Common Core State Standards

Transcription

1 Common Core State Standards 7.NS.1 7.NS.1a Common Core State Standards 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. 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. 7.NS.1d 7.NS.3 7.EE.3 Apply properties of operations as strategies to add and subtract 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 11, 13, and focus on Mathematical Practice 4 Model with mathematics. Exercise 12 focuses on Mathematical Practice 2 Reason abstractly and quantitatively. Exercise 14 focuses on Mathematical Practice 1 Make sense of problems and persevere in solving them and Mathematical Practice 7 Look for and make use of structure. Exercises 18 and focus on Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Printed by: Nicole Sladich Page 1 of 20 Printed on 11/04/ :07 PM

2 1 Launch the Lesson Focus narrowing the scope Objective Add integers. 7.NS.1, 7.NS.1a, 7.NS.1b, 7.NS.1d, 7.NS.3, 7.EE.3 Coherence Previous Students used models to add integers. connecting within and across grades Now Students add integers with the same sign and with different signs. Next Students will use models to subtract integers. Rigor pursuing concepts, fluency, and applications See the Levels of Complexity chart in the Practice and Apply section. Materials/Manipulatives red and yellow counters Integer Counters Master Integer Mat Master Lesson Vocabulary opposites additive inverse 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. Pairs Discussion Have pairs of students generate real-world situations in which they have encountered opposites and write these examples in their graphic organizers on the student page. Have them read aloud the paragraph about additive inverses and summarize to each other what the additive inverse of a number means. 1, 4, 6 Alternate Strategy Have students create a list of words that use the same roots as additive and inverse. For example, they may list addition, add, inverted, inversion, and so on. Have them explain how they can remember what an additive inverse is by looking at each word separately. 1, 3 Resources Common Core Quick Check: Add Integers Manipulative Masters: Integer Counters Manipulative Masters: Integer Mat Printed by: Nicole Sladich Page 2 of 20 Printed on 11/04/ :07 PM

3 2 Teach the Concept Focus: Mathematical Background The Commutative Property of Addition states that the order in which you add does not change the sum. a + b = b + a The Identity Property of Addition states that the sum of any number and zero is that number. a + 0 = a The McGraw-Hill Professional Development Video Library provides short videos that support Glencoe Math. The featured video for mathematical content in this lesson is: Integer Addition 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. Add integers with the same signs. Do both integers have the same sign? yes How does the model help connect to the rule for adding integers with the same sign? The model shows that the sum of 2 negative integers is negative. If you were told that the sum of these numbers is positive, how could you justify why you knew the sum had to be negative? Sample answer: Both integers are negative. The sum of 2 negative integers is negative. Need Another Example? Find 6 + (3) Add integers with the same signs. Do both integers have the same sign? yes How can you use estimation to check your answer? Sample answer: 26 + (17) is about 30 + (20), or 50. The answer is reasonable. What would the second integer need to be in order to have the least positive integer sum possible? 27 Need Another Example? Find 34 + (21) Add integers with different signs. Do the integers have the same sign in Example 3? Example 4? no; no Which integer has the greater absolute value in Example 3? Example 4? 5; 3 Will the sum be positive or negative in Example 3? Example 4? positive; negative Printed by: Nicole Sladich Page 3 of 20 Printed on 11/04/ :07 PM

4 In Example 3, how could you alter one of the integers in the expression so that the sum is 0? Sample answer: Change the second integer to 5. Need Another Example? a. Find 8 + (7). 1 b. Find Add integers with different signs. 5. Do the integers have the same sign? no What do you notice about the addends? They are additive inverses. What is the sum of additive inverses? 0 What property states that the sum of any number and its opposite is always zero? Additive Inverse Property Need Another Example? Find 5 + (5) Add integers with different signs. Do the integers have the same sign? no Which integer has the greater absolute value? 8 Will the sum be positive or negative? Why? Negative; 8 is negative, and it has the greater absolute value. How could you alter one of the integers in the expression so that the sum is positive? Sample answer: Change the second integer to 9. Need Another Example? Find Add integers with different signs. How many integers are being added? 3 Do the integers have the same sign? no What do you notice about the addends? 2 and 2 are additive inverses. What is the sum of additive inverses? 0 What properties allow you to rewrite the sentence so that it is easier to add? Commutative and Associative How could you alter one of the integers in the expression so that the sum is positive? Sample answer: Change the second integer to 15. Need Another Example? Find 11 + (4) + (11) Add integers to solve a real-world problem. Printed by: Nicole Sladich Page 4 of 20 Printed on 11/04/ :07 PM

5 What integer is used to represent up 20 feet? 20 What integer is used to represent down 32 feet? 32 What integer is used to represent up 16 feet? 16 What property allows you to rewrite the addition expression so that it is easier to add? Commutative Property of Addition Will the sum be positive or negative? Explain. positive; 36 is positive, and it has the greater absolute value How could you mentally determine that the sum will be positive, without calculating? Sample answer: The sum of the two positive numbers, 20 and 16, is greater than the absolute value of 32. Need Another Example? Max owes his brother $20. He gives his brother $10 one week and then the next week he gives him $5. Write an addition expression to describe this situation. Then find the sum and explain its meaning ; $5; Max still owes his brother $5. Alternate Teaching Strategy If students have trouble adding integers, Then use one of these reteach options: Quick Review Math Handbook, p. 87 Problem-Based Learning Book 2, Unit A: Add and Subtract with Negative Numbers, Investigation 1: The Two-Color Chip Model Problem-Based Learning Book 2, Unit A: Add and Subtract with Negative Numbers, Inquiry Investigation 2: A Number Line Model Problem-Based Learning Book 2, Unit A: Add and Subtract with Negative Numbers, Investigation 3: Add and Subtract on the Number Line Problem-Based Learning Book 2, Unit A: Add and Subtract with Negative Numbers, Investigation 6: Predict Signs of Sums and Differences Work with students to make an addition table using 3, 2, 1, 0, 1, 2, and 3 across the top and down the side of the table. Help students fill in the table and see the patterns and their relationships to the addition of integers rule. Formative Assessment Use these exercises to assess students' understanding of the concept of the lesson. If they need more help use the Personal Tutors available online. If some of your students are not ready for assignments, use the differentiated activity below. Roundrobin In groups of four, have students complete Exercises 1 3 using number lines. Student 1 draws a number line and places the first addend on the number line. Student 2 will tell whether the second addend indicates movement to the right or left. Student 3 performs the movement on the number line. Student 4 determines the sum. Call on a random student from each group to explain how they solved a particular exercise. 1, 5 Tips for New Teachers Adding Integers with Different Signs Remind students that in adding integers with different signs, the larger absolute value does not always come first. For example, in adding 6 + (11), the larger absolute value is 11, not 6. Resources Virtual Manipulatives Printed by: Nicole Sladich Page 5 of 20 Printed on 11/04/ :07 PM

6 Add Integers with Different Signs Add Integers with the Same Sign Add Integers with the Same Sign Add Integers with Different Signs Add Integers Add Integers Add Integers Add Integers Real-World Example: Add Integers with Different Signs Add Integers with the Same Sign 1 Add Integers with the Same Sign 2 Add Integers with Different Signs 1 Add Integers with Different Signs 2 Add Integers with Different Signs 3 Add Integers with Different Signs 4 Add Three or More Integers with Different Signs Real-World Example: Write an Addition Sentence etoolkit Problem-Based Learning Book 2, Unit A, Student Edition: Add and Subtract with Negative Numbers Problem-Based Learning Book 2, Unit A, Teacher Edition: Add and Subtract with Negative Numbers Hot Topic 1-4: Integer Operations Integer Addition Printed by: Nicole Sladich Page 6 of 20 Printed on 11/04/ :07 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 11, 13, 14, 18, 31, 32 On Level 1 9 odd, 11 14, 18, 31, 32 Beyond Level 11 18, 31, 32 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 20 Printed on 11/04/ :07 PM

8 Common Error When adding integers, students may incorrectly determine the sign of the sum. You may wish to have students imagine a number line in their heads, or draw one on paper, as they determine the sum of two integers. If they are adding a positive integer, they will move to the right. If they are adding a negative integer, they will move to the left. Resources Step by Step Solutions: Add Integers Skills Practice: Add Integers Skills Practice Answers: Add Integers Homework Practice: Add Integers Homework Practice Answers: Add Integers Problem-Solving Practice: Add Integers Problem-Solving Practice Answers: Add Integers Extra Practice: Add Integers Extra Practice Answers, Chapter 3 Homework Practice: Add Integers Problem-Solving Practice: Add Integers Skills Practice: Add Integers Extra Practice: Add Integers esolutions, Glencoe Math Course 2 Printed by: Nicole Sladich Page 8 of 20 Printed on 11/04/ :07 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: dry-erase markers, clear sheet protectors Many students would benefit from having number lines to use as they do independent work. Give each student a copy of a number line sheet in a clear sheet protector so they can reuse the number lines with dry-erase markers. Have students place the sheet in their three-ring binders so they can use it as a resource. The Reteach Worksheet provides additional examples and practice for students who may have difficulty in grasping the math concepts in this lesson. Option 2: Interpersonal Learners Materials: number cubes Provide groups of students with two number cubes, each of which has a positive integer on half of the faces and a negative integer on the other half of the faces. Students take turns rolling the two cubes. The student who rolls writes an addition problem using the two integers from the roll and finds the sum. Each student keeps a running total of his or her sums. The first student to get to a predetermined sum wins. Option 3: Kinesthetic Learners Materials: 10 pennies per student pair Separate students into pairs and give each pair 10 pennies. Determine which side (heads or tails) will represent 1 and which will represent 1. Have one student in each pair toss the 10 coins. The other student adds the negative and positive coin by removing the zero pairs, and writes the sum for that turn. Players take turns tossing and keeping score. The first player to reach an absolute value of 5 wins. Option 4: Intrapersonal Learners Have students complete the following problem: The scoring system for a video game is shown in the table. Assume Dylan starts with zero points. What will his score be after he finds two treasure chests, captures a monster, and is knocked off three cliffs? +8 Action Points find treasure chest +10 knocked off cliff 5 capture monster +3 Have students write their own problems using the table.you may wish to extend this activity by having students create their own games. Make sure they include both positive and negative scoring. Divide students into small groups and have them play each other s games. Option 5: Logical Learners Have students look up the periodic table on the Internet. Have them choose another element and represent it in the same way as in the beginning of the lesson. Printed by: Nicole Sladich Page 9 of 20 Printed on 11/04/ :07 PM

10 The Enrich Worksheet provides students with valuable opportunities for extending this lesson. Resources Reteach: Add Integers Reteach Answers: Add Integers Enrich: Adding a List of Integers Enrich Answers: Adding a List of Integers Reteach: Add Integers Enrich: Adding a List of Integers Printed by: Nicole Sladich Page 10 of 20 Printed on 11/04/ :07 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: Verbal/Linguistic Learners Materials: index cards Have students work in pairs to practice the vocabulary presented in this lesson. Have them make index cards with each vocabulary term from the lessons. Then have them make cards with several integers, one integer for each card. Students should draw one vocabulary card and one integer card. They should use the cards to say a sentence to their partner. For example, if the student draws opposite and 3, the student might say, The opposite of 3 is 3. Pairs should take turns drawing cards and speaking sentences. Option 3: Kinesthetic Learners Materials: counters, integer mat Have students use positive and negative counters to model the addition of integers with different signs. Present the following examples: 5 + (3) (6) Remind students to remove all zero pairs. The remaining counters represent the solution. Word Wall Printable English/Spanish Word Wall cards are available for the following vocabulary words in this lesson: additive inverse opposites Resources Word Wall Cards Interactive Guide for English Learners, Student Edition, Course 2, Lesson 3-2: Add Integers Interactive Guide for English Learners, Teacher Edition, Course 2, Lesson 3-2: Add 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 20 Printed on 11/04/ :07 PM

12 Assessment Formative Assessment Use this activity as a closing formative assessment before dismissing students from your class. Have students find the sum of 3 + (5) and Then have them write a few sentences comparing and contrasting the expressions. 8; 2; Sample answer: In the first expression, I am adding a negative number. Because I am adding a negative number to a number that is already negative, the sum is negative. In the second expression, I am adding a positive number. Because the absolute value of the positive number is greater than the absolute value of the negative number, the sum is positive. Time to update your Foldable! Students should be able to complete the section in their Foldable about adding integers. If they are having difficulty coming up with their own examples, they can use an example or text from this lesson. The online Self-Check Quiz for this lesson allows students to review concepts from each lesson. Exercises 31 and 32 prepare students for more rigorous thinking needed when taking the CCSS assessment. 31. This test item requires students to reason abstractly and quantitatively when problem solving. Depth of Knowledge CCSS Content Standard Mathematical Practices Online Experience DOK1 7.NS.3 MP1 Students will be required to click on the bubbles to select all correct answers. Scoring Rubric 1 point Students correctly answer each part. 32. 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 DOK2 7.NS.3 MP1, MP3 Students will be required to click to plot points on the number line, draw lines on the number line to create the model, and type in boxes to label the diagram and answer the question. Scoring Rubric 2 points Students correctly model the situation and answer the question, explaining their response. Printed by: Nicole Sladich Page 12 of 20 Printed on 11/04/ :07 PM

13 1 point Students correctly model the situation, but fail to explain OR students incorrectly model the situation, but correctly answer the question and explain their response. Quick Check Are students continuing to struggle with adding integers? If Yes Reteach Worksheet If No Differentiated Instruction Option 1 Skills Practice Worksheet Differentiated Instruction Options 2 and 3 Enrich Worksheet Differentiated Instruction Options 4 and 5 Resources Self-Check Quiz: Add Integers Printed by: Nicole Sladich Page 13 of 20 Printed on 11/04/ :07 PM

14 Sketchpad Resources: Adding Integers 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. Adding Integers Students use an animated model to add integers on a number line. They investigate addition of two positive numbers, two negative numbers, and a positive and a negative number. Activity Information Objectives: Add integers by visualizing the operation on a number line; discover rules for various cases of integer addition Suggested Duration: 30 minutes. You may want to combine this activity and the activity Subtracting Integers in a single class period. Sketchpad Level: Beginning Prerequisites: 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 Teacher Sketch: Adding Integers Present.gsp Student Sketch: Adding Integers.gsp Technology Strength: It s important for students to have a mental image of operations on integers. Even strong students who rely on verbal rules make careless mistakes that could be avoided by having an internalized picture. Vocabulary: Integer Using the Sketch Students explore a model that presents integer addition on a number line. Students begin by pressing the Present All button to see addition of two integers modeled on a number line. They then press the Reset button to hide the modeled answer. Students drag the circles to change the integers and press the Show Steps button to see the addition of these two new integers animated step by step. Students continue to reset, change the integers, and then use the model to animate the addition of the two integers, either by pressing the Present All button or by going through the addition step by step. At any time, students can leave the answer showing, drag the circles to change the integers, and explore the different cases of integer addition. Printed by: Nicole Sladich Page 14 of 20 Printed on 11/04/ :07 PM

15 In the Explore More section, students use the model to investigate additive inverses and the commutative property of addition. Resources Discover The Geometer s Sketchpad Adding Integers Present.gsp Adding Integers.gsp Teacher Worksheet: Adding Integers Student Worksheet: Adding Integers Printed by: Nicole Sladich Page 15 of 20 Printed on 11/04/ :07 PM

16 Sketchpad Resources: The Envelope: Adding and Subtracting Integers 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. The Envelope: Adding and Subtracting Integers Students learn about integers and adding and subtracting them, including subtracting negative integers, through a Sketchpad model that both calculates the result of change and shows it on a number line. Activity Information Objectives: Understand how negative integers can represent debts; devise strategies for adding and subtracting positive and negative integers; understand how addition and subtraction can be represented by motion along a number line. Suggested Duration: 30 minutes. Sketchpad Level: Beginning Prerequisites: Familiarity with representing addition and subtraction of positive numbers as movement along a number line 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: Envelope.gsp Technology Strength: This model represents addition and subtraction as actual movement along a vertical number line. Vocabulary: Integers, negative, positive, signed numbers Using the Sketch On page Envelope, students see a calculator that allows them to model real-life situations, as shown in the first illustration. They double-click the green value to enter the amount of change and press the green buttons to determine the kind of change. By adding or subtracting positive or negative values, students represent adding or removing money from the envelope, and adding or removing debts. The result is displayed in the envelope. Students also have the option of seeing the balance in the envelope move along the number line, as shown in the second illutration. On page Practice, students practice adding and subtracting randomly-generated positive and negative integers, as shown in the the third illustration. As they replace the result with their own conjectures, they get feedback from Sketchpad. Printed by: Nicole Sladich Page 16 of 20 Printed on 11/04/ :07 PM

17 Resources Envelope Notes-5.pdf Discover The Geometer s Sketchpad Envelope.gsp Envelope Worksheet-5.pdf Printed by: Nicole Sladich Page 17 of 20 Printed on 11/04/ :07 PM

18 Sketchpad Resources: Right or Left: Adding and Subtracting Integers 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. Right or Left: Adding and Subtracting Integers Students add and subtract integers using animations on a number line and understand that subtracting an integer is the same as adding its opposite. Activity Information Objectives: Understand how to add and subtract integers on a number line; realize that subtracting an integer is the same as adding its opposite, or its additive inverse; recognize that the absolute value of a number is its distance from zero and that the absolute value is always a positive value Suggested Duration: 60 minutes. To complete this activity in a shorter period, do the Summarize portion at the beginning of the next day. Sketchpad Level: Beginning Prerequisites: Understanding that integers are the set of whole numbers and their opposites 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: Right or Left.gsp Technology Strength: This dynamic model helps students visualize integer addition and subtraction. The animation of integer subtraction gives students a way of understanding how the signs of the operands come into play. Vocabulary: Integer, positive, negative, opposite, additive inverse, absolute value Using the Sketch On page Addition, students use the model to explore integer addition. By dragging the sliders, students can set up problems to investigate the addition of two positive integers, two negative integers, and problems with a positive and a negative integer, as shown in the first illustration. The first arrow represents the first integer, the second arrow represents the second integer, and the end of the second arrow indicates the sum. Arrows that represent positive integers point to the right. Arrows that represent negative integers point to the left. On page Subtraction, students use the model to investigate integer subtraction. Students set up problems for subtracting two positive integers, two negative integers, and problems with a positive and a negative integer, as shown in the second illustration. The model shows the arrow representing the second integer being flipped and then added on to the first arrow. This reinforces the idea that Printed by: Nicole Sladich Page 18 of 20 Printed on 11/04/ :07 PM

19 to subtract an integer, you add its opposite. In the Explore More, students can explore the behavior of addition and subtraction on a number line, as shown in the third illustration. They are challenged to determine the location of zero on a number line with the numbers hidden. Resources Discover The Geometer s Sketchpad Right or Left.gsp Right or Left Notes-5.pdf Right or Left Worksheet-5.pdf Printed by: Nicole Sladich Page 19 of 20 Printed on 11/04/ :07 PM

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