Inequality Blueprint Student Packet 7.8: Inequality Blueprint
Objective: Write simple and compound linear inequalities from scenarios.
Name:
Date:
DO NOW (5 min)
Circle the correct inequality symbol for each phrase:
"Greater than" < > ≤ ≥
"At most" < > ≤ ≥
"Minimum of" < > ≤ ≥
"Less than" < > ≤ ≥
EXCELLENCE
At North Star, we are master architects. We aren't just writing numbers; we are defining boundaries . Precision leads to power. Track every keyword and use your blueprint to be 100% accurate.
INEQUALITY BLUEPRINT TABLE
<
Less Than
>
Greater Than
≤
At Most / Max
≥
At Least / Min
ACTIVATE PRIOR KNOWLEDGE (5 min)
Simple Inequality
x > 5
"One Boundary"
Compound Inequality
3 < x < 10
"The Sandwich (Two Boundaries)"
GUIDED PRACTICE (15 min)
MODEL 1: SIMPLE
"Marcus needs to save at least $45 for his shoes."
1. Keywords
At least
2. Symbol Choice
< > ≤ ≥
Final Construction:
x [ ] 45
MODEL 2: COMPOUND
"The temperature must be greater than 60° but no more than 85°."
Low Boundary
60 < t
High Boundary
The Sandwich (Compound):
60 < t [ ] 85
PRIORITY PRACTICE (15 min)
01 LEVEL 1: Simple Structures
A number is less than 12:
A speed of at most 65 mph:
02 LEVEL 2: The Sandwich Zone
"The recipe calls for more than 2 cups but less than 5 cups of flour."
Boundary 1
Boundary 2
Combined (x):
EXIT TICKET (5 min)
A student tries to keep their phone bill between $30 and $55 per month, inclusive.
[ bill (b) ]
"Inclusive" means I must use:
< , >
≤ , ≥
HOMEWORK
7.8: INEQUALITY BLUEPRINT
Part 1: Translation Lab
1. "Taller than 48 inches." (Variable: h)
2. "Capacity is no more than 30." (Variable: s)
Part 2: Compound Zone
3. "Cat weighs between 8 and 12 pounds." (Variable: w)
4. Target heart rate is at least 120 but no more than 160 bpm.
Final Inspection:
Did you use a variable? Did you choose the correct symbol? Double-check Part 2 for the "sandwich" format!
Inequality Blueprint Teacher Guide TEACHER PLAN: 7.8
Inequality Blueprint
North Star Academy
55 MINUTE SESSION
Heart of the Lesson
Inequalities are mathematical boundaries. Translating words to symbols requires identifying the "ceiling" (max) vs the "floor" (min).
Instructional Objective
Students will translate real-world scenarios into simple and compound inequalities using keywords and structured blueprints.
Time Section Teacher Actions & Prompts 5m Do Now Prompt: "Track keywords with your pencils. If I say 'at most', circle the symbol that keeps us at 5 or lower."
|
| 2m | Excellence |
Script: "At North Star, we are architects. We define boundaries with precision. High expectations today."
|
| 8m | Activate Knowledge |
Question: "If a speed limit is 65, can I go 70? No. What symbol represents that boundary?"
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| 15m | Guided Practice |
Model 1: Marcus savings (at least 45). Model 2: Temp sandwich (60 to 85).
|
| 20m | Priority Practice |
L1: Keywords. L2: Sandwich. L3: Budget logic.
|
| 5m | Exit Ticket |
Script: "Silent and solo. Tracking 'inclusive' means we must use the bar."
|
Monitoring Tracker (25 Students)
# Student Name Lap 1: Keywords Lap 2: Sandwich L3: Budget ET Accuracy 1 2 3 4 5 6 7 8 9 10 11
Inequality Blueprint Answer Key Answer Key: 7.8
TEACHER COPY
Section Keys
Do Now
1. > | 2. ≤ | 3. ≥ | 4. <
Guided Practice
Model 1: x ≥ 45
Model 2 (Sandwich): 60 < t ≤ 85
Priority Practice
L1: n < 12 and s ≤ 65
L2: 2 < c < 5
L3: 12t ≤ 50
Exit Ticket
30 ≤ b ≤ 55
Homework Master Key
PROB 1
h > 48
PROB 2
s ≤ 30
PROB 3
8 < w < 12
PROB 4
120 ≤ heart ≤ 160
Mastery Data Sheet Intersection Investigation Student Packet 8.1: Intersection Investigation
Graphing Systems of Equations
Name:
Date:
DO NOW (5 min)
Graph the line: y = 2x - 1
Checklist:
Start at (0, -1)
Slope: Up 2, Right 1
ACTIVATE PRIOR KNOWLEDGE (8 min)
The Big Investigation Question:
If I tell you the coordinate ( 2 , 5 ) is a solution to two different lines, what MUST happen to those lines at that exact spot on the grid?
Write your conclusion here...
GUIDED PRACTICE (15 min)
Line A: y = x + 2
Line B: y = -x + 4
Investigator Prompt:
Where do the two paths meet? Find the exact coordinate.
Final Solution (x, y)
( ___ , ___ )
PRIORITY PRACTICE (20 min)
LEVEL 1: Basic Crossings
y = 2x + 1
y = -x + 4
Sol: (____, ____)
LEVEL 2: Horizontal Investigation
y = -2
y = x - 3
Sol: (____, ____)
LEVEL 3: Origin Investigation
Pro-Level: Cross at the Origin?
y = 2x
Sol: ( __ , __ )
EXIT TICKET (5 min)
Write the intersection coordinate for the two paths below:
( __ , __ )
HOMEWORK: 8.1
1. y = x + 3 | y = 2x + 1
Sol: ( __ , __ )
2. y = -x + 2 | y = x - 4
Sol: ( __ , __ )
"Truth Check: If your coordinate pair (x,y) makes both equations EQUAL, your investigation is complete!"
Intersection Investigation Teacher Guide TEACHER PLAN: 8.1
Intersection Investigation
North Star Academy
55 MINUTE SESSION
Heart of the Lesson
Systems are solved visually by identifying the single coordinate point where two distinct paths cross. Intersection = Truth.
Instructional Objective
Students will graph two linear equations and identify the solution as the unique point of intersection (x, y).
Time Section Teacher Actions & Prompts 5m Do Now Script: "Graphing is our foundation. Plot that -1 on the y-axis first. Now, Up 2, Right 1."
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| 2m | Excellence |
Message: "Precision is our standard. If dots are off, the intersection vanishes."
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| 8m | Activate Knowledge |
Question: "If a point is a solution to TWO lines, what must they do at that spot? Turn & Talk."
|
| 15m | Guided Practice |
Model: Graph y=x+2 and y=-x+4. Circle (1,3). "They meet at (1,3). That's the solution."
|
| 20m | Priority Practice |
L1: Intersecting lines. L2: Horizontal line (y=-2). L3: Origin start (y=2x).
|
Monitoring Tracker (25 Students)
# Student Name Lap 1: Correct Intercepts Lap 2: Intersection L3: Challenge Acc ET Score 1 2 3 4 5 6 7 8 9 10 11 12
Intersection Investigation Answer Key Answer Key: 8.1
Secret Weapon
Section Solutions
Do Now (Graph y=2x-1)
Passes thru (0,-1) and (1,1). Up 2, Right 1.
Guided Practice
Sol: (1, 3)
Priority Practice
L1: ( 1 , 3 )
L2: ( 1 , -2 )
L3: ( 3 , 6 )
Exit Ticket
( -2 , 2 )
Homework Key
Prob 1: y = x + 3 & y = 2x + 1
( 2 , 5 )
Prob 2: y = -x + 2 & y = x - 4
( 3 , -1 )
Classroom Data Snapshot
IEP 1
IEP 2
L1
L2
L3
ET
The Great Swap Student Packet 8.2: The Great Swap
Objective: Solve systems algebraically using substitution.
Name:
Date:
DO NOW (5 min)
If x = 5, find y below:
y = 2x + 3
Architect Workspace
"In math, equal means 100% swappable. Be precise. Be surgical."
ACTIVATE PRIOR KNOWLEDGE (8 min)
The Swap Challenge:
If I tell you that y = 3x, what can you do with every 'y' you find in any other equation?
"I can swap 'y' for ..."
GUIDED PRACTICE
y = 3x
x + y = 12
1
The Swap Operation
x + ( _______ ) = 12
2
Solve for Both (x and y)
( x , y ) = ( ___ , ___ )
PRIORITY PRACTICE
01 LEVEL 1: BASIC SWAP
x = 4
2x + y = 10
( ___ , ___ )
02 LEVEL 2: EXPRESSION SURGERY
y = x + 2
x + y = 8
( ___ , ___ )
03 LEVEL 3: MASTERCLASS
y = 2x | 3x + y = 20
( __ , __ )
EXIT TICKET (5 min)
y = 2x | x + y = 15
( __ , __ )
HOMEWORK: 8.2
y = 4x
Solution
y = x + 5
Solution
The Great Swap Teacher Guide TEACHER PLAN: 8.2
The Great Swap
North Star Academy
55 MINUTE SESSION
Heart of the Lesson
Algebraic substitution allows us to solve complex systems by using the logical principle that equal expressions are interchangeable.
Instructional Objective
Students will solve systems of linear equations algebraically by substituting one variable with its equivalent expression.
Time Section Teacher Actions & Prompts 5m Do Now Script: "Surgery time. Kick out the 'x', bring in the 5. Use parentheses to protect the swap."
|
| 2m | Excellence |
Message: "In math, equal means swappable. Precision is our baseline for algebraic surgery today."
|
| 8m | Activate Knowledge |
Question: "If y = 3x, explain to your partner what you can do with EVERY 'y' you see in a system."
|
| 15m | Guided Practice |
The Swap: Replace 'y' in second equation with (3x). x + (3x) = 12.
Anticipated Error: Students forget to solve for the second variable.
|
| 20m | Priority Practice |
L1: Standard swap (x=4). L2: Expression surgery (y=x+2).
L3: Challenges with distribution (y=2x).
|
Monitoring Tracker (25 Students)
Lesson 8.2: The Great Swap
# Student Name Lap 1: Uses Parentheses Lap 2: Finds 2nd Var L3: Distribution ET Mastered 1 2 3 4 5 6 7 8 9 10 11
The Great Swap Answer Key Answer Key: 8.2
Secret Weapon
Section Keys
Do Now
y = 13
Guided Practice
x=3, y=9 → (3, 9)
Priority Practice
L1: (4, 2)
L2: (3, 5)
L3: (2, 4)
Exit Ticket
(5, 10)
Homework Key
Prob 1
( 2 , 8 )
Prob 2
( 2 , 7 )
Vanishing Variables Student Packet 8.3: Vanishing Variables
Objective: Solve systems using the elimination method.
Name:
Date:
DO NOW (5 min)
Think: What happens to opposites when added?
5 + (-5) = ___
-2x + 2x = ___
10y - 10y = ___
"Elimination is tactical magic. Precision is the path to zero."
ACTIVATE PRIOR KNOWLEDGE (8 min)
The Vanishing Challenge:
If Line A has +4x and Line B has -4x, what happens when we add the equations?
"The x-variable will..."
GUIDED PRACTICE (15 min)
Align → Add → Solve
2x + 3y = 15
1
The Crunch (Add Down)
7x = _____
2
Solve for Partner (x, y)
( ___ , ___ )
PRIORITY PRACTICE (20 min)
01 LEVEL 1: ROOKIE VANISH
x + y = 10
-x + y = 4
( ___ , ___ )
02 LEVEL 2: ELITE ALIGNMENT
3x + 2y = 8
-3x + 5y = 13
( ___ , ___ )
03 LEVEL 3: MASTERCLASS
4x + 2y = 12 | 4x - 2y = 4
Perform Surgical Addition Here
( __ , __ )
EXIT TICKET
2x + y = 10 | 2x - y = 6
( __ , __ )
HOMEWORK: 8.3
x + y = 15
Solution
2x + 4y = 20
Solution
Mastery Checklist:
L1 Done
L2 Done
L3 Done
Vanishing Variables Teacher Guide TEACHER PLAN: 8.3
Vanishing Variables
North Star Academy
55 MINUTE SESSION
Heart of the Lesson
Systems of equations can be simplified by vertically adding them together, forcing one variable to cancel out through opposites.
Instructional Objective
Students will solve systems of linear equations using the elimination method by identifying and adding opposites (+ and -).
Time Section Teacher Actions & Prompts 5m Do Now Script: "Opposites cancel. If I have 5 and take 5 away, what's left? 0. Make zeros happen."
|
| 2m | Excellence |
Message: "Elimination is tactical magic. Precision is our path to zero."
|
| 8m | Activate Knowledge |
Question: "If Line A has +4x and Line B has -4x, what happens to 'x' when you add the equations?"
|
| 15m | Guided Practice |
The Crunch: Add 2x+5x=7x. Constants: 15+6=21. 7x=21 → x=3.
Error: Forgetting to add constants.
|
| 20m | Priority Practice |
L1: Rookie (x+y, -x+y). L2: Elite (3x, -3x). L3: Master (Rearrange).
|
Elimination Tracker (25 Students)
Lesson 8.3: Vanishing Variables
# Student Name Lap 1: Correct Opposites Lap 2: Constant Col Add L3: Accuracy Score ET Accuracy 1 2 3 4 5 6 7 8 9 10 11 12
Vanishing Variables Answer Key Answer Key: 8.3
Secret Weapon
Section Solutions
Do Now
5+(-5)=0 | -2x+2x=0 | 10y+(-10y)=0
Guided Practice
7x = 21 → x = 3
2(3) + 3y = 15 → 6 + 3y = 15 → 3y = 9 → y = 3
Final: (3, 3)
Priority Practice
L1: 2y = 14 → y=7. x+7=10 → x=3.
L2: 7y = 21 → y=3. 3x+2(3)=8 → 3x=2 → x=2/3.
L3: 8x = 32 → x=4. 2(4)+y=9 → y=1.
Exit Ticket
1. Var Y vanishes.
2. 8x = 16 → x = 2.
3. 4(2) + 2y = 12 → y = 2.
(2, 2)
Homework Master Key
System 1: x+y=15 & x-y=5
2x = 20 → x=10
10+y=15 → y=5
( 10 , 5 )
System 2: 2x+4y=20 & -2x+2y=10
6y = 30 → y=5
2x+4(5)=20 → x=0
( 0 , 5 )
Class Progress Recording Sheet Solution Scavenger Student Packet 8.4: Solution Scavenger
Objective: Determine the number of solutions in a system.
Name:
Date:
DO NOW (5 min)
Compare these lines. Predict: will they touch or stay parallel?
y = 2x + 1 | y = 2x + 5
Prediction: ______________
y = x + 3 | y = 4x - 2
Prediction: ______________
"Work smarter. Identify the DNA, and the solution becomes clear."
ACTIVATE PRIOR KNOWLEDGE (8 min)
The DNA Question:
If two lines have the exact same slope but start at different spots , can they ever meet? Why or why not?
"I believe they..."
GUIDED PRACTICE
SCENARIO 1: The Parallel Problem
y = 3x + 10
Slope 1: ________
Slope 2: ________
None
SCENARIO 2: The Doppelgänger
y = x + 4
Same Slope?
Same Y-Int?
Infinite
PRIORITY PRACTICE
01 LEVEL 1: IDENTIFY THE DNA
y = 5x + 2
One
None
Inf
02 LEVEL 2: IDENTICAL DNA
y = 2x + 8
One
None
Inf
03 LEVEL 3: SCAVENGER CHALLENGE
Scramble Alert: Rearrange then classify!
y = 4x + 1
ONE
NONE
INF
EXIT TICKET
Plan A and B both charge $0.10 per min but start at $20 and $10. Will they ever meet?
y = 0.10x + 20
YES
NO
HOMEWORK: 8.4
The Matching Lab
A. y = x + 4 ; y = x - 2
B. y = 3x + 1 ; y = 3x + 1
C. y = 2x + 5 ; y = -x + 5
One Sol
No Sol
Infinite
Solution Scavenger Teacher Guide TEACHER PLAN: 8.4
Solution Scavenger
North Star Academy
55 MINUTE SESSION
Heart of the Lesson
By analyzing slopes and y-intercepts (the 'DNA'), we classify outcomes without fully solving. DNA match = Prediction power.
Instructional Objective
Students will determine if a system has one solution, no solution, or infinitely many solutions based on coefficients.
Time Section Teacher Actions & Prompts 5m Do Now Script: "Don't solve! Look at the DNA. If slopes are identical, can they touch?"
|
| 2m | Excellence |
Message: "Architects work smarter. Precise slope identification is the key to prediction."
|
| 8m | Activate Knowledge |
Question: "If two paths have the SAME SLOPE but start in DIFFERENT SPOTS, will they meet? Explain why."
|
| 15m | Guided Practice |
Case 1: Parallel tracks. Case 2: Doppelgänger. Case 3: Different slopes.
|
| 20m | Priority Practice |
L1: Identifies DNA correctly. L2: Identifies Infinite vs None. L3: Rearrangement logic.
|
Scavenger Tracker (25 Students)
# Student Name Lap 1: Slope Identified Lap 2: NONE vs INF L3: Logic Correct ET Accuracy 1 2 3 4 5 6 7 8 9 10 11 12
Solution Scavenger Answer Key Answer Key: 8.4
Secret Weapon
Section Solutions
Do Now (DNA Prediction)
1. Parallel / No Solution (m=2, m=2)
Guided Cases
Scen 1: NO SOLUTION (Same m, Diff b)
Scen 2: INFINITE SOLUTIONS (Identical)
Scen 3: ONE SOLUTION (Different Slopes)
Priority Practice
L1: NONE (m=5, m=5, b=2 vs -4)
L2: INFINITE (Identical DNA)
L3: ONE (m=-3 vs 4)
Exit Ticket
Answer: NO
Reasoning: Both lines have the same slope (0.10) but different starting intercepts ($20 vs $10). They are parallel and will never cross.
Homework Master Key
Part 1: Matching Game
A → No Solution
B → Infinite
C → One Solution
Part 2: Reasoning Lab
PARALLEL
Lines that maintain distance.
Inequality Mastery Test Unit 7 Mastery Test
Linear Inequalities (7.1 — 7.8)
Name:
Date:
Instructions: Read carefully. For Multiple Choice, circle the correct letter. For Part 3, show all algebraic work.
Which symbol represents "at most"?
A) <
B) >
C) ≤
D) ≥
Which graph represents x > -2?
Choice A
Choice B
Solve: x + 10 ≤ 5
A) x ≤ 15
B) x ≤ -5
C) x ≥ -5
D) x ≤ 5
When do you flip the inequality symbol?
A) When adding a negative number.
B) When multiplying or dividing by a positive number.
C) When multiplying or dividing by a negative number.
Solve: -2x < 10
A) x < -5
B) x > -5
C) x > 5
D) x < 5
Solve: 2x - 5 ≥ 11
A) x ≥ 3
B) x ≥ 16
C) x ≤ 8
D) x ≥ 8
Solve: 3(x + 2) < 15
A) x < 3
B) x < 5
Solve: 5x + 4 > 2x + 10
A) x > 2
B) x < 2
C) x > 6
D) x > 4
Which represents an "OR" inequality?
A) 5 < x < 10
B) x < 2 or x > 8
Translate: "A number is at least 15."
A) n < 15
B) n > 15
C) n ≤ 15
D) n ≥ 15
Solve: -x/4 + 5 ≥ 3
A) x ≥ 8
B) x ≤ 8
C) x ≤ -8
D) x ≥ -8
A number between -5 and 5:
A) -5 < x < 5
B) -5 > x > 5
C) x ≥ 0
D) x ≤ 5
Solve: 2(3x - 4) ≤ 10
A) x ≤ 1
B) x ≤ 3
C) x ≥ 3
D) x ≤ 6
"Salary plus $50 bonus is no more than $500":
A) s + 50 > 500
B) s + 50 ≤ 500
Which graph shows a "Sandwich"?
Inequality Mastery Review Sheet Unit 7 Mastery Review
Lessons 7.1 — 7.8 Cheat Sheet
Blueprint Series
EXAM PREP
Circle Rule
< > are OPEN .CLOSED .
Sign Flip
Flip sign ONLY when mult/div by a negative .
Sandwich
'AND' puts variable between boundaries.
Keyword Lab
<
Fewer, Less, Under
≤
At Most, Maximum, No more
>
More, Greater, Over
≥
At Least, Minimum, No less
Mini-Practice Lab
01. One-Step
x - 5 ≥ 10
Surgical Solve Space
02. Flip Rule
-3x < 12
Surgical Solve Space
Advanced Skills
The Protocol
1
Distribute Parenthesis
2
Combine Like Terms
3
Isolate the Variable
Protocol Example:
2(x + 5) > 20
2x + 10 > 20
2x > 10
x > 5
Solving the Sandwich
10 < 2x ≤ 20
Blueprint Workspace:
Show Step
Final Strategy
Verify your sign immediately. Logical consistency is your greatest tool!
Master Target
I AM READY
Inequality Mastery Test Key Master Key: Unit 7 Test
MASTER COPY
Multiple Choice Solutions
1. C
2. A
3. B
4. C
5. B
6. D
7. A
8. A
9. B
10. D
11. B
12. A
13. B
14. B
15. A
16. A
17. A
18. C
Open Ended Solutions
Item 19: Multi-Step Solve
-2(x - 3) + 4 ≤ 14
-2x + 6 + 4 ≤ 14 (Dist)
-2x + 10 ≤ 14 (Comb)
-2x ≤ 4 (Sub 10)
x ≥ -2 (FLIP!)
Graph: Closed circle @ -2, shade right.
Item 20: Marcus Word Problem
Part A: Inequality
12t + 25 ≤ 85 (85 = budget left)
Part B: Solve
12t ≤ 60 → t ≤ 5
Part C: Result
Max 5 tickets.
Classroom Performance Data Sheet
01
02
03
04
05
06
07
08
09
10
Architect: Record mastery levels (1-4) for each student.
