Mission 24 Unit 9 6.EE.B.6 · 6.EE.C.9

Patterns and Rules

6.EE.C.9 · Unit 10
Today's objective: Write a rule that relates input and output values in a pattern.
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 school cafeteria is setting up tables for a big family night. Each rectangular table seats 4 people. When two tables are pushed together end-to-end, the combined table seats 6 people (not 8, because 2 seats are lost where the tables touch). Three tables in a row seat 8 people. The cafeteria has 15 tables. Your team must find the pattern, write a rule, and figure out how many people can sit at all 15 tables pushed into one long row.

Table Seating Pattern 1 table = 4 seats = 4 2 tables = 6 seats X X = 6 3 tables = 8 seats = 8 Pattern Discovery Tables: 1 → 2 → 3 → 4 → ... → t Seats: 4 → 6 → 8 → 10 → ... → ? +2 each time!

The Problem

How many people can sit at 15 tables pushed into one long row? Use the visual pattern to discover the rule. Then write an equation using a variable. Verify your equation works for 1, 2, 3, and 4 tables before predicting the answer for 15. Finally, explain WHY the pattern works — where do the "lost seats" come from?

Visual Model: Growing Pattern

Finding the Rule T1 4 seats +2 T1 T2 6 seats +2 8 seats +2 10 seats Two Ways to See the Rule Method 1: Constant Rate + Start Starts at 4, adds 2 each time s = 2t + 2 Check: 2(1)+2=4, 2(2)+2=6, 2(3)+2=8 Method 2: Think About Sides Each table adds 2 sides, plus 2 ends s = 2t + 2 2 seats per table + 2 end seats

Step-by-Step Investigation Guide

  1. Draw the pattern. Sketch 1 table (4 seats), 2 tables (6 seats), 3 tables (8 seats), and 4 tables (? seats). Count the seats each time. How many seats does the 4th configuration have? Did you predict correctly?
  2. Make a table of values. Tables (t): 1, 2, 3, 4, 5. Seats (s): 4, 6, 8, 10, 12. Calculate the change between each row. What is the constant change? Why does it stay the same?
  3. Find the rule. The seats go up by 2 each time. If t=1 gives s=4, then s = 2t + 2. Verify: 2(1)+2=4, 2(2)+2=6, 2(3)+2=8. It works! What does the 2 in "2t" represent physically? What does the "+2" represent?
  4. Explain why the rule works. Each table adds 2 long sides for seating. The +2 comes from the 2 end seats (one on each end of the row). If you pushed tables into a ring instead of a row, would the rule change?
  5. Predict for 15 tables. s = 2(15) + 2 = 32 people. Is this enough for 30 parents? For 35? What is the minimum number of tables needed to seat 40 people in a row?

Language Support: Key Vocabulary

Pattern
Something that repeats or changes in a regular way. Here: seats go up by 2.
Rule / Equation
A math sentence that tells you how to find the output from the input. s = 2t + 2.
Variable
A letter that stands for a number that can change. t = tables, s = seats.
Constant
A number that stays the same. The +2 (end seats) is constant no matter how many tables.
Rate of Change
How much the output changes for each unit of input. Here: +2 seats per table.
Verify
To check that your rule gives the correct answer for known values.
Sentence Frames:

"I see a pattern: every time we add 1 table, we add ___ seats."

"The rule is s = ___ because each table contributes ___ seats, plus ___ end seats."

"For 15 tables, I predict ___ seats because ___."

Multiple Representations

The visual pattern shows that each table contributes its 2 long sides for seating. When tables are pushed together, the touching ends are "lost." Only the 2 far ends of the entire row keep their seats. So: s = 2t + 2 (2 seats per table for the sides + 2 end seats).

Tables (t) Seats (s) Change Rule Check: 2t+2
1 4 -- 2(1)+2=4
2 6 +2 2(2)+2=6
3 8 +2 2(3)+2=8
4 10 +2 2(4)+2=10
5 12 +2 2(5)+2=12
15 32 -- 2(15)+2=32

If you plot (t, s) on a coordinate graph — (1,4), (2,6), (3,8), (4,10), (5,12) — the points form a straight line. The line has a slope of 2 (goes up 2 for every 1 to the right) and crosses the s-axis at 2 (the "+2" in the equation). At t=15, reading the graph gives s=32.

Team Roles

Facilitator
Read the problem. Guide the team to draw the first 4 configurations before jumping to the equation. Ask: "Why do we lose 2 seats each time we add a table?"
Model Builder
Draw the table configurations for 1-5 tables. Show the seats as circles. Mark the "lost" seats with an X. Create the data table.
Precision Checker
Verify the rule s = 2t + 2 works for ALL known values (t=1 through t=5). Check the final answer for t=15.
Reporter
Prepare to explain WHERE the rule comes from (not just WHAT it is). Connect the 2t to "2 sides per table" and the +2 to "end seats."

Timed Lab Phases

Ready
Click a phase, then press Start.
03:00
  • Read the scenario. Draw 1 table with 4 seats.
  • Assign roles.
  • Predict: how many seats for 4 tables? Write your guess.
  • Checkpoint: Does everyone understand WHY seats are lost when tables touch?
  • Draw configurations for 1-5 tables.
  • Count seats carefully. Build the data table.
  • Find the constant change (+2).
  • Write the equation s = 2t + 2.
  • Checkpoint: Does your equation pass the "check all known values" test?
  • Calculate seats for 15 tables: s = 2(15) + 2 = 32.
  • Answer: can 30 parents sit? What about 35?
  • Find the minimum tables needed for 40 people.
  • Checkpoint: Can you explain WHY the rule works using the physical model?
  • Reporter: practice explaining the rule AND where it comes from.
  • Team: answer the defense questions.
  • Revise any weak explanations.
  • Checkpoint: Can you explain the +2 without just saying "because the table shows it"?

Challenge

Extension: What if the tables were hexagonal (6 sides) instead of rectangular? Each hexagonal table seats 6 alone. Two hexagons pushed together seat 10 (losing 2 at the joint). Find the new rule and predict seating for 15 hexagonal tables.
What If...?
  • What if you arranged the rectangular tables in a square (like a banquet) instead of a row? How would the pattern change?
  • What if each table seated 6 instead of 4? Write the new rule.

Real-Life Connection: Event planners use exactly this kind of thinking. When a wedding planner arranges long banquet tables, they use rules like s = 2t + 2 to figure out how many tables to rent. Understanding patterns saves time and money.

Defense Preparation

  1. How did you find the rule from the pattern?"We noticed the seats increase by ___ each time, so we tried s = ___t + ___. We checked it for ___."
  2. What does each part of s = 2t + 2 represent in real life?"The 2t represents ___ and the +2 represents ___."
  3. How do you know your rule works for values you did not draw?"We verified the rule for t = 1 through 5, and the pattern is ___. Since the rate of change is constant, we trust it continues."
  4. What is the minimum number of tables to seat 40 people?"We solved 40 = 2t + 2, which gives t = ___. So we need at least ___ tables."
Rubric Quick-Check:
  • Visual pattern drawn for at least 3 configurations
  • Data table with at least 5 rows
  • Rule written as an equation with variables defined
  • Rule verified against all known values
  • Prediction for t=15 stated and explained
  • Physical explanation of WHY the rule works

Exit Product

Submit a one-page Lab Report that includes:
  • Drawings of at least 3 table configurations with seats labeled
  • A data table showing t and s values
  • The equation s = 2t + 2 with a sentence explaining each part
  • The answer for 15 tables with verification
  • An explanation of WHERE the rule comes from (not just WHAT it is)
Self-Assessment:
  • I can see a pattern in a growing visual model
  • I can write a rule using variables from a pattern
  • I can verify my rule against known data
  • I can explain WHY the rule works, not just state it

Work Space

Pattern Drawing:



Table and Rule:



Prediction and Defense: