Students will be able to discover the rules for adding and subtracting integers.
NCTM Standards. Number and Operations. Problem Solving. Communication.
Adding and Subtracting Integers Unit
Grade 7 Math 5 Days Tools: Algebra Tiles Four-Pan Algebra Balance Playing Cards
By Dawn Meginley
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Objectives and Standards Objectives: • Students will be able to add positive integers together. • Students will be able to add negative integers together. • Students will be able to define the additive inverse and apply it in adding integers together and finding zero pairs. • Students will be able to define zero principle and apply it to solve integer addition problems. • Students will be able to define the commutative property of addition and use it to solve problems. • Students will be able to add positive integers to negative integers • Students will be able to subtract positive integers from positive integers. • Students will be able to subtract negative integers from negative integers. • Students will be able to subtract positive integers from negative integers. • Students will be able to subtract negative integers from positive integers. • Students will be able to compare integers. • Students will be able to discover the rules for adding and subtracting integers. NCTM Standards Number and Operations Problem Solving Communication Connections Representations New York State Standards, KIs, and PIs Standard 3 Mathematics--Students will be able to understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry. KI 1—Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument. PI 1C—Make conclusions based on inductive reasoning KI 2—Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas. PI 2A—Understand, represent, and use numbers in a variety of equivalent forms PI 2D—Recognize order relations for integers KI 3—Students use mathematical operations and relationships among them to understand mathematics. PI 3A—Add and subtract integers
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PI 3E—Demonstrate an understanding of operational algorithms PI 3F—Develop an appropriate proficiency with facts and algorithms KI4—Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships. PI 4F—Use concrete materials and diagrams to describe the operation of real-world processes and systems
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Resources http://www.aea1.k12.ia.us/schoolimprove/math.pdf, Math, Key Stone Area Education Agency 1, no author found. http://www.edhelper.com/, edHelper.com, no author found. Lester, Diana. Exploring Integer Addition, 2003 summer workshop on algebra tiles. Lester, Diana. Exploring Integer Subtraction, 2003 summer workshop on algebra tiles. Nelson, Gary T. Four-Pan Algebra Balance, Cuisenaire Company of America Inc, 2001. Scott Foresman Addison Wesley, Transition Mathematics, Feldman, Cathy; Usiskin, Zalman; Davis, Suzanne; Mallo, Sharon; Sanders, Glaydys; Witonsky, David; Flanders, James; Polonsky, Lydia; Porter, Susan; Viktora, Steven, 1998 chapter 5.
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Materials Algebra Tiles—class set Algebra Tiles work mat—class set Four-Pan Algebra Balance—enough for groups of four Four-Pan Algebra Balance Chips and canisters Index cards with , and = written on them, enough for each group to have a set Playing Cards—enough for pairs of students Adding and Integers using Algebra Tiles—worksheet Subtracting Integers using Algebra Tiles—worksheet Chart Paper and markers Red and Green Crayons or markers
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Unit Overview Day 1-Students will use the Four-Pan Algebra Balance to compare integers in groups of four students. Day 2-Students will use the four-Pan Algebra Balance to add integers, in groups of four. Students will begin to look at patterns to find rules for adding integers Day 3- Students will use algebra tiles to add integers. They will begin to discover the rules for adding integers. Day 4-Students will use algebra tiles to subtract integers. They will discover the rules for subtracting integers. They will make posters of the rules for adding and subtracting integers and share them with the class. Day 5-Students will play addition math war to practice the rules they discovered for adding and subtracting integers.
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Lesson: Day One-Comparing integers Time: 45 minutes Objectives: • The students will be able to compare integers. Materials: • Four-Pan Algebra Balance • Index cards with , and = written on them • Chips for the Four-Pan Algebra Balance Activities: 1. On the board will be the following problem of the Day, for the students to complete as they sit down. ß Using < , > , or = to compare the following numbers • 10 2 • 3 6 • 0 4 • 5 4 • 6 9 ß We will then go over the problems reviewing < , > , and =. 2. I will model to the class how to use the Four-Pan Algebra Balance to compare integers. ß I will write 3 = 3 on the chalkboard. Then I will place 3 chips in the yellow pan the left side then 3 chips in the yellow pan the right side. I will ask the class how the balance shows us that 3 really equals 3. Then I will hold up the equal index card. ß Next I will write -4 = -4 on the chalkboard. I will place 4 chips in each red pan the balance and I will hold up the equal card again. I will ask the class what the different colored cups represent. ß Then I will write 2 < 4 on the chalkboard and place the corresponding chips in the pans. We will note that the balance is pointing towards the smaller number. ß Have each group of students use the Four-Pan Algebra Balance to make a model of the following equations, while noticing which way the pointer is pointing: • 7>3 • 4 -3 • -1 > -4 • -6 < -1 3. In groups have the students find ten different number sentences involving inequalities. ß Group member 1: Place chips in the left side of the Four-Pan Algebra, in one tray at a time. ß Group member 2: Place chips in the right side of the Four-Pan Algebra, in one tray at a time. ß Group member 3: Hold up the corresponding index card with , or = written on it. ß Group member 4: Record the number sentences created. ß Repeat 10 times. For 10 different number sentences. Closure: • Discuss as a class, how the Four Pan Algebra Balance helps us determine which number is greater or smaller when given 2 numbers. Also, discuss how we can tell which number is greater or smaller without using the balance. Homework: • Unit one practice from pg 49 in The Four-Pan Algebra Balance Student Handouts: • Unit one practice from pg 49 in The Four-Pan Algebra Balance
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Lesson: Day Two- Using the Four-Pan Algebra Balance to add integers. Time: 45 minutes Objectives: • Students will be able to add positive integers together. • Students will be able to add negative integers together. • Students will be able to add positive integers to negative integers • Students will be able to define the additive inverse and apply it in adding integers together and finding zero pairs. • Students will be able to define zero principle and apply it to solve integer addition problems. • Students will be able to define the commutative property of addition and use it to solve problems. Materials: • Four-Pan Algebra Balance • Four-Pan Algebra Balance Chips and canisters Activities: 1. On the board will be the following problem of the Day, for the students to complete as they sit down. ß The bills gained 5 yards in the first down, and lost 3 yards in the second down. How many yards, altogether, did the Bills have going into the third down? ß We will go over the Problem of the Day by drawing a picture. 2. In notes we will define the following terms, commutative property of addition, zero pairs, additive inverse, and the zero principle. ß I will tell the class that -3 + 5 = 5 + -3, is an example of the commutative property of addition. Changing the order of addition does not change the value (a + b = b + a). ß I will also explain about Zero pairs. Zero pairs represent additive inverses, a number and its opposite that equal zero. For example +1 and -1 are opposites that equal zero when put together. (a + - a = 0). ß Then I will explain the Zero principle, when adding or subtracting zero to an integer its value does not change (a + 0 = a). 3. I will model adding integers together, using the balance. ß I will write 5 + -3 on the board. Then I would ask the class what the colored pans mean, to remind them of what they learned last class. I would model the sentence by placing 5 chips in the yellow pan and 3 chips in the red pan, all on the left side of the balance. Then I would point out that the balance is off centered, and ask the class what I could do to balance it. I am looking for a student to say place 5 chips in the opposite yellow pan and 3 chips in the opposite red pan. I will then write -3 + 5 = 5 + -3. I will remind
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the class that these pairs of numbers are equal by the commutative property of addition. Next I will remove one chip from the yellow tray and one chip from the red tray. Again reminding the class that these two chips together make a zero pair and removing zero pairs will not change the value of our sentence. I will continue removing zero pairs until one of the trays is empty. Last I would write what I see on the board -3 + 5 = 2, because there will be 2 chips left in the yellow pan on the right side of the balance. ß I will write 3 + 1 on the board and model in the same fashion as above. This time I would not have to remove zero pairs because there are not any chips in both pans to make zero pairs. I would just have 4 chips in the yellow pan making the sentence to be 3+1=4 ß I will write -2 + -4 on the board and model in the same fashion as above. This time I would not have to remove any zero pairs because there are not any chips in both pans to make a zero pair. I would just have 6 chips in the red pan, making the sentence to be -2 + -4 = -6. 4. In small groups have students complete Unit 3 Practice from pg 50 in The Four-Pan Algebra Balance. Closure: • In journals have students answer the question on the bottom of page 50, "Look for patterns in your practice problems. Use them to write rules for adding integers without the help of the balance." Homework: • Write a paragraph on why you like the algebra-pan balance for adding integers or why you disliked it. Please write about any confusion that you may have had using the balance or why this helped you learn. Student Handouts: • None
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Lesson: Day Three-Adding integers using algebra tiles Time: 45 minutes Objectives: • Students will be able to add positive integers together. • Students will be able to add negative integers together. • Students will be able to define the additive inverse and apply it in adding integers together and finding zero pairs. • Students will be able to define zero principle and apply it to solve integer addition problems. • Students will be able to define the commutative property of addition and use it to solve problems. • Students will be able to add positive integers to negative integers Materials: • Class set of Algebra tiles • Class set of Algebra tile mats (I use a piece of construction paper laid horizontally with one line down the middle) • Overhead algebra tiles • Adding integers using algebra tiles—worksheet • Red and green colored markers, crayons, ect. • Chart paper and markers. Activities: 1. On the board will be the following problem of the Day, for the students to complete as they sit down (5 min). ß An underwater diver is at 8 feet below sea level and he descends 5 more feet. Where is the diver located now? ß We will go over the Problem of the Day by drawing a picture or using the rules on chart paper, from last class. 2. Pass out algebra tiles and mats to each student. Explain the red ones are negative integers and the other colored ones are positive integers, (my positive integers are green). Explain to the class that today we are only using the integer tile, the small square of both colors. 3. Ask the class how they would make a zero pair using algebra tiles (answer: one green tile and one red tile of the same size.) 4. Pass out Handout tiled Adding integers using algebra tiles (see attached). 5. I will model how to do question #1 from worksheet, using overhead algebra tiles. ß Place 3 green integers on the left side of the mat (keep them in a group of 3), then add 4 more green integers on the left side of the mat (keep them in a group of four). Now on the right side of the mat add 3 green integers and 4 green integers, but this time I will push my integers together. Now count how many integers we have on the right side of the mat, altogether. Therefore 3 + 4 = 7. Ask
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the class what part of their mat represents the equal sign (the dividing line on the mat represents the equal sign). Make sure you also draw this on the overhead, so students see what they should be drawing on their worksheet. 6. Have the students answer questions 2-4 on their worksheet, using the algebra tiles. Make sure that students are also drawing what they are doing using red and green crayons. (See answer key for what student should be drawing). 7. Have the students create a rule for adding two positive integers together. Have the students pair share their results, than record them on chart paper, with the person that they pair-shared with. (See answer key to see what the rules are). 8. I will model how to do question #5 from the worksheet, using over head algebra tiles. ß I will place 2 red integers on the left side of my mat (grouped) and then add 5 more red integers to that side (grouped). I will now do the same for the left side of my mat, but this time I will push all the integers together. Now count how many integers we have on the right side of the mat., we have 7 red ones. Therefore -2 + -5 = -7 9. Have the students answer questions 6-8 on their worksheet, using the algebra tiles. Make sure that students are also drawing what they are doing using red and green crayons. 10. Have the students write a rule for adding two negative integers together. Have the students pair share their results, then with their partner record them on chart paper. 11. I will model how to do question #9 from the worksheet, using over head algebra tiles. I will place 2 red integers on the left side of the mat and I will add 4 green integers to the left side of the mat. On the right side, I will do the same, but I will push the integers together. This time we can not count the integers because they are not alike, some are red and some are green. In order to get them all the same I will need to find and remove zero pairs from the right side. I will continue to do this until no more zero pairs exist (Remember zero pairs are one green integer and one red integer). I will have 2 green integers remaining on the right making -2 + 4 = 2. 12. Have the students answer questions 10-12 on their worksheet, using the algebra tiles. Make sure that students are also drawing what they are doing using red and green crayons. 13. Have the students write a rule for adding a negative integer and a positive integer together. Have the students pair share their results, then record their rules on chart paper. Closure:
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We will discuss the importance of using models to learn new things. We will discuss the importance of learning the rules from working with the models. (In real life and on tests we can not use balances of Algebra Tiles to add integers).
Homework: • Have students write a paragraph on which model they like better, the Four-Pan Algebra balance or the Algebra Tiles. Which made adding integers clearer? Which helped them see the rules easier? Or were they both the same and why. Or did neither help them see the rules and why. Student Handouts: • Adding and Integers using Algebra Tiles—worksheet
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Name _______________________________
Class _____________
Adding Integers Using Algebra Tiles Addition of Integers Use the algebra tiles to find each sum. Make a picture of your work. 1. 3+4= 2. 1+3=
3.
2+7=
4.
5+2=
Write a rule for adding two positive integers together: ________________________________________________________________________ ________________________________________________________________________ 5.
-2 + -5 =
6.
-3 + -1 =
7.
-4 + -2 =
8.
-6 + -1 =
Write a rule for adding two negative integers together: ________________________________________________________________________ ________________________________________________________________________ 9.
-2 + 4 =
10.
-6 + 1 =
11.
3 + (-4) =
12.
5 + (-2) =
Write a rule for adding a positive and a negative integer: ________________________________________________________________________ ________________________________________________________________________
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Name _________KEY______________________
Class _____________
Adding Integers Using Algebra Tiles Addition of Integers Use the algebra tiles to find each sum. Make a picture of your work. 1. 3+4=7 2. 1+3=4
3.
2 + 7 =9
4.
5+2=7
Write a rule for adding two positive integers together: _Add both numbers together and keep the positive sign_______________________ 5.
-2 + -5 = -7
6.
-3 + -1 = -4
7.
-4 + -2 = -6
8.
-6 + -1 = -7
Write a rule for adding two negative integers together: Add both numbers together and keep the negative sign ___________________________________________________________________ 9.
-2 + 4 = 2
10.
-6 + 1 = -5
11.
3 + (-4) = -1
12.
5 + (-2) = 3
Write a rule for adding a positive and a negative integer: _Subtract the smaller number from the larger number and keep the sign of the larger number._____________________________________________________________
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Lesson: Day Four-Using Algebra Tiles to subtract integers and students sharing rules that they have discovered on adding and subtracting integers Time: 45 minutes Objectives: • Students will be able to subtract positive integers from positive integers. • Students will be able to subtract negative integers from negative integers. • Students will be able to subtract positive integers from negative integers. • Students will be able to subtract negative integers from positive integers. • Students will be able to discover the rules for adding and subtracting integers and use them in real life context. Materials: • Class set of Algebra tiles • Overhead algebra tiles • Chart paper and markers—from last class to continue rules • Red and green crayons or markers • Subtracting Integers using Algebra Tiles--Worksheet Activities: 1. On the board will be the following problem of the Day, for the students to complete as they sit down (5 min). ß The temperature at 8pm was -6 degrees by 11pm the temperature had dropped 12 degrees. What was the temperature at 11pm? ß Go over the different strategies your class tried to solve this problem. 2. Pass out the algebra tiles the Exploring Integer Multiplication Worksheet. 3. I will model how to do question #1, for subtracting integers. ß We do not use the mat since we need to take away integers. I will place six red integers in my workspace, than I will take away 2 red integers leaving me with four red integers. Therefore -6 – (-2) = 4. 4. I will model question #2. ß I will place 2 green integers on my workspace. I need to remove 5 green integers. I do not have 5 green integers. I will ask the class if I can do anything to get more green integers. That's right I can add zero pairs to get green integers, because adding zero to a number will not change its value. I will add one green and one red integer to my workspace. I will have to add two more sets of zero pairs to have enough green integers to remove. Now I can take away 5 green integers. What is left on my workspace is 3 red integers. Therefore 2 – 5 = -3. 5. Have the Students answer questions 3-6 using their algebra tiles.
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6. Have the students write a rule for subtracting integers. Have the students pair share their results, then write the rule they created on their chart paper from last class. 7. In pairs, have the students present the rules that the have created with the class. They should have their rules already written on chart paper. Closure: • We will hang all of the group's rules up on the wall and see if there are any similar rules between the groups. We will pick out those rules and create a new chart paper with a class set of rules on it, to display in the room. Homework: • Worksheet titled, Integers, from Edhelper.com Student Handouts: • Worksheet titled, Integers, from Edhelper.com • Subtracting integers using algebra tiles--worksheet
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Name _______________________________
Class _____________
Subtracting Integers Using Algebra Tiles Use your algebra tiles to find each difference. Make a picture of your work. 1.
-6 – (-2) =
2.
2-5=
3.
4 - (-3) =
4.
-6 – (-4) =
5.
6 – (-4) =
6.
-8 – 5 =
Write a rule for subtracting integers: ________________________________________________________________________ ________________________________________________________________________
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Name ___________KEY____________________
Class _____________
Subtracting Integers Using Algebra Tiles Use your algebra tiles to find each difference. Make a picture of your work. 1.
-6 – (-2) = -4
2.
2 - 5 = -3
3.
4 - (-3) = 7
4.
-6 – (-4) = -2
5.
6 – (-4) =10
6.
-8 – 5 = -13
Write a rule for subtracting integers: If you have two negatives in a row, without any numbers between them, the two negative signs become a positive sign. Then you follow the rules from before.
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Name _____________________________
Date ___________________
Integers Complete. 1.
-
2.
45 + -7
3.
-
5 + -3
4.
-
5.
26 - -33
6.
30 + 21
7.
25 + 3
8.
-
9.
-
47 + 2
10.
28 - 29
11.
40 - -4
12.
16 + 32
13.
-
4 - 39
14.
-
15.
8 + -14
16.
-
17.
23 - -48
18.
-
19.
-
7 + -41
20.
-
21.
2 - 39
22.
4 + 19
23.
-
3 + -45
24.
-
25.
33 - 38
26.
16 - -34
27.
41 + 44
28.
-
29.
9 - 42
30.
38 - 33
31.
34 + 20
32.
35 + 21
33.
-
44 - -33
34.
19 + 46
35.
48 + 8
36.
42 - 14
37.
49 + -35
38.
-
39.
-
40.
-
20
16 - -37
41 - 17
34 - -39
11 - 38
11 + -8 1 - 7 41 - -48 6 + 18
5 - -43
7 + 46
23 - -6 5 + 46
Name _____Key________________________
Date ___________________
Integers Complete. 1.
-
2.
45 + -7= 38
3.
-
5 + -3 = -8
4.
-
34 - -39 = -73
5.
26 = -7
6.
30 + 21 = 51
7.
25 + 3 = 28
9. 11.
16 - -37 = -53
-
33
8.
-
11 - 38 = 49
-
47 + 2 = - 45
10.
28 - 29 = -1
40 - -4 = 44
12.
16 + 32 = 48
-
13.
4 - 39 = -43
14.
15.
8 + -14 =-6
16.
17.
23 - -48 = 71
18.
-
-
11 + -8 = -19 -
1 - 7 = -8 -
41 - -48 =7
19.
7 + -41 =- 48
20.
6 + 18 = 12
21.
2 - 39 = -37
22.
4 + 19 = 23
23.
3 + -45 = -48
24.
5 - -43 = 38
25.
33 - 38 = -5
26.
16 - -34 = 50
27.
41 + 44 = 85
28.
7 + 46 = 39
29.
9 - 42 = -33
30.
38 - 33 =5
21
-
-
-
-
31.
34 + 20 = 64
35 + 21 = 56
33.
44 - -33 = -11
34.
19 + 46 = 65
35.
48 + 8 = 56
36.
42 - 14 = 28
37.
49 + -35 = 14
38.
23 - -6 = -17
40.
-
39.
22
-
32.
-
41 - 17 = -58
-
5 + 46 =41
Lesson: Students will practice adding integers by playing addition math war Time: 45 minutes Objectives: • The students will be able to add integers using the rules that they have created. Materials: • Deck of playing cards, for each pair of students Activities: 1. On the board will be the following problem of the Day, for the students to complete as they sit down (5 min). • Sally was on floor two in an elevator. She needs to go done 6 more floors to get off. What floor did she get off at? • We will go over the Problem of the Day. 2. We will review the rules that we created for adding and subtracting integers. 3. We will play Addition Math War. • Each pair of students will be given a deck of playing cards. Each student in the pair will be dealt half of the cards. The red cards are negative integers. The black cards are positive integers. The A = 1, the number cards are equal to their number, J = 11, Q = 12, and K = 13. To play each player turns over their top two cards and adds them together. Whoever has the larger total points wins all the cards. If the total is equal than there is a war and another card is flipped over and added to their previous total until someone has a greater value. Whoever has the larger total wins all the cards. The game is played until someone wins all the cards. Closure: For a better understanding of adding and subtracting integers we will discuss the similarities between positive numbers and addition signs and negative numbers and subtraction. I will make sure that every student understands that - - = + . Homework: None Student Handouts: None
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