Proportions in Production

Overview

Today students use the real-world context of YKTA in Huntsville, Alabama, to compute how many parts can be produced given a Takt time and available production time. They will connect proportional relationships to unit rates and use those ideas to solve multi-step ratio and percent problems.

Learning intentions

Students will be able to:

• Recognize proportional relationships and represent them using equivalent ratios or equations.
• Compute unit rates from ratios of fractions and explain what the unit rate means in context.
• Calculate parts per hour using Takt time and total available time.
• Solve real-world problems involving proportional reasoning and check that results make sense.

Success criteria

I can…

• Determine whether two quantities are in a proportional relationship by testing equivalent ratios.
• Find the parts per hour from a given Takt time (in minutes or fractions of an hour).
• Use an equation (like total = rate × time) to model a proportional situation.
• Solve a real-world production problem accurately and justify my method.

Curriculum links

• Ratios and proportional relationships: Compute unit rates associated with ratios of fractions (7.RP.A.1).
• Ratios and proportional relationships: Recognize and represent proportional relationships (7.RP.A.2, 7.RP.A.2a).
• Ratios and proportional relationships: Represent proportional relationships by equations (7.RP.A.2c).
• Ratios and proportional relationships: Use proportional relationships to solve multistep ratio and percent problems (7.RP.A.3).

Lesson structure (60 minutes)

1. 0–8 min · Hook (Real-world story + quick think). Teacher shows a short prompt on the board: “A plant makes car parts for Mazda–Toyota using Takt time. If one part is completed every __ minutes, how many parts fit in __ hours?” Students do a 2-minute think, 2-minute pair share, and 1–2 students share reasoning.

2. 8–18 min · Mini-lesson (Unit rate + proportional relationships). Teacher reviews:

• Proportional relationship means a constant unit rate.
• Unit rate can come from a complex fraction (for example, parts per hour from minutes per part).
• How to test proportionality using equivalent ratios in a table. Teacher models one example: “Takt time = 2 minutes per part.” Students help rewrite: 1 part / 2 minutes, then invert to get 30 parts per hour. They record the steps and the meaning of “parts per hour.”

3. 18–30 min · Guided practice (YKTA parts per hour). Teacher provides a small data set and works through it with the class:

• Scenario A: Takt time = 1/3 hour per part (or 20 minutes per part). Total available time = 2 hours.
• Scenario B: Takt time = 1/4 hour per part (or 15 minutes per part). Total available time = 3.5 hours. For each scenario, teacher asks students to complete:
• Unit rate (parts per hour)
• Number of parts produced = unit rate × hours available Students work in pairs on a table of equivalent ratios (time per part and parts per hour) and decide whether the relationship is proportional.

4. 30–42 min · Guided practice (Equation representation + proportional check). Teacher asks: “How can we write an equation that matches the proportional situation?” Students derive and use: P = r × T, where r = parts per hour and T = total hours. Teacher emphasizes that r is constant, so doubling time doubles parts. Students confirm proportionality by checking that r stays the same when they compute from different time amounts.

5. 42–52 min · Independent practice (Real-world problem solving + multistep). Students complete two tasks individually (show work required):

• Task 1 (proportional rate): At YKTA, Takt time is 18 minutes per part. There are 4 hours available. Compute parts produced using unit rate.
• Task 2 (multistep percent): The plant must produce 5% more parts to replace a predicted defect rate. Using your Task 1 result, compute the final number of parts needed. Students are reminded to (a) convert minutes to fractions of an hour when helpful, (b) compute unit rate, (c) use that rate to scale by time, and (d) apply percent correctly.

6. 52–58 min · Assessment (Quick check + reasoning). Teacher collects 3–5 solutions quickly (document camera or spot-check). Students answer a one-question prompt on paper: “If Takt time stays the same, are parts proportional to time? Explain using a ratio or equation.” Teacher uses responses to identify misconceptions about proportionality or unit conversion.

7. 58–60 min · Exit ticket (1-minute targeted check). Exit ticket: “Takt time = 12 minutes per part. Find parts per hour, then compute parts in 2.5 hours.” Students submit; teacher uses results for next-day grouping.

Resources

• Teacher-created YKTA scenario worksheet (table + two independent tasks)
• Student graph paper or lined paper for tables and equations
• Timer or slide showing minute-by-minute pacing
• Fraction-to-decimal conversion reference (optional handout)
• Calculators allowed for checking, but not for the entire unit-rate step (teacher discretion)
• Whiteboard or document camera for modeling one example

Assessment

• Formative: teacher circulates during guided practice, checking unit-rate setup and proportionality tests using equivalent ratios.
• Formative: during guided practice, teacher listens for correct meaning of the unit rate (“parts per hour”) and constant rate reasoning.
• Summative (exit ticket): parts per hour from Takt time and parts from total available time; includes unit rate + scaling accuracy.

Differentiation

• Support: provide a partially completed table for Task 1 showing “minutes per part → parts per hour” conversion steps; include sentence starters like “Because r is constant, parts are proportional to time.”
• Support: offer an additional example using whole numbers first (e.g., 30 minutes per part) before moving to fractional hours.
• Extension for early finishers: change one condition (e.g., a break reduces available time by 20 minutes) and have students update both the unit-rate application and final percent adjustment.
• EAL/SEN: allow use of a ratio diagram (time per part with arrows to parts per time) and encourage highlighting the “constant rate” in every solution step.