Enhanced Instructional Transition Guide

Transcription

1 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Unit 02: Numerical Operations: Integers and Positive Rational Numbers (16 days) Possible Lesson 01 (6 days) Possible Lesson 02 (10 days) POSSIBLE LESSON 01 (6 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students use manipulatives and number lines to investigate integer operations (addition, subtraction, multiplication, and division). TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strikethrough (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to: 7.2C Use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms. Supporting Standard Underlying Processes and Mathematical Tools TEKS: 7.13 Underlying processes and mathematical tools. The student applies mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to: page 1 of 86

2 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days 7.13A Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics. 7.13B Use a problemsolving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Underlying processes and mathematical tools. The student communicates about mathematics through informal and mathematical language, representations, and models. The student is expected to: 7.14A Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models. 7.14B Evaluate the effectiveness of different representations to communicate ideas Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: 7.15A Make conjectures from patterns or sets of examples and nonexamples. 7.15B Validate his/her conclusions using mathematical properties and relationships. Performance Indicator(s): page 2 of 86

3 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Grade 07 Unit 02 PI 02 Construct a threecolumn chart to explain and demonstrate the connection between a pictorial model and a symbolic representation to the algorithm of all four operations using integers. Sample Performance Indicator: Brittany told a student who was absent from class, A positive number and a negative number will always be negative, regardless of the operation. Jordan told the absent student, A positive number and a negative number can sometimes be positive. Reed told the absent student A negative number and a negative number always equal a negative number. Create a three column chart of the pictorial model, symbolic representation, and connect them to the algorithm using all four operations involving integers. Use the chart to write a justification to determine if the statements made by Brittany, Jordan, and Reed are always true in all operations. Standard(s): 7.2C, 7.13A, 7.13B, 7.14A, 7.14B, 7.15A, 7.15B ELPS ELPS.c.1C, ELPS.c.3J Key Understanding(s): Different models may be used to add, subtract, multiply, and divide integers and connect the actions to the algorithms. When given a model or realworld problem using integers, an expression or equation may be written to represent and solve the problem situation. When using a problem solving model to perform integer operations in a problem situation, conjectures can be communicated and validated. Misconception(s): Some students may think the sum of two integers is always greater than the two addends. Some students may think the difference of two integers is always less than the minuend (e.g., 9 ( 2) = 7, where 9 is the minuend, 2 is the subtrahend, and 7 is the difference). Some students may think the product of two negative integers is negative. Some students may think the quotient of two negative integers is negative. Vocabulary of Instruction: absolute value integer zero pair page 3 of 86

4 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days conjectures negative number simplifying an expression Materials: twocolor counters (20 per student) math journal (1 per student) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Integers and Football KEY Integers and Football Integers and Games KEY Integers and Games Adding Integers TwoColor Counters KEY Adding Integers TwoColor Counters Subtracting Integers TwoColor Counters KEY Subtracting Integers TwoColor Counters Integers and Sea Otter KEY Integers and Sea Otter page 4 of 86

5 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Adding Integers Number Lines KEY Adding Integers Number Lines Subtracting Integers Number Lines KEY Subtracting Integers Number Lines Subtracting Integers Using Models KEY Subtracting Integers Using Models Adding Integers and Connecting Patterns KEY Adding Integers and Connecting Patterns Adding and Subtracting Integers Practice KEY Adding and Subtracting Integers Practice Multiplication and Division of Integers KEY Multiplication and Division of Integers Multiplying Integers and Connecting Patterns KEY Multiplying Integers and Connecting Patterns Dividing Integers and Connecting Patterns KEY Dividing Integers and Connecting Patterns Note Page for Integer Operations Extending Integers KEY page 5 of 86

6 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Extending Integers Integer Practice KEY Integer Practice GETTING READY FOR INSTRUCTION Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1 Topics: Engage 1 Introduction to integers Spiraling Review ATTACHMENTS Students use logic, reasoning, and prior knowledge to review integers. Instructional Procedures: 1. Display teacher resource: Integers and Football. Facilitate a class discussion to review representing integers. Ask: Teacher Resource: Integers and Football KEY (1 per teacher) Teacher Resource: Integers and Football (1 per teacher) What does the term gained mean with respect to integers? Lost? (gainedpositive; lostnegative) What other terms are used to represent a positive integer? Answers may vary. Deposit; page 6 of 86

7 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures addition; etc. What other terms are used to represent a negative integer? Answers may vary. Withdrawal; less than; etc. How would you represent gaining 12 yards as an integer? (12) How would you represent losing 6 yards as an integer? ( 6) Notes for Teacher Topics: ATTACHMENTS Adding and subtracting integers with models Explore/Explain 1 Students use twocolor counters to explore the relationship of adding and subtracting integers. Instructional Procedures: 1. Place students in pairs and distribute 20 twocolor counters to each student. 2. Display the expression 1 + ( 1) for the class to see. Instruct students to lay 1 two color counter on yellow and another twocolor counter on red. Explain to students that the yellow side of the twocolor counters will be used to represent the integer 1, while the red side of the twocolor counters will be used to represent the integer ( 1). Facilitate a class discussion to define the term zero pairs. Ask: How would you model 1 + ( 1) using two color counters? (use 1 yellow and 1 red) What would be a real life example of 1 + ( 1)? Answers may vary. I made 1 point and I Teacher Resource: Integers and Games KEY (1 per teacher) Teacher Resource: Integers and Games (1 per teacher) Teacher Resource: Adding Integers TwoColor Counters KEY (1 per teacher) Handout: Adding Integers TwoColor Counters (1 per student) Teacher Resource: Subtracting Integers TwoColor Counters KEY (1 per teacher) Handout: Subtracting Integers Two Color Counters (1 per student) MATERIALS twocolor counters (20 per student) page 7 of 86

8 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day lost 1 point, so, I have 0 points; etc. Suggested Instructional Procedures Notes for Teacher math journal (1 per student) Explain to students that 1 + ( 1) is called a zero pair. Why do you think 1 + ( 1) is called a zero pair? Answers may vary. A number and its opposite that combine to a value of zero; etc. What is the sum of the counter set of 1 and ( 1)? (1 + ( 1) = 0) 3. Display teacher resource: Integers and Games. Instruct student pairs to create a model with their twocolor counters that demonstrates the problem situation. Allow 3 minutes for students to create their model. Monitor and assess student pairs to check for understanding. Facilitate a class discussion about integers and using models. Ask: What number would you use to represent the points Devin lost? ( 4) How would you model a negative number with twocolor counters?(red counters) What number would you use to represent the points Devin gained? (6) How would you model a positive number with twocolor counters?(yellow counters) What equation would you record to model a loss of 4 points and a gain of 6 points? (( 4) + 6 = 2) How would you use two color counters to model the equation ( 4) + 6 = 2? (I would place 4 negative counters and 6 positive counters on the integer mat. There are 4 zero pairs made when I combine 4 negatives with 4 positives which leave 2 positives.) TEACHER NOTE For the twocolor counters use the yellow side to represent positive and the red side to represent negative. Emphasize combine when placing the objects representing addends together to model addition. Emphasize remove when modeling subtraction problems using twocolor counters. TEACHER NOTE Make sure to record the addition problem directly under the original subtraction problem for students to notice the relationship between the subtraction and addition problem. page 8 of 86

9 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher 4. Distribute handout: Adding Integers TwoColor Counters to each student. Instruct student pairs to model the addition problems with twocolor counters, create a sketch of the model with pictures of the two color counters or ( ) and (+), and record each solution process on their handout. Allow time for student pairs to complete the addition problems. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. 5. Display the expression ( 4) + ( 3) for the class to see. Ask: How would you model the expression ( 4) + ( 3) with two color counters? (I would place 4 negative counters and 3 negative counters on the integer mat for a total of 7 negative counters.) page 9 of 86

10 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher How is the expression 8 4 different from the problems you have been modeling? Answers may vary. I am now subtracting instead of adding; etc. How would you model the expression 8 4 with two color counters? (I would place 8 positive counters on the integer mat and then remove 4 positive counters leaving 4 positive counters.) 6. Display the expression 7 8 for the class to see. Facilitate a class discussion about subtracting integers. Ask: How would you begin to model the expression 7 8 with two color counters? (Place 7 positive counters on the integer mat.) What does subtract 8 mean? (remove 8 positive counters) How do you model removing 8 positive counters when you only have 7 positive counters?explain. Answers may vary. I will have to create a zero pair and then I will have 8 positive counters to remove; etc. page 10 of 86

11 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What remains when the 8 positive counters are removed from the integer mat? (1 negative counter) What integer value is remaining on the integer mat? (( 1)) 7. Distribute handout: Subtracting Integers TwoColor Counters to each student. 8. Display the expression ( 6) ( 4) for the class to see. Facilitate a class discussion about subtracting negative integers. Instruct students to model the subtraction problems with twocolor counters, create a sketch of the model with pictures of the two color counters or ( ) and (+), and record each solution process on their handout: Subtracting Integers TwoColor Counters throughout the discussion. Ask: How would you begin to model the expression ( 6) ( 4) with two color counters? (Place 6 negative counters on the integer mat.) Explain to students that subtract negative 4 means the opposite of removing negative 4 and that the opposite of removing negative 4 is adding positive 4. Ask: If you have 6 negative counters and you add 4 positive counters, how many zero pairs page 11 of 86

12 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures are created? (4 zero pairs) What remains when the zero pairs are removed from the integer mat? (2 negative counters) What integer value is remaining on the integer mat? (( 2)) What addition equation can be created using 6 negatives that would yield the same solution as ( 6) ( 4)? (( 6) + 4 =( 2)) How are these problems similar? Different? Answers may vary. Both have negative 6. The second integers in each problem are opposites, and one problem has a subtraction symbol and the other problem has an addition symbol; etc. Why do both ( 6) ( 4) and ( 6) + 4 equal ( 2)? Answers may vary. Both expressions equal ( 2) because subtracting negative 4 from negative 6 is read as the opposite of removing negative 4 from negative 6. The opposite of removing negative 4 is adding positive 4; etc. Notes for Teacher 9. Display the expression ( 7) ( 4) for the class to see. Ask: How would you begin to model the expression ( 7) ( 4) with two color counters? (Place 7 negative counters on the integer mat.) What does subtract negative 4 mean? (the opposite of removing 4 negative counters) What is the opposite of removing 4 negative counters? (adding 4 positive counters) If you have 7 negative counters and you add 4 positive counters, how many zero pairs are created? (4 zero pairs) What remains when the zero pairs are removed from the integer mat? (3 negative counters) What integer value is remaining on the integer mat? (( 3)) page 12 of 86

13 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures What addition equation can be created using 7 negatives that would yield the same solution as ( 7) ( 4)? (( 7) + 4 = ( 3)) How are these problems similar? Different? Answers may vary. Both have negative 7. The second integers in each problem are opposites, and one problem has a subtraction symbol and the other problem has an addition symbol; etc. Why do both ( 7) ( 4) and ( 7) + 4 equal ( 2)? Answers may vary. Both expressions equal ( 2) because subtracting negative 4 from negative 7 is read as the opposite of removing negative 4 from negative 7. The opposite of removing negative 4 is adding positive 4; etc. Notes for Teacher 10. Display the expression 7 ( 4) for the class to see. Ask: How is this problem different from the expression you just modeled? (The first number is positive 7 instead of ( 7).) How would you begin to model the expression 7 ( 4) with two color counters? (Place 7 positive counters on the integer mat.) What does subtract negative 4 mean? (the opposite of removing 4 negative counters) What is the opposite of removing 4 negative counters? (adding 4 positive counters) If you have 7 positive counters and you add 4 positive counters, how many zero pairs are created? (0 zero pairs) What is on the integer mat when you add 4 positive counters to the 7 positive counters? (11 positive counters) What integer value is remaining on the integer mat? (11) What addition equation can be created using 7 positives that would yield the same solution as 7 ( 4)? (7 + 4 = 11) page 13 of 86

14 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures How are these problems similar? Different? Answers may vary. Both have positive 7. The second integers in each problem are opposites, and one problem has a subtraction symbol and the other problem has an addition symbol; etc. Why do both 7 ( 4) and equal 11? Answers may vary. Both expressions equal 11 because subtracting the opposite of negative 4 from positive 7 is read as the opposite of removing negative 4 from positive 7. The opposite of removing negative 4 is adding positive 4; etc. Notes for Teacher 11. Display the expression ( 8) 2 for the class to see. 12. Instruct students to individually evaluate the expression by modeling the solution process with twocolor counters, creating a pictorial record of the process, and recording the symbolic mathematical representation for each step in the process in their math journal. Monitor and assess students to check for understanding. Facilitate a class discussion to summarize subtracting integers with twocolor counters using the problem ( 8) 2. Ask: How would you begin to model the expression ( 8) 2 with twocolor counters? (Place 8 negative counters on the integer mat.) What does subtract 2 mean? (remove 2 positive counters) How would you model removing 2 positive counters when you only have 8 negative counters? Explain. Answers may vary. I will have to create 2 zero pairs, and then I will have 2 positive counters to remove; etc. What remains when the 2 positive counters are removed from the integer mat? (10 negative counters) What integer value is remaining on the integer mat? (( 10)) page 14 of 86

15 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2 Topics: Adding and subtracting integers with a number line Explore/Explain 2 Students use a number line to explore the relationship of adding and subtracting integers. Instructional Procedures: 1. Distribute 20 twocolor counters to each student. Facilitate a class discussion to connect the zero pair twocolor counter model to a number line. Ask: How would you model 1 + ( 1) using the twocolor counters? (use 1 yellow and 1 red) What would be a real life example for 1 + ( 1)? Answers may vary. I made 1 point, and I lost 1 point, so I have 0 points; etc. What happens on the number line if you begin at 0, move 1, and then move ( 1)? (I will end where I began, at 0.) Spiraling Review ATTACHMENTS Teacher Resource: Integers and Sea Otter KEY (1 per teacher) Teacher Resource: Integers and Sea Otter (1 per teacher) Teacher Resource: Adding Integers Number Lines KEY (1 per teacher) Handout: Adding Integers Number Lines (1 per student) Teacher Resource: Subtracting Integers Number Lines KEY (1 per teacher) Handout: Subtracting Integers Number Lines (1 per student) Teacher Resource (optional): Subtracting Integers Using Models KEY (1 per teacher) Handout (optional): Subtracting Integers page 15 of 86

16 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Using Models (1 per student) MATERIALS What happens on the number line if you begin at 0, move ( 1), and then move 1? (I will end where I began, at 0.) twocolor counters (20 per student) ADDITIONAL PRACTICE Handout (optional): Subtracting Integers Using Models may be used as additional practice, as needed. 2. Display teacher resource: Integers and Sea Otter. Instruct students to use twocolor counters and a number line to represent the problem situation. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion about adding integers using a number line. Ask: How would you use twocolor counters to model this situation? page 16 of 86

17 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher How would you use a number line to model this situation? page 17 of 86

18 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What is the distance between ( 5) and 0 on a number line? (5) Explain to students that the distance between any integer and zero on a number line is also referred to as the absolute value. Since the absolute value represents distance on a number line, the absolute value of a number will always be positive. 3. Collect the twocolor counters from students. page 18 of 86

19 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures 4. Place students in pairs and distribute handout: Adding Integers Number Lines to each student. Instruct student pairs to model the addition problems with number lines and record their solution process and answer. Allow time for student pairs to complete the addition problems. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: Notes for Teacher How would you model the expression ( 4) + ( 3) using a number line? (Begin at 0 and move in the direction of the sign of the integer in front of the addition symbol, ( 4), then move in the direction of the sign of the integer following the addition symbol, ( 3).) 5. Facilitate a class discussion about the generalization for adding integers on a number line. Ask: When adding integers on a number line, which direction would you move? (Begin at 0 and move in the direction of the sign of the integer in front of the addition symbol, then move in the direction of the sign of the integer following the addition symbol.) 6. Distribute handout: Subtracting Integers Number Line to each student. Demonstrate the solution process for the first 2 problems. Instruct students to replicate the model throughout the demonstration. Facilitate a class discussion about the solution process. Ask: page 19 of 86

20 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What relationship exists between addition and subtraction? (They are opposite operations.) When you encountered the addition symbol to model adding integers on a number line, you began at 0 and first moved in the direction of the integer in front of the addition symbol and then moved in the direction of the integer following the addition symbol. What do you think you would do when you encounter the subtraction symbol to model subtracting integers on a number line? (I will begin at 0 and move in the direction of the sign of the integer before the subtraction symbol, and then I will move in the opposite direction of the sign of the integer following the subtraction symbol.) How would you model the expression ( 5) ( 3) using a number line? (Step 1: Begin at 0 and move in the direction of the sign of the integer that is in front of the subtraction symbol, ( 5). Step 2: Begin at ( 5) and move in the opposite direction of the sign of the integer following the subtraction symbol, ( 3) move in a positive direction 3 units. page 20 of 86

21 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Step 3: The nonoverlapping section of the rays is the difference between both integers. The number line model for the expression ( 5) ( 3) is the same as the number line model for the expression ( 5) + 3.) What addition equation can be created using 5 negatives that would yield the same solution as ( 5) ( 3) = ( 2)? (( 5) + 3 = ( 2)) 7. Display the expression 4 7 for the class to see. Ask: How would you model the expression 4 7 using a number line? (Step 1: Begin at 0 and move in the direction of the sign of the integer that is in front of the subtraction symbol, 4. page 21 of 86

22 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Step 2: Begin at 4 and move in the opposite direction of the sign of the integer following the subtraction symbol, 7 move in a negative direction 7 units. Step 3: The nonoverlapping section of the rays is the difference between both integers. The number line model for the expression 4 7 is the same as the number line model for the expression 4 + ( 7).) What addition equation can be created using 4 positive that would yield the same solution as 4 7 = ( 3)? (4 + ( 7) = ( 3)) 8. Instruct student pairs to complete the remainder of handout: Subtracting Integers Number page 22 of 86

23 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Lines. Allow time for students to complete the number lines. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. 9. Facilitate a class discussion about the generalization for subtracting integers on a number line. Ask: Notes for Teacher When subtracting integers on a number line, which direction would you move? (Begin at 0 and move in the direction of the sign of the integer in front of the subtraction symbol, then move in the opposite direction of the sign of the integer following the subtraction symbol.) 3 Topics: Developing the rules for addition and subtraction of integers Elaborate 1 Students use patterns to discover the relationships and rules for adding and subtracting integers. Instructional Procedures: 1. Distribute handout: Adding Integers and Connecting Patterns to each student. Instruct students to complete the tables and record their observations. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion about the patterns for adding integers. Ask: When integers with the same sign are added, what inferences can you make? Answers may vary. I ended up combining sets, so the sum has the same sign as the integers; etc. Spiraling Review ATTACHMENTS Teacher Resource: Adding Integers and Connecting Patterns KEY (1 per teacher) Handout: Adding Integers and Connecting Patterns (1 per student) Teacher Resource: Adding and Subtracting Integers Practice KEY (1 per teacher) Handout: Adding and Subtracting Integers Practice (1 per student) page 23 of 86

24 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Does this pattern always work? Explain your reasoning. (Yes) Answers may vary. There will be no zero pairs if the signs are the same; on the number line, if the directions from zero are the same the sum will be the same direction from zero; etc. When integers with different signs are added, what inferences can you make? Answers may vary. When integers having different signs are added, sets of zero pairs are produced. The sign of the sum depends on which integer has more twocolor counters; when integers having different signs are added on the number line, there will always be portions of the number line that show both directions, ( ) and (+). This corresponds to the same distance from zero in a positive and negative direction; the sign of the sum depends on which integer is the farthest from zero, and in which direction; etc. When a number and its opposite are added, what do you notice? Why do you think this happens? What did you call these numbers? (The sum is 0.) Answers may vary. Both numbers are equal distances from 0 and have opposite signs; these numbers are opposites and form a set of zero pairs; etc. Notes for Teacher 2. Facilitate a class discussion about the patterns for subtracting integers. Ask: When integers with the same sign are subtracted, what inferences can you make? Answers may vary. When integers having the same signs are subtracted, add the opposite and then follow the addition rules; etc. When integers with different signs are subtracted, what inferences can you make? Answers may vary. When integers having different signs are subtracted, add the opposite of the second number and follow the addition rules; etc. page 24 of 86

25 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures 3. Place students in pairs and distribute handout: Adding and Subtracting Integers Practice to each student. Instruct student pairs to complete the handout together to provide additional integer practice and solidify the rules for addition and subtraction of integers. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions, as needed. Notes for Teacher 4 5 Topics: Adding and subtracting integers Multiplying and dividing integers with twocolor counters Explore/Explain 3 Students demonstrate their knowledge of adding and subtracting integers. Students use twocolor counters to explore the relationship of multiplying and dividing integers. Instructional Procedures: 1. Distribute 20 twocolor counters to each student and display the following model for the class to see. Spiraling Review ATTACHMENTS Teacher Resource: Multiplication and Division of Integers KEY (1 per teacher) Handout: Multiplication and Division of Integers (1 per student) MATERIALS twocolor counters (20 per student, 20 per teacher) math journal (1 per student TEACHER NOTE 2. Facilitate a class discussion about using twocolor counters to model multiplication problems. It is important for students see the relationship between 4( 2) and ( 2)(4) to use as a reference page 25 of 86

26 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Instruct students to replicate the model with twocolor counters, create a sketch of the model with pictures of the two color counters or ( ) and (+), and record an equation to represent the model in their math journal. Ask: Notes for Teacher when working with multiplication of integers. What equation(s) would you record to represent this model? ( = 8 or 4(2) = 8) What does the equation 4(2) = 8 mean? (There are 4 groups with 2 positive counters in each group and the product is 8. If I count the counters by groups, there would be 2, 4, 6, 8.) 3. Display the following model for the class to see: Ask: What equation(s) would you record to represent this model? (4 + 4 = 8 or 2(4) = 8) What does the equation 2(4) = 8 mean? (There are 2 groups with 4 positive counters in each group and the product is 8. If I count the counters by groups, there would be 4, 8.) 4. Display the following expressions for the class to see: 4(2) and 2(4). Ask: page 26 of 86

27 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures How are the expressions 4(2) and 2(4) related? Answers may vary. Both have the same factors and the same product; the factors switched positions, and due to commutative property of multiplication the products are the same; etc. Notes for Teacher 5. Display the following model for the class to see: Ask: What equation(s) would you record to represent this model? (( 2) + ( 2) + ( 2) + ( 2) = ( 8) or 4( 2) = ( 8)). What does the equation 4( 2) = ( 8) mean? (There are 4 groups with 2 negative counters in each group and the product is ( 8). If I count the counters by groups, there would be ( 2), ( 4), ( 6), ( 8).) 6. Display the following model for the class to see: page 27 of 86

28 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Ask: What equation(s) would you record to represent this model? (( 4) + ( 4) = ( 8) or 2( 4) = ( 8)) What does the equation 2( 4) = ( 8) mean? (There are 2 groups with 4 negative counters in each group and the product is ( 8). If I count the counters by groups, there would be ( 4), ( 8).) 7. Display the following expressions for the class to see: 4( 2) and 2( 4). Explain to students that although one factor is positive and one factor is negative, the commutative property of multiplication still applies because the expressions 4( 2) and 2( 4) both equal ( 8). 8. Display the following model for the class to see: Explain to students that this model represents taking the opposite of 4 groups with 2 positive counters in each group and can be represented with the expression ( 4)2. Remind students that the opposite of a positive is a negative. page 28 of 86

29 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Ask: What equation represents the opposite of 4 groups with 2 positive counters in each group? (( 4)2 = ( 8)) 9. Display the following model for the class to see: Explain to students that this model represents the opposite of 2 groups with 4 positive counters in each group, which becomes 2 groups with 4 negative counters in each. Ask: What equation(s) would you record to represent this model? (( 2)4 = ( 8)) 10. Display the following expressions for the class to see: ( 4)2 and ( 2)4. Ask: page 29 of 86

30 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures How are the expressions ( 4)2 and ( 2)4 related? Answers may vary. Both the same product; etc. Notes for Teacher 11. Facilitate a class discussion about multiplying two negative integers. Ask: How would you model the opposite of 4 groups with 2 negative counters in each group? (I would place 2 negative counters in 4 groups and then flip the counters over to the positive side.) What equation would you record to represent this model? (( 4)( 2) = 8) How would you model the opposite of 2 groups with 4 negative counters in each group? (I would place 4 negative counters in 2 groups and then flip the counters over to the positive side.) page 30 of 86

31 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What equation would you record to represent this model? (( 2)( 4) = 8) 12. Display the following expressions for the class to see: ( 4)( 2) and ( 2)( 4). Ask: How are the expressions ( 4)( 2) and ( 2)( 4) related? Answers may vary. Both have the same factors and the same product; the factors switched positions, and due to commutative property of multiplication the products are the same; etc. 13. Place students in pairs and distribute handout: Multiplication and Division of Integers to each student. Instruct student pairs to complete problems 1 5. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: What generalizations can be made about multiplying integers? Answers may vary. The product of two positive integers is always positive; the product of two negative integers is always positive; the product of a positive integer and a negative integer is always negative; etc. page 31 of 86

32 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures 14. Facilitate a class discussion on the relationship between multiplication and division. Ask: Notes for Teacher How are multiplication and division related? Answers may vary. They are inverse operations; a multiplication problem involves factor x factor = product. A division problem involves the product from a multiplication problem (dividend) divided by 1 of the factors (divisor) which will equal the other factor (quotient); etc. What multiplication equation would you use to verify the quotient for the equation 24 3 = 8? (3 8 = 24) How would you model the equation 3 8 = 24 in terms of groups and number of objects per group? (I would create 3 groups with 8 positive counters in each group for a total of 24 positive counters.) How would you model the equation 24 3 = 8 in terms of groups and number of objects per group? (I would separate 24 positive counters equally in 3 groups with 8 positive counters in each group.) 15. Facilitate a class discussion about using twocolor counters to model division problems. Instruct students to replicate the model with twocolor counters, create a sketch of the model with pictures of the two color counters or ( ) and (+), and record an equation to represent the model in their math journal. Ask: How would you model 8 positive counters in 4 groups? (I would separate 8 positive counters equally in 4 groups with 2 positive counters in each group.) page 32 of 86

33 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What equation would you record to represent this model? (8 4 = 2) What multiplication equation would you use to verify the quotient for 8 4 = 2? (4 2 = 8) How would you model the equation 8 2 = 4? (I would separate 8 positive counters equally into 2 groups with 4 positive counters in each group.) What multiplication equation would you use to verify the quotient for 8 2 = 4? (2 4 = 8) How would you model 8 negative counters in 4 groups? (I would separate 8 negative counters equally in 4 groups with 2 negative counters in each group.) What equation would you record to represent this model? (( 8) 4 = ( 2)) page 33 of 86

34 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What multiplication equation would you use to verify the quotient for ( 8) 4 = ( 2)? (4 ( 2) = ( 8)) How would you model the expression ( 8) 2? (I would separate 8 negative counters equally into 2 groups with 4 negative counters in each group.) What multiplication equation would you use to verify the quotient for ( 8) 2 = ( 4)? (2 ( 4) = ( 8)) How would you model the opposite of 8 positive counters in 4 groups? (I would separate 8 positive counters equally into 4 groups with 2 positive counters in each group and then flip the counters over to the negative side.) page 34 of 86

35 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What equation would you record to represent this model? (8 ( 4) = ( 2)) What multiplication equation would you use to verify the quotient for 8 ( 4) = ( 2)? (( 4) ( 2) = 8) How would you model the expression 8 ( 2)? (I would separate 8 positive counters equally into 2 groups with 4 positive counters in each group and then flip the counters over to the negative side.) What multiplication equation would you use to verify the quotient for 8 ( 2) = ( 4)? (( 2) ( 4) = 8) How would you model the opposite of 8 negative counters in 4 groups? (I would separate 8 negative counters equally into 4 groups with 2 negative counters in each group and then flip the counters over to the positive side.) page 35 of 86

36 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher What equation would you record to represent this model? (( 8) ( 4)= 2) What multiplication equation would you use to verify the quotient for ( 8) ( 4) = 2? (( 4) 2 = ( 8)) How would you model the expression ( 8) ( 2)? (I would separate 8 negative counters equally into 2 groups with 4 negative counters in each group and then flip the counters over to the positive side.) What multiplication equation would you use to verify the quotient for ( 8) ( 2) = 4? (( 2) 4 = ( 8)) 16. Instruct student pairs to complete problems 6 10 on handout: Multiplication and Division of page 36 of 86

37 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Integers. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: Notes for Teacher What generalizations can be made about dividing integers? Answers may vary. The quotient of two positive integers is always positive; the quotient of two negative integers is always positive; the quotient of a positive integer and a negative integer is always negative; etc. Topics: ATTACHMENTS Developing the rules for multiplication and division of integers Explore/Explain 4 Students use patterns to discover the relationships and rules for multiplying and dividing integers. Instructional Procedures: 1. Place students in pairs and distribute handout: Multiplying Integers and Connecting Patterns to each student. Instruct student pairs to complete the handout. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion about the patterns for multiplying integers and the rules they have discovered. Ask: What patterns did you notice when multiplying the integers? Answers may vary. When both factors are positive, the product is positive. When both factors are negative, the product is positive. If multiplying a pair of factors and one of the factors is negative, the product is Teacher Resource: Multiplying Integers and Connecting Patterns KEY (1 per teacher) Handout: Multiplying Integers and Connecting Patterns (1 per student) Teacher Resource: Dividing Integers and Connecting Patterns KEY (1 per teacher) Handout: Dividing Integers and Connecting Patterns (1 per student) Handout: Note Page for Integer Operations (1 per student) page 37 of 86

38 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher negative; etc. 2. Distribute handout: Dividing Integers and Connecting Patterns to each student. Instruct student pairs to complete the handout. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion about the patterns for dividing integers and the rules they have discovered. Ask: What patterns did you notice when dividing the integers? Answers may vary. When the dividend and divisor are positive, the quotient is positive. When the dividend and divisor are negative, the quotient is positive. If the divisor or dividend is negative, the quotient is negative; etc. 3. Distribute handout: Note Page for Integer Operations to each student. Facilitate a class discussion summarizing the rules for integer operations. 5 Topics: Extending Integers Elaborate 2 Students apply integer rules to solve equations with more than 2 numbers and reallife problem situations. Instructional Procedures: Spiraling Review ATTACHMENTS Teacher Resource: Extending Integers KEY (1 per teacher) Handout: Extending Integers (1 per student) page 38 of 86

39 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures 1. Place students in pairs and distribute handout: Extending Integers to each student. Instruct student pairs to complete the handout. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Notes for Teacher Teacher Resource (optional): Integer Practice KEY (1 per teacher) Handout (optional): Integer Practice (1 per student) ADDITIONAL PRACTICE Handout (optional): Integer Practice may be used as additional practice, as needed. 6 Evaluate 1 Instructional Procedures: 1. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. Performance Indicator(s): page 39 of 86

40 Enhanced Instructional Transition Guide / Unit 02: Suggested Duration: 6 days Suggested Day Suggested Instructional Procedures Notes for Teacher Grade 07 Unit 02 PI 02 Construct a threecolumn chart to explain and demonstrate the connection between a pictorial model and a symbolic representation to the algorithm of all four operations using integers. Sample Performance Indicator: Brittany told a student who was absent from class, A positive number and a negative number will always be negative, regardless of the operation. Jordan told the absent student, A positive number and a negative number can sometimes be positive. Reed told the absent student A negative number and a negative number always equal a negative number. Create a three column chart of the pictorial model, symbolic representation, and connect them to the algorithm using all four operations involving integers. Use the chart to write a justification to determine if the statements made by Brittany, Jordan, and Reed are always true in all operations. Standard(s): 7.2C, 7.13A, 7.13B, 7.14A, 7.14B, 7.15A, 7.15B ELPS ELPS.c.1C, ELPS.c.3J 03/27/13 page 40 of 86

41 Integers and Football KEY The star running back for the home football team had these statistics for the last game of the season. Gained 12 yards = yards rushing Lost 6 yards = ( 6) 6 yards rushing Gained 8 = yards rushing Lost 4 = ( 4) 10 yards rushing Gained 3 = yards rushing Lost 10 = ( 10) 3 yards rushing 2012, TESCCC 05/24/12 page 1 of 1

42 Integers and Football The star running back for the home football team had these statistics for the last game of the season. Gained 12 yards Lost 6 yards Gained 8 Lost 4 Gained 3 Lost , TESCCC 05/24/12 page 1 of 1

43 Integers and Games KEY Devin started a board game with 0 points and lost 4 points. In the next play, he gained 6 points. Use models to show his new score. + = Gain 1 Point = Lose 1 Point = Zero Pair = 0 Points Loses 4 points Gains 6 Loses 4 points combined with gained 6 points ( 4) + 6 Combine 4 lost points with 4 gained points to create 4 zero pairs. 2 gained points left ( 4) + 6 = , TESCCC 04/05/13 page 1 of 1

44 Integers and Games Devin started a board game with 0 points and lost 4 points. In the next play, he gained 6 points. Use models to show his new score. 2012, TESCCC 04/05/13 page 1 of 1

45 Adding Integers TwoColor Counters KEY Use twocolor counters and an equation to model each expression. TwoColor Expression Counters 1. ( 4) + ( 3) Equation ( 4) + ( 3) = ( 7) = ( 5) ( 5) = ( 2) 4. ( 6) + ( 3) ( 6) + ( 3) = ( 9) 5. ( 4) ( 4) = 0 6. ( 5) ( 5) + 1 = ( 4) 7. ( 8) ( 8) + 10 = ( 8) 2 + ( 8) = ( 6) 9. ( 7) ( 7) + 2 = ( 5) 10. ( 2) + ( 2) ( 2) + ( 2) = ( 4) 2012, TESCCC 05/24/12 page 1 of 1

46 Adding Integers TwoColor Counters Use twocolor counters and an equation to model each expression. TwoColor Expression Counters 1. Equation Sentence (4) + (3) (5) 4. (6) + (3) 5. (4) (5) (8) (8) 9. (7) (2) + (2) 2012, TESCCC 05/24/12 page 1 of 1

47 Subtracting Integers Two Color Counters KEY Use two color counters and an equation to model each expression. Expression Two Color Counters Equation ( 6) ( 4) = ( 2) ( 6) ( 4) Related to the addition equation: ( 6) + 4 = ( 2) ( 7) ( 4) = ( 3) ( 7) ( 4) Related to the addition equation: ( 7) + 4 = ( 3) 7 ( 4) = 11 7 ( 4) Related to the addition equation: = 11 ( 8) 2 ( 8) 2 = ( 10) Related to the addition equation: ( 8) + ( 2) = ( 10) 2012, TESCCC 05/24/12 page 1 of 1

48 Subtracting Integers Two Color Counters Use two color counters and an equation to model each expression. Expression Two Color Counters Equation ( 6) ( 4) ( 7) ( 4) 7 ( 4) ( 8) , TESCCC 05/24/12 page 1 of 1

49 Integers and the Sea Otter KEY A sea otter is on the surface of the water. He dives down 3 meters with one kick of his legs. After another kick he goes down another 2 meters. Use a model to show how far below the surface of the water the sea otter dove. TwoColor Counters ( 5) Equation ( 3) + ( 2) = ( 5) Number Line 1 5 meters below the surface 3 Down 3 meters water surface 0 2 Down 2 meters 1 ( 3) 2 3 ( 2) , TESCCC 05/24/12 page 1 of 1