SFUSD Mathematics Core Curriculum Development Project

1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding

Grade 7 Unit 7.1: Operations with Rational Numbers 2 Number of Days Lesson Reproducibles Number of Copies Materials 1 Entry Task Constructing a Number Line Frayer Model 6 Lesson Series 1 CPM CCC2 Lesson 2.1.1 (3 pages) Resource Page 2.1.1 HW: CPM CCC2 Lesson 2.1.1 CPM CCC2 Lesson 2.1.2 (4 pages) Resource Pages 2.1.2 (3 pages, one-sided) HW: CPM CCC2 Lesson 2.1.2 CPM CCC2 Lesson 2.2.1 (2 pages) HW: CPM CCC2 Lesson 2.2.1 CPM CCC2 Lesson 2.2.2 (4 pages) HW: CPM CCC2 Lesson 2.2.2 CPM CCC2 Lesson 2.2.3 (3 pages) Resource Pages 2.2.3B (2 pages) HW: CPM CCC2 Lesson 2.2.3 1 per student 1 per student 1 per student Sticky notes Calculators Computer with projector for animation: www.cpm.org/technology Course 2 (otherwise use Resource Pages 2.2.1A and 2.2.1B, scissors, envelopes or small bags) Integer tiles (or counters with 2 colors) 1 Apprentice Task Cecil Steps Up to the Challenge Poster paper and markers 10 Lesson Series 2 CPM CCC2 Lesson 2.2.4 (3 pages) HW: CPM CCC2 Lesson 2.2.4 CPM CCC2 Lesson 2.2.5 (3 pages) Resource Page 2.2.5 HW: CPM CCC2 Lesson 2.2.5 CPM CCC2 Lesson 2.2.6 (3 pages) HW: CPM CCC2 Lesson 2.2.6 CPM CCC2 Lesson 3.1.1 (3 pages) HW: CPM CCC2 Lesson 3.1.1 CPM CCC2 Lesson 3.1.2 (3 pages) HW: CPM CCC2 Lesson 3.1.2 CPM CCC2 Lesson 3.2.1 (4 pages) HW: CPM CCC2 Lesson 3.2.1 1 per student 4 number cubes (or 1 number cube rolled 4 times) Integer tiles (or counters with 2 colors) Poster paper and markers Paperclips (2 per student) Place markers (pennies or beans)

3 CPM CCC2 Lesson 3.2.2 (2 pages) HW: CPM CCC2 Lesson 3.2.2 CPM CCC2 Lesson 3.2.3 (2 pages) HW: CPM CCC2 Lesson 3.2.3 CPM CCC2 Lesson 3.2.4 (5 pages) Resource Page 3.2.4 HW: CPM CCC2 Lesson 3.2.4 CPM CCC2 Lesson 3.2.5 (2 pages) Resource Page 3.2.5 HW: CPM CCC2 Lesson 3.2.5 1 Expert Task Multiplication and Division Rational Number Story: Activity Elements Graphic Organizer (2 pages) 4 Lesson Series 3 CPM CCC2 Lesson 3.3.1 (3 pages) HW: CPM CCC2 Lesson 3.3.1 CPM CCC2 Lesson 3.3.2 (3 pages) HW: CPM CCC2 Lesson 3.3.2 CPM CCC2 Lesson 3.3.3 (4 pages) HW: CPM CCC2 Lesson 3.3.3 1 per student Integer tiles (or counters with 2 colors) 1 Milestone Task Distances Between Houses (2 pages) 1 per student Rulers (optional)

4 Unit Overview Big Idea A rational number is a number that can be written as a fraction with integers in the numerator and the denominator (denominator not equal to zero). Properties of arithmetic and operations extend to rational numbers, which include zero and positive and negative: integers, fractions, mixed numbers, terminating decimals, and repeating decimals. Unit Objectives Students will be able to understand the concept of opposites and additive inverse. Students will be able to understand properties: additive inverse, distributive, multiplicative inverse, commutative property, and identity property. Students will be able to connect concepts to real-world contexts. Students will be able to add, subtract, multiply, and divide integers (including negatives). Students will be able to add, subtract, multiply, and divide all rational numbers. Students will be able to apply four operations with rational numbers to real-world problems. Students will be able to convert rational numbers to decimals. Language Objective: Writing in completes sentences students will justify their mathematical process using academic vocabulary. Using Think-Pair-Share (T-P-S) students will articulate their questions and/or understanding. Unit Description Students will begin with a review of ordering integers and placing them on a number line. Then they will learn to add, subtract, multiply, and divide simple integers, and apply these to real-world situations. Then they will apply what they have learned to more complex rational numbers (negative fractions and decimals), which they will then use to explore a variety of math activities and real-world problem solving. The Number System CCSS-M Content Standards Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7.NS.1b Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

5 7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing real world contexts. 7.NS.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics 6 Students have been working with addition and subtraction of unlike fractions, multiplication and division of fractions, and long division. Students have had exposure to the concept of positive and negative numbers. In sixth grade, students develop understanding about the relationship between multiplication and division, as well as between fractions and decimals. Students apply this knowledge through real-world problem-solving tasks and demonstrate these skills using equal groups, arrays, area models, number lines, multiplication tables and word problems. Students will be able to take any two or more rational numbers and add, subtract, multiply, and divide them and apply to real-world problems. Students will describe situations in which opposite quantities combine to zero and are additive inverses. Students will understand that subtraction of rational numbers is the same as adding the additive inverse. Students will use properties of numbers as strategies to add or subtract rational numbers. Students will use properties of numbers as strategies to multiply or divide rational numbers, particularly the Distributive Property. Students understand that integers can be divided, provided the divisor is not zero, and every quotient of integers is a rational number. Students will convert a rational number to a decimal using long division and know that the decimal form will terminate with a zero or eventually repeat. Students will understand the rules for multiplying signed numbers. [(-1)(-4) = +4] Later in seventh grade, students will know that proportional relationships are written as rational numbers. They will know probability can be written as a rational number. In eighth grade, students will learn that slope can be written as a rational number to be used to graph lines. They will learn that there are numbers that are not rational and approximate by using rational numbers. They will learn that negative exponents create rational numbers.

Entry Task Unit Design Apprentice Task All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Expert Task Lesson Series 1 Lesson Series 2 Lesson Series 3 Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit? Milestone Task 7 Total: 24 days 1 day 6 days 1 day 10 days 1 day 4 days 1 day

8 Entry Task Constructing a Number Line Apprentice Task Shell Center Positive and Negative Integers in Context Expert Task Multiplication and Division Rational Number Story Milestone Task Distance Between Houses CCSS-M Standards 6.NS.6, 6.NS.7 7.NS.1, 7.NS.2 7.NS.1, 7.NS.2 7.NS.1, 7.NS.2, 7.NS.3 Brief Description of Task You provide a list of rational numbers (including positive and negative, integers, fractions, and decimals) and students place them on a number line. Students activate their prior knowledge about the concept of opposites by completing a Frayer Model. Students will be able to apply four operations with integers. Students write word problems that involve operations with rational numbers. Using the Math Story Rubric and Graphic Organizer from Illuminated Mathematics, students will be able to multiply and divide rational numbers. Students will be able to solve realworld mathematical problems involving the four operations with rational numbers. Source SFUSD Teacher Created Shell Centre Using Positive and Negative Integers in Context Illuminated Mathematics, NCTM Presentation 2012 Illustrative Mathematics Lesson Series 1 Lesson Series 2 Lesson Series 3 CCSS-M Standards 7.NS.1a, 7.NS.1b, 7.NS.2b, 7.NS.2d 7.NS.1c, 7.NS.1d, 7.NS.2a, 7.NS.3 7.NS.2c, 7.NS.3 Brief Description of Lessons Students will learn how to convert fractions to decimals and add integers using different representations. Students learn multiplication as repeated addition, which is then expanded to repeated addition of negative numbers. They subtract positive and negative rational numbers and make connections between addition and subtraction. They continue to make sense of these operations through multiple representations. Students learn to divide rational numbers and solve problems involving four basic operations on rational numbers. Sources CPM Core Connections Course 2 CPM Core Connections Course 2 CPM Core Connections Course 2

Entry Task Constructing a Number Line What will students do? 9 Mathematics Objectives and Standards Math Objectives: Correctly place rational numbers on a number line. Understand that opposite numbers are located equidistant from 0 on the number line and on opposite sides. CCSS-M Standards Addressed: 6.NS.6c, 6.NS.6b Time: 1 day Potential Misconceptions: Students may not know that -1.125 is the same as -1⅛ or that 6/4 is the same as 1 and ½. Students may have trouble ordering negative numbers (for instance, they place -3¾ to the right of -3. Framing Student Experience Launch: To find out what students already know, students will make a number line given limited instructions. Rulers may or may not be used; this is up to you. Have students work in pairs. During: Students will place the following numbers on their number line: -9/4, 1.5, 8, -6, 3⅓ -12/4, -1.125, 2.3 (with a bar over the 3), 6/4, 0.25. Walk around and observe results, but do not help with number placement. Students complete a Frayer Model (QTEL) on the concept of opposites. Divide an 8½ by 11 sheet of paper into four quadrants (2 x 2). Draw an oval in the center with the target word (Opposites). Label the four quadrants as described: 1. Top Left Characteristics 2. Top Right Examples 3. Bottom Left Non-Characteristics 4. Bottom Right Non-Examples After completing the Frayer Model individually, students pair-share their responses with their elbow partner, and then selected pairs share with the class. Finally, lead the students in a discussion of how two numbers can be opposites. Closure/Extension: Display a number line and elicit discussion about where the various numbers go. You can give sticky notes to students to place numbers on a number line on the white board. During the discussion, discuss the meaning of opposite: What is the opposite of 1.5? Where would I place it on the number line?

Constructing a Number Line 10 Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. Structures for Student Learning: Academic Language Support: How will students do this? Vocabulary: number line, intervals, scale, tick marks, zero, opposites, positive, negative, fractions, decimals Sentence frames: I think this number should be placed here because. I think this number should be placed between and because Differentiation Strategies: Students will work with partners and justify their reasoning for placing numbers in certain locations on the number line. Participation Structures: Pairs of students can meet with another pair before whole-group debriefing.

Lesson Series #1 Lesson Series Overview: Students learn how to convert fractions to decimals and add integers using different representations. 11 CCSS-M Standards: 7.NS.1a, 7.NS.b, 7.NS.2b, 7.NS.2d Time: 6 days Note: Only assign selected homework problems or this unit will take much more time than anticipated. If you do not have time to look through the problems, we recommend you assign Review &Preview problems or Glencoe workbook or other source for homework, based on your students needs. Lesson Overview Days 1 3 Description of Lesson: Students convert fractions to decimals using long division and develop understanding of why decimals repeat or terminate. Resources CPM CCC2 Lesson 2.1.1, 2.1.2 CPM CCC2 Lesson 2.1.1 Resource Page CPM CCC2 Lesson 2.1.2 ABC Resource Pages Notes: Calculators are needed for problem 2-1 to convert fractions to decimals. Lesson Overview Days 4 6 Description of Lesson: Students compose and decompose numbers, which will lead to work with adding integers. Students will also do addition of integers on a number line as well as with tiles. Notes: For problem 2-31, access the Acrobat Number Line activity at www.cpm.org/technology (choose course 2). This activity sets up many problems that follow in Lesson 2.2.1 through 2.2.4 and in the subsequent lesson series. Resources CPM CCC2 Lesson 2.2.1 www.cpm.org/technology (Acrobat Number Line animated activity) CPM CCC2 Lesson 2.2.2 CPM CCC2 Lesson 2.2.3 CPM CCC2 Lesson 2.2.3 AB Resource Pages (Win-A-Row game) CPM CCC2 Lesson 2.2.3 Resource Page (tile templates) Lesson 2.2.3 requires integer tiles (or use tile templates).

Apprentice Task Cecil Steps Up To The Challenge What will students do? 12 Mathematics Objectives and Standards Math Objectives: Students will add positive and negative rational numbers. They will use a number line representation to show the process. CCSS-M Standards Addressed: 7.NS.1 Potential Misconceptions: Addition rules for negative numbers are confused with multiplication/division rules. Framing Student Experience Launch: Read introduction and answer questions. During: Students work in teams of 2-4 students. Closure/Extension: Groups share their routine with the class. Pick a random student from the group to explain the strategies they used to calculate the distance that Cecil traveled. This can also be done as a Gallery Walk.

13 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them 4. Model with mathematics 6. Attend to precision 8. Look for and express regularity in repeated reasoning Structures for Student Learning: Academic Language Support Vocabulary: Tightrope, routine, handbook, color-code Cecil Steps Up To The Challenge How will students do this? Sentence frames: I think this number (or move) should be placed here because.. I think this number or move should be placed between and because.. Differentiation Strategies: Have multiple tightrope pieces available that show 1.2 feet, 3 and ½ feet and 5 feet. You might choose to use calculators. Participation Structures: Collaborative group work or partner work

Lesson Series #2 14 Lesson Series Overview: Students learn multiplication as repeated addition, which is then expanded to repeated addition of negative numbers. They subtract positive and negative rational numbers and make connections between addition and subtraction. They continue to make sense of these operations through multiple representations. CCSS-M Standards: 7.NS.1c, 7.NS.1d, 7.NS.2a, 7.NS.3 Time: 10 days Note: Only assign selected homework problems or this unit will take much more time than anticipated. If you do not have time to look through the problems, we recommend you assign Preview and Review problems or Glencoe workbook or other source for homework, based on your students needs. Lesson Overview Day 1 Description of Lesson: Students understand multiplication of rational numbers as repeated addition, continuing using multiple representations (tightrope walker, tiles, arrow diagrams). CPM CCC2 Lesson 2.2.4 Resources Notes: Use CPM Foundations for Algebra Year 2 GC 31 for distributive property practice. Lesson Overview Days 2 3 Description of Lesson: Students will understand multiplication of positive fractions, decimals, and percents. They will then learn/review the standard algorithm for multiplying fractions. Resources CPM CCC2 Lesson 2.2.5 CPM CCC2 Lesson 2.2.5 Resource Page CPM CCC2 Lesson 2.2.6 Notes: Lesson Overview Days 4 5 Description of Lesson: Students learn that parentheses can be used to group members, identify terms, and simplify numerical expressions. CPM CCC2 Lesson 3.1.1 CPM CCC2 Lesson 3.1.2 Resources Notes: Materials: 4 number cubes

15 Lesson Overview Days 6 7 Description of Lesson: Students develop an understanding of subtraction with tiles and number lines and how addition and subtraction are connected. Resources CPM CCC2 Lesson 3.2.1 CPM CCC2 Lesson 2.2.3 Resource Page (tile templates) CPM CCC2 Lesson 3.2.2 Notes: Lesson Overview Days 8 9 Description of Lesson: Students will extend multiplication to negative integers, making sense of signed numbers, especially (-)(-) = (+). They then learn and make sense of multiplying rational numbers, in particular with negative fractions and decimals. Resources CPM CCC2 Lesson 3.2.3 CPM CCC2 Lesson 2.2.3 Resource Page (tile templates) CPM CCC2 Lesson 3.2.4 CPM CCC2 Lesson 3.2.4 Resource Page Notes: Lesson Overview Day 10 Description of Lesson: Practice adding, subtracting, multiplying, and adding positive and negative integers. Resources CPM CCC2 Lesson 3.2.5 CPM CCC2 Lesson 3.2.5 Resource Page Notes:

Expert Task Multiplication and Division Rational Number Story What will students do? 16 Mathematics Objectives and Standards Math Objectives: Students will be able to apply the four operations with rational numbers where one has to be negative and not an integer. CCSS-M Standards Addressed: 7.NS.1d, 7.NS.3 Potential Misconceptions Students may invent a problem that s harder than they can solve. Framing Student Experience Launch: Use the Math Story Rubric and Graphic Organizer from Illuminated Mathematics. Students will be able to multiply and divide rational numbers. Students can brainstorm into activity by thinking of who, what, when, where, and how their word problem takes place. During: Students will tend to create problems around addition and subtraction. You should require them to also create a problem involving multiplication or division. Closure/Extension: Have students share their stories with other students and have pairs of students solve each other s problems.

17 Multiplication and Division Rational Number Story How will students do this? Focus Standards for Mathematical Practice: 4. Model with mathematics. Structures for Student Learning: Academic Language Support: Vocabulary: profit, gain, elevation, strokes under par, balance, overdrawn. Sentence frames: Once upon a time there was... Differentiation Strategies: Students can just use integers or whole numbers. Students can use voice recognition software to read their story into a computer. Include a graphic organizer in the lesson to assist with common uses of negative numbers to help students get started. Participation Structures: Students can work individually or in a partnership. There will be a peer review, which utilizes a rubric. (See folder for PDF.)

Lesson Series #3 18 Lesson Series Overview: Students learn to divide rational numbers and solve problems involving four basic operations on rational numbers. CCSS-M Standards: 7.NS.2c, 7.NS.3 Time: 4 days Note: Only assign selected homework problems or this unit will take much more time than anticipated. If you do not have time to look through the problems, we recommend you assign Preview and Review problems or Glencoe workbook or other source for homework, based on your students needs. Lesson Overview Days 1 2 Description of Lesson: Rational number is formally defined; students divide fractions, mixed numbers and decimals. Students make sense of and learn a standard algorithm for decimal division. CPM CCC2 Lesson 3.3.1 CPM CCC2 Lesson 3.3.2 Resources Notes: Lesson Overview Days 3 4 Description of Lesson: Students determine which of the four basic operations are commutative or associative. They use the four operations of rational numbers to solve problems. CPM CCC2 Lesson 3.3.3 Resources Notes: Make sure to do problems 3-118 and 3-119 on the second day.

Milestone Task Distances Between Houses What will students do? 19 Mathematics Objectives and Standards Math Objectives: Students apply properties of operations as strategies to add or subtract rational numbers. Students apply properties of operations as strategies to multiply and divide rational numbers. CCSS-M Standards Addressed: 7.NS.1, 7.NS.2, 7.NS.3 Potential Misconceptions: Students may not connect West and East with Negative and Positive numbers. Students may not connect the number on the street with numbers on the number line. Students may not see distance as an absolute value and might just add the location of the houses on the number line instead. Students may not realize that the school is located at 0 on the number line. If they do not realize this, they will miss all questions. Teacher should be prepared to issue partial points for partial solutions. Framing Student Experience Launch: Do Now activity at your discretion to focus students. A possible do now might be to show pictures of what a neighborhood block looks like. (You can find many examples on Google images). Some students may have a hard time visualizing houses on either side of the school. In the Resources folder there are also some real-life activities on temperature, elevators and interstate that reinforce the ideas of finding distance on a number line. Students need to connect that the school is at 0 on the number line; otherwise they will miss the point of the task. During: Students work independently. Closure/Extension: If you have not already used temperature, elevators and interstate activities (from the Resources folder), you may use them to reinforce conceptual understanding.

20 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Attend to precision. Structures for Student Learning: Academic Language Support Distances Between Houses How will students do this? Vocabulary: positive, negative, east, west, intervals, ticks (marks on a number line) Sentence frames: I would place house here because Differentiation Strategies: Provide students with number lines broken into fractions and/or fraction bars divided into fourths. Have students fill out a graphic organizer with school and student houses on a number line with East and West and blocks from home but without the structures named so that students are not confused about which house goes where. Ask students to place the students houses and the school on a number line in the front of the class with East and West. Have students change it until everyone (including you) agrees the structures are in the correct order. Participation Structures: Students will work individually.