G6 m3-a-lesson 1-t

Lesson 1: Positiveand Negative Numbers ontheNumberLine—Opposite
Direction and Value
Date: 1/20/15
11
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 1
Lesson 1: Positive and Negative Numbers on the Number
Line—Opposite Direction and Value
Student Outcomes
 Students extend their understandingof the number line,which includes zero and numbers to the right, that
are above zero, and numbers to the left, that are below zero.
 Students use positiveintegers to locatenegative integers, moving in the opposite direction from zero.
 Students understand that the set of integers includes the set of positivewholenumbers and their opposites,as
well as zero. They also understand that zero is its own opposite.
Classwork
OpeningExercise (3 minutes): NumberLine Review
Display two number lines (horizontal and vertical),each numbered 0–10. Allow students
to discussthefollowingquestions in cooperativelearning groups of 3 or 4 students each.
Students should remain in the groups for the entire lesson.
Questions to Discuss:
 What is the startingposition on both number lines?
 0 (zero)
 What is the lastwhole number depicted on both number lines?
 10 (ten)
 On a horizontal number line,do the numbers increaseor decrease as you move
further to the right of zero?
 Increase.
 On a vertical number line,do the numbers increaseor decrease as you move
further above zero?
 Increase.
Exploratory Challenge (10 minutes): Constructingthe NumberLine
The purpose of this activity is to let students constructthe number line (positiveand negative numbers and zero) usinga
compass.
 Have students draw a line, placea pointon the line, and label it0.
 Have students use the compass to locateand label the next point 1 thus creatingthe scale. Students will
continue to locate other whole numbers to the rightof zero usingthe sameunit measure.
 Usingthe same process, have students locatethe opposites of the whole numbers. Have students label the
firstpointto the left of zero, −1.
Scaffolding:
 Create two floor models
for vertical and horizontal
number lines using
painter’s tape for visual
learners.
 Have a student model
movement alongeach
number linefor kinesthetic
learners.
 Use pollingsoftwarefor
questions to gain
immediate feedback while
accessingprior knowledge.

Direction and Value
Date: 1/20/15
12
 Introduce to the class thedefinition of the opposite of a number.
 Sample Student Work
Given a nonzero number, 𝑎, on a number line,the oppositeof 𝑎, labeled −𝑎, is the number such that:
 0 is between 𝑎 and −𝑎.
 The distancebetween 0 and 𝑎 is equal to the distancebetween 0 and −𝑎.
The opposite of 0 is 0.
 The set of whole numbers and their opposites,includingzero, arecalled integers. Zero is its own opposite. The
number linediagramshows integers listed in order from leastto greatest usingequal spaces.
Monitor student constructions,makingsurestudents are payingcloseattention to the direction and sign of a number.
Example 1 (5 minutes): Negative Numberson the NumberLine
Students will usetheir constructions to model the location of a number relativeto zero by usinga curved arrow starting
at zero and pointingaway from zero towards the number. Pose questions to the students as a whole group, one
question at a time.
Startingat 0, as I move to the righton a horizontal number line,the values get larger. These numbers are called positive
numbers because they are greater than zero. Notice the arrowis pointingto the rightto show a pos itivedirection.
 How far is the number from zero on the number line?
 3 units
 If 0was a mirror facingtowards the arrow,what would be the direction of the arrow in the mirror?
 To the left.
0 1 2 3 4 5 6 7 8 9 10 11 12
MP.6

Direction and Value
Date: 1/20/15
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 Would the numbers get larger or smaller as we move to the left of zero?
 Smaller.
Startingat 0, as I move further to the left of zero on a horizontal number line,the values get smaller. These numbers are
called negativenumbers becausethey are less than zero. Notice the curved arrowis pointingto the left to showa
negative direction. The position of the pointis now, negative 3, written as −3.
 Negative numbers are less than zero. As you look to the left, the values of the numbers decrease.
 What is the relationship between 3 and −3 on the number line?
 3 and −3 are located on opposite sides of zero. They are both the same
distance from zero. 3 and −3 are called opposites.
 As we look further right on the number line,the values of the numbers increase.
For example, −1 < 0 < 1 < 2 < 3.
 This is also truefor a vertical number line. On a vertical number line,positive
numbers are located above zero. As we look upward on a vertical number line,
the values of the numbers increase. On a vertical number line,negative numbers
are located below zero. As we look further down on a vertical number line,the
values of the numbers decrease.
 The set of whole numbers and their opposites,includingzero, arecalled integers.
Zero is its own opposite. A number linediagramshows integers listed in
increasingorder fromleft to right usingequal spaces. For example:
−4, −3, −2, −1, 0, 1, 2, 3, 4.
 Allowstudents to discussthe example in their groups to solidify thei r
understandingof positiveand negative numbers.
Possiblediscussion questions:
 Where are negative numbers located on a horizontal number line?
 Where are negative numbers located on a vertical number line?
 What is the opposite of 2?
 What is the opposite of 0?
 Describethe relationship between 10 and −10.
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
4
3
2
1
0
-1
-2
-3
-4
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Direction and Value
Date: 1/20/15
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Example 2 (5 minutes): UsingPositive Integersto Locate Negative Integerson the NumberLine
Have students establish elbowpartners and tell them to move their finger alongtheir number lines to answer the
followingset of questions. Students can discussanswers with their elbow partners. The teacher circulates around the
room and listens to the student-partner discussions.
 Describeto your elbow partner how to find 4 on a number line. Describehow to
find −4.
 To find 4, start at zero and move right to 4. To find −4, start at zero and
move left to −4.
Model how the location of a positiveinteger can be used to locatea negative integer by
moving in the opposite direction.
 Explain and showhow to find 4 and the oppositeof 4 on a number line.
 Start at zero and move 4 units to the right to locate 4 on the number line.
To locate −4, start at zero and move 4 units to the left on the number line.
 Where do you startwhen locatingan integer on the number line?
 Always start at zero.
 What do you notice about the curved arrows that represent the location of 4 and −4?
 They are pointing in opposite directions.
Exercises1–6 (13 minutes)
The teacher will createand display onelargehorizontal and vertical number line on the board, numbered −12 to 12, for
the class. Distributean index card with an integer on itto each group.1 Students will work in groups first,completing
the exercises in their student materials. Concludethe exerciseby havingstudents locateand label their integers on the
displayed teacher’s number lines.
Exercises 1–6
1. Complete the diagrams. Countby 𝟏’𝒔to label thenumber lines.
2. Plot your point on both number lines.
(Answers may vary.) −𝟒
1 Depending onthe class size, label enoughindex cards for tengroups (one card per group). Varythe numbers using positives and
negatives, such as −5 through 5, including zero. If eachgroupfinds andlocatesthe integer correctly, eachgroupwillhave a cardthat
is the opposite of another group’s card.
−5 −4 −3 −2 −1 0 1 2 3 4 5
Scaffolding:
 As an extension activity,
have students identify the
unit differently on
different number lines,
and ask students to locate
two whole numbers other
than 1 and their opposites.

Direction and Value
Date: 1/20/15
15
3. Show and explain how to find the oppositeofyour number on both number lines.
I found my point by startingat zero andcounting four units totherightto end on 𝟒. Then I started on 𝟎and moved
to theleft four units to end on −𝟒.
4. Mark the opposite on both number lines.
(Answers may vary.) −𝟒
5. Choose agroup representativeto place theoppositenumber on theclassnumber lines.
6. Which group had the oppositeofthe numberon your index card?
(Answers may vary.) Jackie’s group. They had −𝟒.
Closing(2 minutes)
Give an example of two oppositenumbers and describetheir locations on a horizontal and vertical number line.
 6 and −6 are the same distancefrom zero but on opposite sides. Positive6 is located 6 units to the rightof
zero on a horizontal number lineand 6 units above zero on a vertical number line. Negative 6 is located 6
units to the left of zero on a horizontal number lineand 6 units below zero on a vertical number line.
Exit Ticket (7 minutes)
𝟒
𝟑
𝟐
𝟏
𝟎
−𝟏
−𝟐
−𝟑
−𝟒
−𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓

Direction and Value
Date: 1/20/15
16
Name ___________________________________________________ Date____________________
Lesson 1: Positive and Negative Numbers on the Number Line—
Opposite Direction and Value
Exit Ticket
1. If zero lies between 𝑎 and 𝑑, give one set of possiblevalues for 𝑎, 𝑏, 𝑐, and 𝑑.
2. Below is a listof numbers in order from leastto greatest. Use what you know about the number lineto c omplete
the listof numbers by fillingin theblanks with the missingnumbers.
−6, −5, ________, −3, −2, −1, _________,1, 2, _________,4, _________,6
3. Complete the number linescale. Explain and showhow to find 2 and the oppositeof 2 on a number line.
2
𝑎 𝑏 𝑐 𝑑

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Date: 1/20/15
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4
3
2
1
0
-1
-2
-3
-4
Exit Ticket Sample Solutions
1. If zero liesbetween 𝒂 and 𝒅, give oneset ofpossiblevaluesfor 𝒂,𝒃, 𝒄, and 𝒅.
𝒂:−𝟒; 𝒃: −𝟏; 𝒄:𝟏; 𝒅: 𝟒
2. Below isalist ofnumbersin order from least to greatest. Use what you know about thenumber line to complete
the list ofnumbers by filling in the blankswiththe missing numbers.
−𝟔,−𝟓,−𝟒, −𝟑,−𝟐,−𝟏,𝟎, 𝟏,𝟐, 𝟑,𝟒,𝟓, 𝟔
3. Complete thenumber linescale. Explainand show how to find 𝟐and the oppositeof 𝟐on anumber line.
I would start at zero and move 𝟐units to the left to locatethenumber −𝟐on thenumber line. So to locate 𝟐, I
would start at zero and move 𝟐units to the right (theoppositedirection).
Problem Set Sample Solutions
1. Create ascale for the number linein order toplot the points −𝟐,𝟒, and 𝟔.
a. Graph each point and itsoppositeon thenumberline.
b. Explain how you found the oppositeofeach point.
To graph each point,I started at zero andmoved rightor left based on thesignand number (to
theright for a positivenumber and to theleft for a negativenumber). To graph theopposites, I
started at zero, but this timeI moved in theoppositedirection thesamenumber of times.
2. Carlosusesavertical number line tograph the points −𝟒, −𝟐, 𝟑, and 𝟒. He noticesthat −𝟒iscloser
to zero than −𝟐. He isnot sure about hisdiagram. Use what you know about avertical number line
to determineifCarlosmade amistake or not. Support yourexplanationwith anumber line diagram.
Carlos madea mistakebecause −𝟒is less than −𝟐, so it should befurther down thenumber line.
Starting at zero, negativenumbers decreaseas welook further to theleft of zero. So, −𝟐lies before
– 𝟒on a number line since −𝟐 is 𝟐units below zero and −𝟒is 𝟒units below zero.
−𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
𝒂 𝒃 𝒄 𝒅

Direction and Value
Date: 1/20/15
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3. Create ascale in order to graph thenumbers −𝟏𝟐through 𝟏𝟐on anumber line. What doeseach tick mark
represent?
Each tick mark represents 𝟏 unit.
4. Choose an integer between −𝟓and −𝟏𝟎. Label it 𝑹on thenumber lineabove and completethefollowing tasks.
(Answers may vary. Refer to thenumber lineabovefor samplestudent work.)−𝟔,−𝟕,−𝟖, or −𝟗
a. What isthe opposite of 𝑹? Label it 𝑸.
𝟔
b. State apositive integergreater than 𝑸. Label it 𝑻.
𝟏𝟏
c. State anegative integergreater than 𝑹. Label it 𝑺.
−𝟑
d. State anegative integerlessthan 𝑹. Label it 𝑼.
−𝟗
e. State an integer between 𝑹and 𝑸. Label it 𝑽.
𝟐
5. Will the oppositeofapositive number always, sometimes, or never be apositivenumber? Explain yourreasoning.
The oppositeofa positivenumber willnever bea positivenumber. For two nonzero numbers to be opposites, 𝟎has
to bein between both numbers, andthedistancefrom 𝟎 toonenumber has to equal thedistancebetween 𝟎and the
other number.
6. Will the oppositeofzero always, sometimes, or never be zero? Explain your reasoning.
The oppositeof zero will always bezero becausezero is its own opposite.
7. Will the oppositeofanumber always, sometimes, or never be greaterthan thenumber itself? Explain. Provide an
example to support your reasoning.
The oppositeof a number willsometimes begreater thanthenumber itselfbecauseit depends on thegivennumber.
For example, ifthenumber given is −𝟔, then theoppositeis 𝟔, which is greater than−𝟔. Ifthenumber is 𝟓, then
theoppositeis −𝟓, which is not greater than 𝟓. Ifthenumber is 𝟎, then theoppositeis 𝟎, whichis never greater
than itself.
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
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