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Ocean Springs School District 6th Grade – 2nd 9 Weeks Instructional Map Common Core Standard Topic 6 Standard Expectations(s) Students will be able to Clarity of the Standard Vocabulary Example: Jason thinks that ¾ and .34 are equivalent. Is he correct? Explain why or why not. Proper fraction Improper fraction Mixed number Termination decimal Repeating decimal Resources Mathematical Practices Suggested number of days Decimals, Fractions, and Mixed Numbers 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Additional cluster I Can: 6.NS.3 Fluently add multi-digit decimal numbers. 6.NS.3 Fluently subtract multi-digit decimal numbers. 6.NS.3 Fluently multiply multi-digit decimal numbers. 6.NS.3 Fluently divide multi-digit decimal numbers. Example: enVision (TopicSection) 6-1 Circle all of the following that are equivalent to 37 divided by 5 A. 7.2 B. 7 2 5 C. 37 5 D. .01 6-2 E. 7.4 6-3 6-4 6.MP.2. Reason abstractly and quantitatively. 6.MP.7. Look for and make use of structure. 6.MP.8. Look for and express regularity in repeated reasoning. Topic 6 7 days 6-5 Topic 7: Adding and Subtracting Fractions and Mixed Numbers 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than I Can: Example 1: What is the least common multiple (LCM) of 18 and 24? 6.NS.4 Determine the greatest common Example 2: Mr. Barrett’s class made a banner that is 6 ½ feet long and Ms. factor (GCF) of two Like Denominators Common Multiples Least Common Multiple Unlike Denominators Least Common Denominators 7-1 7-2 6.MP.1. Make sense of problems and persevere in solving them. Topic 7 8 days or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). Additional cluster numbers less than or Alvarend’s class made a banner that is 7 1 feet long . How much longer is Ms. 5 equal to 100. Alvarend’s banner? 6.NS.4 Justify the LCM of two whole numbers using math. 6.NS.4 Prove that two whole numbers have a common factor based on the distributive property. 7-3 7-4 7-5 7-6 6.MP.2. Reason abstractly and quantitatively . 6.MP.3. Construct viable arguments and critique the reasoning of others. 6.MP.4. Model with mathematics. 6.MP.6. Attend to precision 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. MAJOR cluster Example : I Can: 6.RP.1 Write a ratio to describe a situation. 6.RP.1 Write a ratio a as a to b, , or b a:b. or describe it in words. A ratio is a comparison of two quantities which can be written as a to b, a , or a:b. b A rate is a ratio where two measurements are related to each other. When discussing measurement of different units, the word rate is used rather than ratio. Understanding rate, however, is complicated and there is no universally accepted definition. When using the term rate, contextual understanding is critical. Students need many opportunities to use models to demonstrate the relationships between quantities before they are expected to work with rates numerically. A comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4 and can be regrouped into 4 black circles to 2 white circles (4:2) and 2 black circles to 1 white circle (2:1). 7-7 6.MP.2. Reason abstractly and quantitatively . 6.MP.6. Attend to precision Students should be able to identify all these ratios and describe them using “For every…., there are …” Topic 8: Multiplying Fractions and Mixed Numbers This topic does not have any 6th grade CCSS as its focus. It is taught as preparation for Topic 9 and reviews multiplication of fractions and estimation. I Can: Fluently multiply common fractions. Fluently multiply common fractions with whole numbers. 8-1 Example 1: Eric says that if you multiply a whole number by a fraction, you get something smaller than the whole number you started with. a. Give an example that supports Eric’s statement. b. Give an example that disproves Eric’s statement. c. Explain what kind of fraction will always make his statement true. 8-2 8-3 8-4 Fluently change mixed numbers into improper fractions. 8-5 6.MP.2. Reason abstractly and quantitatively . 6.MP.3. Construct viable arguments and critique the reasoning of others. Topic 8 5 days 6.MP.7. Look for and make use of structure. I can estimate fractional products. Topic 9: Dividing Fractions and Mixed Numbers 6.NS.1 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent 1 I Can: 6.EE.3 Apply the properties of operations to generate equivalent expressions. 1 Example 1 1 4 cups of nuts fill 2 of the container. How many cups fill it entirely? Common Misconception: Students may believe that dividing by ½ is the same as dividing in half. Dividing by half means to find how many one-halves there are in a quantity. Dividing in half means to take a quantity and split it into two equal parts. Thus 7 divided by ½ = 14 and 7 divided in half equals 3½. Reciprocals 9-1 9-2 9-3 6.MP.2. Reason abstractly and quantitatively. 6.MP.3. Construct viable arguments and critique the reasoning of Topic 9 7 days expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. MAJOR cluster 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. MAJOR cluster 6.NS.6 Identify when two expressions are equivalent (i.e., when the two expressions name the same others. Students also write contextual problems for fraction division problems. For 2 1 example, the problem, ÷ can be illustrated with the following g word 3 6 problem: 2 1 Susan has of an hour left to make cards. It takes her about of an hour to 3 6 make each card. About how many can she make? 9-4 9-5 6.MP.4. Model with mathematics. 6.MP.6. Attend to precision. 6.MP.7. Look for and make use of structure. I Can: 6.EE.7 Write an equation from realworld situations and then use inverse operations to solve the equation. 6.EE.7 Write and solve equations of the form x+p and px=q (in which p,q, and x are nonnegative rational numbers) I Can: 6.NS.2 Apply the rules of division to solve multi-digit Students also write contextual problems for fraction division problems. For 2 1 example, the problem, ÷ can be illustrated with the following g word 3 6 problem: 2 1 Susan has of an hour left to make cards. It takes her about of an hour to 3 6 make each card. About how many can she make? 9-6 7 6.MP.1. Make sense of problems and persevere in solving them. 6.MP.2. Reason abstractly and quantitatively. Solve for n: n = 42 6 6.MP.4. Model with mathematics. Find the pattern in this set of numbers and find the next number: 13, 11 2 3 1 , 10 , 9, ____ 3 9-7 6.MP.3. Construct viable arguments and critique the reasoning of number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for problems. others. 6.MP.7. Look for and make use of structure. 6.MP.8. Look for and express regularity in repeated reasoning. MAJOR cluster Topic 10: Number and Fraction Concepts 6.NS.5 . Understand that positive and negative numbers are used together to describe quantities having opposite directions or values 6. NS. 6a a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, I Can: 6.NS.5 Understand that positives and negatives represent opposites. 6.NS.7c Define and identify absolute value. 6.NS.7c Use a number line to find the absolute value of any number. 6.NS.7c Understand absolute value as the distance from zero on the number line 5. For example, 25 feet below sea level can be represented as -25; 25 feet above sea level can be represented as + 25 and zero would be sea level Example: Write a real-life example of each of the following numbers: -13 +72 -26 6a. Students recognize that a number and its opposite are equidistant from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. Opposites Integers Absolute Value 10-1 6.MP.1. Make sense of problems and persevere in solving them. 6.MP.2. Reason abstractly and quantitatively. 6.MP.4. Model with mathematics. For example, – 4 could be read as “the opposite of 4” which would be negative 4. In the example, – (–6.4) would be read as Topic 10 10 days e.g., – (–3) = 3, and that 0 is its own opposite 6.NS.7.c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute as magnitude for a positive or negative quantity in a real-world situation. 6.NS.7c Apply absolute value to real world situations. 6.NS.7c Recognize the symbol \\ as representing absolute value Example: What is the opposite of 2 ½ ? Explain your answer. Solution:- 2 ½ because it is the same distance from 0 on the opposite side. 7c. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. Example 1: Which numbers have an absolute value of 7 Solution: 7 and –7 since both numbers have a distance of 7 units from 0 on the number line. All are MAJOR clusters 6. NS. 7 “the opposite of the opposite of 6.4” which would be 6.4. Zero is its own opposite. In real-world contexts, the absolute value can be used to describe size or magnitude. For example, for an ocean depth of 900 feet, write | –900| = 900 to describe the distance below sea level. I Can: 6.NS.7 Students use inequalities to express the relationship Understand ordering and 6.NS.7a. Compare between two rational numbers, understanding that the value of absolute value of negative and positive numbers is smaller moving to the left on a number line. Common rational numbers. numbers models to represent and compare integers include number line a. Interpret statements 6.NS.7a Understand models, temperature models and the profit-loss model. On a number of inequality as ordering or rational line model, the number is represented by an arrow drawn from zero statements about the numbers to the location of the number on the number line; the absolute 6.NS.7a Order relative position of value is the length of this arrow. The number line can also be two numbers on a number rational numbers on viewed as a thermometer where each point of on the number line is a number line line. a specific temperature. In the profit-loss model, a positive b. Write, interpret, and number corresponds to profit and the negative number corresponds 10-2 6.MP.1. Make sense of problems and persevere in solving them. 6.MP.2. Reason abstractly and quantitatively. 6.MP.3. Construct viable arguments explain statements of order for rational numbers in real-world contexts MAJOR clusters 6.NS.7b Write an inequality to show the relationship between rational numbers in real world situations. 6.NS.7b Explain in my own words how to compare rational numbers (written or spoken) to a loss. Each of these models is useful for examining values but can also be used in later grades when students begin to perform operations on integers. and critique the reasoning of others. a. Example :Write a statement to compare – 4 . and –2. Explain your answer. Solution: – 4 . < –2 because – 4 . is located to the left of –2 on the number line Operations with integers are not the expectation at this level. In working with number line models, students internalize the order of the numbers; larger numbers on the right of the number line and smaller numbers to the left. They use the order to correctly locate integers and other rational numbers on the number line. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between two numbers 6. MP.5 Use appropriate tools strategically. b. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right Example :The balance in Sue’s checkbook was –$12.55. The balance in John’s checkbook was –$10.45. Write an inequality to show the relationship between these amounts. Who owes more? Example : A meteorologist recorded temperatures in four cities around the world. List these cities in order from coldest temperature to warmest temperature: Albany 5° Anchorage -6° Buffalo -7° Juneau -9° Reno 12° 6. NS.6c Find and position integers and other rational numbers on a horizontal line Rational Number I Can: 6.NS.6 Plot all integers and other 6.NS.6 In elementary school, students worked with positive fractions, decimals and whole numbers on the number line and in quadrant 1 of the coordinate plane. In 6th grade, students extend the number line to represent all rational numbers and recognize 10-3 6.MP.1. Make sense of problems and persevere in solving them. diagram. MAJOR cluster rational numbers on number lines 6.NS.6 Order rational numbers on a number line horizontally 6.MP.2. Reason abstractly and quantitatively. that number lines may be either horizontal or vertical (which facilitates the movement from number lines to coordinate grids. Students recognize that a number and its opposite are equidistant from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. 6.MP.3. Construct viable arguments and critique the reasoning of others. 6c. Students place the following numbers would be on a number line: 3 11 –4.5, 2, 3.2, –35 , 0.2, –2, 2 6. MP.5 Use appropriate tools strategically. 8. Look for and express regularity in repeated reasoning. Sections 10-4, 10-5, 10-6, 10-7 WILL NOT be taught. Please skip to Section 10-8 6.NS. 7d Distinguish comparisons of absolute value from statements about order. I Can: 6.NS.7d Distinguish comparisons of absolute value by When working with positive numbers, the absolute value (distance from zero) of the number and the value of the number is the same; therefore, ordering is not problematic. However, negative numbers have a distinction that students need to understand. As the negative number increases (moves to the left on a number line), the value of the number decreases. For example, –24 is less than –14 because –24 is located to the left of –14 on the number line. 10-8 6.MP.2. Reason abstractly and quantitatively. 6.MP.3. Construct viable arguments MAJOR cluster 6. NS. 6b 6. NS.6c b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line statements of order. 6.NS.7d Understand that absolute value deals only with distance from zero (regular whole numbers are their true values) I Can: 6.NS.6c Plot all integers and other rational numbers on coordinate planes 6.NS.6c Find the position of integer pairs and other rational numbers on a coordinate plane. 6.NS.8 Graph points in all four quadrants of the coordinate plane. 6.NS.8 Solve realworld problems by and critique the reasoning of others. However, absolute value is the distance from zero. In terms of absolute value (or distance) the absolute value of –24 is greater than the absolute value of –14. For negative numbers, as the absolute value increases, the value of the negative number decreases. 6. MP.5 Use appropriate tools strategically. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30dollars. 6. MP.6 Attend to precision. Students worked with Quadrant I in elementary school. As the xaxis and y-axis are extending to include negatives, students begin to with the Cartesian Coordinate system. Students recognize the point where the x-axis and y-axis intersect as the origin. Students identify the four quadrants and are able to identify the quadrant for an ordered pair based on the signs of the coordinates. For example, students recognize that in Quadrant II, the signs of all ordered pairs would be (–, +). Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the ycoordinates differ only by signs, which represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been reflected across both axes. Example1: Graph the following points in the correct quadrant of the coordinate plane. ( ½ , -3½ ) ( - ½ , -3) (0.25, 0.75) Coordinate Plane x-axis y-axis Quadrant Ordered Pair Origin 10-9 6.MP.1 Make sense of problems and persevere in solving them. 6.MP.2. Reason abstractly and quantitatively. 6. MP.6 Attend to precision. 6.MP.7 Look for and make use of structure. diagram; find and position pairs of integers and other rational numbers on a coordinate plane 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. graphing points in all four quadrants of a plane(maps, shapes, pictures) 6.NS.8 Solve mathematical problems by graphing points in all four quadrants of a plane (maps, shapes, pictures) 6.NS.8 Calculate distances between two points with the same first or second coordinates using absolute values, given only coordinates. If the points are reflected across the x-axis, what are the coordinates of the reflected points? What similarities are between coordinates of the original point and the reflected point? 6. NS.8 Example 1: What is the distance between (–5, 2) and (–9, 2)? All are MAJOR clusters 6. G.3 G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use I Can: 6.G.3 Draw polygons on a coordinate plane 6.G.3 Students are given the coordinates of polygons to draw in the coordinate plane. If both x-coordinates are the same (2, -1) and (2, 4), then students recognize that a vertical line has been created and the distance between these coordinates is the distance between -1 and 4, or 5. If both the y-coordinates are Perimeter 10-10 coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. ADDITIONAL cluster given coordinates for the vertices 6.G.3 Evaluate the length of polygons by using grid models. 6.G.3 Determine the length of the sides of polygons in a coordinate plane given the same first or second coordinate. 6.G.3 Apply the technique of using coordinate planes to find the length of the side of polygons in realworld situations. the same (-5, 4) and (2, 4), then students recognize that a horizontal line has been created and the distance between these coordinates is the distance between -5 and 2, or 7. Using this understanding, student solve real-world and mathematical problems, including finding the area and perimeter of geometric figures drawn on a coordinate plane. Example 1:If the points on the coordinate plane below are the three vertices of a rectangle, what are the coordinates of the fourth vertex? How do you know? What are the length and width of the rectangle? Find the perimeter of the rectangle. Solution: To determine the distance along the x-axis between the point (-4, 2) and (2, 2) a student must recognize that -4 is |-4| or 4 units to the left of 0 and 2 is |2| or 2 units to the right of zero, so the two points are total of 6 units apart along the x-axis. Students should represent this on the coordinate grid and numerically with an absolute value expression, |-4| + |2| . The length is 6 and the width is 5. The fourth vertex would be (2, -3). The perimeter would be 5 + 5 + 6 + 6 or 22 units.