Algebra 1 - Unit 1 Day 5 Notes: Identifying Rules for Patterns

Warm-up

• Brandee's scores: 70, 92, 91, 71, 89, 84, 0, 85
• a) Average of the scores:
  • Sum = 70 + 92 + 91 + 71 + 89 + 84 + 0 + 85 = 582
  • Number of scores = 8
  • Average = $\frac{582}{8} = 72.75$
• b) Does this average truly represent her abilities?
  • Consider outliers and the impact of a missing/makeup test
  • A single very low score (0) can drag down the average and may not reflect steady performance.
• Brandee made up the missed test and earned a 60 on the makeup.
• c) New average after makeup test:
  • If the missed test (0) is replaced by 60, new sum = 582 - 0 + 60 = 642
  • New average = $\frac{642}{8} = 80.25$
• Difference due to the 0 score:
  • Old average: 72.75; New average: 80.25
  • Difference = $80.25 - 72.75 = 7.50$
• Does the new average better represent Brandee’s ability?
  • The makeup test improves the average and reduces the distortion caused by the zero, giving a fairer indication of ability, but it depends on policies about makeup scores and consistency over time.

Learning Targets

• Understand that explicit rules can be used to find any term in a sequence, without knowing the one before it.
• Make use of structure to generalize a formula for the nth term of a sequence.
• Recognize and describe geometric sequences by identifying a common ratio between terms.
• Distinguish between arithmetic and geometric growth.

Problem 1 – Seeing Squares

• Goal: write a rule describing a visual pattern of small squares.
• Information given: Figures 1–4 show increasing numbers of small squares.
• What to determine:
  • How many squares are in each figure?
  • How does the pattern grow?
  • Sketch/describe Figure 5 and count its squares.
  • Find a faster way to determine the number of squares (not by counting each one).
  • Determine Figure 38’s number of squares; show the rule works for N = 1, 2, 3, 4.

Problem 1 – Observations and Rule

• Figure counts (based on pattern in the problem):
  • Figure 1: 1 small square
  • Figure 2: 4 small squares
  • Figure 3: 9 small squares
  • Figure 4: 16 small squares
• Pattern growth:
  • The difference between successive figures is 3, 5, 7, … (increases by 2 each step)
  • This suggests a rule of the form a_n = n^2 for the number of squares in Figure n
• Figure 5: would have 25 small squares (since 5^2 = 25)
• A faster method:
  • Use the explicit rule a_n = n^2 to compute any figure quickly
• Figure 38:
  • Number of small squares = $38^2 = 1444$
• Verification for n = 1, 2, 3, 4:
  • a1 = 1^2 = 1; a2 = 2^2 = 4; a3 = 3^2 = 9; a4 = 4^2 = 16

Tying it all together (Explicit Rules)

• LT# Look for a relationship between the term number (Figure #) and the term value (number of squares) and describe it with an equation.
• Explicit rule for this pattern: the number of small squares in Figure n is given by
  • $a_n = n^2$
• This allows you to find any term directly without referencing previous terms.

Check Your Understanding – Sequence in Figure 1–6 (Problem 1)

• 1, 4, 9, 16, 36, 49
• a) Find the 5th term and the 8th term of the sequence.
  • 5th term is the 5th entry: in the given sequence it is 36
  • 8th term: if the rule is "squares of natural numbers with 5 skipped" this would be
  • bases: 1,2,3,4,6,7,8,9,…
  • 8th term = 9^2 = 81
  • A simple explicit rule that matches the given terms is: let the base sequence be the natural numbers with 5 skipped:
  • Let bn = n for n ≤ 4, and bn = n+1 for n ≥ 5
  • Then an = bn^2
  • This yields a5 = (5+1)^2 = 36 and a8 = (8+1)^2 = 81
• b) Relationship between term number and term value?
  • For n ≤ 4: a_n = n^2 (1, 4, 9, 16)
  • For n ≥ 5: a_n = (n+1)^2 (36, 49, 64, 81, … skipping 25 = 5^2 observed in the given list)
• c) Explicit rule for the nth term:
  • Piecewise form:
  • an = egin{cases} n^2, & n efore 5 \ (n+1)^2, & n ext{ from } 5 ext{ onward} \ ext{(interpreting the sequence as squares with 5 skipped)} \ ext{Or equivalently: } an = bn^2, ext{ where } bn = egin{cases} n, & n [2pt] ext{for } n ≤ 4 \ n+1, & n ≥ 5 \ \ \ \ \ \ \ \ \ \ \ \ \end{cases}
  • This captures the observed terms (1,4,9,16,36,49,…) and predicts a_8 = 81.

Problem 2 – How Many Teams? (March Madness)

• Important information in the problem:
  • There are 64 teams in round 1, paired to play against one opponent based on ranking.
  • It is a single-elimination tournament: the winner of each game advances; losers are removed.

Problem 2 – Questions and Answers

• 1) How many teams will be in round 2? How do you know?
  • Round 1 eliminates half the teams: 64 → 32 teams in round 2
• 2) How many teams will be in round 3?
  • Round 2 eliminates half again: 32 → 16 teams in round 3
• 3) How many rounds will there be before the championship game?
  • The typical sequence: Round of 64, Round of 32, Sweet 16 (Round of 16), Elite Eight, Final Four, Championship Game
  • This means there are 5 rounds before the championship game (the 6th round is the Championship Game itself)
• 4) Does the same number of teams get eliminated every round? Explain.
  • No. Each round halves the number of teams remaining (until you reach 1 champion). The number eliminated per round is not constant.
• 5) Describe how the number of teams changes after each round.
  • After each round, the number of teams remaining is divided by 2: 64 → 32 → 16 → 8 → 4 → 2 → 1
• 6) Plot points to show the relationship between the round and the number of teams remaining. What do you notice about the shape?
  • Relationship is exponential/halving per round: a rapid drop that forms a downward curve when plotted (discrete steps).
• 7) General rule for number of teams after r rounds:
  • T(r) = rac{64}{2^r} = 64 imes 2^{-r}
  • This holds while the result is an integer; it matches the halving pattern of a single-elimination bracket.

Problem 3 – Gramma Fletcher’s Cookies

• Gramma Fletcher makes 243 cookies in week 1, then each week makes two-thirds of the previous week's amount.
• a) How many cookies in week 2? In week 4?
  • Week 2: 243 imes rac{2}{3} = 162
  • Week 4: Week 2 → Week 3 → Week 4: 162 → 108 → 72
  • So week 4: 72 cookies
• b) In which week will she make 32 cookies?
  • General term: a_n = 243 imes igl(rac{2}{3}igr)^{n-1}
  • Solve 243 imes igl(rac{2}{3}igr)^{n-1} = 32
  • igl(rac{2}{3}igr)^{n-1} = rac{32}{243} = igl(rac{2}{3}igr)^5
  • Therefore n - 1 = 5 \ n = 6
  • Week 6 will have 32 cookies
• c) Can the number of cookies per week be described as a geometric sequence? Explain why or why not.
  • Yes. Each week, multiply the previous week’s amount by the constant ratio r = rac{2}{3}
  • Therefore the sequence is geometric with common ratio r = rac{2}{3}

QuickNotes – Tying it all together

• Geometric sequences have a common ratio (multiplier).
• The previous term is multiplied by the same amount each time.
• If the common ratio is between 0 and 1, terms decrease.
• If the common ratio is greater than 1, terms increase.

Problem 1 (Page 18) – Determine sequence type

• Determine if each sequence is arithmetic, geometric, or neither. Provide a reason.
• a) 15, 10, 5, 0, -5, …
  • Difference between consecutive terms is -5; therefore arithmetic (d = -5).
• b) 2, 4, 8, 16, 32
  • Each term is multiplied by 2; therefore geometric (r = 2).
• c) 3, 3, 3, …
  • Constant sequence; it is both arithmetic (d = 0) and geometric (r = 1).
• d) 1, 4, 9, 16, 25, 36, …
  • These are perfect squares, not produced by a constant addition or a constant multiplication; therefore neither arithmetic nor geometric.

Problem 2 – Learning Opportunity (Page 19)

• Note: Some LOs are incomplete. Complete 1.02, 1.03 on Delta Math and upload 1.04 Review packet before doing 1.05.
• This section reinforces progression from explicit rules to modeling terms in sequences and applying to real-world patterns (patterns, averages, tournaments).

Connections and takeaways

• Explicit rules enable finding any term without recourse to prior terms (useful for quick computation and verification).
• Distinguishing arithmetic vs geometric growth helps model real-world processes (e.g., cookie production, population growth, investment) and choose the correct formula.
• Real-world contexts (grading, sports tournaments, patterns in visuals) provide concrete applications of sequence rules and help with interpretation of results and limitations of models.

Formulas and key equations to remember

• Arithmetic sequence (a_1 = a, common difference d):
  • a_n = a + (n-1)d
• Geometric sequence (a_1 = a, common ratio r):
  • a_n = a imes r^{n-1}
• Number of teams remaining after r rounds in a single-elimination bracket (starting with 64):
  • T(r) = rac{64}{2^r} = 64 imes 2^{-r}
• Pattern for Seeing Squares (Figure n) – explicit rule:
  • a_n = n^2
• Gramma Fletcher cookies (week n):
  • a_n = 243 imes igl(rac{2}{3}igr)^{n-1}$$