Lesson 12 · Three Rich Problems

Rich Task Session · Mia

Each problem below takes longer than a worksheet question. There is no obvious first step written in — you decide where to start. You may try something that doesn't work; that is completely normal and useful. The goal is not to find the answer quickly. It is to think carefully, notice things, and be surprised. Give each problem at least 15 minutes before asking for a hint.

Problem A · ~15 minutes

The Two Clues

A mystery sequence has exactly two known terms: the 3rd term is 19 and the 6th term is 46.

Part 1. Could this be an arithmetic sequence? If so, find $d$ and $a_1$, and write out the first 8 terms. Check that the 3rd and 6th terms match.

Part 2. Could this be a geometric sequence? Set up the equation you would need to solve. You do not need to find a decimal answer — just decide: does a "nice" whole-number ratio $r$ work?

Part 3. Can you invent a completely different sequence (not arithmetic, not geometric) that also passes through 19 at position 3 and 46 at position 6? Write it out. There is more than one answer.

Stuck on Part 1? Between position 3 and position 6, how many steps are there? How much did the sequence grow in total over those steps?

Problem B · ~20 minutes

The Savings Race

Three friends all want to save money for a joint stable trip that costs 225 € total.

Rider	Starts with	Saves per week
Mia	15 €	18 € per week
Lea	45 €	12 € per week
Sophie	0 €	25 € per week

Part 1. Write an expression for how much each person has saved after $w$ weeks.

Part 2. Is there a week when Mia and Lea have exactly the same amount? If so, when and how much do they each have?

Part 3. Is there a week when all three have exactly the same amount? Investigate and explain your answer.

Part 4. Together, they need 225 €. Write an inequality for when their combined total is at least 225 €. In which week can they first afford the trip?

Stuck on Part 2? Set the two expressions equal. Solve for $w$. Does $w$ come out as a whole number?

Problem C · ~20 minutes

Every Score Is Possible

At a dressage competition, a rider starts at 0 points.

Each correctly executed movement earns 8 points.
Each error costs 11 points.

So a rider who does $c$ correct movements and $e$ errors ends with score $8c - 11e$.

Part 1. Find a combination $(c, e)$ — both non-negative whole numbers — that gives a final score of exactly 5. Show your working.

Part 2. Find a combination that gives a score of exactly 13.

Part 3. Find a combination that gives a score of exactly 100.

Part 4. Try 5 more target scores of your choice (pick anything from 1 to 200). For each one, find a valid $(c, e)$ pair.

Part 5 (big question). Based on your experiments, write a conjecture: do you think every positive whole number score is achievable? Or can you find a score that is impossible? Explain your reasoning.

Stuck on Part 1? Start by trying small values of $e$ (0, 1, 2, …) and see if $8c = 5 + 11e$ ever gives a whole number $c$.

Show answers

Problem A · Part 1

Yes. 3 steps from $a_3$ to $a_6$: $46 - 19 = 27$, so $d = 9$ and $a_1 = 19 - 2 \times 9 = 1$. Sequence: 1, 10, 19, 28, 37, 46, 55, 64 — 3rd term 19 ✓, 6th term 46 ✓

Problem A · Part 2

Equation: $19r^3 = 46$, so $r^3 = \frac{46}{19} \approx 2.42$ and $r \approx 1.34$ — no "nice" whole-number ratio. A geometric sequence exists, but it is messy.

Problem A · Part 3

Open — many answers. Check only two things: position 3 holds 19 and position 6 holds 46. Example: 7, 12, 19, 25, 33, 46, 50, 61 (neither arithmetic nor geometric).

Problem B · Part 1

Mia: $15 + 18w$ · Lea: $45 + 12w$ · Sophie: $25w$

Problem B · Part 2

$15 + 18w = 45 + 12w$ → $6w = 30$ → $w = 5$. Yes: in week 5 both have 105 €.

Problem B · Part 3

No. Mia = Sophie: $15 + 18w = 25w$ → $w = \frac{15}{7} \approx 2.14$ — not a whole week. Lea = Sophie: $45 + 12w = 25w$ → $w = \frac{45}{13} \approx 3.46$ — not a whole week. All three are never equal on a whole week.

Problem B · Part 4

$(15 + 18w) + (45 + 12w) + 25w = 60 + 55w \geq 225$ → $55w \geq 165$ → $w \geq 3$. First affordable at the end of week 3 (exactly 225 €).

Problem C · Part 1 (score 5)

$c = 2$, $e = 1$: $8 \times 2 - 11 \times 1 = 16 - 11 = 5$ ✓

Problem C · Part 2 (score 13)

$c = 3$, $e = 1$: $8 \times 3 - 11 \times 1 = 24 - 11 = 13$ ✓

Problem C · Part 3 (score 100)

$c = 18$, $e = 4$: $8 \times 18 - 11 \times 4 = 144 - 44 = 100$ ✓

Problem C · Part 4

Open — for each target $T$, check the pair satisfies $8c - 11e = T$ with $c, e$ non-negative whole numbers.

Problem C · Part 5 (conjecture)

Every positive whole number score is achievable — there is no impossible score. Any well-argued conjecture based on her experiments counts; the reasoning matters more than the verdict.

Facilitator notes (parent / tutor — read after she has worked)

Problem A

The surprise: the two clues pin down exactly one clean arithmetic sequence, but the geometric one that fits has an ugly ratio ($r \approx 1.34$), so it doesn't feel "natural."

Key discussion: two data points do not uniquely determine a sequence — you also need to know the type. Ask: "How many data points would you need to be sure?" (It depends on what you're allowed to assume about the type.)

Problem B

The surprise: the one pair that ever matches (Mia and Lea, week 5) doesn't include Sophie — all three are never equal at a whole week. Great opportunity to discuss that "setting equal" gives a solution, but the solution must make sense in context (whole weeks).

Problem C

The mathematical surprise: every positive whole number is achievable. This feels like it shouldn't be true — but it is, and the reason connects to deep number theory: since $\gcd(8, 11) = 1$, Bézout's identity says every integer can be written as $8c - 11e$. (The Chicken McNugget theorem covers the sum-only case: the largest number not expressible as a sum of 8s and 11s is $8 \times 11 - 8 - 11 = 69$.) She doesn't need the proof — the experience of trying and always succeeding is the lesson.

Worked hunt for 13 (shows the productive-struggle path): $e = 0$: $8c = 13$ — no. $e = 1$: $8c = 24$, $c = 3$ ✓.

Good follow-up question: what is the highest score achievable with at most 10 errors?

What's next → Category W — Slope. You'll learn to measure how steeply a line rises or falls, find it from a graph or two points, and graph lines yourself. The connection to sequences: the common difference $d$ of an arithmetic sequence is exactly the slope of the line it lies on.

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Focus 25:00