Mathematics/Grade 6 Unit 1: Understanding Positive and Negative Numbers Grade/Subject Grade 6/ Mathematics Unit Title Unit 1: Understanding Positive and Negative Numbers Overview of Unit In this unit students will apply and extend previous understanding of numbers to the system of rational numbers. They will develop an understanding of numbers less than zero and their real-world applications. Pacing 21 days Background Information For The Teacher In previous grades students used number lines as a tool and in fifth grade, students learned to graph points on the coordinate plane in the first quadrant to solve real-world and mathematical problems. In this unit, students will extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. Prior to the Common Core Standards, sixth grade students were responsible for operating (addition and subtraction) with integers. There has been a shift in learning where students are now developing a strong understanding of positive and negative numbers and how they are used in the real-world. All activities used throughout this unit should be related to the real world in order to help students model with mathematics. A student-centered approach will foster students’ understanding of the concepts. Through student discourse, and justification of their answers, students should regularly be required to construct viable arguments and critique the reasoning of others in order to further develop their understanding. Essential Questions (and Corresponding Big Ideas ) Revised March 2017 1 Mathematics/Grade 6 Unit 1: Understanding Positive and Negative Numbers How do I plot and order whole numbers, fractions, and decimals on a number line? Use inequalities to compare two numbers in the same form or different forms. Proper fractions are between 0 and 1 on the number line. Why are there negative numbers? When zero is used as the origin it is useful in describing a unit of space. To show a mathematical relation of loss. When will I ever need to use negative numbers? Negative numbers can be used to represent quantities less than zero or quantities with an associated direction. When will I ever use greatest common factor and least common multiple? The greatest common factor and least common multiple are used when operating with fractions. Core Content Standards Explanations and Examples 6.EE.1. Write and evaluate numerical 6.EE.1. Examples: expressions involving whole-number • Write the following as a numerical expressions using exponential notation. exponents. o The area of a square with a side length of 8 m (Solution: 82m2) 3 3 o The volume of a cube with a side length of 5 ft: (Solution: 5 ft ) This standard concentrates on whole number exponents with a focus o Yu-Lee has a pair of mice. The mice have 2 babies. The babies grow up and on understanding the meaning of exponents and exponential notation 3 such as 32 = 3 × 3. Students find the value of an expression using have two babies of their own: (Solution: 2 mice) exponential notation such as 43 = 64. Students write and evaluate • Evaluate: numerical expressions such as: 5 + 24 ∗ 6. 3 o 4 (Solution: 64) 4 What the teacher does: o 5 + 2 ● 6 (Solution: 101) Plan experiences for students to investigate that an 2 7 – 24 ÷ 3 + 25 (Solution: 67) exponent is a notation representing repeated multiplication such as 10 * 10 * 10 as 103. The exponent is the number that tells how many factors of 10 there are. The expression What the students do: 103 is called the exponential expression. Exponential Understand the meaning of exponents and exponential notation such as 33 = 3 ∗ 3. notation was developed to write repeated multiplication Use appropriate terminology to explain how to evaluate an expression. more efficiently. Evaluate numerical expressions containing exponents. Use patterns to discover why for example 20 is one and 350 Discover that nay base to the zero power is 1. is one. Allow students to use manipulatives to show 32 = 9 square units. Ask students to model the meaning of 33 = 27 cubic Misconceptions and Common Errors: Revised March 2017 2 Mathematics/Grade 6 Unit 1: Understanding Positive and Negative Numbers units with manipulatives. Expand to problems such as 5 + 24 and 7 ∗ 43. Some students interpret 32 as 3 X 2 = 6. This is a common error. Use a number line representation to model the expression. Also, writing the expanded notation of 32 =3 x 3 helps students. 6.NS.4. Find the greatest common factor of 6.NS.4 two whole numbers less than or equal to 100 What is the greatest common factor (GCF) of 24 and 36? How can you use and the least common multiple of two whole factor lists or the prime factorizations to find the GCF? numbers less than or equal to 12. Use the Solution: 22 3 = 12. Students should be able to explain that both 24 and 36 distributive property to express a sum of two have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers factor.) with no common factor. For example, express What is the least common multiple (LCM) of 12 and 8? How can you use 36 + 8 as 4(9+2). The emphasis for this standard is finding factors and multiples of a multiple lists or the prime factorizations to find the LCM? given number(s). Students need to know that numbers being Solution: 23 3 = 24. Students should be able to explain that the least multiplied are the factors, and the product is the multiple. Explore two different methods for factoring. common multiple is the smallest number that is a multiple of 12 and a multiple Introduce the distributive property as an application of factors. When of 8. To be a multiple of 12, a number must have 2 factors of 2 and one factor you add two numbers that have a common factor such as 36 and 8, of 3 (2 x 2 x 3). To be a multiple of 8, a number must have 3 factors of 2 (2 x 2 x you can remove the greatest common factor, 4, and distribute it to the remaining factors. Such as: 36 + 8 = (4 X 9) + (4 X 2) = 4(9 + 2) 2). Thus the least common multiple of 12 and 8 must have 3 factors of 2 and one factor of 3 ( 2 x 2 x 2 x 3). What the teacher does: Explore finding the greatest common factor (GCF) of two Rewrite 84 + 28 by using the distributive property. Have you divided by the whole numbers less than or equal to 100. The greatest common factor is the largest number that is a factor of two largest common factor? How do you know? whole numbers. For example, 30 and 18 can be found first Given various pairs of addends using whole numbers from 1-100, students by o Listing the factors of 30 (1, 2, 3, 5, 6, 10, 15, 30) should be able to identify if the two numbers have a common factor. If they do, o Listing the factors of 18 (1, 2, 3, 6, 9, 18) and noticing the they identify the common factor and use the distributive property to rewrite largest number in each list is the greatest common factor of (6) the expression. They prove that they are correct by simplifying both Examine another way for students to find the GCF. List the expressions. prime factors of two numbers and then multiply the common factors. For example, 40 = (2X2X2X5) and 16 = (2X2X2X2), and then multiply prime common factors o 27 + 36 = 9 (3 + 4) (2X2X2= 8). Ensure students understand the process of prime 63 = 9 x 7 factorization. Prime factorization is finding which prime Revised March 2017 3 Mathematics/Grade 6 Unit 1: Understanding Positive and Negative Numbers numbers multiply together to make the original number 63 = 63 such as the number 24 can be written as a product of prime numbers: 24 = 2 X 2 X 2 X 3. This called the prime factorization of 24. Model prime factorization for numbers o 31 + 80 1-100 using factor trees. There are no common factors. I know that because 31 is a prime number, it only has 2 factors, 1 and 31. I know that 31 is not a factor of 80 because 2 x 31 is 62 and 3 x 31 is 93. What the students do: Understand that a factor is a whole number that divides without a remainder into another number. Understand that a multiple is a whole number that is a product of the whole number and any other factor. Calculate GCF and LCM for given sets of numbers. Apply knowledge of factors and multiples to problem solving. Investigate the distributive property using sums and its use Communicate, using precise mathematical language, how the distributive property makes use of factors and multiples. in adding numbers 1-100 with a common factor. For example, 20 + 24 = 4(5 + 6). Use a common factor and the distributive property to find the sum of 36 and 8. 36 + 8 = 4(9) + 4(2) 44 = 4 (9 + 2) 44 = 4(11) 44 = 44 Misconceptions and Common Errors: Emphasize the least common multiple as it relates to this Some students may confuse the concepts of factors and multiples.