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Rigorous Curriculum Design Showcase Grade: 6th to 8th Subject: Math Summary of the CCSS Standards 6th and 7th Grade • Expressions and Equations – Apply and extend previous understandings of arithmetic to algebraic expressions – Reason about and solve one-variable equations and inequalities. – Represent and analyze quantitative relationships between dependent and independent variables. – Use properties of operations to generate equivalent expressions. – Solve real-life and mathematical problems using numerical and algebraic expressions and equations. The Common Core State Standards define the rigorous skills and knowledge in English Language Arts (ELA) and Mathematics that students need to succeed in college and workforce training programs. Summary of the CCSS Standards 6th and 7th Grade • Ratios and Proportional Relationships – Understand ratio concepts and use ratio reasoning to solve problems. – Analyze proportional relationships and use them to solve real-world and mathematical problems. • The Number System – Apply and extend previous understandings of multiplication and division to divide fractions by fractions and to operations involving rational numbers. – Analyze proportional relationships and use them to solve real-world and mathematical problems. – Multiply and divide multi-digit numbers and find common factors and multiples. The Common Core State Standards define the rigorous skills and knowledge in English Language Arts (ELA) and Mathematics that students need to succeed in college and workforce training programs. Summary of the CCSS Standards 6th and 7th Grade • Geometry – Solve real-world and mathematical problems involving area, surface area, volume, and angle measurements – Draw, construct and describe geometrical figures and describe the relationships between them. • Statistics and Probability – Develop understanding of statistical variability. – Summarize and describe distributions. – Use random sampling to draw inferences about a population and draw informal comparative inferences about two populations. – Investigate chance processes and develop, use, and evaluate probability models. The Common Core State Standards define the rigorous skills and knowledge in English Language Arts (ELA) and Mathematics that students need to succeed in college and workforce training programs. Summary of the CCSS Standards 8th Grade • The Number System – Know that there are numbers that are not rational, and approximate them by rational numbers. • Expressions and Equations – Work with radicals and integer exponents. – Understand the connections between proportional relationships, lines, and linear equations. – Analyze and solve linear equations and pairs of simultaneous linear equations. The Common Core State Standards define the rigorous skills and knowledge in English Language Arts (ELA) and Mathematics that students need to succeed in college and workforce training programs. Summary of the CCSS Standards 8th Grade • Functions – Define, evaluate, and compare functions. – Use functions to model relationships between quantities. • Geometry – Understand congruence and similarity using physical models, transparencies, or geometry software. – Understand and apply the Pythagorean Theorem. – Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. The Common Core State Standards define the rigorous skills and knowledge in English Language Arts (ELA) and Mathematics that students need to succeed in college and workforce training programs. Unit Overviews – 6th Grade • Unit One – Rational Numbers This unit combines the skill of dividing fractions with graphing skills both on a number line and on a coordinate grid in a meaningful way for the students. • Unit Two – Ratios and Proportions Students use reasoning about multiplication and division to solve ratio and rate problems. By viewing equivalent ratios and rates, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems they can use multiplication and division to solve, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates. Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Unit Overviews – 6th Grade • Unit Three – Evaluating Expressions Students understand the use of variables in mathematical expressions. They will write, use, solve and evaluate expressions that correspond to given situations. Students will understand that expressions in different forms can be equivalent and use the properties of operations to rewrite expressions in equivalent forms. • Unit Four – Solving Equations and Inequalities Students write equations that correspond to given situations and use formulas to solve problems. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations and inequalities. Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Unit Overviews – 7th Grade • Unit One – Operations with Rational Numbers Students develop a unified understanding of numbers, recognizing fractions, decimals (finite or repeating), and percent as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationship between addition and subtraction and multiplication and division. • Unit Two – Percent Problems Students extend their learning of ratios and develop understanding of proportionality to solve single-and multi-step problems. Students use their understanding if ratios and proportionality to solve a wide variety of percent problems. Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Unit Overviews – 7th Grade • Unit Three – Expressions, Equations, and Inequalities Students use the arithmetic of rational numbers to formulate, use, and solve expressions and equations with one variable. • Unit Four – Functions and Proportionality Students extend their understanding of ratios and develop understanding of proportionality to solve single or multi-step problems and to solve a wide variety of percent problems (discounts, interest, taxes, and percent increase or decrease). Students solve problems about scale drawings by relating corresponding lengths between objects or by using the fact that relationships of lengths within objects are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of steepness of the related line, called the slope, They distinguish proportional relationships from other relationships. Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Unit Overviews – 8th Grade • Unit One – Geometry Analyze two- and three- dimensional space and figures using distance, angle, similarity, and congruence. • Unit Two– Exponents, Roots, and Pythagorean Theorem Students understand the statement of the Pythagorean Theorem and its converse and can explain why the Pythagorean Theorem holds, for example, by decomposing a square two different ways. They apply the Pythagorean Theorem to find distances between points on a coordinate plane. Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Unit Overviews – 8th Grade • Unit Three – Solve Equations Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use properties of equality and the concept of logical equivalence they maintain the solutions of the original equation. • Unit Four– Functions Students grasp the concept of a function as a rule that assigns to each input exactly an output. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in different representations Big Ideas: The three or four main ideas, conclusions, or generalizations that teachers want their students to discover and state in their own words by the end of the unit. These are concepts students should remember long after instruction ends. Authentic Performance Tasks (APTs) 7th Grade Example Video Instructions 1)Brief introduction on topic and why you are creating the video 2) Define positive and negative integers on a number line 3) Define opposites and additive inverses 4) State the rule for adding and subtracting integers when the signs are the same and show at least three examples 5) State the rule for adding and subtracting integers when the signs are not the same and show at least three different examples 6) Explain how to solve problems of the form -7-(-12), giving two more examples 7) State the rule for multiplying and dividing integers when the signs are the same and show at least three examples 8) State the rule for multiplying and dividing integers when the signs are not the same and show at least three examples An APT is a performance assessment that involves a real-world context, where the tasks are either replicas of or similar to the kinds of problems faced by adults, consumers, and professionals in the field. Engaging Scenario - Example 7th Grade Your friend has not been at school for the past few days due to a terrible cold. You, being a great friend, don’t want them to fall behind. You have decided to create a video presentation on negative integer rules that your friend can watch before they come back to school. You will create a 5 minute video using the Educreation application on the web illustrating the number line, opposites, and additive inverses. The video will show the negative integer rules for adding, subtracting, multiplying, and dividing integers with multiple examples for your friend to see how you applied the rules. Make sure you point out some common mistakes that people make when doing these problems. The Engaging Scenario is an interesting and compelling hypothetical situation provided to students at the start of each unit. It is designed to capture students’ imaginations and help them to buy into the lesson ahead. Scoring Guides – Example 7th Scoring Guide Task 1 Advanced Meets all of the "Proficient" criteria plus: Correctly shows some common mistakes that students can make and how to avoid them Proficient Brief introduction on topic and why you are creating the video (or visual aide) Defines positive and negative integers on a number line Defines opposites and additive inverses States the rule for adding and subtracting integers when the signs are the same and show at least three examples States the rule for adding and subtracting integers when the signs are not the same and show at least three different examples Explains how to solve problems of the form -7-(12), giving two more examples States the rule for multiplying and dividing integers when the signs are the same and show at least three examples States the rule for multiplying and dividing integers when the signs are not the same and show at least three examples Progressing Beginning Meets __6-7__of the Meets fewer "Proficient" criteria than__6__ of the "Proficient" criteria A scoring guide is a written list of specific criteria describing different levels of student proficiency on a standards‐based assessment task.