Exercises: The problem-solving cycle step by step

The Junior Cycle Mathematics specification says that for CBA 1 you follow the Problem-solving cycle over a three-week period. The five stages are:

1. Define a problem

2. Decompose it into manageable parts and/or simplify it using appropriate assumptions

3. Translate the problem to mathematics if necessary

4. Engage with the problem and solve it if possible

5. Interpret any findings in the context of the original problem

Aoife is investigating the problem "How many weeks does she need to save €240 for a bike?" Below are six things she wrote in her notes.

• A: "I will ignore any birthday money and assume she saves the same amount every week."

• B: "Let $w$ be the number of weeks. Then $15w \ge 240$."

• C: "The task is to find the smallest number of whole weeks needed to save €240."

• D: "I solved $15w \ge 240$ and got $w \ge 16$, and checked by substitution."

• E: "So Aoife needs 16 weeks, but if she misses a week she will take longer, so 16 weeks is the minimum."

• F: "I split the job into: how much she saves each week, the target, and the number of weeks."

1. a) Match each note (A to F) to the stage (1 to 5) it belongs to. Two of the stages appear twice. Give a short reason for each match.
2. b) The specification says the work is done over a three-week period. Write a simple plan: for Week 1, Week 2 and Week 3, say which stages you would mainly work on and what you would have written down by the end of the week. (Your teacher decides the exact dates and timing, so this is only your draft plan.)
3. c) In one sentence, explain why stage 5 says "in the context of the original problem" and not just "write down the answer".

The 2nd Year students in a school in Navan are going to Croke Park. The school must hire coaches. Each coach seats 52 passengers and costs €480 to hire. There are 138 students and 9 members of staff going.

1. a) Define the problem. Write the question you must answer as one precise sentence.
2. b) List the information you are given (with units) and the quantity you must find.
3. c) State two assumptions you are making. For example, is everyone going? Does a coach need a spare seat?
4. d) Solve the problem. Show the calculation for the number of passengers, the number of coaches needed (before rounding), and the final cost.
5. e) Interpret your answer in context. Why can you not just round $2.83$ to the nearest whole number? What does your answer mean for the cost?

Cian wants to estimate how many litres of paint are needed to paint the walls of his classroom (not the ceiling or floor) with 2 coats.

He decides on these assumptions: the room is $8\ \text{m} \times 6\ \text{m}$ and $3\ \text{m}$ high; there is one door of area $2\ \text{m}^2$; there are four windows, each $1.5\ \text{m} \times 1.2\ \text{m}$; one litre of paint covers $10\ \text{m}^2$ per coat; paint is sold in 5-litre tins.

1. a) Decompose the problem into at least four smaller parts. Write them as a numbered list (for example, "Part 1: total wall area").
2. b) Carry out each part and find the number of tins Cian should buy. Show your working.
3. c) Which assumption would change the answer the most if it were wrong (for example, the paint coverage or the window sizes)? Justify your choice with a quick calculation, such as "if coverage were only $8\ \text{m}^2$ per litre, then...".
4. d) Write one more assumption that Cian should state in his report.

Seán is designing a square patio in Kilkenny. The patio is an $n \times n$ square of grey slabs, surrounded by a border one slab wide made of red slabs. All slabs are the same size.

The first values he found by drawing are:

$n$ (grey slabs along one side)	Red slabs in the border
1	8
2	12
3	16

1. a) Draw (or describe in words) the patio for $n = 4$ and count the red slabs.
2. b) Extend the table to $n = 6$. Find the pattern in how the number of red slabs changes and explain it in words.
3. c) Write an expression for the number of red slabs in terms of $n$.
4. d) Justify your expression with a diagram-based argument: for example, split the border into four equal parts and explain how many slabs are in each part. Check that this gives the same expression as part c).
5. e) Use your expression to find the number of red slabs if $n = 10$. If red slabs cost €2.40 each, what does the border cost for $n = 12$?
6. f) State one assumption you made in setting up this investigation.

A Clare minor hurling league has $n$ teams. In a round-robin, every team plays every other team exactly once. The organisers want to know how many matches there are in total.

Some small cases have been counted:

Number of teams $n$	Number of matches
2	1
3	3
4	6

1. a) Use a systematic method, such as listing the pairs of teams A, B, C, D, E, to find the number of matches for $n = 5$, $6$ and $7$. Show how you did it.
2. b) Look at the numbers $1, 3, 6, 10, \dots$ How much is added each time $n$ goes up by 1? Write a rule in words.
3. c) Generalise. Each team plays $n - 1$ matches. Explain why the total number of matches is $\dfrac{n(n-1)}{2}$, and in particular why you must divide by 2.
4. d) Verify your formula for $n = 5$ and $n = 7$ using your table from part a).
5. e) A league has 12 teams and the organisers can fit in one round of matches per weekend, with 6 matches per weekend. How many weekends does the league take? Interpret your answer in context.

In Cork, two taxi companies charge as follows:

• Company A: €4.00 fixed charge plus €1.20 per km.

• Company B: €2.50 fixed charge plus €1.50 per km.

Ruairí wants to know which company is cheaper for his journeys.

1. a) Mathematise: write an expression for the cost $C_A$ and the cost $C_B$ of a journey of $d$ km.
2. b) Complete this table (give each cost in euro to 2 decimal places):

$d$ (km)	2	4	6	8	10
$C_A$ (€)
$C_B$ (€)

1. c) Solve an equation to find the distance at which both companies charge the same.
2. d) Interpret: which company should Ruairí choose for a 3.5 km trip? For a 12 km trip? Support each answer with a calculation.
3. e) Write one limitation of this model (for example, something real taxis do that your expressions ignore).
4. f) Self-assessment. Use this simple rubric (not the official features of quality; those are in the Assessment Guidelines for Mathematics from your teacher) to rate your own write-up. For each criterion, write the level (1, 2 or 3) and quote one piece of evidence from your answer.

• Translating to maths: 1 = expressions missing or wrong; 2 = expressions correct but variables not defined; 3 = expressions correct and variables and units clearly defined.

• Working and accuracy: 1 = few steps shown, errors in the table; 2 = steps shown with one or two errors; 3 = clear steps and an accurate table and equation solution.

• Interpretation: 1 = only numbers given; 2 = decisions given but with little explanation; 3 = decisions explained with calculations and a limitation stated.

Orla is investigating: "How many diagonals does a polygon with $n$ sides have?" (A diagonal joins two vertices that are not next to each other.) She counted:

$n$	Shape	Diagonals
3	triangle	0
4	quadrilateral	2

From these two cases she wrote in her draft: "I think the number of diagonals is $2n - 6$."

1. a) Test Orla's claim. Count the diagonals of a pentagon ($n = 5$) and decide whether $2n - 6$ works. Explain why testing a rule with only the two cases used to find it is not enough.
2. b) By counting from each vertex, complete the table for $n = 6, 7, 8$ and explain your method.
3. c) Generalise and justify: from each vertex you can draw $n - 3$ diagonals. Explain why the total number of diagonals is $\dfrac{n(n-3)}{2}$ (why is it $n-3$, and why divide by 2?).
4. d) Verify your formula using at least three values from your table.
5. e) Use the formula to find the number of diagonals of a polygon with 20 sides, and find the number of sides of a polygon that has exactly 54 diagonals. Show your equation and explain how you decided which solution makes sense.
6. f) Write two sentences that Orla could add to her report to explain what she did wrong at first and how she corrected it.

Brian, a farmer near Tralee, has 40 m of fencing. He wants to make a rectangular pen for his sheep with the largest possible area. He uses all the fencing for the four sides of the rectangle. Let $w$ be the width in metres.

1. a) Explain why the length is $20 - w$ metres and the area is $A = w(20 - w)$ square metres.
2. b) Copy and complete the table:

$w$ (m)	2	4	6	8	10	12	14	16	18
$A$ (m²)	36	64			100			64	36

1. c) Describe the pattern in the areas. Why are the areas for $w = 6$ and $w = 14$ the same? Test $w = 9.5$ and $w = 10.5$ and say what this suggests about the maximum.
2. d) Write a findings and conclusion paragraph (about 100 to 150 words) for the report. It should: answer the original question; refer to evidence in your table; interpret the result in context (give the dimensions of the pen); and state at least two assumptions or limitations (for example, a gate, sheep welfare, ground that is not flat).
3. e) Self-assessment. Rate your paragraph using this rubric (not the official features of quality; check the Assessment Guidelines for Mathematics from your teacher). For each criterion, write the level and quote the words or numbers that show it.

• Answers the question: 1 = no clear answer; 2 = answer given but dimensions or area not stated; 3 = clear answer with the dimensions and the maximum area.

• Uses evidence: 1 = no reference to the table; 2 = refers to the table in general; 3 = refers to specific values and the pattern or symmetry.

• Limitations and assumptions: 1 = none; 2 = one, with little explanation; 3 = at least two, each explained.

• Communication: 1 = hard to follow; 2 = understandable but with missing units or steps; 3 = clear sentences, correct units and logical order.

Below is an extract from a draft report written by a student, Fionn.

Problem: How many squares are there in an $n \times n$ grid?

Method: I counted squares in a $1 \times 1$, $2 \times 2$ and $3 \times 3$ grid.

Data:

| $n$ | Squares |

|---|---|

| 1 | 1 |

| 2 | 4 |

| 3 | 9 |

Conclusion: An $n \times n$ grid has $n^2$ squares. So a chessboard ($8 \times 8$) has 64 squares. I am sure because the pattern is obvious.

1. a) Fionn counted only the smallest squares. In a $2 \times 2$ grid, count all the squares of every size. Do the same for a $3 \times 3$ grid. Show how you count (by size).
2. b) Use this rubric (not the official features of quality; check the Assessment Guidelines for Mathematics from your teacher) to evaluate Fionn's draft. For each criterion, give a level (1 = not yet, 2 = partly, 3 = fully) and one sentence of evidence.

• Problem is clearly defined (including what counts as a square)

• Assumptions and method are stated and suitable

• Data are accurate and complete

• Conclusion is justified by the evidence

• Communication (clear, organised, uses correct units and notation)

1. c) Find the correct total for a $4 \times 4$ grid and for an $8 \times 8$ chessboard. Explain the pattern you used, for example, how many squares of each size.
2. d) Rewrite Fionn's problem statement and conclusion so that they would reach a higher level on your rubric. Include a sentence stating how you checked your result and one limitation.
3. e) List three specific improvements Fionn should make before submitting, in order of importance.