) to compare and order rational numbers (including positive and negative) (Holt Text:4-4) to know absolute value of rational numbers to determine the absolute value of rational numbers( including positive and negative) AECSD Grade_6_Math_Curr Rev 7-08.doc 32

33 6A: Enriched Objectives and Resources to add integers (POST MARCH) (Holt 11-4) (see Holt: Chapter 11 Resource Book) to subtract integers (POST MARCH) (Holt: 11-5) (see Holt: Chapter 11 Resource Book) Resources: Supplement: Holt Text Skills Bank page 762; Coach Workbook (Rational numbers; define and locate on number line) Supplement: Holt Grade 7 Text 2-11; Coach Workbook (Comparing, ordering rational numbers) Supplement: Holt Text Skills Bank page 762; Coach Workbook (Absolute Value) Review Template (No Calculators): Compare and order whole number, decimals, and fractions Compare using >, <, or =. 1. 1/2 7/8 2. 5/8 2/ /8 2 5/ /7 1 4/ / / / / AECSD Grade_6_Math_Curr Rev 7-08.doc 33

34 Unit 9 Data and Statistics (D.2) ~ 2 weeks February State Standards (Shaded statements are identified as Post-March Indicators): 6.S.5 Determine the mean, mode and median for a given set of data 6.S.6 Determine the range for a given set of data 6.S.7 Read and interpret graphs 6.S.8 Justify predictions made from data Local Standards (Stricken text is covered in a different unit): L.6.S.2 Calculate and use the mean, mode, median, and range of a set of data; read, interpret, and predict from (with justification) graphs Big Ideas: Graphs make it easier to compare data and find possible patterns. Mean, median and mode are all measures of central tendency but they tell us different things. Essential Questions: Why might a graph showing data be better than a list of data? Why might someone want to create a misleading graph? Must the mean, the median, or the mode of a data set a member of the set? Explain. How are the mean, median and the mode similar? Different? Why would you use one before another? Prior knowledge: to read and interpret line and bar graphs to calculate the mean of a set of data Unit Objectives: to identify types of graphs (line, bar, circle, pictograph, histogram, Venn diagram) to identify common ways that a graph can suggest misleading relationships to read and interpret different types of graphs to make predictions based on data analysis to determine the mean, mode and median for a given set of data to determine the range for a given set of data 6A: Enriched Objectives and Resources to learn the effect of additional data and outliers (Holt: 6-3) to make and analyze stem-and-leaf plots (Holt: 6-9) to choose an appropriate way to display data (POST MARCH) (Holt: 6-10) AECSD Grade_6_Math_Curr Rev 7-08.doc 34

35 Resources: Holt: 6-2 (Mean, median, mode, range) 6-4 (Bar graphs) 6-5 (Histogram) Holt Text, page (Line graphs) 6-8 (Misleading graphs) Supplement: Holt Text: Extension pages (Venn Diagrams) Holt Text: Hands on Lab pages (Circle Graphs) Holt Text: page 251, 759 (pictograph) Optional: M States Project Review Template (No Calculators): Read and interpret line and bar graphs ***Find and scan in graphs Mean for a set of data Calculate the mean for each set of data. 1. 8,10,7.6.5,7, , 18,10, 13, 15, 19, 22, , 2.2, 3.1, 2.4, 1.7, Problem solving: 1. Fifteen customers have dessert with lunch. Seven order ice cream and 10 order apple pie. How many order both? (Answer: 2 - Use a Venn diagram) Review Unit for State Assessment (Test dates: March) 1 week first week in March AECSD Grade_6_Math_Curr Rev 7-08.doc 35