Common Core State Standards

Transcription

1 Common Core State Standards Common Core State Standards 7.NS.2 7.NS.2a 7.NS.2c 7.NS.3 7.EE.3 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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. Apply properties of operations as strategies to multiply and divide rational numbers. Solve real-world and mathematical problems involving the four operations with rational numbers. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $ If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. Mathematical Practices 1, 3, and 4 are aspects of mathematical thinking that are emphasized in every lesson. Students are given opportunities to be persistent in their problem solving, to express their reasoning, and apply mathematics to realworld situations. Exercises 9 and 12 focus on Mathematical Practice 4 Model with mathematics. Exercises 10, 13, 15, and 33 focus on Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Exercise 11 focuses on Mathematical Practice 8 Look for and express regularity in repeated reasoning. Exercise 14 focuses on Mathematical Practice 1 Make sense of problems and persevere in solving them. Printed by: Nicole Sladich Page 1 of 18 Printed on 11/04/ :08 PM

2 1 Launch the Lesson Focus narrowing the scope Objective Multiply integers. 7.NS.2, 7.NS.2a, 7.NS.2c, 7.NS.3, 7.EE.3 Coherence connecting within and across grades Previous Students used models to multiply integers. Now Students multiply integers with the same sign and with different signs. Next Students will use properties to prove the rules for multiplying integers. Rigor pursuing concepts, fluency, and applications See the Levels of Complexity chart in the Practice and Apply section. As students enter the classroom, have them complete the Common Core Quick Check to review and assess the skills presented in the previous lesson. Engage Explore Explain Elaborate Evaluate Ideas for Use You may wish to launch the lesson using a whole group, small group, think-pair-share activity, or independent activity. Numbered Heads Together Have students work in groups of 3 4 to complete Exercises 1 3. Each student is responsible to ask for support and to ensure that every team member understands. Call on one team member to explain the group s responses to the class. 1, 5 Alternate Strategy Draw a number line on the floor. Have students take 5 steps to the left of zero four times. Ask how far they are from zero and how to represent this number. Then have them complete Exercise 2 by stepping along the number line. 1, 4 Use the video Skydiving with the opening activity. Resources Common Core Quick Check: Multiply Integers Lesson Video: "Skydiving" Printed by: Nicole Sladich Page 2 of 18 Printed on 11/04/ :08 PM

3 2 Teach the Concept Focus: Mathematical Background Number lines are a good way to model multiplying integers. Multiplication can be shown as repeated addition. The Commutative Property of Multiplication is related to the Commutative Property of Addition. The order in which you multiply or add does not change the product or sum. The McGraw-Hill Professional Development Video Library provides short videos that support Glencoe Math. The featured video for mathematical content in this lesson is: Solving Integer Problems For classroom presentation, use the Step-by-Step Examples to demonstrate the examples from the Student Edition. Engage Explore Explain Elaborate Evaluate Ask the scaffolded questions for each example to differentiate instruction. 1. Multiply integers. How is multiplication like addition? Multiplication is repeated addition. Do the integers have the same sign? no Is the sign of two integers with different signs positive or negative? negative How would a number line illustrate this product? Starting from zero, three equal jumps of 5 to the left will land on 15 on a number line. Generate a different multiplication expression in which the product is also 15. Sample answer: 3(5) Need Another Example? Find 5(4) Multiply integers. Do the integers have the same sign? no Write an addition sentence that represents this situation. Sample answer: (6) + (6) + (6) + (6) + (6) + (6) + (6) + (6) = 48 Is the product of two integers with different signs positive or negative? negative Rewrite the problem using the Commutative Property of Multiplication. Does the order in which you multiply the numbers matter? 8(6); No, the product is the same. Generate a different multiplication expression in which the product is also 48. Sample answer: 6(8) Need Another Example? Find 3(9) Multiply integers. Printed by: Nicole Sladich Page 3 of 18 Printed on 11/04/ :08 PM

4 Do the integers have the same sign? yes What is the sign of both integers? negative Is the product of two integers with the same sign positive or negative? positive Generate your own multiplication expression involving two negative numbers in which the product is 24. Sample answer: 3(8) Need Another Example? Find 6(8) Multiply integers. What is the base? 4 What is the exponent? 2 How many times will 4 be used as a factor? two times Is the product of two integers with the same sign positive or negative? positive Is the square of any negative number always positive? yes Need Another Example? Find (8)² Multiply more than two integers. How many factors are being multiplied? three What is the sign of each integer? Each integer is negative. What property allows you to group the factors in any way? Associative Property What is the sign of the product of 3 and 4? positive What is the sign of the product of 12 and 2? negative Write a generalization for the sign of the product of 3 negative integers. Explain. The product of 3 negative integers will be negative because the product of the first 2 integers will be positive and a positive integer multiplied by the third negative integer will be negative. Need Another Example? Find (2)(5)(6) Multiply integers. Does descending refer to going up or down? down What integer represents descending at a rate of 90 feet per minute? 90 For how long is the submersible descending? 7 minutes How can you find the product? Multiply 7 by 90. Will the product be positive or negative? negative Printed by: Nicole Sladich Page 4 of 18 Printed on 11/04/ :08 PM

5 1. 2. If the submersible ascends 150 feet after it descended for the 7 minutes, what will be the new depth of the submersible? Explain. 480 ft; = 480 Need Another Example? A mine elevator descends at a rate of 200 feet per minute. Write a multiplication expression to represent the total distance descended after four minutes. Then find the product and explain its meaning. 4(200); 800; In four minutes, the elevator will descend 800 feet. Use the animation Multiply Integers with Example 6. Alternate Teaching Strategy If students have trouble multiplying integers, Then use one of these reteach options: Quick Review Math Handbook, pp Have them use counters or a diagram to find the product of a negative and a positive integer and the product of two negative integers. Formative Assessment Use these exercises to assess students' understanding of the concepts in this lesson. below. If some of your students are not ready for assignments, use the differentiated activities Roundrobin In groups of 4, have students analyze Exercises 1 3. Have Student 1 identify which numbers are negative, Student 2 identify whether the product will be negative or positive, Student 3 explain why the product will be negative or positive, and Student 4 give the product. 1, 7 Pairs Discussion Have the students discuss why the sign of the product is determined based upon whether the number of negative factors is odd or even. Ask them what the product will be if one of the factors is 0. 1, 7 Tips for New Teachers Here is a practical example for remembering the multiplication rules for integers. Good guys leave town Result + Good guys stay in town Result Bad guys leave town Result + Bad guys stay in town Result + Multiplying Integers Remind students that repeated addition can be used to check their work. For example, 3(5) = (5) + (5) + (5) or 15. Teach with Tech Instant Messaging Have students send multiplication problems involving integers to each other using instant messaging. Resources Virtual Manipulatives Printed by: Nicole Sladich Page 5 of 18 Printed on 11/04/ :08 PM

6 Multiply Integers Multiply Integers with Different Signs Multiply Integers with Different Signs Multiply Integers with the Same Sign Multiply Integers with the Same Sign Multiply Integers Real-World Example: Multiply Integers with Different Signs Multiply Integers with Different Signs Multiply Integers with Different Signs Real-World Example: Multiply Integers with Different Signs Multiply Integers with the Same Sign Multiply Integers with the Same Sign Multiply Integers with the Same Sign etoolkit Hot Topic 1-4: Integer Operations Solving Integer Problems Printed by: Nicole Sladich Page 6 of 18 Printed on 11/04/ :08 PM

7 3 Practice & Apply Engage Explore Explain Elaborate Evaluate Independent Practice and Extra Practice The Independent Practice pages are meant to be used as the homework assignment. The Extra Practice page can be used for additional reinforcement or as a second-day assignment. Levels of Complexity The levels of the exercises progress from 1 to 3, with Level 1 indicating the lowest level of complexity. Suggested Assignments You can use the table below that includes exercises of all complexity levels to select appropriate exercises for your students' needs. Differentiated Homework Options Approaching Level 1 9, 11 13, 34, 35 On Level 1 7 odd, 9 13, 34, 35 Beyond Level 9 15, 34, 35 can provide additional remediation support with personalized instruction and practice. Create Your Own Homework Online esolutionsmanual can be used to create worksheets for the suggested assignments above, or to create your own worksheets for differentiated homework or review. The following worksheets are also available for additional homework practice: The Skills Practice editable worksheet provides students with additional practice in the skills taught in this lesson. The Homework Practice editable worksheet helps students practice skills in this lesson and use those skills to solve problems. The Problem-Solving Practice editable worksheet helps students become better problem solvers by providing frequent opportunities to solve real-world problems. The Extra Practice worksheet provides additional practice for the skills taught in this lesson. The Step-by-Step Solutions provide a worked out solution for selected exercises from the Independent Practice in the Student Edition. Common Error Students may have trouble multiplying three or more factors. These students may find Printed by: Nicole Sladich Page 7 of 18 Printed on 11/04/ :08 PM

8 it easier to multiply first, ignoring the signs. Then they can apply the rules to find the product s sign. You may wish to point out that if the number of negative integers being multiplied is odd (1, 3, 5, and so on), the product will be negative. Common Error In Exercise 10, students may incorrectly interpret the expression. Remind them that a late fee charge is an amount that Kaitlyn will owe, which can be represented by a negative integer. Find the Error In Exercise 33, Jamar did not include the negative signs when he multiplied. The product of three negative factors is always negative. Resources Step by Step Solutions: Multiply Integers Skills Practice: Multiply Integers Skills Practice Answers: Multiply Integers Homework Practice: Multiply Integers Homework Practice Answers: Multiply Integers Problem-Solving Practice: Multiply Integers Problem-Solving Practice Answers: Multiply Integers Extra Practice: Multiply Integers Extra Practice Answers, Chapter 3 Homework Practice: Multiply Integers Problem-Solving Practice: Multiply Integers Skills Practice: Multiply Integers Extra Practice: Multiply Integers esolutions, Glencoe Math Course 2 Printed by: Nicole Sladich Page 8 of 18 Printed on 11/04/ :08 PM

9 Additional Activities for Differentiated Instruction Differentiated Instruction activities for Approaching Level, On Level, and Beyond Level Learners can also be found at point-of-use in the Launch the Lesson and Teach the Concept sections. The following are additional activities that can be used with this lesson. Option 1: Visual/Spatial Learners Materials: highlighters To help students identify which values should be substituted for variables in an expression, suggest that they use different colored highlighters to organize the information. It may be helpful to provide an enlarged photocopy of the exercises so that students can use the highlighters directly on the homework problems. Option 2: Interpersonal Learners Materials: index cards, markers Have students create flash cards to help them remember the signs for the product and quotient of two integers. Have them write the problem on the front of the card and the answer on the back. Make one card for each of the following: positive positive negative negative positive negative negative positive positive negative negative positive Students can take turns quizzing each other using the cards. The Reteach Worksheet provides additional examples and practice for students who may have difficulty in grasping the math concepts in this lesson. Option 3: Verbal/Linguistic Learners Materials: poster board, markers Working in groups, students should create real-world problems to illustrate each of the different rules for multiplication of integers. Each member of the group chooses one of the problems and displays it on a poster. Posters can be displayed. Option 4: Logical Learners Display the following two expressions on the board: (3) 15 (5) 10 Share each of the statements shown below and have students determine whether John s or Sarah s statement is correct. Have them explain their reasoning to the class. John: (3) 15 is greater than (5) 10 because the exponent is greater. Sarah: (5) 10 is greater because it is positive and (3) 15 is negative. Sarah; a negative number raised to an even power is positive and a negative number raised to an odd power is negative. So, (5)10 is greater than (3)15. The Enrich Worksheet provides students with valuable opportunities for extending this lesson. Resources Reteach: Multiply Integers Reteach Answers: Multiply Integers Enrich: Integer Maze Enrich Answers: Integer Maze Printed by: Nicole Sladich Page 9 of 18 Printed on 11/04/ :08 PM

10 Reteach: Multiply Integers Enrich: Integer Maze Printed by: Nicole Sladich Page 10 of 18 Printed on 11/04/ :08 PM

11 Additional Activities for English Learners Differentiated Instruction activities for English Learners can also be found at point-of-use in the Launch the Lesson and Teach the Concept sections. The following is an additional activity that can be used with this lesson. Option 1: Use Chapter 3, from the Interactive Guide for English Learners. Resources Interactive Guide for English Learners, Student Edition, Course 2, Lesson 3-4: Multiply Integers Interactive Guide for English Learners, Teacher Edition, Course 2, Lesson 3-4: Multiply Integers Interactive Guide for English Learners, Course 2, Teacher Edition Interactive Guide for English Learners, Course 2, Student Edition Visual Kinesthetic Vocabulary, Course 2, Student Edition Visual Kinesthetic Vocabulary, Course 2, Teacher Edition Printed by: Nicole Sladich Page 11 of 18 Printed on 11/04/ :08 PM

12 Assessment Formative Assessment Use this activity as a closing formative assessment before dismissing students from your class. Remind students that a previous lesson was about adding integers. Have them write about how that lesson's concepts helped them with this lesson's material. See students' work. Time to update your Foldable! Students should be able to complete the section in their Foldable about multiplying integers. If they are having difficulty coming up with their own examples, they can use an example from this lesson. The online Self-Check Quiz for this lesson allows students to review concepts from each lesson. Exercises 34 and 35 prepare students for more rigorous thinking needed when taking the CCSS assessment. 34. This test item requires students to analyze and solve complex real-world problems through the use of mathematical tools and models. Depth of Knowledge CCSS Content Standard Mathematical Practices Online Experience DOK3 7.NS.2, 7.NS.2c, 7.NS.3, 7.EE.3 MP1, MP4 Students will drag points and arrows to model the situation on the number line and type their response in the box. Scoring Rubric 2 points Students correctly model the situation and type a correct expression. 1 point Students correctly type an expression OR correctly model the situation. 35. This test item requires students to explain and apply mathematical concepts and solve problems with precision, while making use of structure. Depth of Knowlege CCSS Content Standard Mathematical Practices Online Experience DOK1 7.NS.2, 7.NS.2a MP1 Students will be required to type their response in the box. Scoring Rubric 1 point Students correctly answer the question. Printed by: Nicole Sladich Page 12 of 18 Printed on 11/04/ :08 PM

13 Quick Check Are students continuing to struggle with multiplying integers? If Yes Reteach Worksheet If No Differentiated Instruction Options 1 and 2 Skills Practice Worksheet Differentiated Instruction Option 3 Enrich Worksheet Differentiated Instruction Option 4 Resources Self-Check Quiz: Multiply Integers Printed by: Nicole Sladich Page 13 of 18 Printed on 11/04/ :08 PM

14 Sketchpad Resources: Multiple Models of Multiplication The Geometer s Sketchpad, the world s leading software for teaching mathematics, gives students at all levels a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. Multiple Models of Multiplication Students work with four different models of multiplication and use each model to solve problems and investigate properties of multiplication. Students compare the four models, particularly with regard to how they make sense of negative operands. Activity Information Objectives: Develop a deeper understanding of multiplication; investigate various rules and properties of multiplication; compare different ways of understanding and using multiplication Suggested Duration: 45 minutes. Sketchpad Level: Beginning Prerequisites: Some familiarity with negative numbers Preparation: Review the Activity Notes. For a student-pairs activity, preview the student sketch and make a copy of the worksheet for each student. For a whole-class presentation, preview the presentation sketch. Materials: None Student Sketch: Multiplication Models.gsp Technology Strength: Manipulating models gives students ownership of their learning and challenges them to explain mathematically the behavior they observe. The mental images students develop provide a more solid foundation than a verbal rule for the idea that the product of two negative numbers is positive. Vocabulary: Scale, map, commutative property Using the Sketch Students explore four models that present multiplication in different ways multiplication as jumping, as grouping, as area, and as scaling. Each model is presented on a separate page. Students use each model to solve problems and investigate properties of multiplication. On the Jumping page, students drag the circles labeled drag to adjust the number of jumps and the size of each jump, and then press the Jump! button to see the animated rabbit. They repeat the process using different factors. Next, students go to the Grouping page and adjust the number of groups and the size of each group and observe the results. On the Area page, students adjust the width and height and observe how the area changes. Then on the Scaling page, students drag marker a for the Printed by: Nicole Sladich Page 14 of 18 Printed on 11/04/ :08 PM

15 starting value and marker b for the scaling factor and observe how changing the scale factor changes the spacing on a number line. Finally, students use what they have observed to compare the four models, particularly with regard to how they make sense of negative operands. Resources Discover The Geometer s Sketchpad Multiplication Models.gsp Student Worksheet: Multiple Models of Multiplication Teacher Worksheet: Multiple Models of Multiplication Printed by: Nicole Sladich Page 15 of 18 Printed on 11/04/ :08 PM

16 Sketchpad Resources: Magic Multiplying Machine: Exploring Multiplication The Geometer s Sketchpad, the world s leading software for teaching mathematics, gives students at all levels a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. Magic Multiplying Machine: Exploring Multiplication Students explore multiplication by dragging values for factors and observing the behavior of the product on a number line. Students discover the special roles that 0 and 1 play in multiplication and the rules regarding the sign of a product. Activity Information Objectives: Understand that any number multiplied by 0 equals 0, that any number multiplied by 1 is that number, and that any number multiplied by -1 is the opposite of that number; realize that multiplication of two factors does not always result in a product that is greater than either factor; understand that multiplying a positive by a negative results in a negative product and that multiplying a negative by a negative results in a positive product Suggested Duration: 45 minutes. Sketchpad Level: Beginning Prerequisites: Previous work with positive and negative numbers and with decimals Preparation: Review the Activity Notes. Preview the student sketch. Work through the steps on the worksheet and make a copy of the worksheet for each student. Materials: None Student Sketch: Magic Multiplying Machine.gsp Technology Strength: Students explore properties of multiplication dynamically by dragging values for factors along a number line and observing the effect on the product. Vocabulary: Multiplication, product, factor, positive, negative, multiplicative identity Using the Sketch Students explore properties of multiplication by dragging markers a and b to change the values of the factors in a multiplication problem and observing the effect on the product, as shown in the first illustration. Through a series of problems, they discover special roles that 0, 1, and 1 play in multiplication. Students also learn that multiplication of two factors does not always result in a product that is greater than either factor, as shown in the second illustration, and the rules regarding the sign of a product. In the Explore More, they are challenged to determine the location of zero and one on a number line with the numbers hidden, as shown in the third illustration. Printed by: Nicole Sladich Page 16 of 18 Printed on 11/04/ :08 PM

17 Resources Discover The Geometer s Sketchpad Magic Multiplying Machine Notes-5.pdf Magic Multiplying Machine Worksheet-5.pdf Magic Multiplying Machine.gsp Printed by: Nicole Sladich Page 17 of 18 Printed on 11/04/ :08 PM

18 Printed by: Nicole Sladich Page 18 of 18 Printed on 11/04/ :08 PM