Integer Explorer Student Packet Unit 13: Integers • Lesson 13.1
Positive & Negative Numbers
Name:
Date: Team:
DO NOW
Prior Knowledge Review
Target Time: 5 min
Carefully answer each question to activate your understanding of number order and direction.
1. What whole number comes immediately before 3 ?
2. Compare: Is 7 greater than or less than 4 ?
3. Complete the forward pattern: 8, 9, , (Rule: +1)
4. On a thermometer, does the temperature level move UP or DOWN when it becomes colder?
UP DOWN
5. Have you ever seen or heard of a number below zero ? Where was it?
Sentence Starter: I have seen or heard of a number below zero when...
FLUENCY
Mental Math Sprint (Multiplication: 3s)
Goal: Fast & Accurate
\(3 \times 0 =\)
\(3 \times 1 =\)
\(3 \times 2 =\)
\(3 \times 3 =\)
\(3 \times 4 =\)
\(3 \times 5 =\)
\(3 \times 6 =\)
\(3 \times 7 =\)
\(3 \times 8 =\)
\(3 \times 9 =\)
\(3 \times 10 =\)
\(3 \times 12 =\)
Today's Mathematical Mission
I can locate and identify positive and negative integers on a number line, recognize zero as the neutral reference point, and describe opposites as values at equal distances from zero in opposite directions.
13.1 Guided Notes: The Integer Line
Teacher Model & Direct Instruction
Negative Integer
A whole number less than zero . Located to the left (or below ) zero. Written with a minus sign: -4.
Positive Integer
A whole number greater than zero . Located to the right (or above ) zero. Can be written as +4 or simply 4.
Zero (The Origin)
The neutral center point. Zero is neither positive nor negative . It serves as our starting reference location: distance is measured from 0.
Opposites
Two numbers that are the exact same distance from zero, but on opposite sides of the number line (e.g., \(-3\) and \(+3\)).
Horizontal Number Line Anatomy
← Left: Less than Zero (Negative) • Origin (Zero) Right: Greater than Zero (Positive) →
⟨ ⟩
-5
-4
-3
-2
-1
0
+1
+2
+3
+4
+5
-8 can be read as:
"Negative eight"
"The opposite of positive 8"
+8 or 8 can be read as:
"Positive eight"
"The opposite of negative 8"
Class Stamp
How to Locate Any Integer on a Number Line
1. Always begin at 0 (the neutral origin).2. Look at the sign : a minus sign (-) means travel LEFT ; a plus or no sign means travel RIGHT .3. Count the exact number of spaces (units) to plot your point.
Check for Understanding (Turn & Talk)
A. To plot -4, start at and move spaces to the .
B. What number is neither positive nor negative?
Page 2
Investigation
13.1 Explore Activity: The Symmetry of Opposites
Partner or Independent
Follow the steps below to discover how numbers on opposite sides of zero relate to one another in distance and direction.
Part 1: Construct & Observe
Step A: Fill in all missing integer labels on the number line below from \(-7\) to \(+7\).
⟨ ⟩
__
__
__
__
__
__
__
0
1
2
3
4
5
__
__
Step B: Draw a circle around 4. Draw a circle around the number that is the exact same distance from 0 on the other side.
Step C: Put a box around -6 and a box around 6.
Part 2: Analyzing Distance & Direction
1. Distance from Zero: How many units away from 0 is 4? How many units away is -4? What do you notice about their distances?
2. Direction from Zero: What do you notice about the directions you must travel from 0 to reach -6 versus +6?
3. Symmetry (Paper Fold Idea): If you folded this paper vertically right through 0, what would happen to the points -4 and +4?
Part 3: Synthesize & Deepen
Define in your own words:
Two numbers are opposites when they:
Brain Teaser:
The opposite of -2 is:
What is the opposite of the opposite of +5?
Page 3
Practice
13.1 Leveled Independent Practice
Tiered Mastery
Reference Number Line for Questions 1–4:
⟨ ⟩
-8
F
G
0
H
J
8
Level 1: Core Identification
1. Which point represents the integer -4?
[ ] Point F [ ] Point G [ ] Point H [ ] Point J
2. What integer is represented by Point F ?
[ ] -8 [ ] -6 [ ] +6 [ ] -2
Level 2: Opposites & Distance
3. Which two points represent a pair of opposites ?
[ ] F and H [ ] F and J [ ] G and J [ ] G and F
4. What is the opposite of Point G ?
[ ] Point F [ ] Point H [ ] Point J [ ] 0
5. Plot a point K at -3 and point M at its opposite on the line below:
-5 | | | | 0 | | | | +5
Level 3: Vertical & Real-World Application
Thermometer Reading:
The temperature in Anchorage is \(6^\circ\text{F}\) below zero.
Write as an integer:
Its opposite temperature:
Sea Level Reference:
A submarine is at \(150\text{ ft}\) below sea level (\(-150\)). A bird flies at \(150\text{ ft}\) above sea level (\(+150\)).
What represents sea level?
Page 4
Zero Line Exit Ticket 13.1 Exit Ticket Pacing: 5 Minutes
Positive & Negative Numbers
Name:
Date: Score: / 4
Directions: Examine the number line below. Pay close attention to unit intervals and distance from zero to answer questions 1–3.
⟨ ⟩
A
-3
B
0
C
D
1. Which pair of letters represents two numbers that are true OPPOSITES? 1 pt
Remember: Opposites must have the exact same distance from zero.
A. Point A and Point B
B. Point A and Point C
C. Point A and Point D
D. Point B and Point C
E. Point B and Point D
F. Point C and Point D
2. Mathematical Argument Critique: 1 pt
Nyla claims: "Point A represents the opposite of \(-5\)."
Do you agree or disagree with Nyla? Explain your mathematical reasoning.
Circle one: I AGREE I DISAGREE
3. Which statement accurately justifies the relationship between Point A and Point D? 1 pt
A. Point A is the opposite of Point D because they are on different sides of zero.
B. Point A is the opposite of Point D because both points are exactly 5 units away from zero.
C. Point A is not the opposite of Point D because Point A has a negative sign and Point D does not.
D. Point A is not the opposite of Point D because they are unequal distances from zero.
4. Precision of Language: 1 pt
Complete the sentence accurately using math vocabulary:
An integer is any number that is a...
Lesson 13.1 Formative Check • Integers and Their Opposites MIT Math • Grade 6
Integer Voyage Homework Sheet Unit 13 • Homework 13.1
Integer Voyage Homework
Name:
Date: Team:
Part 1: Number Line Location
Use the number line below to answer questions 1–3.
⟨ ⟩
-10
A
B
C-5
D
0
E
+5
1. Which point represents the integer -7?
Point:
2. What integer is represented by Point E ?
Value:
3. What is the opposite of Point D ?
Opposite:
Part 2: Draw & Plot
4. On the blank line below, place 0 in the center. Then plot and label the integers: -4, +3, and the opposite of +3.
⟨ ⟩
Part 3: Opposites & Patterns
5. Fill in each blank with the correct value:
Opposite of 6:
Opposite of -8:
Opposite of -25:
Opposite of 25:
Opposite of -(-5):
Opposite of -(-25):
Part 4: Real-World Integer Modeling
6. A winter storm drops the temperature to \(8^\circ\text{F}\) below zero .
Write as an integer:
7. A scuba diver swims \(42\text{ feet}\) below sea level .
Write as an integer:
Lesson 13.1 Homework • Independent Practice MIT Math • Grade 6
Integer Launch Slide Deck Unit 13 • Lesson 13.1 Grade 6 Mathematics
Zero & Beyond
Unlocking Positive & Negative Numbers on the Number Line
Mission: Locate integers and uncover the symmetry of opposites.
MIT Math
Launch • Notice & Wonder Warm-Up
Think Fast
What happens when the weather gets colder than freezing?
Does temperature stop at zero, or does it keep dropping?
Real-World Examples
• \(-5^\circ\text{F}\) in a winter blizzard
• \(-200\text{ ft}\) below sea level
• \(-\$15\) owed to a friend
Today, we learn how mathematics visualizes numbers below zero!
Direct Instruction The Number Line Anatomy
Zero is the Balance Point (The Origin)
← NEGATIVE (Less than 0) • ZERO (Neutral Center) POSITIVE (Greater than 0) →
-4
-3
-2
-1
0
+1
+2
+3
+4
Crucial Rule: Zero is neither positive nor negative. It is where all measurements begin!
Big Concept What are Opposites?
Opposites = Same Distance, Opposite Sides
-3
Start at 0 and move 3 units LEFT .
Distance from 0 = 3 units
+3
Start at 0 and move 3 units RIGHT .
Distance from 0 = 3 units
Both numbers are exactly 3 spaces away from zero !
If you folded the number line at zero, opposite numbers would touch!
Teacher Model (I Do) Locating -4
"How do I locate and plot -4?"
1
Start at Zero (0): Place your pencil right at the origin.
2
Check the Sign: The minus sign (-) commands us to go LEFT .
3
Count the Units: Move 1, 2, 3, 4 units left and plot your point at -4.
Never start counting from the ends—always start at zero!
Collaborative Discourse Turn & Talk (2 Min)
Prompt for Partners:
"Marcus says:-6 is greater than 0 because 6 is bigger than 0.'
Sentence Starter: "Marcus is incorrect because \(-6\) is located to the..."
Be prepared to share out with precise math vocabulary!
Quick Check All Fingers on Chest
Which of the following is true about -5 and +5?
A. They are on the same side of zero.
B. They are both exactly 5 units away from zero on opposite sides.
C. Zero is greater than \(+5\).
D. The opposite of \(-5\) is \(-5\).
Hold up: 1 = A • 2 = B • 3 = C • 4 = D
Core Class Stamp Work Time
Stamp of the Day
Integers represent values above (right) and below (left) of zero.
To locate any integer: Start at ZERO , follow the SIGN for direction, and count the SPACES for distance!
Next Step: Independent Practice in Your Packet Finish Strong with the Exit Ticket!
Integer Foundations Teacher Guide Unit 13: Integers • Lesson 13.1 Teacher Guide
Zero & Beyond: Instructional Facilitation
CCSS: 6.NS.C.5, 6.NS.C.6a
Essential Question
How does zero function as a neutral balance point to define positive and negative numbers?
Core Conceptual Goal
Internalize integers as directed distances from zero; master opposites as equal distance in opposite directions.
Upcoming Bridge
Prepares students directly for comparing integers (\(a < b\) means \(a\) is to the left) and absolute value (\(|x|\)).
Master Lesson Arc & Time Pacing (60 Minutes)
1:10 PM – 2:10 PM
1:10 – 1:17 (7m)
Do Now & Fluency Sprint (Lap 1):
Execute homework collection routine. Check headings. Students complete 5-item Do Now and 3s fluency. Give positive public praise.
1:17 – 1:22 (5m)
Hook & Real-World Framing (Slide 2):
Temperature discussion: What happens below freezing? Introduce terminology: above/below zero, elevation, debt.
1:22 – 1:35 (13m)
Direct Instruction & Number Line Anatomy (Slides 3–5):
Define Integer, Positive, Negative, Zero, Opposites. Model locating \(-4\) and \(+3\). Conduct CFU Poll and Turn & Talk.
1:35 – 1:45 (10m)
Explore Activity: Symmetry of Opposites (Packet p. 3):
Students complete number line labeling and paper-fold visualization. Debrief: "Opposites are equal distance from zero in opposite directions."
1:45 – 1:57 (12m)
Tiered Independent Practice (Lap 2):
Students work through Levels 1–3. Teacher pulls targeted small group; Paras provide tier-1 accommodation.
1:57 – 2:05 (8m)
Class Stamp & Formative Exit Ticket (5m silent):
Deliver final Stamp. Collect 13.1 Exit Ticket silently at door. Assign Homework 13.1.
Inclusion & Para Protocol
Co-Teacher / Paraprofessional Deployment Plan
Ms. Dixon (Prompting & Scaffolding):
Jhon & Austin: Prompt immediately at 1:10 to write heading on homework and packet.Read-Aloud Support: Read Do Now questions aloud to Nia and Austin.Targeted Prompt: "Where is zero? (Center). Which direction is negative? (To the left)."
Ms. Dennison (Accountability & Check-ins):
Norjae: Prompt at transition to begin Do Now without delay.Lap 1 Patrol: Verify student agenda entries and homework completion.Physical Manipulative: Provide laminated desktop number lines with dry-erase markers for students requiring tactile tracking.
Teacher Guide • Page 1
Direct Scripting, CFUs, & Scaffolding Bridges
Teacher Delivery Playbook
Teacher Script (TWS & Think-Aloud)
Framing the Integer Line: "Look at our number line. Zero is in the exact center. To the right are our familiar counting numbers—these are positive numbers. To the left of zero, numbers continue below zero—these are negative numbers. Notice zero has no sign: it is neither positive nor negative."
Think-Aloud for Locating \(-4\): "Watch where I begin. I do NOT start at the left edge. I put my finger right at ZERO. The negative sign in \(-4\) gives me a command: move LEFT. Now I count the units: 1, 2, 3, 4. I land at \(-4\) and plot my point. If I wanted its opposite, I return to zero and travel 4 units in the opposite direction—RIGHT—to land on \(+4\)!"
Formative Error Analysis & In-the-Moment Moves
Misconception 1: "Is \(-6\) bigger than \(0\) because 6 is bigger than 0?"
Teacher Correction: Re-anchor on direction. "On any horizontal number line, numbers further to the left are always smaller. Since \(-6\) is 6 spaces to the left of 0, \(-6\) is less than 0. Having \(-6\) dollars means you owe money; 0 means you owe nothing."
Misconception 2: "The opposite of \(-5\) is Point A (which is at \(-5\))." (Exit Ticket Q2 / Nyla's Error)
Teacher Correction: Prompt student to state the sign and direction. "Point A is located at \(-5\). It cannot be the opposite of itself! Its opposite must be on the OTHER side of zero at \(+5\)."
Misconception 3: "Counting from the wrong point (e.g., counting tick marks from the left edge)."
Teacher Correction: Physically tap the origin. "Anchor at Zero! Zero is your home base. Repeat with me: Start at Zero, follow the sign, count the spaces."
Vertical Scaffolding
How to Scaffold Up in Upcoming Lessons
Lesson 13.2: Comparing & Ordering
Leverage the Left-to-Right principle established today: \(a < b\) if and only if \(a\) lies to the left of \(b\). Students will easily see that \(-7 < -2\) because \(-7\) is farther left.
Lesson 13.3: Absolute Value (\(|x|\))
Directly build on today's definition of opposites ("same distance from zero"). Define absolute value as simply that distance: \(|-4| = 4\) and \(|4| = 4\). Distance is never negative!
Lesson 13.4: Integer Addition (Vectors)
Use the "Start at 0, move with sign" rule. Adding a negative means walking left; adding a positive means walking right. Combining opposites (\(4 + (-4) = 0\)) creates a zero pair.
Lesson 13.5: Rational Coordinate Plane
Intersect today's horizontal number line (x-axis) with the vertical thermometer line (y-axis). Zero becomes the origin \((0,0)\), maintaining quadrant symmetry.
Teacher Guide • Page 2
