Foundations of Fractions (LCM, GCF, Mixed Numbers, Improper Fractions)

Transcription

1 Grade 6 Mathematics, Quarter 2, Unit 2.1 Foundations of Fractions (LCM, GCF, Mixed Numbers, Improper Fractions) Overview Number of instructional days: 12 (1 day = minutes) Content to be learned Find the greatest common factor (GCF) of two or more numbers. Find the least common multiple (LCM) of two or more numbers. Solve real-world problems involving GCF and LCM. Convert between and solve problems involving proper fractions, improper fractions, and mixed numbers. Order and compare fractions (proper, improper, mixed). Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Justify conclusions (by showing thinking/processes used to find LCM, GCF, and comparing/ordering fractions). Analyze situations (to determine when to use GCF vs. LCM). Attend to precision. Use clear definitions in discussion (by using GCF and LCM in discussion with others and in one s own reasoning). Calculate accurately and efficiently (the GCF and LCM in real-world problems). Make sense of problems and persevere in solving them. Check solutions with another method (by using multiple strategies/representations for GCF, LCM, and comparing/ordering fractions). Ask, Does my solution make sense? (when comparing and ordering fractions or when finding the GCF/LCM of two or more numbers. Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Cumberland, Lincoln, and Woonsocket Public Schools C-21

2 Grade 6 Mathematics, Quarter 2, Unit 2.1 Foundations of Fractions (LCM, GCF, Mixed Numbers, Improper Fractions) (8 days) Essential questions How is a factor different from a multiple? How can you decide if finding common multiples or common factors is helpful in solving problems? How do you find the common factors and greatest common factor of two numbers? What strategies can you use to order and compare fractions (proper, improper, mixed)? How do you find the common multiples and the least common multiple of two numbers? What are the differences between a proper and improper fraction? What is the relationship between improper fractions and mixed numbers? Given two unequal fractions with different denominators, which fraction is greater? How do you know? Written Curriculum Grade-Level Expectations M(N&O) 6 4 Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple. (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) M(N&O) 6 2 Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents (e.g.,3 3, 4 3 ), integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1 100) using number lines or equality and inequality symbols. (State) Clarifying the Standards Prior Learning In grades 3 4, students compared and ordered fractions using models, number lines, and explanations. Fourth-grade students explored the properties of factors and multiples. Then in grade 5, students further investigated multiples and factors, including prime and composite numbers. Current Learning In grade 6, students convert between and solve problems involving proper fractions, improper fractions, and mixed numbers. These processes are introduced, reinforced, and mastered at this level. Students also use their prior knowledge of factors and multiples to find and solve problems involving greatest common factor and least common multiple. Students also compare and order fractions (proper, improper, and mixed). Future Learning In grades 7 8, students will solve problems involving proportional reasoning. In grades 9 10, students will solve problems involving ratios, rates, and proportional relationships. C-22 Cumberland, Lincoln, and Woonsocket Public Schools

3 Grade 6 Mathematics, Quarter 2, Unit 2.2 Adding and Subtracting Fractions Overview Number of instructional days: 9 (1 day = minutes) Content to be learned Demonstrate a conceptual understanding of addition and subtraction of positive fractions (proper, improper, mixed). Solve problems involving single or multiple operations on fractions (proper, improper, and mixed) including order of operations and apply it in real-world situations. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Make conjectures (by estimating sums and differences). Justify conclusions (by showing thinking/processes used to solve problems). Clarify and improve arguments (by using different representations to clarify a solution). Attend to precision. Calculate accurately and efficiently (the sums and differences of problems in real-world situations and problems involving fractions). Use clear definitions in discussion (to use math vocabulary in discussion and in one s own reasoning). Communicate precisely. Model with mathematics. Apply math to solve problems involving addition and subtraction of fractions in everyday life, society, and the workplace. Essential questions How do you add and subtract fractions with like and unlike denominators? How could you explain to a peer that = 4 is not correct? 0 How do you add and subtract mixed numbers with unlike denominators? How are addition and subtraction of fractions related? In what real-world situation do you need to apply addition and subtraction of fractions? Explain. When solving a real-world problem, how do you know when to add or subtract to arrive at a correct solution? Cumberland, Lincoln, and Woonsocket Public Schools C-23

4 Grade 6 Mathematics, Quarter 2, Unit 2.2 Adding and Subtracting Fractions (9 days) Grade-Level Expectations Written Curriculum M(N&O) 6 3 Demonstrates conceptual understanding of mathematical operations by adding and subtracting positive fractions and integers; and multiplying and dividing fractions and decimals. (Local) M(N&O) 6 4 Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple. (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) Clarifying the Standards Prior Learning In grades K 2, students developed a conceptual understanding of addition and subtraction of whole numbers through solving problems a variety of ways, including through part-part-whole relationships. They also began formal addition and subtraction. In grade 3, students looked at the inverse relationship between addition and subtraction. In grades 4 and 5, students learned how to accurately solve problems involving the addition and subtraction of positive fractions with like and unlike denominators using number lines, models, and explanations. In grade 4, students began solving problems involving the addition and subtraction of positive proper fractions with like denominators. In grade 5, students solved problems involving the addition and subtraction of positive proper fractions with unlike denominators. Current Learning In grade 6, students continue to develop their conceptual understanding of the addition and subtraction of positive fractions, including mixed numbers and improper fractions. Students also accurately solve problems involving single or multiple operations on fractions (proper, improper, and mixed). Problem types must include adding mixed numbers where an improper fractions results as part of the answer, and when subtracting, requires regrouping. These concepts should be mastered by the end of this unit. Students continue to use their estimation and mental-computation skills to verify the accuracy of their results. The order of operations, with and without parentheses, will be reinforced throughout the unit as students solve problems. Future Learning In grades 7 12, the addition and subtraction of positive fractions will be expanded to the addition and subtraction of negative fractions and will show up in probability (e.g., determining probabilities), algebra (e.g., solving equations), and geometry (e.g., finding perimeter). In grades 9 10, students will solve problems that involve proportional relationships, percents, ratios, and rates. In grades 11 12, students will solve problems involving scientific notation and will interpret rational exponents and their relationship to radicals. C-24 Cumberland, Lincoln, and Woonsocket Public Schools

5 Adding and Subtracting Fractions (9 days) Grade 6 Mathematics, Quarter 2, Unit 2.2 Additional Research Findings According to Principles and Standards for School Mathematics, students in the middle grades should continue to refine their understanding of addition, subtraction, multiplication, and division as they use these operations with fractions, decimals, percents, and integers. Students in grades 6 8 will continue to build their understanding of the inverse relationship between addition and subtraction (as well as the inverse relationship between multiplication and division) (p. 218). The book also states, teachers can help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation. Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation (Principles, p. 218). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-25

6 Grade 6 Mathematics, Quarter 2, Unit 2.2 Adding and Subtracting Fractions (9 days) C-26 Cumberland, Lincoln, and Woonsocket Public Schools

7 Grade 6 Mathematics, Quarter 2, Unit 2.3 Multiplying and Dividing Fractions Overview Number of instructional days: 7 (1 day = minutes) Content to be learned Demonstrate a conceptual understanding of multiplication and division of positive fractions. Solve and apply problems involving single or multiple operations on fractions (proper, improper, and mixed) including order of operations. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Make conjectures (by estimating products and quotients). Justify conclusions (by showing thinking/processes used to solve problems). Clarify and improve arguments (by using different representations to clarify a solution). Attend to precision. Calculate accurately and efficiently (the products and quotients of real-world problems involving fractions). Communicate precisely (use math vocabulary in discussion and in one s own reasoning). Model with mathematics. Apply math to solve problems (involving multiplication and division of fractions) in everyday life, society, and the workplace. Essential questions How are multiplication and division of fractions related? How would you represent 1/2 of 1/3 on a number line? How do you multiply and divide proper fractions, improper fractions, and mixed numbers? In what real-world situation would you apply multiplication and division of fractions? Explain. Why is the quotient of two positive fractions larger than the dividend? Why is the product of two positive fractions less than each of the two fractions you multiplied? Describe and illustrate how to multiply fractions by fractions, fractions by mixed numbers, and fractions by whole numbers. When solving real-world problems, how do you know when to multiply or divide? Cumberland, Lincoln, and Woonsocket Public Schools C-27

8 Grade 6 Mathematics, Quarter 2, Unit 2.3 Multiplying and Dividing Fractions (7 days) Written Curriculum Grade-Level Expectations M(N&O) 6 3 Demonstrates conceptual understanding of mathematical operations by adding and subtracting positive fractions and integers; and multiplying and dividing fractions and decimals. (Local) M(N&O) 6 3 Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of a power by representing the relationship between the base (whole number) and the exponent (whole number) (e.g.,3 3, 4 3 ); and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal, or fraction. (State) M(N&O) 6 4 Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple. (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) Clarifying the Standards Prior Learning Multiplication was introduced in third grade. Students learned multiplication of whole numbers based on repeated addition and used models, number lines, and explanations. In grade 4, students learned division of whole numbers (with no remainders) based on repeated subtraction and were introduced to the inverse relationship between multiplication and division of whole numbers. In grade 5, students continued their work with division and the meaning of a remainder with respect to whole numbers using models, explanations, or solving problems. As early as grade 3, students began to solve problems involving multiplication of whole numbers. In grade 4, they solved problems using division with single-digit divisors. Their work with solving division problems was expanded to two-digit divisors in grade 5. Ongoing practice with multiplication continued from grades 3 5. Current Learning In grade 6, students are introduced to multiplication and division of positive fractions, including mixed numbers and improper fractions. Students also accurately solve problems involving single or multiple operations on fractions (proper, improper, and mixed). These concepts should be mastered by the end of this unit. Students continue to use their estimation and mental computation skills to verify the accuracy of their results. Future Learning In grades 7 12, the multiplication and division of positive fractions will be expanded to the multiplication and division of negative fractions and will show up in probability (e.g., determining probabilities), algebra (e.g., solving equations), and geometry (e.g., finding area). C-28 Cumberland, Lincoln, and Woonsocket Public Schools

9 Multiplying and Dividing Fractions (7 days) Grade 6 Mathematics, Quarter 2, Unit 2.3 Additional Research Findings According to Principles and Standards for School Mathematics, multiplying and dividing fractions can be challenging for many students because of problems that are primarily conceptual rather than procedural. From their experience with whole numbers, many students appear to develop the belief that multiplication makes bigger and division makes smaller a mistaken expectation about the magnitude of a computational result is likely to interfere with students making sense of multiplication and division of fractions (p. 218). It is imperative that students master these concepts for further growth in mathematics. The basic operations form the foundation for future math courses. Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-29

10 Grade 6 Mathematics, Quarter 2, Unit 2.3 Multiplying and Dividing Fractions (7 days) C-30 Cumberland, Lincoln, and Woonsocket Public Schools

11 Grade 6 Mathematics, Quarter 2, Unit 2.4 Ratios, Rates, and Percentages Overview Number of instructional days: 12 (1 day = minutes) Content to be learned Demonstrate conceptual understanding of rational numbers with respect to ratios and rates. Demonstrate understanding of the relative magnitude of numbers by ordering and comparing across number formats (fractions, decimals, and whole number percents from 1 100). Solve problems involving percent of a whole accurately. Convert between fractions, decimals, and percents (not evident in GLEs). Mathematical practices to be integrated Make sense of problems and persevere in solving them. Make conjectures (using benchmark percents to estimate the percent of a whole number). Decide on a solution pathway before jumping into the work (when determining rates and unit rates). Check solution with another method (when determining rates and unit rates). Ask, Does my answer make sense? (when finding the percent of a whole; when converting between fractions, decimals, and percents; and when finding rates and unit rates). Attend to precision. Specify units of measure (when finding rates and unit rates). State the meaning of symbols (when writing ratios, rates, and percents). Use = consistently and appropriately (when determining whether two ratios are proportional; when converting between fractions, decimals, and percents; and when comparing rates and unit rates. Essential questions How are fractions and ratios similar? How are they different? How can you find a percent of a whole? How is a percent a ratio? What strategy do you use to order and compare percents? If 5% is $4.50 what is 100%? Given a fraction, a percent, and a decimal of different values, how would you order them from least to greatest? Cumberland, Lincoln, and Woonsocket Public Schools C-31

12 Grade 6 Mathematics, Quarter 2, Unit 2.4 Ratios, Rates, and Percentages (12 days) Written Curriculum Grade-Level Expectations M(N&O) 6 1 Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a : b, and a b, where b 0); and rates (e.g., a out of b, 25%) using models, explanations, or other representations. (State) M(N&O) 6 2 Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents (e.g.,3 3, 4 3 ), integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1 100) using number lines or equality and inequality symbols. (State) M(N&O) 6 4 Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple. (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) Clarifying the Standards Prior Learning In grades K 5, students progressively learned about fractions starting with the concept of (1/2) as a fair share in kindergarten moving on to benchmark fractions such as fourths and thirds in grade 5 using models, explanations, and other representations. In grade 5, they continued these concepts and began to look at benchmark percents within number formats (fractions to fractions, decimals to decimals, and percents to percents) using models or number lines. Current Learning In grade 6, students are introduced to the concepts of ratio and rate using models, explanations, and other representations. Students expand their work of understanding the relative magnitude of benchmark percents to encompass all whole number percents. Students are introduced to problem solving involving percent of a whole. Although not evident in the GLEs (grade 6 or prior), students must demonstrate an understanding of converting between fractions, decimals, and percents in order to be successful in this unit. Future Learning In grades 7 and 8, students will demonstrate conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of the whole when the wholes vary in magnitude; as a way of expressing multiples of a number using models, explanations, and other representations; and as a way of describing change (percent increase and decrease). These concepts will show up again in grades 11 and 12 when students are working with why certain real numbers are rational. C-32 Cumberland, Lincoln, and Woonsocket Public Schools

13 Ratios, Rates, and Percentages (12 days) Grade 6 Mathematics, Quarter 2, Unit 2.4 Students will continue to develop their conceptual understanding of the relative magnitude of numbers across number formats in grade 7 and will begin to look at irrational numbers in grade 8. In grades 10 12, students will expand this understanding to real numbers. In grades 7 and 8, students will take their knowledge of ratios and rates and begin to accurately solve problems involving discounts, tax, tips, rates, and proportional reasoning. Grade 8 also will add percent increase and decrease, interest rates, and markups. These concepts will continue in grades Additional Research Findings According to Principles and Standards for School Mathematics, by solving problems that require multiplicative comparisons, students will gain extensive experience with ratios, rates, and percents, which helps form a solid foundation for their understanding of, and their facility with, proportionality. The study of rational numbers in the middle grades should build on students prior knowledge of whole-number concepts and skills and their encounters with fractions, decimals, and percents in lower grades and in everyday life Students facility with rational numbers and proportionality can be developed in concert with their study of many topics in the middle-grades curriculum (p. 215). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-33

14 Grade 6 Mathematics, Quarter 2, Unit 2.4 Ratios, Rates, and Percentages (12 days) C-34 Cumberland, Lincoln, and Woonsocket Public Schools