Mission 15 · Unit 6

Inequalities

6.EE.B.8 · Unit 7
Today's objective: Write and graph inequalities to represent real-world constraints.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The 6th-grade class is planning a field trip to the science museum. The bus holds a maximum of 48 passengers. There are already 6 adults confirmed. Students sign up in groups. Your team must write and graph inequalities to figure out how many student groups of different sizes can fit, and advise the trip organizer on the sign-up limit.

SCHOOL BUS MAX CAPACITY 48 ADULTS 6 6 + s ≤ 48

The Investigation

The Problem: The bus holds at most 48 people. 6 adults are already going. Let s = the number of students. Write an inequality to show how many students can go. Then graph the solutions on a number line. Finally, determine: if students sign up in groups of 7, what is the maximum number of full groups that can go?

Visual Model: Number Line for 6 + s ≤ 48

Number Line: s ≤ 42 0 10 20 30 40 42 50 All values from 0 to 42 are solutions Closed circle at 42 means 42 IS included (≤ means "or equal to") Groups of 7: 7, 14, 21, 28, 35, 42 ✔ (6 full groups)

Step-by-Step Investigation Guide

  1. Identify the constraint. The bus holds at most 48 people. Write this as: total people ≤ 48. Guiding question: What does "at most" mean in math language?
  2. Write the inequality. Total people = 6 adults + s students. So: 6 + s ≤ 48. Guiding question: Why do we use ≤ instead of = or <?
  3. Solve for s. Subtract 6 from both sides: s ≤ 42. Guiding question: What operation undoes adding 6?
  4. Graph on a number line. Draw a number line. Place a closed circle at 42. Shade to the left toward 0. Guiding question: Why closed circle and not open? Why shade left?
  5. Test solutions. Is s = 42 a solution? Check: 6 + 42 = 48 ≤ 48. Yes! Is s = 43 a solution? Check: 6 + 43 = 49. Is 49 ≤ 48? No! Guiding question: Why is testing the boundary value important?
  6. Answer the group question. If groups have 7 students, how many full groups fit? 42 ÷ 7 = 6 groups. Guiding question: What if groups had 8 students instead?

Language Support: Key Vocabulary

Inequality
A math sentence that uses <, >, ≤, or ≥ instead of =.
At most
The biggest it can be. "At most 48" means 48 or less. Use ≤.
At least
The smallest it can be. "At least 10" means 10 or more. Use ≥.
Solution
A value that makes the inequality true.
Number line
A line with numbers in order, used to show solutions visually.
Constraint
A limit or rule. The bus capacity is a constraint.
Sentence Frames:

"The inequality is _____ because the bus can hold at most _____ people."

"I used a closed/open circle because _____."

"The value s = _____ is/is not a solution because _____."

Multiple Representations

Approach 1: Algebraic

6 + s ≤ 48
s ≤ 48 - 6
s ≤ 42
The number of students must be 42 or fewer.

Approach 2: Number Line

Draw a line from 0 to 50. Place a filled circle at 42. Shade from 0 to 42. Every shaded value is a valid number of students.

Approach 3: Table of Values

Test: s=40 → 46 ≤ 48 ✔
s=42 → 48 ≤ 48 ✔
s=43 → 49 ≤ 48 ✘
s=50 → 56 ≤ 48 ✘

Team Roles

Facilitator Read the bus scenario aloud. Make sure everyone identifies the constraint (at most 48). Keep the team on schedule.
Model Builder Draw the number line on team paper. Mark 0 and 42. Use the correct circle type (closed). Shade the solution region.
Precision Checker Test at least 3 values: one below 42, one equal to 42, and one above 42. Confirm the circle type is correct. Check the group division.
Reporter Prepare the defense: state the inequality, explain the number line, and give the group recommendation with evidence.

Timed Lab Phases

Launch Phase
03:00

Read the field trip scenario. Assign roles. Identify the constraint and the variable.

  • What is the maximum number of people allowed?
  • How many adults are already confirmed?
  • What does s represent?
Checkpoint: Everyone can say the constraint in their own words.

Write the inequality 6 + s ≤ 48. Solve for s. Model Builder draws the number line.

  • What operation isolates s?
  • Should the circle at 42 be open or closed? Why?
  • Which direction do you shade?
Checkpoint: Inequality written, solved, and graphed on a number line.

Test boundary values. Calculate maximum number of groups of 7.

  • Does s = 42 satisfy 6 + s ≤ 48?
  • Does s = 43 satisfy it?
  • 42 ÷ 7 = ? How many full groups?
Checkpoint: Three test values checked. Group answer calculated.

Reporter prepares the defense with claim, evidence, and recommendation.

  • "The maximum number of students is _____ because _____."
  • "We used a closed circle because _____."
  • "We recommend _____ groups of 7 because _____."
Checkpoint: Defense has inequality, number line explanation, and group recommendation.

Challenge Extensions

Extension Problem: The museum requires at least 3 adults per 15 students for safety. With 6 adults, write a new inequality for the maximum students based on this chaperone rule. Is this more or less restrictive than the bus capacity?

What If?

  • What if the bus held 54 passengers? How does the inequality change?
  • What if each student also brings a backpack that takes half a seat? Rewrite the constraint.
  • Write an inequality for the minimum number of students needed for the trip to happen (at least 20).

Real-Life Connections

Inequalities set limits everywhere: elevator weight limits, age requirements, spending budgets, speed limits, and food serving sizes.

Defense Preparation

  1. What inequality did you write? What does each part represent?
  2. Why is the circle at 42 closed instead of open?
  3. Name two values that are solutions and one that is not. How do you know?
  4. How many full groups of 7 students can go? Show the math.
Sentence Starters:
  • "Our inequality is _____ which means _____."
  • "We used a closed circle because the symbol ≤ includes _____."
  • "S = 42 is a solution because _____."

Rubric

Criteria Excellent (4) Proficient (3) Developing (2)
Inequality Correct inequality with context explanation Correct inequality Inequality attempted
Number line Correct circle, shading, and labels Minor labeling gaps Number line attempted
Testing values 3+ values tested at boundary 2 values tested 1 value tested
Recommendation Group answer with clear reasoning Correct answer given Attempted

Exit Product

Your team submits: A Field Trip Inequality Report that includes:
  • The inequality 6 + s ≤ 48 with an explanation of each part
  • Solution: s ≤ 42 with algebraic steps shown
  • A number line graph with correct circle type and shading
  • At least 3 tested values (solution, boundary, non-solution)
  • A recommendation: how many groups of 7 can go, with evidence

Self-Assessment Checklist