Home
Forums Grades Pre K-2
Grades 3-5
Grades 6-8
Grades 9-12
Introduction to Integers (Never Neglect
Negatives)
Objectives:
§
To identify, compare, and order
integers.
§
To relate familiar situations to
integers.
§
To develop and appropriately use
the mathematical terms: integer, positive integer, negative integer, opposite(s).
Download as Microsoft Word Documents:
Instructional Plan Template
Mathematics Governor’s Institute 2005
Names
of group members: Susan Davis.
Mary Beth Eyet, Ellen Frank
Topic/Theme: Introduction
to Integers / Never Neglect Negative Numbers
Level: 5th
grade
Time Element: 2-3 days
NCTM Standards Addressed:
- Explore
numbers <0 by extending the number line and through familiar
applications.
PA Math Standards Addressed:
- 2.1.5
F Use simple concepts of
negative numbers.
- 2.5.5 B
Use appropriate mathematical terms, vocabulary,
language and graphs to explain mathematical
concepts.
Math Assessment Anchors Addressed:
- M5.A.1.4 Use simple applications of
negative numbers.
- M5.A.1.4.1 Identify negative numbers on a number
line.
- R5.A.2 Understand nonfiction text appropriate to
grade level.
- R5.A.2.1 Identify the meaning of
vocabulary from various subject
areas.
Objectives:
- To
identify, compare, and order integers.
- To
relate familiar situations to integers.
- To
develop and appropriately use the mathematical terms:
integer, positive integer, negative integer, opposite(s).
Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):
II. Whole class discussion
about negative numbers
Pose the question: Where do you see negatives numbers in
the real world?
Allow students to ponder momentarily.
Accept all responses. List them on chart paper. Allow
discussion but do not disqualify any responses at this point in time.
(Some responses may include temperature, sports scores,
banking, etc)
Give the following assignment: Look in a newspaper or
magazine for a negative number used in the context of the real world. Read it,
think about it, and be prepared to explain the use of the negative number to
the group. The use of the negative number found could be on the list we have
compiled or have a new use. Newspapers and magazines are available in class if
needed.
Give a day or two for students to complete this task. Advise them not to just cut out a random negative number. It should be included in either an article or graph in the magazine or newspaper. If possible, show one that you have found as a model. If they don’t understand the negative number in the article, they should ask for help or look for another example that makes sense to them.
III. Vocabulary lesson (integer, positive, negative,
opposite)
Pose the question: What is an integer? Accept any
responses.
Show a variety of whole numbers. Explain that an integer
is any whole number and its opposite. Show this
briefly on the number line. Give an example such as 2 and -2. Point out where these integers are located
compared to zero on the number line.
Ask students the opposite of different words as a review
of the meaning of opposites:
(inside – outside) (over – under) (first – last) (hot – cold)
For an online matching of opposites go to: http://www.primaryresources.co.uk/online/antonyms.htm
Have them come up with some word opposites to share with
each other. Next, give each student a
card with an integer (depending on the number of students in the class) Make
sure each positive integer has an opposite.
Allow students to find their opposites by moving around
the room. Adapt it to fit a class that may not be able to move around. For
example, one student can stand and state the integer and then his/her opposite
can stand and the class can check the opposites. After completing this,
continue
by showing on the number line that an integer and its opposite are the same
distance from zero on the number line. The numbers to the right of zero are
positive and the numbers to the left of zero are negative.
Add each word to the word wall. Review the four words
before moving on:
integer, opposite, positive integer,
negative integer
Refer to attachments #2 for
student vocabulary cards and #3 for classroom word-wall cards.
IV. Comparing and Ordering Negative Integers
With the number line in clear view show how the negative
numbers go backward from 0. The farther from zero, the smaller the number is.
The negative integers that may appear greatest are really the least.
Give a list of integers for students to order from least
to greatest. They can do this individually.
-5,
-6, -3, -9
Have a volunteer order them. Ask how they know that -9 is the smallest. The student should respond that it is the
farthest from zero. Continue in this manner until students are comfortable. The
integers can also be ordered from greatest to least.
Next, give out integer cards (again depends on number of
students in the room.) Hold up a random
integer (focus on negatives) and ask that all students who have an
integer larger to stand. Check the responses. Hold up another integer. Ask that
all students who have a smaller integer to stand. Continue in this manner. Allow students to
lead as well. Adapt this as needed for your particular group.
Play “Tug-of-War with
Integers.” See game instructions
(attachment #4), game board (attachment #5) and game cards (attachment #6).
Materials/ Chart
paper, markers
Resources Magazines and newspapers
Number
cards with integers
Attachment
#1 “Life on
Attachment
#2 – Student Vocabulary Cards
Attachment
#3 – Classroom Word-wall Cards
Attachment
#4 – Tug-of-War Game Instructions
Attachment
#5 - “Tug-of-War” with Integers Game board
Attachment
#6 - “Tug-of-War” Game Cards
Attachment
#7 - “Link to
Attachment
#8 - “Link to
Attachment
#9 - Independent Student Project
Attachment
#10 – Student Project Scoring Rubric
Scott Foresman/Addison Wesley, Grade 5, Vol. 4
Harcourt
PA Edition, Grade 5, Vol. 3
Math
Talk: Mathematical Ideas in Poems for Two Voices
by Theoni Pappas
Mathematical
Fractals, Googols, and Other Mathematical Tales
by Theoni Pappas
Interdisciplinary
Connections:
·
Reading Choral Reading “Life on
·
Link to
Technology
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/vlibrary.html
Click on 3-5 numbers and operations to locate the number line bars and number line bounce activities
Line Jumper
http://www.primaryresources.co.uk/online/antonyms.htm
The Number Line
http://www.learningwave.com/chapters/integers/numline.html
Number Lines
http://www.amblesideprimary.com/ambleweb/mentalmaths/numberlines.html
Printable Number Lines
http://lessonplancentral.com/lessons/Math/Math_Worksheets/Number_Lines/
Other Connections:
Science – Use of thermometer
Music – Notes above and
below middle C
Sports – yardage in
football and scoring
Assessment
Strategies:
·
Formative
Evaluation (checking student understanding during the lesson):
On-going informal student observations during the following activities:
During activity I:
student participation in the choral reading
During activity II: the list on
chart paper of student responses will help check understanding
During activity III: finding the
correct opposites will help check understanding
During activity IV: students
ordering the integers and standing at the appropriate time will check
understanding
Summative Evaluation (How will it be determined that the objectives were achieved?):
Objective 1: To identify, compare and order integers
To see if this objective has been met, students will
complete the following task (s)
(with the use of a number line or
other representations)
Start on the ground floor
Go up 4 floors
Go down to the basement
Go up 6 floors
Go down 3 floors
Where are you now?
Ellen was playing a game on her computer. In the first
round, she scored 5 points. In the second round she scored 4 points. In the
third round she scored -6 points. In the fourth and final round she scored -2
points. What was Ellen’s final score?
A vine grew to a height of 6 feet in one year before the
gardener cut off 2 feet. The vine then grew another 3 feet and the gardener cut
off 4 feet. What is the height of the vine now?
The temperature at
Objective 2: To relate familiar situations with integers
To assess this objective, students will verbally share with
the class what they found in a newspaper or magazine that relates negative
integers with the real world. These will be added to the initial list created
at the beginning. Any items that are incorrect from the beginning of the list
will be taken off at this point.
Objective 3: To develop and appropriately use the
vocabulary terms; integer, positive integer, negative integer, opposite
To see if this objective has been achieved, students will
describe in written form a real world example using the appropriate vocabulary.
Direction:
Write about a situation that can be represented by a negative integer
Rubric:
4- Demonstrates a thorough understanding of negative
integers by describing a situation using accurate vocabulary
3 – Demonstrates a general understanding of negative
integers by describing a situation with minor errors or omissions.
2 – Demonstrates a partial understanding of negative
integers by describing a situation.
1- Demonstrates minimal understanding of negative integers
by describing a situation.
0 – Insufficient evidence of understanding
Correctives/Remediation:
Floor or walk on number line
Desk reference number line
Permanent classroom size number line
Calculator
Extensions/Enrichment:
Independent student project (refer to
attachment numbers 9 and 10)
ESL : Ask a student for whom English is not his/her first language to share
“positive” and “negative” in his/her language
Special
Accommodations (special needs students)
·
Description
of the Special Needs student selected:
This female student is an excellent reader. However, her
spelling is extremely poor. She has attempted to compensate for her inability by
memorizing spelling words. Each night, she takes a page from a book and studies
all the spelling of the words on the page. This has helped her.
·
Accommodations
to use with this student:
This student should be asked to pair up with a
struggling reader before the choral reading. This will help build self-esteem
and appeal to her love of reading.
Personal set of vocabulary flash cards to
refer to for spelling as well as reminders to refer to word wall
This student will be selected for a solo part
in the choral reading
GOVERNOR’S INSTITUTE FOR MATH 2005
Susan
Davis
Mary
Beth Eyet
Ellen Frank
Grade 5
Life on
Characters: Narrator,
a group of Positives, a group of Negatives, Zero the Hero
Narrator: A debate has been going
on between the Positive Numbers who live on
Positives: We are positively fine as we
are. Things are just perfect here! Look how evenly we are spaced. Positively perfect!
Negatives: There’s plenty of room on the
lane to the left. We won’t be crowding
you at all! No!
No! No! Not
at all!
Positives: But we are all very orderly
and we like it that way. Positively love
it!
Negatives: We can be orderly, too – in
fact – just as orderly as you are!
Positives: There you go being copy-cats
again. We have a “1,” then you have to
have a “-1.” We have a “2” and you have
to have a “-2.” Now you’re going to put
yourselves in the same order, too!
Negatives: You SAID you liked things
orderly. Think how easy we’ll be to find
if we use the same order as you. –1 will
move in next to Zero. Then –2 will move
in next to –1. We won’t make a mess of
(Zero the Hero enters.)
Zero: What’s going on
here? I can hear you all the way from
Alphabet Alley! I was visiting my twin
sister “O.” This was a quiet
neighborhood when I left.
Narrator: The Positives and
Negatives all began talking at once, giving their arguments. Finally, Zero the Hero said:
Zero: Stop! I have decided! We will let the Positives have this side of
Narrator: And that’s what
happened. The Negatives really didn’t
mess up
The End
Adapted from “The Day the Number Line Fell Apart” from Factals, Googols and Other Mathematical Tales
by Theoni Pappas, Wide World Publishing/Tetra, 1993.
#1
|
integers |
The set of whole
numbers and their opposites. |
|
negative integer |
Any integer less
than zero. |
|
positive integer |
Any integer
greater than zero. |
|
opposites |
Two numbers that
are the same distance, but in opposite directions, from zero on a number
line. |
#2
integer
positive
integer
negative
integer
opposite(s)
#
3
Rules and Procedures
Instructional Objective: To
relate familiar situations to integers and movement on the number-line.
Materials: Whole
class version: Student
version:
Blackboard,
chalk Tug-of-War
game board
String
(approx. 20 feet) One marker or counter
One
bandanna Game
cards
Game
cards
Note: For
the following instructions, the whole class version is followed by the student
version in parentheses.
Object of the Game: Imagine
that one team (person) is pulling the rope toward +10, while the other team
(person) is pulling the rope toward -10.
The bandanna (counter), beginning on zero, represents the flag in the
middle of the rope in a real Tug-of-War Game.
The
winner is the first team (person) who pulls the bandanna (counter) past
the +10 or -10. (You do not need an
exact count to win.)
Set-up: For
the whole class version, hang the string horizontally across a blackboard
(about head-level.) Then make a number
line along the string by writing zero in the middle and numbering + 1 to +10 to
the right and -1 to -10 on the left.
Hang the bandanna on the string with a loose knot exactly at zero.
(Place counter at zero on game board.)
Shuffle
deck of game cards and place face down on a table.
Important: Both
teams (or players) move the same marker for this game.
Procedure:
o
Divide
the class up the middle into 2 groups.
Decide which team (player) will play first. Choose which end of the number-line (game
board) you will be playing from.
- One player from Team #1
will draw the top card from the pile.
Read it aloud. Move the
flag (counter) the correct number of spaces and in the appropriate
direction according to the situation given on the card, for example, for
“You put $5.00 in your piggy
bank,” you would move the bandanna (counter) 5 to the right. Be sure the whole class is in
agreement before the other team (player) plays.
- A player from Team #2 draws
a card, reads it aloud, and moves appropriately beginning from the location of the bandanna at the end of
the previous turn. Do not return the marker to
zero first.
- Play continues with the
flag moving back and forth until one team pulls it beyond +10 or -10.
#4
. . . . . . . . . . . . . .
. . . . . . .
-10 –9
-8 –7 –6
–5 –4 –3 –2 –1 0 +1
+2 +3 +4
+5 +6 +7
+8 +9 +10
#5
|
Game Cards for Tug-of-War with Integers |
The Steelers lost 2 yards in the last pass play. |
|
The scuba diver dives 9 feet below the surface of
the water. |
The Lions were off-sides. Five yard penalty. |
|
May loses 2 inches from her
waist because she exercises. |
You take the elevator up 7 floors to reach your
apartment. |
|
The Eagles gain 6 yards for a first down. |
When the sun came up the temperature rose 6 degrees. |
|
Jessie’s temperature goes up 2 degrees. She has to go to bed. |
It’s Daylight Savings time. Set your clock ahead one
hour. |
|
Your Dad scored an “eagle” on the 14th
hole. (An “eagle” is 2 strokes under
par.) |
The rescue worker climbs down 9 rungs on a ladder to
save a cat. |
|
Sam climbs up 6 stairs to
reach the deck. |
The insect dives 4 inches below the surface of the
water. |
|
The Frisbee reaches a height of 3 feet off the
ground. |
The temperature has dropped 8 degrees. |
|
John was told to move the clock forward 5 minutes. |
When you hit a “bogey” you are 1 stroke over
par. You just hit a “bogey!” |
|
Raymond hikes 2 miles up the mountain. |
Oh no! Ben
has gained 5 pounds on vacation! |
|
Kara withdraws $7 from her checking account. |
Your Gram gives you $5.00 to put in your Savings
Account. |
|
Your temperature has dropped 2 degrees. You may go to school tomorrow! |
To reach the kite in the tree, Charlie Brown has to
climb up 6 rungs of a ladder. #6 |
Name
_________________________________________________________________________
|
Negative |
Zero |
Positive |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#7
STRATEGY – USE CONTEXT
To use context,
look for words and phrases that help you understand the meanings of words,
sentences, paragraphs, and situations. The words below can help you decide if an
integer is negative, zero, or positive.
|
Negative |
Zero |
Positive |
|
withdraw, loss,
spend, lose, lost |
no change, break even |
deposit, profit,
earn, gain, win |
|
below, below sea level, down |
sea level |
above, above sea level, up |
|
drop, behind |
even |
rise, ahead |
|
before |
now |
after |
#8
At
the age of 24, Tiger Woods became the first golfer to be under par in every PGA
tour event for an entire year. In golf, the object of the game is to get the
ball into the hole in as few strokes (hits) as possible. The standard number of
strokes to get the ball into the hole is called par. A golfer’s score is sometimes shown as the number of strokes
above (+) or below (-) par.
|
Tournament Scorecard 82nd PGA
Championship – |
||||||
|
Player |
Rounds |
Total Score |
Par |
|||
|
1 |
2 |
3 |
4 |
|||
|
Sergio Garcia |
74 |
69 |
73 |
69 |
285 |
-3 |
|
Phil Mickelson |
70 |
70 |
69 |
70 |
279 |
-9 |
|
Loren Roberts |
74 |
72 |
71 |
73 |
290 |
+2 |
|
Brian
Watts |
72 |
74 |
73 |
70 |
289 |
+1 |
|
Tiger
Woods |
66 |
67 |
70 |
67 |
270 |
-18 |
PURPOSE:
To explore integers by playing a
tossing game.
GROUPING: small groups
MATERIALS: plastic counters, cup, yardstick
·
Make
a scorecard, using the tournament scorecard above as a model.
·
Mark
a starting point, and place the cup three feet from the starting point.
·
Take
turns tossing a counter from the starting point into the cup. Par is 3. After
each toss, pick up the counter and toss again from where the counter has
landed. The number of tosses needed to “sink” the counter in the cup is
recorded.
·
After
4 rounds, the total score for each player is calculated and written as a
positive or negative integer with respect to par.
Source:
Harcourt Math PA Teacher Edition – grade 5. #9
STUDENT
________________________________________
PERFORMANCE
RUBRIC – INTEGERS
INDEPENDENT
PROJECT
|
Task |
Possible points |
Points earned |
|
Create a scorecard, using model provided. |
5 |
|
|
Follow directions appropriately for game setup. |
5 |
|
|
Calculate scores correctly. |
5 |
|
|
Record total score as positive or negative integer. |
5 |
|
|
TOTAL POINTS |
20 |
|
#
10
Home
Forums Grades Pre K-2
Grades 3-5
Grades 6-8
Grades 9-12