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Addition and Subtraction of Integers

Objectives:

· Students will be able to add integers.

· Students will be able to subtract integers.

· Students will be assessed on their ability to add and subtract integers.

Download as Microsoft Word Document:

Instructional Plan

Instructional Plan Template

Mathematics Governor’s Institute 2005

Names of group members: Rich Blessing, Nick Gildow, Joan Horwatt, Mary Oles, Kate Reardon, Dani Retallack

Topic/Theme: Addition and Subtraction of Integers

Level: Pre-Algebra

Time Element: 150 minutes (Broken into 3 – 50 minute classes)

NCTM Standards Addressed:

Develop meaning for integers and represent and compare quantities with them; understand the meaning and effects of arithmetic operations with fractions, decimals and integers

PA Math Standards Addressed: 2.2.8 B; 2.7.8 B

Math Assessment Anchors Addressed: M8.A 3.3

Reading Assessment Anchors Addressed: 1.1.8B; 1.1.8 F

Objectives:

• Students will be able to add integers.
• Students will be able to subtract integers.
• Students will be assessed on their ability to add and subtract integers.

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

Day 1 – 50 minutes

Pre-instructional Phase:

Have students refer to a number line (either a classroom, personal, or on the overhead).

· If possible, have a volunteer demonstrate in front of the class while peers work on number lines at their seats.

• Have students choose any positive number
• Ask students to show adding one to that number on the number line. Then, show subtracting one.
• Students will notice that they end up back on their original number. Thus, the one that was added and the one that was subtracted yield a net of 0.
• Use this as a teachable moment to introduce the chips and zero pairs.

• Explain that each black chip will represent adding one and each red chip will represent subtracting one.
• Black chips and red chips cancel each other out because + 1 – 1 = 0.

• State objective.

Instructional Phase:

Model:

• Spread a small handful of randomly mixed chips out on the overhead.
• Separate the chips into two groups: black and red.
• Begin taking away pairs (1 black and 1 red at a time)
• Once all pairs have been taken away, you should be left with either no chips or chips of all the same color. (Depending on the prior knowledge of your students, you may want to vary your terminology – plus/minus, positive/negative)

• Ex. 1:
• Chips drawn: + + + - -
• Chips left after all pairs have been taken away: +
• There is one plus remaining so my sum is 1.
• Ex. 2:
• Chips drawn: + + + - - - - - -
• Chips left after all pairs have been taken away: - - -
• There are three minuses remaining so my sum is -3.
• This may require further explanation: If we begin at zero (since there are no positive numbers), and then subtract three on the number line = -3.
• Ex. 3:
• Chips drawn: + + - -
• Chips left after all pairs have been taken away:
• There are no chips left so my sum is 0.
• Explain expectations for chip use, then pass out chips to the students (each partner set or team will need at least 20 of each color).
• Depending on the needs of the students you may want to allow them a minute to play/explore with the manipulatives so that they are less tempted to use them inappropriately later.
• With their partners/team, have students take turns adding a small handful of chips.

Model:

• Again on the overhead, spread out a small handful of chips and separate them into colored groups.
• This time, use your hand or a sheet of paper to cover up the reds.
• Ask students the value of the black chips and write it on the overhead.
• Ex: + + + + + = 5
• Now, uncover the reds and cover the blacks.
• Ask students the value of the red chips.
• Ex: - - - - = -4
• Write this value next to the 5.
• Ex: 5 -4
• Ask students what operation they were using before to calculate the answer: addition
• Ex: 5 + -4 =
• Another teachable moment – Does this look like anything we’ve ever seen? Do we put two operations right next to each other? No.
• Ex: 5 + (-4) =
• On the number line, show beginning at +5 and then moving negatively 4 times = 1.
• Some students will notice right away that adding a negative is just like subtracting the second number. Do not allow these students to ruin it for everyone else. Instead of acknowledging that adding a negative is subtracting, challenge the student’s thinking by asking something like “Are you sure that you have enough data to make such a statement?” or “That is a great observation, but are you sure that it would work for all numbers?” This will enrich the students learning and help him/her understand that it’s important to test theories before jumping to overgeneralizations. It will also allow the students who haven’t noticed the relationship some more time for discovery.

Pass out worksheet (see attached document) and model (see directions):

• Allow students time to practice transitioning from chips to integers using the worksheet.
• Hint: It is much faster to allow students to draw +’s and –’s rather than black and red chips. However, color-coding may accommodate some students’ needs.
• Circulate throughout the room, stopping at each team to take a formative assessment of their progress. Try not to get stuck at any one group for too long. Keep moving! Stop as needed to address the class.

Post-instructional Phase:

Pass out Rock, Paper, Scissors worksheet (see attached file) and model (see directions):

• Allow students to play the game.
• Circulate throughout the room, stopping at each team to take a formative assessment of their progress. Try not to get stuck at any one group for too long. Keep moving! Stop as needed to address the class.
• Encourage students to move away from using the chips. As students feel comfortable, they can put their chips away and record their wins and losses with integers.

Debrief the game: observations, strategies, questions about integer addition, etc.

If you would like a closer look at the progress the students made, collect their game sheets to look over.

Restate the objective and summarize the lesson.

If you feel that the students have caught on to integer addition, administer the mini-integer addition summative assessment (see attached file and directions).

Day 2 – 50 minutes

Pre-instructional Phase:

Pass out graphic organizer (see attached file):

• Introduce the term integers.
• Help the students practice reading integers. No, it is not ‘integgers’! J
• Begin in the examples box.
• Students list examples of integers they saw yesterday.
• Characteristics and Facts:
• List observations from yesterday here.
• Non-examples box:
• What numbers do not meet the characteristics?: fractions, decimals, etc.
• Definition: The set of natural numbers, their opposites, and zero.

Review chips

• Black chip = +1
• Red chip = -1
• Black and red pair = +1 + (-1) = 0

State objective.

Instructional Phase:

Model:

• Ask a student for two integers between -10 and +10 on the number line.
• Ex: -4 – 3
• Put out 4 red chips: - - - -
• Explain that we are trying to take away 3 plusses but there are no positives to take away.
• Add 3 positives: - - - - + + +
• Ask if the value has changed? Yes, now -1 instead of -4 so that can’t be right.
• Have to add zeros instead of positives: - - - - - + + + - - -
• Now, the value has not changed = -4
• Take away 3 positives: - - - - - - -
• Difference = -7
• Ex: 5 – (-1)
• + + + + +
• Explain that we are trying to take away 1 negative but there are no negatives to take away.
• Add zero pairs: + + + + + - +
• Take away 1 negative: + + + + + +
• Difference = 6
• Again, some students will pick up on the subtracting a negative relationship right away. Encourage this discovery by all students before announcing ‘rules’.
• Model several examples. Scaffold several examples.

Post-instructional Phase:

Pass out Integer Wars worksheet (see attached file) and model (see directions):

• Allow students to play the game.
• Circulate throughout the room, stopping at each team to take a formative assessment of their progress. Try not to get stuck at any one group for too long. Keep moving! Stop as needed to address the class.
• Encourage students to move away from using the chips. As students feel comfortable, they can put their chips away and work with the integers.

Debrief the game: observations, questions about integer subtraction, etc.

If you would like a closer look at the progress the students made, collect their game sheets to look over.

Restate objective and summarize the lesson.

If you feel that the students have caught on to integer subtraction, administer the mini-integer subtraction summative assessment (see attached file and directions).

Day 3 – 50 minutes

Pre-instructional Phase:

Review graphic organizer.

Review adding and subtracting – answer questions.

Instructional Phase:

Play extended version of Integer Wars (see attached sheet).

Post-instructional Phase:

Give summative assessment.

As time allows, play extension of Integer Wars.

Materials/Resources:

• 2 color chips
• Calculators, which will generate random numbers (alternative 2 different colored dice)
• Overhead
• Handouts:
• Recording Sheet; Rules for Rock, Paper, Scissors; Integer War rules; Frayer worksheet 9(graphic organizer)
• Playing cards or number cards

Interdisciplinary Connections:

· Reading - Graphic organizer, application problems

· Technology: Calculators - generate random numbers

Assessment Strategies:

· Formative Evaluation (checking student understanding during the lesson):

o Students’ oral responses and questions during the lesson presentation.

o Teacher will monitor group work

o Quiz on addition of integers (attached)

o Quiz on subtraction of integers (attached)

· Summative Evaluation (How will it be determined that the objectives were achieved?):

o Test on addition and subtraction of integers (attached)

Correctives/Remediation:

• Walk This Way game
• Number line on desk
• Additional practice with manipulatives

Extensions/Enrichment:

• Play integer war giving students the option of choosing operation, which will give them the greatest number.
• It’s All in the Cards game

Special Accommodations (special needs students)

• Description of the Special Needs student selected:

o This student is truly dyslexic. He has great difficulty with reading and writing material. He is unable to put sounds together to create words when he reads. When he writes, he is unable to hear the sounds in order to transcribe them into words. He is extremely clumsy and has real coordination problems. His mathematic skills are excellent.

• Accommodations to use with this student:

• Use a tape recorder to record lessons.
• Have a copy of the class notes given to the student prior to teaching the lesson.
• Have a peer read questions/ directions.
• Cut down on written assignments.
• Have him repeat directions orally.
• Enlarge or color code print
• Give test orally or send to learning support for testing
• Involve student in group work.
• Chunk the work into smaller segments.
• Use larger sized graph paper to do assignments.
• Make more space between items.
• Give preferential seating to alleviate clumsiness.
• Make greater use of graphic organizers.
• Give him more time.
• Use a white board to write class assignments.

Name _______________________________

Date _______________________________

Directions: Record what you draw out of the cup in the box and write an equation to match.

_____+_____= ____

_____ + _____ = ____

_{_______} + ______ = _______

______ + ______ = ______

______ + ______ = ______

Integer Rock, Paper, Scissors

In this activity, the students will play a special version of the game Rock, Paper Scissors and will record answers using positive or negative symbols and their corresponding integer. The game is played following the regular rules, where rock smashes scissors, paper covers rock, and scissors cuts paper. Ties are not used as part of this variation of the game and do not get recorded in any way. This variation requires the winner to record positive (+) sign(s) in his/her column and the loser of the round to record negative (–) sign(s) in his/her column. One symbol is recorded when rock smashes scissors, two symbols are recorded when paper covers rock, and three symbols are recorded when scissors cut paper as indicated in the box below.

Symbol	Win	Loss
Rock	+	-
Paper	++	- -
Scissors	+++	- - -

For example, player 1 has displayed the symbol for paper, and player two has displayed the symbol for rock. Since paper covers rock, player one is the winner and records two positive (+) symbols in his/her column and player two is the loser and must record one negative (-) symbol in his/her column as shown below. On the second turn, player one displays paper and player two displays scissors. Since scissors beats paper, player 2 is the winner and records three positive (+) symbols and player one is the loser and records two negative (-) symbols for losing with paper. The players continue to do this five times and then record the integer representing their positive or negative signs in the integer column.

Round	Player 1	Integer	Player 2	Integer
1	++	+2	-	-1
2	- -	-2	+ + +	+3
3
4
5
Total	-----------------		-----------------

When five sets have been completed, each player finds the sum of his/her column. The player with the greatest sum (or least if this is determined at the beginning) is the winner.

Integer Rock, Paper, Scissors

Symbol	Win	Loss
Rock	+	-
Paper	++	- -
Scissors	+++	- - -

Round 1-

Round	Player 1	Integer	Player 2	Integer
1
2
3
4
5
Total	----------		-----------

Round 2-

Round	Player 1	Integer	Player 2	Integer
1
2
3
4
5
Total	---------		-----------

Integer War: Subtraction

In this game you will use your newly learned skills to play a game of War using a random number generator on your calculator.

Materials: TI-83/84 Calculator (for each player)

Players: 2-4

Procedure:

Set-up:

1. Turn on your TI-83/84 Calculator
2. Press MATH and select PRB (on the right). Select randomInt(
3. Type in the following on the home screen. randomInt( -10, 10, 2)

Playing the game (each player should follow these steps):

1. Complete the steps above on your calculator.
2. Press ENTER once, after randomInt( -10, 10, 2) is typed in. The calculator should output two numbers between –10 and 10. Example: {-3, 5}
3. Write each number in the appropriate place in the Score Card provided.
4. Complete the subtraction problem that you have created – without your calculator!
5. Compare the answer from your subtraction problem (the difference) to the differences of your opponents’ numbers.
6. The player that has that greatest difference receives a point. (note: 2 is greater than –5)
7. To play another round each player should just press ENTER once more to get two new numbers.
8. The first player to receive 5 points wins the game.

Extensions:

Once students have mastered subtraction they can play variations of this game:

1. The set up stays the same, the only difference is that students may add or subtract the two output numbers (in the order in which they are given) to come up with the largest result.
2. For even more of a challenge, allow students to add or subtract the two output numbers in any order to come up with the largest result..

Score Card for Integer War

Round Number	Calculator Output	Subtraction Problem	Difference	Point Received?
Example	{-6, 9}		-15
1
2
3
4
5
6
7
8
9
10
11
12

Total Points:_______

Score Card for Integer War-Extension

Round Number	Calculator Output	Addition or Subtraction Problem	Difference	Point Received?
Example 1	{-6, 9}		15
Example 2	{3, 8}		11
2
3
4
5
6
7
8
9
10
11
12

Total Points:_______

It’s all in the Cards

During this activity you will have the opportunity to practice adding and subtracting integers.

Materials : 1 deck of playing cards (remove the face cards from the deck; keep Aces to be used as ones)

1 number cube. The number cube will indicate the operation to be performed. If an EVEN number is rolled, the operation is addition; if an ODD number is rolled, the operation is subtraction.

Players: 2

Procedures:

1. Shuffle the deck of cards and place between the 2 players.

2. Decide who is rolling the number cube.

3. Two cards are drawn from the top of the deck with the number side showing.

4. The number cube is rolled.

5. The first student to give the correct answer takes both cards.

6. The student with the most cards at the end of the activity is the winner.

Adaptation:

If necessary, there could be a third student with a calculator to verify answers.

Walk This Way

During this activity you will have the opportunity to practice

adding and/or subtracting integers

Materials: Walk on number line.

The number line could be made from laminating oak tag if it is to be used

multiple times, or from old computer paper for 1 time use. Write the numbers in the center of each card or sheet with a large black number line. Since this is a beginning activity, you would probably want to limit the numbers from -10 to +10.

Procedure: Student begins at zero. The teacher (or another student) gives them a simple equation involving addition or subtraction of integers. The student walks to the first number and then continues in the same direction or turns around and walks in the other direction in order to solve the problem.

Example: -5 + 7 =

The student walks to negative 5. The student then turns around and walks 7 places to the right (+7) landing on +2.

Addition of Integers Quiz

NAME_________________________________

ADD

1. 12 + (-4) = __________

2. 5 + (-9) = __________

3. –3 + (-6) = __________

4. –5 + 5 = __________

5. 8 + 2 = __________

6. –7 + 10 = __________

7. 1 + (-3) = __________

8. –9 + (-2) = __________

9. –4 + 7 = __________

10. 9 + (-5) + (-7) = __________

Subtraction of Integers Quiz

NAME___________________________________

SUBTRACT

1. -9 – 3 = __________

2. -4 – 6 = __________

3. 8 – (-2) = __________

4. -5 – (-7) = __________

5. 6 – (-6) = __________

6. -10 – (-3) = __________

7. -8 – 8 = __________

8. 12 – 7 = __________

9. 4 – (-4) = __________

10. - 7 – 10 = __________

Test on Addition and Subtraction of Integers

NAME_________________________________________

ADD OR SUBRACT AS INDICATED

1. 9 – (-4) = __________

2. -7 + 3 = __________

3. -6 + (-9) = __________

4. -8 – 2 = __________

5. -4 – (-4) = __________

6. 5 + (-3) = __________

7. -9 – 9 = __________

8. 12 - 8 = __________

9. 7 – 9 = __________

10. -12 + 12 =__________

APPLICATIONS

11. Suppose the temperature is -6˚.

a. What will the temperature be if it falls 7˚? _________

b. What will the temperature be if it rises 15˚? _________

12. Ed plays fullback for his football team. Sometimes he gains yardage (+5 means a 5-yard gain). Sometimes he loses yardage (-5 means a 5-yard loss). Find Ed’s total yardage in each game below.

a. Game one: +4 +6 +9 +2 -8 -6 +10

b. Game two: +6 -3 0 +15 -5 +9 +11

a._______________ b._______________

EXTRA CREDIT

13. Suppose you are in a building in which the floors are

numbered from 0 to 15. The building has an underground parking garage with ten levels, which are numbered from –1 to –10. Which floor is farther from floor –2? (Explain how you got your answer)

a. floor 7 or floor –10 ___________

b. floor 2 or floor –8 ___________

c. floor 1 or floor –5 ___________

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