Topic 1: Numeration Weeks: 1 Domain: The Number System Cluster: Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Knowledge Targets “I Can” Statements Standard Interpretations Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation with speed and accuracy. I can fluently add multi-digit decimals using the standard algorithm. I can fluently subtract multi-digit decimals using the standard algorithm. I can fluently multiply multi-digit decimals using the standard algorithm. I can fluently divide multi-digit decimals using the standard algorithm. The use of estimation strategies supports student understanding of operating on decimals. Example: First, students estimate the sum and then find the exact sum of 14.4 and 8.75. An estimate of the sum might be 14 + 9 or 23. Students may also state if their estimate is low or high. They would expect their answer to be greater than 23. They can use their estimates to self-correct. Answers of 10.19 or 101.9 indicate that students are not considering the concept of place value when adding (adding tenths to tenths or hundredths to hundredths) whereas answers like 22.125 or 22.79 indicate that students are having difficulty understanding how the four-tenths and seventy-five hundredths fit together to make one whole and 25 hundredths. Students use the understanding they developed in 5th grade related to the patterns involved when multiplying and dividing by powers of ten to develop fluency with operations with multi-digit decimals. Reasoning Target Performance Target
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
Cluster: Reason about and solve one –variable equations and inequalities. 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Knowledge Targets “I Can” Statements Standard Interpretations Recognize solving an equation or inequality as a process of answering “which values from a specified set, if any, make the equation or inequality true?” Know that the solutions of an equation or inequality are the values that make the equation or inequality true. Use substitution to determine whether a given number in a specified set makes an equation or inequality true. I can understand solving an equation or inequality as a process of answering questions. I can recognize that the correct answer to an equation or inequality makes it true. I can use substitution to check the answer or determine if a given number is a solution. Beginning experiences in solving equations should require students to understand the meaning of the equation as well as the question being asked. Solving equations using reasoning and prior knowledge should be required of students to allow them to develop effective strategies such as using reasoning, fact families, and inverse operations. Students may use balance models in representing and solving equations and inequalities. Consider the following situation: Joey had 26 papers in his desk. His teacher gave him some more and now he has 100. How many papers did his teacher give him? This situation can be represented by the equation 26 + n = 100 where n is the number of papers the teacher gives to Joey. This equation can be stated as “some number was added to 26 and the result was 100”. Students ask themselves “What number was added to 26 to get 100?” to help them determine the value of the variable that makes the equation true. Students could use several different strategies to find a solution to the problem. Reasoning: 26 + 70 is 96. 96 + 4 is 100, so the number added to 26 to get 100 is 74. Use knowledge of fact families to write related equations: n + 26 = 100, 100 - n = 26, 100 - 26 = n. Select the equation that Reasoning Target Performance Target
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
Cluster: Reason about and solve one-variable equations and inequalities. 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Knowledge Targets “I Can” Statements Standard Interpretations Recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. I can write an expression to represent a or mathematical situation. I can represent a number with a variable. I can justify that a variable can represent an unknown number or any number in a specified set. See Topic 2 6.EE.6 Reasoning Target Relate variables to a context. Write expressions when solving a real-world or mathematical problem. Performance Target Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 3: Operations with Decimals Estimate Compatible Numbers world
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Reason about and solve one-variable equations and inequalities
6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Knowledge Targets “I Can” Statements Standard Interpretations Recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. I can write an expression to represent a or mathematical situation. I can represent a number with a variable. I can justify that a variable can represent an unknown number or any number in a specified set. See Topic 2 6.EE.6 Reasoning Target Relate variables to a context. Write expressions when solving a real-world or mathematical problem. Performance Target Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Reason about and solve one-variable equations and inequalities. world
money is divided. I ended up giving each pair of jeans another $0.86. Each pair of jeans costs $18.86 (15+3+0.86). I double check that the jeans cost $18.86 each because $18.86 x 3 is $56.58.” Julio gets paid $20 for babysitting. He spends $1.99 on a package of trading cards and $6.50 on lunch. Write and solve an equation to show how much money Julio has left. (Solution: 20 = 1.99 + 6.50 + x, x = $11.51) Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
should be able to identify if the two numbers have a common factor. If they do, they identify the common factor and use the distributive property to rewrite the expression. They prove that they are correct by simplifying both expressions. 27 + 36 = 9 (3 + 4) 63 = 9 x 7 63 = 63 31 + 80 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. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 5: Number and Fraction Concepts Divisible Prime Number Composite Number Prime Factorization Greatest Common Factor Fractions Numerator Denominator Equivalent Fractions
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 8: Multiplying Fractions and Mixed Numbers
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.6 Understand rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Knowledge Targets “I Can” Statements Standard Interpretations Identify rational number as a point on the number line. Identify the location of zero on a number line in relation to positive and negative numbers. I can understand a rational number as a point on the number line. Number lines can be used to show numbers and their opposites. Both 3 and -3 are 3 units from zero on the number line. Graphing points and reflecting across zero on a number line extends to graphing and reflecting points across axes on a coordinate grid. The use of both horizontal and vertical number line models facilitates the movement from number lines to coordinate grids. Example: Graph the following points in the correct quadrant of the coordinate plane. If you reflected each point across the x-axis, what are the coordinates of the reflected points? What similarities do you notice between coordinates of the original point and the reflected point? Reasoning Target Performance Target Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
the number line. I can interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. I can distinguish comparisons of absolute value by statements of order. I can understand that absolute value deals only with distance from zero. 5 > 3 5 is greater than 3 Case 2: One positive and one negative number 3 > -3 positive 3 is greater than negative 3 negative 3 is less than positive 3 Case 3: Two negative numbers -3 > -5 negative 3 is greater than negative 5 negative 5 is less than negative 3 Continued on next page Comparative statements generate informal experience with operations and lay the foundation for formal work with operations on integers in grade 7. Example: One of the thermometers shows -3°C and the other shows -7°C. Which thermometer shows which temperature? Which is the colder temperature? How much colder? Write an inequality to show the relationship between the temperatures and explain how the model shows this relationship. Students recognize the distance from zero as the absolute value or magnitude of a rational number. Students need multiple experiences to understand the relationships between numbers, absolute value, and statements about order. Example: The Great Barrier Reef is the world’s largest reef system and is located off the coast of Australia. It from statements about order and apply to real-world contexts. Performance Target
Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 10: Integers Opposites Integers Absolute Value Rational Number Coordinate Plane X-Axis Y-Axis Quadrants Ordered Pair Origin
Examples: On a bicycle you can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance you can travel in 1 hour and the amount of time required to travel 1 mile)? Solution: You can travel 5 miles in 1 hour written as and it takes of a hour to travel each mile written as . Students can represent the relationship between 20 miles and 4 hours. A simple modeling clay recipe calls for 1 cup corn starch, 2 cups salt, and 2 cups boiling water. How many cups of corn starch are needed to mix with each cup of salt? Performance Target Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
Graph: Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 12: Ratios, Rates, and Proportions Ratio
represent the relationship between distance and time. Knowledge Targets “I Can” Statements Standard Interpretations Define independent and dependent variables. Use variables to represent two quantities in a real-world problem that change in relationship to one another. I can use variables to represent two quantities in a real situation that changes in relationship to one another. I can distinguish between dependent and independent variables. I can write an equation to express one quantity (dependent variable) in terms of the other quantity (independent variable). I can relate the data between the dependent and independent variable using graphs and tables and relate these to the equation. See Topic 14 6.EE.9 Reasoning Target Write an equation to express one quantity (dependent) in terms of the other quantity (independent). Analyze the relationship between the dependent variable and independent variable using tables and graphs Relate the data in a graph and table to the corresponding equation. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 15: Equations and Graphs T-Table Linear Equation
the appropriate unit fraction edge lengths. Cubes Tool on NCTM’s Illuminations (). In addition to filling boxes, students can draw diagrams to represent fractional side lengths, connecting with multiplication of fractions. This process is similar to composing and decomposing two dimensional shapes. Examples: The model shows a cubic foot filled with cubic inches. The cubic inches can also be labeled as a fractional cubic unit with dimensions of ft3. The models show a rectangular prism with dimensions 3/2 inches, 5/2 inches, and 5/2 inches. Each of the cubic units in the model is in3. Students work with the model to illustrate 3/2 x 5/2 x 5/2 = (3 x 5 x 5) x 1/8. Students reason that a small cube has volume 1/8 because 8 of them fit in a unit cube. Reasoning Target Apply volume formulas for right rectangular prisms to solve real- world and mathematical problems involving rectangular prisms with fractional edge lengths. Performance Target Model the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Knowledge Targets “I Can” Statements Standard Interpretations
sure that students ask questions that have specific numerical answers. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Develop understanding of statistical variability. 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Knowledge Targets “I Can” Statements Standard Interpretations
Cluster: Develop understanding of statistical variability. 6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number Knowledge Targets “I Can” Statements Standard Interpretations Recognize there are measures of central tendency for a data set, e.g., mean, median, mode. Recognize there are measures of variances for a data set, e.g., range, interquartile range, mean absolute deviation. Recognize measures of central tendency for a data set summarizes the data with a single number. Recognize measures of variation for a data set describes how its values vary with a single number. I can use mean to describe data. I can understand that there is variation in data. I can distinguish between “measure of center” and “measure and variation”. When using measures of center (mean, median, and mode) and range, students are describing a data set in a single number. The range provides a single number that describes how the values vary across the data set. The range can also be expressed by stating the minimum and maximum values. Example: Consider the data shown in the dot plot of the six trait scores for organization for a group of students. How many students are represented in the data set? What are the mean, median, and mode of the data set? What do these values mean? How do they compare? What is the range of the data? What does this value mean? Reasoning Target Performance Target
Quartile 1 (Q1) – (132 + 133) ÷ 2 = 132.5 months Median (Q2) – 139 months Quartile 3 (Q3) – (142 + 143) ÷ 2 = 142.5 months Maximum – 150 months This box plot shows that ¼ of the students in the class are from 130 to 132.5 months old ¼ of the students in the class are from 142.5 months to 150 months old ½ of the class are from 132.5 to 142.5 months old the median class age is 139 months. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning