Patterns, Sequences & Algebraic Thinking — Free IB MYP2 Mathematics Questions

1QuestionRecognizing Geometric SequencesConcept Practice

2 marks~3 minCriterion B

The side lengths of squares in a pattern form a geometric sequence (a sequence where each term is multiplied by the same number): 1 cm, 2 cm, 4 cm, 8 cm.

a

State the common ratio of this sequence. [1]

b

Show, using two different pairs of consecutive terms, that the ratio is the same throughout the sequence. [1]

Solutions

2QuestionCalculating Common RatiosConcept Practice

2 marks~3 minCriterion A

The side lengths of four squares form a geometric sequence (each term is multiplied by the same number):

Square 1: 2 cm
Square 2: 6 cm
Square 3: 18 cm
Square 4: 54 cm

a

State the name of the value that links consecutive terms in a geometric sequence. [1]

b

Calculate the common ratio of this sequence. [1]

Solutions

3QuestionEvaluating Terms Using nth Term RuleConcept Practice

2 marks~3 minCriterion A

A tile pattern follows the rule

T_n = 2 \times 3^{n-1}

, where

T_n

is the number of tiles in term

n

.

Term 1: 2 tiles Term 2: 6 tiles Term 3: 18 tiles Term 4: 54 tiles

a

Calculate the number of tiles in term 5. [1]

b

Calculate the number of tiles in term 6. [1]

Solutions

4QuestionUsing nth Term to Predict Future TermsConcept Practice

3 marks~5 minCriterion C

A tile pattern is plotted on a graph. The points lie on a straight line passing through

(1, 5)

and

(3, 11)

, where the horizontal axis shows the step number

n

and the vertical axis shows the number of tiles.

a

Identify the gradient (rate of change) of the line. [1]

b

Calculate the

n

th term rule for the pattern. [1]

c

A classmate says: "Step 20 will have twice as many tiles as step 10." Explain whether your classmate is correct. [1]

Solutions

5QuestionCreating Expressions from PatternsConcept Practice

2 marks~3 minCriterion A

You run a lemonade stand. Each cup costs 0.50 dollars to make, and a permit costs a fixed 2.00 dollars.

Number of cups sold ( $n$ )	1	2	3	4
Total cost ( $C$ in dollars)	2.50	3.00	3.50	4.00

a

State an expression for the total cost

C

in terms of

n

. [1]

b

Explain why this expression would no longer be valid if the cost per cup increased to 0.75 dollars. [1]

Solutions

6QuestionForming Rules from Visual SequencesConcept Practice

2 marks~3 minCriterion B

The diagram shows the first three patterns in a sequence of square tile arrangements.

Pattern 1:

1 \times 1

(1 tile) Pattern 2:

2 \times 2

(4 tiles) Pattern 3:

3 \times 3

(9 tiles)

a

Identify the number of tiles in Pattern 4. [1]

b

Write an expression for the number of tiles in Pattern

n

. [1]

Solutions

7QuestionUsing Letters to Represent NumbersConcept Practice

2 marks~3 minCriterion C

A school canteen sells bottles of juice and bags of crisps. Layla does not know the individual prices, so she uses

j

to represent the cost of one juice bottle and

c

to represent the cost of one bag of crisps.

a

Identify what the expression

3j + 2c

represents in this context. [1]

b

Explain one reason why using variables

j

and

c

is helpful here. [1]

Solutions

8QuestionCreating Patterns Using RulesConcept Practice

2 marks~3 minCriterion B

A gardener plants trees in rows. The first row has 3 trees, and each row after that has 2 more trees than the row before it.

a

State a rule for the number of trees in row

n

. [1]

b

Calculate the number of trees in the 5th row. [1]

Solutions

9QuestionFinding the Next Term in a SequenceConcept Practice

2 marks~3 minCriterion A

A sequence of shapes follows a repeating pattern:

square, triangle, square, triangle, square, triangle, …

a

Identify the repeating unit of this pattern. [1]

b

State the 7th shape in the sequence. [1]

Solutions

10QuestionIntroduction to Arithmetic SequencesAssessment Practice

5 marks~8 minCriterion D

A student saves money each week. In week 1 they save 5 dollars, week 2 they save 8 dollars, week 3 they save 11 dollars, and week 4 they save 14 dollars. This is an arithmetic sequence (a number pattern with a constant difference between terms).

a

Identify the first term and the common difference of this sequence. [1]

b

Calculate the amount saved in week 10. Show all working. [2]

c

The student plans to buy birthday gifts in weeks 6 and 7, increasing their spending. Explain whether the arithmetic sequence model remains reliable for predicting savings beyond week 5, giving two reasons. [2]

Solutions

11QuestionFinding Common DifferencesAssessment Practice

4 marks~6 minCriterion C

A student records weekly savings (in dollars) over four weeks:

5, \ 8, \ 11, \ 14

.

a

State whether these four values form an arithmetic sequence. Give one reason for your answer. [1]

b

The student saves

0

dollars in week 5 due to an unexpected expense. Calculate the differences between consecutive terms for weeks 1–5 and explain why the sequence is no longer arithmetic. [2]

c

Identify one assumption the arithmetic sequence model makes about savings behaviour and explain why this assumption may not hold in real life. [1]

Solutions

12QuestionSolving Problems with nth Term in ContextAssessment Practice

4 marks~6 minCriterion D

A student models plant height using the formula

h = 3n + 2

, where

h

is the height in centimetres and

n

is the week number. The measured heights are:

Week 1: 5 cm, Week 2: 8 cm, Week 3: 11 cm, Week 4: 13 cm, Week 5: 14 cm

a

Calculate the predicted heights for weeks 4 and 5 using the formula. [1]

b

Describe how the predicted heights compare with the actual heights for weeks 4 and 5, and identify the trend in the difference. [2]

c

Explain whether the formula is suitable for predicting the plant's height at week 10, identifying one limitation of assuming constant growth. [1]

Solutions

13QuestionUsing nth Term to Predict Future TermsAssessment Practice

8 marks~12 minCriterion B

The first four figures of a tile pattern are shown below.

Figure 1: 1 tile Figure 2: 3 tiles Figure 3: 7 tiles Figure 4: 13 tiles

a

State the number of tiles in Figure 5 and Figure 6. [2]

b

Calculate the values of

a

,

b

, and

c

in the general rule

T(n) = an^2 + bn + c

, where

T(n)

is the number of tiles in Figure

n

. Show your working. [4]

c

Explain how the constant second difference in the sequence confirms that the rule for

T(n)

is quadratic. Use your rule to find

T(10)

. [2]

Solutions

14QuestionEvaluating Expressions Derived from PatternsAssessment Practice

4 marks~6 minCriterion D

A student designs a tile walkway where the number of tiles in figure

n

is given by

3n + 2

. Due to material constraints, any figure can have a maximum of 20 tiles.

a

Calculate the number of tiles in figure 6. [1]

b

Explain whether the expression

3n + 2

is valid for figure 10. [2]

c

Compare the mathematical prediction for figure 10 with the real-world constraint. [1]

Solutions

15QuestionCreating Patterns Using RulesAssessment Practice

8 marks~12 minCriterion D

A gardening club plants seeds and records how long each takes to sprout.

Depth (cm)	1	2	3	4
Days until sprouting	12	10	8	6

The club writes a rule:

\text{days} = 14 - 2 \times \text{depth}

a

Calculate the predicted sprouting time for a seed planted at a depth of 2.5 cm. [2]

b

The ruler has a possible measurement error of

\pm 0.5

cm. Calculate the range of possible sprouting times for a depth of 2.5 cm. [2]

c

The club wants to use the same rule for depths greater than 5 cm. Compare the rule's prediction at 7 cm with what might actually happen, and explain one reason why the rule may no longer be reliable at that depth. [4]

Solutions

16QuestionDescribing Patterns Using Words and SymbolsAssessment Practice

6 marks~9 minCriterion C

A student predicts the number of tiles in a growing garden path using the rule

n^2

, where

n

is the step number, giving 1, 4, 9, 16, … The builder's actual sequence is 1, 4, 8, 13, …

a

Calculate the student's predicted number of tiles at step 5, and the builder's actual number at step 5. [2]

b

Explain why the rule

n^2

becomes less accurate as the step number increases. In your answer, refer to the differences between the predicted and actual values. [2]

c

Compare the rule

n^2

with the rule

T_n = \dfrac{n^2 + 3n - 2}{2}

, and state which better fits the builder's sequence. Show a check for one value of

n

. [2]

Solutions

Patterns, Sequences & Algebraic Thinking — Free MYP2 Mathematics Practice Questions