Global Beats Slides Grade 8 Mathematics
Linear Functions in Context
Global Beats Math:
Linear Functions & Rates on Tour
Join musicians Zulu, Bongi, and Tshidi as they model studio rates, tour logistics, and festival sales with algebra.
Rate of Change
Unit rate as slope \(m\)
Initial Value
Starting fee as \(y\)-intercept \(b\)
Linear Systems
Break-even points \(y_1 = y_2\)
Lesson Warm-up
Meet the Artists & The Formula
\( y = mx + b \)
Percussionist
Zulu
Records polyrhythmic djembe tracks. Models studio hourly rates and equipment setup fees.
\( C = mh + b \)
Tour Arranger
Bongi
Coordinates sound systems for festivals. Compares audio supplier plans to find break-even days.
Comparing Rates
Vocal Producer
Tshidi
Manages concert merchandise and audio distribution. Balances sales using 2-variable systems.
Linear Systems
Slope \(m\): Rate of change per unit (cost/hr, fee/day)
Intercept \(b\): Initial value (deposit, fixed setup fee)
Worked Example 1 • Zulu's Studio Budget
Solving for Recording Time
Rate of Change & Initial Value
Studio Scenario
Zulu records at Soweto Sound Lab. Setup fee is $150 plus an hourly rate of $65/hour.
Total Budget: $540
\( C = 65h + 150 \)
Question: How many hours can Zulu record?
Step-by-Step Algebraic Solution
1. Substitute \(C = 540\): \( 540 = 65h + 150 \)
2. Subtract initial fee ($150): \( 390 = 65h \)
3. Divide by rate (\(65\)): \( h = \frac{390}{65} = 6 \text{ hours} \)
Conclusion: Zulu can afford exactly 6 hours of recording time.
Key Concept: The slope is the variable rate ($65/hr); the y-intercept is the fixed start value ($150).
Worked Example 2 • Bongi's Tour Sound Rig
Comparing Two Linear Models
Finding Break-Even
Supplier A: Nairobi Sound Plan 1
$200 upfront deposit plus $40 per festival day:
\( y_A = 40x + 200 \)
Supplier B: Dakar Stage Co. Plan 2
$50 low deposit plus $65 per festival day:
\( y_B = 65x + 50 \)
Break-Even Condition: At what days \(x\) do both cost the same?
\( 40x + 200 = 65x + 50 \)
Subtract \(40x\):
\( 200 = 25x + 50 \)
Subtract 50:
\( 150 = 25x \)
Break-Even Point:
\( x = 6 \text{ days} \) ($440)
Bongi's Rule: If tour < 6 days, Dakar Stage is cheaper. If tour > 6 days, Nairobi Sound saves money!
Worked Example 3 • Tshidi's Merchandise Table
Solving a 2-Variable Linear System
System of Equations
The Merch Table
Tshidi sells tour recordings after the festival:
Total Items Sold: 120 items
Vinyl: $25 | Cassettes: $15
Total Revenue: $2,280
Find vinyl (\(v\)) and cassettes (\(c\)) sold.
System: (1) \( v + c = 120 \) | (2) \( 25v + 15c = 2280 \)
Substitute \(v = 120 - c\):
\( 25(120 - c) + 15c = 2280 \)
Distribute & isolate \(c\):
\( 3000 - 10c = 2280 \rightarrow 10c = 720 \rightarrow c = 72 \)
Cassettes (\(c\))
72 sold
Vinyl (\(v = 120 - 72\))
48 sold
Verification: \( 48(\$25) + 72(\$15) = \$1,200 + \$1,080 = \$2,280 \) ✓
Class Discussion & Transition
Festival Profit Challenge
Ready for Worksheet
Concert Collaboration
Zulu, Bongi, and Tshidi co-host the World Rhythm Gala . Fixed stage rental is $1,400. Tickets sell for $35 each.
Profit Function: \( P = 35t - 1400 \)
Prompt 1
What real-world quantity does the initial value -1400 represent?
Prompt 2
What does the slope 35 tell us about the rate of change per ticket?
Prompt 3
How many tickets must the trio sell to break even (\(P = 0\))?
Now let's apply these strategies on the Global Beats Worksheet !
CCSS.MATH 8.EE & 8.F
Global Beats Worksheet Grade 8 Mathematics • Linear Functions & Modeling
Global Beats: Rhythms & Rates
Name: __________________________
Date: ____________ Period: _______
Mission: Musicians Zulu (percussionist), Bongi (arranger), and Tshidi (vocalist/producer) are on an international world music tour. Apply linear equations, slope, and systems to analyze their production costs and logistics.
1
Zulu's Studio Recording Sessions
Slope-Intercept Form
Zulu books time at Johannesburg Sound Studio to record djembe tracks. The studio charges a one-time sound engineer setup fee of $120 , plus an hourly tracking fee of $45 per hour .
(a) Write a linear cost equation \(C\) for \(h\) hours:
\( C = \)
(b) Identify slope \(m\) and state its real-world meaning:
\(m = \) ______ | Rate of change per: ______________
(c) Identify \(y\)-intercept \(b\) and explain its meaning:
\(b = \) ______ | Initial fee before hours start: _______
(d) Total invoice is $390 . Hours recorded:
\( 390 = 45h + 120 \rightarrow h = \) ________ hrs
Show algebraic steps for 1(d):
2
Bongi's Sound Equipment Rental
Comparing Functions
Bongi rents mobile stage rigs for a regional tour and compares quotes from two audio suppliers:
Supplier A: Cape Sound Logistics
\( C_A = 50d + 180 \)
$180 deposit + $50/day
Supplier B: Durban Audio Rig
\( C_B = 30d + 300 \)
$300 deposit + $30/day
(a) Set the equations equal to find the break-even days (\(d\)) where both cost the same. Show steps:
Break-Even Days: \( d = \) ________ days | Cost at Break-Even: $________
(b) Bongi's tour is 10 days . Calculate each supplier's cost and state the cheaper option:
Supplier A: $________ | Supplier B: $________ | Choose: Supplier ________ (Save $________)
Global Beats Mathematics • Page 1 of 2 • Continue to page 2 for Systems of Equations
Global Beats Worksheet • Page 2
Name: __________________________ Period: _______
3
Tshidi's Festival Merchandise System
Systems of Equations
At the Pan-African Music Festival, Tshidi sells thumb pianos (kalimbas, \(k\)) for $30 each and tour vinyl records (\(v\)) for
Global Beats Answer Key Teacher Edition • Scoring & Solutions Guide
Global Beats: Answer Key & Rubric
Total Points: 25 pts
1. Zulu's Studio Recording Sessions (8 Points) CCSS 8.F.B.4
(a) Equation: \( C = 45h + 120 \) (2 pts: 1 pt for slope term, 1 pt for initial value)
(b) Slope & Meaning: \( m = 45 \)
Context Meaning: The studio charges $45 for each additional hour of recording time (unit rate of change). (2 pts)
(c) \(y\)-Intercept & Meaning: \( b = 120 \) (or point \((0, 120)\))
Context Meaning: The fixed, non-refundable setup and sound engineering fee charged before any recording hours begin. (2 pts)
(d) Solving for \(h\) when \(C = \$390\):
\( 390 = 45h + 120 \)
\( 390 - 120 = 45h \rightarrow 270 = 45h \)
\( h = \frac{270}{45} = 6 \text{ hours} \)
Final Answer: Zulu recorded for exactly 6 hours. (2 pts)
2. Bongi's Sound Equipment Rental (7 Points) CCSS 8.EE.C.8
(a) Break-Even Analysis:
Set \( C_A = C_B \): \( 50d + 180 = 30d + 300 \)
\( 50d - 30d = 300 - 180 \)
\( 20d = 120 \rightarrow d = 6 \text{ days} \)
Total Cost: \( C = 50(6) + 180 = 300 + 180 = \$480 \) (or \( 30(6) + 300 = \$480 \))
Break-even at \( d = 6 \) days, where both cost $480. (4 pts)
(b) Evaluation for 10-day tour:
Supplier A: \( 50(10) + 180 = \$680 \)
Supplier B: \( 30(10) + 300 = \$600 \)
Recommendation: Choose Supplier B (Durban Audio Rig). Bongi saves $80 ($680 - $600). (3 pts)
Global Beats Answer Key • Page 1 of 2 • Scoring & Teacher Guide
Global Beats Answer Key • Page 2 Solutions Teacher Reference
3. Tshidi's Merchandise System (6 Points) CCSS 8.EE.C.8
(a) System Formulation:
Quantity Equation: \( k + v = 85 \)
Revenue Equation: \( 30k + 20v = 2050 \)
(2 pts: 1 pt each equation)
(b) Step-by-Step Algebraic Solution:
From Eq 1, isolate \( v \): \( v = 85 - k \)
Substitute into Eq 2: \( 30k + 20(85 - k) = 2050 \)
Distribute: \( 30k + 1700 - 20k = 2050 \rightarrow 10k + 1700 = 2050 \)
Subtract 1700: \( 10k = 350 \rightarrow k = 35 \) kalimbas
Back-substitute: \( v = 85 - 35 = 50 \) vinyl records
Final Answer: Tshidi sold 35 kalimbas and 50 vinyl records. (3 pts)
(c) Verification Check:
Items: \( 35 + 50 = 85 \) ✓ | Revenue: \( 30(35) + 20(50) = 1050 + 1000 = \$2050 \) ✓
(1 pt)
