Common Core State Standards

Transcription

1 Common Core State Standards Common Core State Standards 7.NS.2 7.NS.2b 7.NS.2c 7.NS.3 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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. 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. 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 14, 19, and 23 focus on Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Exercises focus on Mathematical Practice 7 Look for and make use of structure. Exercise 36 focuses on Mathematical Practice 5 Use appropriate tools strategically. Printed by: Nicole Sladich Page 1 of 18 Printed on 11/04/ :08 PM

2 1 Launch the Lesson Focus narrowing the scope Objective Divide integers. 7.NS.2, 7.NS.2b, 7.NS.2c, 7.NS.3 Coherence Previous Students multiplied integers. connecting within and across grades Now Students will divide integers. Next Students will multiply and divide rational numbers. 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. Roundrobin Have students work in small groups to complete Exercises 1 5. Each student is responsible for reading aloud the exercise and providing a response. The rest of the group listens carefully and provides guidance or asks questions, as needed. 1, 7 Alternate Strategy Have students use counters or number lines to model each multiplication sentence and related division sentence. 1, 4 Resources Common Core Quick Check: Divide Integers Printed by: Nicole Sladich Page 2 of 18 Printed on 11/04/ :08 PM

3 2 Teach the Concept Focus: Mathematical Background Division is the inverse operation of multiplication. You can rewrite a division equation as a multiplication equation. The rules for dividing integers are related to the rules for multiplying integers. 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. Divide integers. What is the dividend? 80 What is the divisor? 10 Do the dividend and divisor have the same sign? no Will the quotient be positive or negative? negative How can you check your answer? Multiply. 8(10) = 80 Need Another Example? Find 51 (3) Divide integers. What is the dividend? 55 What is the divisor? 11 Do the dividend and divisor have the same sign? no Will the quotient be positive or negative? negative Write a different division expression in which the quotient is -5. Sample answer: 55 (11) Need Another Example? Find Divide integers to solve a real-world problem. What are you asked to find? the constant rate of change in centimeters each hour Printed by: Nicole Sladich Page 3 of 18 Printed on 11/04/ :08 PM

4 How does the table show how the height of the candle changes? Every hour, the height of the candle decreases by 2 centimeters. How long will it take for the candle to completely burn all the way out? 6 h Need Another Example? Use the table to find the constant rate of change in inches per month. 1 in. per month 4. Divide integers. Do the dividend and divisor have the same sign? yes What is the sign of the dividend and divisor? negative Will the quotient by positive or negative? positive How can you use multiplication to check your answer? Multiply 2 by 7. The answer is 14. So, my answer is correct. How could you alter one of the integers so that the quotient is negative? Sample answer: Change 14 to 14. Need Another Example? Find 12 (2) Divide integers. Do the dividend and divisor have the same sign? yes What is the sign of the dividend and divisor? negative Will the quotient by positive or negative? positive How can you use multiplication to check your answer? Multiply 9 by 3. The answer is 27. So, my answer is correct. How could you alter one of the integers so that the quotient is negative? Sample answer: Change 27 to 27. Need Another Example? Find Evaluate expressions. What integer should replace x in the expression? 4 Will the quotient by positive or negative? Explain. positive; The dividend and the divisor have the same Printed by: Nicole Sladich Page 4 of 18 Printed on 11/04/ :08 PM

5 sign, and the quotient of two integers with the same sign is positive. How can you use multiplication to check your answer? Multiply 4 by 4. The answer is 16. So, my answer is correct. How could you alter the integers and still have a positive quotient? Sample answer: Change 16 to 16 and change 4 to 4. Need Another Example? Evaluate 18 x if x = Solve a real-world problem. What is the problem asking you to determine? the average change in koala population per year What was the original population? 1,000,000 koalas What was the population after the 10 years? 100,000 koalas What do the variables N and P in the expression represent? N is the new population. P is the original population. What operation should be performed first to simplify the expression? subtraction Without solving the problem, will your answer be positive or negative? Explain. negative; Since the population decreased, the change in population per year will be represented by a negative number. Need Another Example? An October newspaper headline read that the total value of stocks held by Japanese financial institutions dropped from $987 billion to $900 billion in 6 weeks. Find the average change in value per week. Then explain its meaning. $14.5 billion; The total value of the stocks dropped $14.5 billion per week. Alternate Teaching Strategy If students struggle with dividing integers, Then use one of these reteach options: Quick Review Math Handbook, pp Use two-color counters to model division of smaller numbers such as Have students place 10 negative counters on a mat. Separate counters into 5 equal groups. This shows 10 5 = 2 because there are 2 negative counters in each group. 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 Pairs Discussion Provide students with positive and negative counters or blank number lines to help them complete the exercises. 1, 5 Trade-a-Problem Have students create problems involving a combination of multiplication and division operations with integers and trade them with a partner. Have them check their answers and discuss how to determine the sign of the final result without calculating. 1, 7 Teach with Tech Podcast Have students summarize rules for adding, subtracting, multiplying, and dividing integers in the form of a song, rap, or poem. Have them record the result using software and share it by way of Printed by: Nicole Sladich Page 5 of 18 Printed on 11/04/ :08 PM

6 podcast. Resources Virtual Manipulatives Divide Integers with Different Signs Divide Integers with Different Signs Divide Integers Divide Integers with the Same Sign Divide Integers with the Same Sign Evaluate Expressions Real-World Example: Multiply Integers with Different Signs Divide Integers with Different Signs Divide Integers with Different Signs Divide Integers with Different Signs Divide Integers Real-World Example: Dividing Integers with Different Signs 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 15, 17 19, 37, 38 On Level 1 13 odd, 15 19, 37, 38 Beyond Level 15 23, 37, 38 practice. can provide additional remediation support with personalized instruction and 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. Printed by: Nicole Sladich Page 7 of 18 Printed on 11/04/ :08 PM

8 Common Error Students may have trouble with Exercises 15 and 16 because of the negative sign in front of the variable. They may think the value is negative. Tell students to think of this as the opposite value of the variable, or 1 times the variable value. Common Error Students may have trouble multiplying three or more factors with the same sign. These students may find it easier to multiply first, ignoring the signs. Then they can apply the rules to find the product s sign. You may wish to points out that if the number of negative integers being multiplied is odd (1, 3, 5, and so on), the product will be negative. Resources Step by Step Solutions: Divide Integers Skills Practice: Divide Integers Skills Practice Answers: Divide Integers Homework Practice: Divide Integers Homework Practice Answers: Divide Integers Problem-Solving Practice: Divide Integers Problem-Solving Practice Answers: Divide Integers Extra Practice: Divide Integers Extra Practice Answers, Chapter 3 Homework Practice: Divide Integers Problem-Solving Practice: Divide Integers Skills Practice: Divide Integers Extra Practice: Divide 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: 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. Option 2: Interpersonal Learners Provide groups of students with two sets of cards and a counter. One set has digits 0 9 written on it. The second set contains the operations +,,, and. Have one student make expressions by drawing two digit cards, flipping the counter to determine if each is positive or negative, and then drawing an operation card. The other members race to solve the number sentence correctly. Whoever solves first gets a point. Encourage students to create more complex expressions as they continue to play. The Reteach Worksheet provides additional examples and practice for students who may have difficulty in grasping the math concepts in this lesson. Option 3: Naturalistic Learners Materials: poster board, markers, reference materials Have students work in pairs. Assign each pair an animal and have them find its estimated population for today and 10 years ago. Have students use the expression, where N is the new population and P is the previous population to determine the average change in numbers for their animals. On poster board, students make a table like the one shown below. Each pair of students should complete the table on the animal they researched. Option 4: Intrapersonal Learners Write the following questions on the board: Can a square number be negative? Explain. No; Sample answer: A negative number multiplied by a negative number, such as 3 (3) = 9, has a positive product, so the square of a number cannot be negative. Can the square root of a number be negative? Explain. Yes; Sample answer: The square root of 9 can be -3 because 3 (3) = 9. Do the values of x and y need to be positive or negative so that the quotient of x ( y) is negative? Sample answer: Either x or y needs to be negative, but not both. What combination of positive and negative values can you substitute for x, y, and z so that the value of this expression is positive: x 2 ( y ) ( z)? Explain. Sample answer: One combination is that x is negative, y is negative, and z is positive. This would give a positive value for x 2 (negative negative = positive), Printed by: Nicole Sladich Page 9 of 18 Printed on 11/04/ :08 PM

10 a negative value for y, and a negative value for z: positive negative negative = positive. Have students write their answers on paper. Have them explain their reasoning and give examples. Option 5: Logical Learners Have each student pick a country and find the highest and lowest temperatures recorded in that country. Students can use the expression in Got It? Exericse h to rewrite Celsius temperatures in degrees Fahrenheit. Use the expression, where F represents the number of degrees Fahrenheit, to find the temperature in degrees Celsius. Draw a thermometer on the board and have students write their countries high and low temperatures on the thermometer. The Enrich Worksheet provides students with valuable opportunities for extending this lesson. Resources Reteach: Percent of Change Reteach Answers: Percent of Change Enrich: A Taxing Exercise Enrich Answers: A Taxing Exercise Reteach: Percent of Change Enrich: A Taxing Exercise 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 are additional activities that can be used with this lesson. Option 1: Use Chapter 3, from the Interactive Guide for English Learners. Option 2: Logical Learners To scaffold this lesson, remind students of the relationship between multiplication and division. Display the following example on the board. Example: Write two multiplication problems related to the division problem 12 6 = 2. Solution: 2 6 = 12 and 6 2 = 12 To help students see this relationship involving negative integers, have them solve several problems like the ones shown below. Write two multiplication problems related to each division problem = = 8; 2 4 = ( 2) = 4 4 (2) = 8; 2 (4) = = = 15; 3 5 = = = 15; 3 (5) = 15 Write two division problems related to each multiplication problem = = 5; 10 5 = ( 2) = (2) = 5; 10 5 = = = 3; 12 3 = (4) = (4) = 3; 12 (3) = 4 Knowing how multiplication and division are related will also help students check their work. Resources Interactive Guide for English Learners, Student Edition, Course 2, Lesson 3-5: Divide Integers Interactive Guide for English Learners, Teacher Edition, Course 2, Lesson 3-5: Divide 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. Have students find. 7 Time to update your Foldable! Students should be able to complete the section in their Foldable about dividing 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 37 and 38 prepare students for more rigorous thinking needed when taking the CCSS assessment. 37. This test item requires students to explain and apply mathematical concepts and solve problems with precision, while making use of structure. Depth of Knowledge CCSS Content Standard Mathematical Practices Online Experience DOK1 7.NS.2, 7.NS.2c, 7.NS.3 MP1 Students will be required to type their response in the box. Scoring Rubric 1 point Students correctly answer the question. 38. This test item requires students to reason abstractly and quantitatively when problem solving. Depth of Knowledge CCSS Content Standard Mathematical Practices Online Experience DOK2 7.NS.2, 7.NS.2c, 7.NS.3 MP1 Students will be required to type their responses in the boxes. Scoring Rubric 2 points Students find the number of incorrect responses for each student and answer the question. 1 point Students find the number of incorrect responses for each student but fail to 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 dividing integers? If Yes Reteach Worksheet If No Differentiated Instruction Options 1 and 2 Skills Practice Worksheet Differentiated Instruction Option 3 Enrich Worksheet Differentiated Instruction Options 4 and 5 Resources Self-Check Quiz: Divide Integers OR students find the number of incorrect responses for 3 of the students and may or may not answer the question. Printed by: Nicole Sladich Page 13 of 18 Printed on 11/04/ :08 PM

14 Sketchpad Resources: Dividing Real Numbers 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. Dividing Real Numbers Students run a multiplication machine in reverse to solve division problems, to observe the relationship between multiplication and division, and to investigate properties of division. Activity Information Objectives: Develop a deeper understanding of division by comparing it directly with multiplication using a visual model; understand properties of division Suggested Duration: 45 minutes. The activity can be shortened considerably by skipping steps 2 6, as described in the Activity Notes. Sketchpad Level: Intermediate Prerequisites: Familiarity with negative numbers Preparation: Review the Activity Notes. For either a student-pairs activity or a whole-class presentation, preview the student sketch and make a copy of the worksheet for each student. Materials: None Student Sketch: Division Machine.gsp Technology Strength: In this activity, to answer questions about the properties of division, students must translate the questions, which are in mathematical terms, into the terms and behavior of the model, interpret the behavior, and express their conclusions mathematically. Vocabulary: None Using the Sketch Students explore a multiplication model for a b = c by dragging markers that determine a and b. They then investigate a division model and observe the relationship between multiplication and division by pressing the Multiply/Divide Toggle button. Students can drag markers a and c to control the divisor and the dividend and investigate different properties of division. Students can construct their own number line models by following worksheet steps 2 6 or they can go to page 2 of the sketch, which has the number lines already constructed. If they choose to construct their own models, students use custom tools to construct the number lines associated with the markers and bars so that they can use real numbers. The last two pages of the sketch describe how to make the custom tools Printed by: Nicole Sladich Page 14 of 18 Printed on 11/04/ :08 PM

15 used during the construction section of the activity. Resources Discover The Geometer s Sketchpad Division Machine.gsp Student Worksheet: Dividing Real Numbers Teacher Worksheet: Dividing Real Numbers Printed by: Nicole Sladich Page 15 of 18 Printed on 11/04/ :08 PM

16 Sketchpad Resources: Magic Dividing Machine: Exploring Division 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 Dividing Machine: Exploring Division Students run a multiplication machine in reverse to solve division problems. In doing so, they observe the relationship between multiplication and division. Students use another division machine to investigate properties of division on a number line. Activity Information Objectives: Understand that any number divided by one equals that number, that any number divided by itself equals one, and that you cannot divide by zero; realize that dividing by a positive number does not always result in a smaller number; understand that dividing a positive by a negative results in a negative quotient and that dividing a negative by a negative results in a positive quotient; recognize that multiplication and division are inverse operations 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 Dividing Machine.gsp Technology Strength: Students investigate the relationship between multiplication and division by switching back and forth between two models and observing the changes that occur. Students explore properties of division dynamically by dragging values for the dividend and the divisor on a number line and observing the effect on the quotient. Vocabulary: Division, quotient, dividend, divisor, product, factor, positive, negative, inverse operation Using the Sketch On page Inverse Operations, students explore a multiplication model in which a and b are factors and c is the product, as shown in the first illustration. Students drag marker a to change the number of rectangles in the blue bar and drag marker b to change the width of each rectangle. The product, marker c, is the total length of the blue bar. Students then press Inverse Operation to switch to the division model. They now control marker a, the divisor, and marker c, the dividend. Marker b becomes the quotient. Students can press Show Numbers to see the numerical values of the markers, as shown in the second illustration. By switching between the two models, students gain a better understanding of the relationship between multiplication and division. Printed by: Nicole Sladich Page 16 of 18 Printed on 11/04/ :08 PM

17 On page Division Machine, students investigate properties of division by dragging values for the dividend and the divisor on a number line and observing the effect on the quotient, as shown in the third illustration. In the Explore More, they are challenged to determine the location of zero and one on a number line with the numbers hidden. Resources Magic Dividing Machine Worksheet-5.pdf Discover The Geometer s Sketchpad Magic Dividing Machine.gsp Magic Dividing Machine Notes-5.pdf Printed by: Nicole Sladich Page 17 of 18 Printed on 11/04/ :08 PM

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