Patterns, Sequences & Algebraic Thinking — Free IB MYP3 Mathematics Questions

1QuestionComparing Arithmetic and Geometric SequencesConcept Practice

2 marks~3 minCriterion C

Compare an arithmetic sequence and a geometric sequence by describing the primary distinction between them.

Then provide one example of each type of sequence. [2]

Solutions

2QuestionCalculating Common RatiosConcept Practice

2 marks~3 minCriterion A

A geometric sequence has the following terms:

1, \ 2, \ 4, \ 8, \ldots

Calculate the common ratio of this geometric sequence. [2]

Solutions

3QuestionFinding Common DifferencesConcept Practice

2 marks~3 minCriterion B

The diagram shows four equilateral triangles with side lengths forming a sequence.

Side length (cm):

2, \ 5, \ 8, \ 11

a

Calculate the common difference of this arithmetic sequence. [1]

b

Explain how you can confirm that the sequence is arithmetic. [1]

Solutions

4QuestionSolving Problems with nth Term in ContextConcept Practice

2 marks~3 minCriterion B

The diagram shows rows of unit squares placed side by side.

Arrangement 1: perimeter

= 4

cm
Arrangement 2: perimeter

= 6

cm
Arrangement 3: perimeter

= 8

cm

a

Write down the

nth

term formula for the perimeter of arrangement

n

. [1]

b

Calculate the perimeter of the 10th arrangement. [1]

Solutions

5QuestionEvaluating Terms Using nth Term RuleConcept Practice

2 marks~3 minCriterion A

A sequence of regular pentagons has side lengths defined by the nth term rule

T_n = 2n + 1

cm.

Calculate the side length of the 5th pentagon in this sequence. [2]

Solutions

6QuestionAnalyzing and Explaining Real-World PatternsConcept Practice

2 marks~3 minCriterion A

A rectangular garden has a length of 7 cm and a width of 4 cm.

Calculate the perimeter of the rectangle, showing all working. [2]

Solutions

7QuestionPatterns in Nature and ArchitectureConcept Practice

2 marks~3 minCriterion B

The diagram shows a nautilus shell with four spiral radii:

r_1 = 1

cm,

r_2 = 1.6

cm,

r_3 = 2.6

cm,

r_4 = 4.2

cm.

Calculate the ratio

\dfrac{r_2}{r_1}

and the ratio

\dfrac{r_3}{r_2}

. Using your results and the term "ratio", describe the mathematical relationship between each radius and the next. [2]

Solutions

8QuestionEvaluating Expressions Derived from PatternsConcept Practice

2 marks~3 minCriterion A

A square has a side length of

(2n + 1)

cm.

Calculate the perimeter of the square when

n = 3

. [2]

Solutions

9QuestionCreating Expressions from PatternsConcept Practice

2 marks~3 minCriterion B

The diagrams show a sequence of square grids.

Grid 1	$1 \times 1$	containing $1$ unit square.
Grid 2	$2 \times 2$	containing $4$ unit squares.
Grid 3	$3 \times 3$	containing $9$ unit squares.

a

State the number of unit squares in the 4th grid. [1]

b

Write an expression for the total number of unit squares in the

n

th grid. [1]

Solutions

10QuestionCreating Patterns Using RulesConcept Practice

2 marks~3 minCriterion B

The diagram shows the first three terms of a pattern of squares. The side lengths are 1 unit, 2 units, and 3 units respectively.

a

State the rule for the side length of the

n

th square. [1]

b

A student claims the 10th square has a side length of 12 units. Determine whether this claim is correct. Justify your answer. [1]

Solutions

11QuestionCreating Patterns Using RulesConcept Practice

2 marks~3 minCriterion A

A gardener plants seeds in a triangular arrangement. The number of seeds in each row follows this pattern:

Row 1: 1 seed, Row 2: 3 seeds, Row 3: 6 seeds, Row 4: 10 seeds.

a

Calculate the number of seeds in the fifth row. [1]

b

Explain why using this pattern rule is more reliable than guessing when planning how many seeds to buy. [1]

Solutions

12QuestionIdentifying Pattern Rules in DiagramsConcept Practice

2 marks~3 minCriterion B

The diagrams show the first three terms of a dot pattern sequence.

Pattern 1:

1

dot
Pattern 2:

4

dots in a

2 \times 2

square
Pattern 3:

9

dots in a

3 \times 3

square

a

Describe how the number of dots changes from one pattern to the next. [1]

b

Write an algebraic expression for the number of dots in Pattern

n

. [1]

Solutions

13QuestionFractal Patterns and RepetitionConcept Practice

2 marks~3 minCriterion A

The diagram shows a Sierpinski triangle after 2 iterations. At iteration 0, there is 1 shaded triangle. At each iteration, every shaded triangle is divided into 4 smaller congruent triangles and the central triangle is removed, leaving 3 shaded triangles in its place.

Calculate the total number of shaded triangles after 2 iterations. [2]

Solutions

14QuestionCalculating Common RatiosAssessment Practice

6 marks~9 minCriterion D

A conservationist tracks a rabbit population using the geometric sequence formula

u_n = u_1 \cdot r^{n-1}

where

u_1 = 50

rabbits and

r = 1.8

.

a

Calculate the predicted population after 5 years. [2]

b

After 5 years, the actual counted population is 320 rabbits. Explain two reasons why using a constant common ratio may not accurately model a real rabbit population. [2]

c

The conservationist later discovers the initial count of 50 rabbits was incorrect; the true starting population was 60 rabbits. Calculate the corrected prediction for year 5, and interpret what this tells you about the effect of an initial counting error on the model. [2]

Solutions

15QuestionEvaluating Terms Using nth Term RuleAssessment Practice

4 marks~6 minCriterion C

The

n

th term of a sequence is given by

4n - 3

.

a

Calculate the 5th term of the sequence. [1]

b

A student claims the 10th term is 43. Calculate the correct value and explain the error the student made. [2]

c

Another sequence has

n

th term

4n + 5

. Analyse how the two sequences are related and justify which sequence will always produce greater values. [1]

Solutions

16QuestionEvaluating Terms Using nth Term RuleAssessment Practice

7 marks~11 minCriterion D

A conservation agency monitors a reintroduced bird species. The predicted population after

n

years is given by the rule

P = 3n + 50

.

a

Calculate the predicted population for years 1, 5, and 10. Show your working. [3]

b

Explain why a linear model may not accurately predict the population after many years. [2]

c

The agency observes that the actual population in year 10 is 74 birds. Interpret what this difference tells you about the model's predictions. [2]

Solutions

17QuestionPatterns in Music and RhythmAssessment Practice

6 marks~9 minCriterion C

A musician writes a rhythm pattern of four notes: a quarter note, an eighth note, another eighth note, and a quarter note. A quarter note lasts 1 beat; an eighth note lasts

\frac{1}{2}

beat.

a

Calculate the total number of beats in the pattern. [1]

b

At a tempo of 60 bpm each beat lasts 1 second, and at 120 bpm each beat lasts 0.5 seconds. Calculate the duration of the pattern in seconds at each tempo. [2]

c

The musician claims: "This pattern always lasts 4 beats, so its duration never changes." Analyse this claim, using your results from part (b) to support your reasoning. [3]

Solutions

18QuestionCultural Patterns in Art and DesignAssessment Practice

4 marks~6 minCriterion D

An Islamic geometric pattern uses a regular octagon tile. The interior angle of a regular

n

-sided polygon is given by:

\text{Interior angle} = \frac{180(n-2)}{n}

a

Calculate the interior angle of a regular octagon. [1]

b

Explain whether regular octagons can tile perfectly around a single point, using your answer to part (a). [2]

c

Describe one limitation of this formula when applied to real handcrafted octagon tiles. [1]

Solutions

19QuestionUsing Letters to Represent NumbersAssessment Practice

4 marks~6 minCriterion C

A school fair charges a base entry fee of 5 dollars. Game tokens cost 2.50 dollars each. Tickets bought in advance receive a 10% discount on the total cost.

A student models the total ticket cost as

C = 5 + 2n

where

n

is the number of game tokens purchased.

a

Identify one difference between the student's model and the actual pricing. [1]

b

Explain how this difference affects the cost calculated by the model compared to the actual cost. [1]

c

Write an improved equation for

C

that correctly reflects the actual token price and the advance-purchase discount. Justify why your equation is a more accurate model. [2]

Solutions

20QuestionCreating Expressions from PatternsAssessment Practice

2 marks~3 minCriterion D

At a party with

n

attendees, each person shakes hands with every other person exactly once. The total number of handshakes is given by

\frac{n(n-1)}{2}

For a party of 1000 attendees, explain one reason why this formula may not reflect the actual number of handshakes that take place. [2]

Solutions

21QuestionCreating Patterns Using RulesAssessment Practice

4 marks~6 minCriterion D

A student plants a sunflower and records its height each week using the rule

\text{Height (cm)} = 10 + 3w

where

w

is the number of weeks after planting.

Actual heights recorded:
Week 1: 13 cm, Week 2: 16 cm, Week 3: 18 cm, Week 4: 21 cm, Week 5: 23 cm

a

Calculate the predicted height at week 6. [1]

b

Compare the predicted weekly increase with the actual weekly increases shown in the data. [1]

c

Explain two reasons why the linear rule may not accurately model the sunflower's height over time. [2]

Solutions

22QuestionIdentifying Patterns in Number SeriesAssessment Practice

4 marks~6 minCriterion C

A population of bacteria starts with 3 bacteria at hour 0 and doubles every hour.

A student claims the sequence

3, 6, 12, 24, \ldots

is generated by adding 3 to the first term, then multiplying each term after that by 2.

a

Calculate the bacterial population at hour 4 using the biological doubling rule. [1]

b

Explain whether the student's rule produces the same values as the doubling rule for the first five terms. Show your working. [2]

c

Justify why assuming this pattern continues indefinitely may not reflect real bacterial growth. [1]

Solutions

23QuestionIdentifying Pattern Rules in DiagramsAssessment Practice

6 marks~9 minCriterion D

A construction company designs square patios using a pattern of square tiles. For a patio with side length

n

tiles (where

n \geq 2

), the number of tiles required is:

T = n^2 + (n-1)^2

The diagram below shows the first three patio sizes (side lengths 2, 3, and 4 tiles).

a

Calculate the number of tiles needed for a patio with a side length of 5 tiles. [2]

b

The company orders 5% extra tiles to cover manufacturing waste. Explain how this affects the total number of tiles ordered for the side-length-5 patio, and why the formula

T = n^2 + (n-1)^2

would give an inaccurate estimate for a rectangular patio. [2]

c

A customer requests a patio measuring 5 tiles by 6 tiles. The company uses the formula with

n = 5

to estimate the tile order. Analyse whether this estimate is sufficient, and justify one change the company should make to their method. [2]

Solutions

24QuestionIdentifying Pattern Rules in DiagramsAssessment Practice

5 marks~8 minCriterion C

A student claims that the total number of dots in a square grid with side length

n

is given by the formula

n^2 + 2n

.

The actual dot counts are:

n = 1

: 1 dot,

n = 2

: 4 dots,

n = 3

: 9 dots,

n = 4

: 16 dots

a

Calculate the value of

n^2 + 2n

for each of

n = 1, 2, 3, 4

. [2]

b

Compare your results from part (a) with the actual values above. [1]

c

The student's claim is incorrect. Identify the correct formula for the total number of dots and justify why it fits the actual values. [2]

Solutions

Patterns, Sequences & Algebraic Thinking — Free MYP3 Mathematics Practice Questions