Number Concepts & Systems — Free IB MYP3 Mathematics Questions

1QuestionIntegers and Their UsesConcept Practice

2 marks~3 minCriterion A

A weather station records the daily high temperature as

12^\circ\text{C}

and the daily low temperature as

-3^\circ\text{C}

.

a

Calculate the temperature range for that day. [1]

b

Explain why representing temperatures as integers is more precise than using descriptive terms such as "hot" or "cold". [1]

Solutions

2QuestionRational Numbers as Terminating and Repeating DecimalsConcept Practice

2 marks~3 minCriterion B

The decimal expansions of

\dfrac{1}{7}

,

\dfrac{2}{7}

,

\dfrac{3}{7}

,

\dfrac{4}{7}

,

\dfrac{5}{7}

, and

\dfrac{6}{7}

share a surprising pattern.

a

Calculate the decimal expansion of each fraction. [1]

b

Describe how the six decimal expansions are related to one another. [1]

Solutions

3QuestionDefining Rational NumbersConcept Practice

2 marks~3 minCriterion A

A right-angled triangle has two legs, each measuring 1 cm.

a

Calculate the length of the hypotenuse. [1]

b

Explain whether the hypotenuse length is a rational number, using the definition of rational numbers in your answer. [1]

Solutions

4QuestionEstimating Sums and DifferencesConcept Practice

2 marks~3 minCriterion B

Each addition problem below has been rounded to the nearest ten to produce an estimated sum.

$47 + 32$	exact sum $79$	estimated sum $80$
$51 + 29$	exact sum $80$	estimated sum $80$
$48 + 33$	exact sum $81$	estimated sum $80$
$46 + 31$	exact sum $77$	estimated sum $80$

Compare the estimated sum to the exact sum in each problem. [1]

Explain why the estimated sum is sometimes higher, sometimes lower, and sometimes equal to the exact sum. [1]

Solutions

5QuestionEstimating Products and QuotientsConcept Practice

2 marks~3 minCriterion A

A rectangle has side lengths of

4.8

cm and

6.2

cm.

(a) Round each side length to the nearest whole number and calculate the estimated area of the rectangle using

\text{Area} = \text{length} \times \text{width.}

Give your answer in cm². [2]

Solutions

6QuestionDecimal Place Value and PrecisionConcept Practice

2 marks~3 minCriterion A

A rectangle has side lengths of

5.3

cm and

8.47

cm.

a

State the number of decimal places in each measurement. [1]

b

Compare the precision of the two measurements and explain which is more precise. [1]

Solutions

7QuestionExpanded and Standard FormsConcept Practice

3 marks~5 minCriterion B

The bar graph below shows the value of the digit 3 in four place value positions.

Place value	ones	tens	hundreds	thousands
Value of digit 3	$3$	$30$	$300$	$3000$

a

Describe the pattern shown in the graph between place value position and the value of the digit 3. [1]

b

Explain why the value of a digit changes as it moves from one place value position to the next. [1]

c

Using the pattern from the graph, calculate the value of the digit 3 in the ten-thousands place. [1]

Solutions

8QuestionApplications of Square Numbers in GeometryConcept Practice

2 marks~3 minCriterion A

A square tile has a side length of 7 cm.

a

Calculate the area of the tile. [1]

b

A second square tile has an area of 49 cm². Explain whether the two tiles are the same size. [1]

Solutions

9QuestionFinding Factors of a NumberConcept Practice

2 marks~3 minCriterion B

The factor pairs of 24 are shown below.

Row 1:

1,\ 2,\ 3,\ 4

Row 2:

24,\ 12,\ 8,\ 6

Each number in Row 1 is paired with the number directly below it in Row 2.

a

Describe the pattern you observe in the product of each factor pair of 24. [1]

b

Apply this pattern to find all factor pairs of 36. [1]

Solutions

10QuestionFinding Factors of a NumberConcept Practice

2 marks~3 minCriterion A

A rectangle has an area of 36 square units. Its length

x

and width

y

are both whole numbers, so

x \times y = 36

.

List all possible pairs

(x, y)

where

x \geq y

. [2]

Solutions

11QuestionComparing and Ordering IntegersConcept Practice

2 marks~3 minCriterion A

A thermometer shows three temperatures:

-5°C

,

0°C

, and

3°C

.

a

Compare the three temperatures and identify the coldest and the warmest. [1]

b

List all three temperatures in ascending order. Explain how you know which temperature is least. [1]

Solutions

12QuestionEven Odd and Prime NumbersAssessment Practice

3 marks~5 minCriterion B

The table below shows the first five prime numbers and their squares.

Prime number:

2, \ 3, \ 5, \ 7, \ 11

Square:

4, \ 9, \ 25, \ 49, \ 121

a

State the definitions of an even number and an odd number. [1]

b

Explain why

2

is the only even prime number. [1]

c

Using the table, explain why the square of every odd prime is odd, while the square of

2

is even. [1]

Solutions

13QuestionIntegers and Their UsesAssessment Practice

5 marks~8 minCriterion D

A student builds a simple financial model to predict weekly profit:

\text{Profit} = \text{Income} - \text{Expenses}

In one week, the student records an income of 250 dollars and expenses of 320 dollars.

a

Calculate the profit for this week. [1]

b

Explain what a profit of

-70

dollars means in this financial context. [2]

c

The model uses only whole dollar amounts. Analyse how this restriction could affect the accuracy of the profit prediction, using a numerical example to support your answer. [2]

Solutions

14QuestionReal Number System OverviewAssessment Practice

4 marks~6 minCriterion C

A student measures the length of a table using a ruler marked in centimetres and records the length as

1.83

m.

a

Justify whether

1.83

is a rational or irrational number. [1]

b

Explain why the ruler's precision of

\pm 0.5

cm means the true length cannot be known exactly. [1]

c

The true length lies somewhere in the interval

1.825

m to

1.835

m. Analyse whether the true length must be rational, and what this tells us about using measurements in mathematics. [2]

Solutions

15QuestionDefining Rational NumbersAssessment Practice

5 marks~8 minCriterion D

A student measures the diagonal of a

1\,\text{m} \times 1\,\text{m}

square tile using a ruler accurate to the nearest

0.01\,\text{m}

. The student records the diagonal as

1.41\,\text{m}

.

a

Calculate the exact length of the diagonal using the Pythagorean theorem. [1]

b

Explain why

1.41\,\text{m}

is a rational number but the exact diagonal length is not. [2]

c

A builder uses

1.41\,\text{m}

when cutting ten tiles in a row. Analyse whether this approximation is appropriate, and describe one consequence of using it. [2]

Solutions

16QuestionVisualizing Rational vs Irrational on a Number LineAssessment Practice

4 marks~6 minCriterion C

A student claims: "I can mark

\sqrt{2}

exactly on a number line by constructing a right triangle with two legs of length 1 unit. This means

\sqrt{2}

must be a rational number."

a

Explain why the student's conclusion is incorrect. [2]

b

Explain one limitation of using geometric constructions to decide whether a number is rational or irrational. [2]

Solutions

17QuestionEstimating Products and QuotientsAssessment Practice

4 marks~6 minCriterion D

A student estimates the cost of 48 concert tickets, each priced at 35 dollars, by rounding 48 up to 50 and calculating

50 \times 35 = 1750

dollars. The actual cost is

48 \times 35 = 1680

dollars.

a

Justify whether 1750 dollars is a valid estimate, and explain why it is an overestimate. [2]

b

Describe a real-world situation in which using this overestimate could cause a problem, and explain how the problem arises. [2]

Solutions

18QuestionRounding Decimals to Given PlacesAssessment Practice

5 marks~8 minCriterion C

A student claims that

0.449

rounded to one decimal place is

0.5

, arguing it is closer to

0.5

than to

0.4

.

a

State the correct value of

0.449

rounded to one decimal place and identify the digit that determines this. [1]

b

Using a number line from

0.4

to

0.5

, mark the positions of

0.44

,

0.45

, and

0.449

. Explain why

0.449

rounds to your answer from part (a). [2]

c

The student's reasoning contains a flaw. Justify why the student's argument is incorrect, using distance calculations to support your answer. [2]

Solutions

19QuestionReading and Writing Numbers in WordsAssessment Practice

5 marks~8 minCriterion C

A city council publishes its annual budget as "three million, four hundred fifty-two thousand, seven hundred dollars." The actual spreadsheet records the amount as 3,452,070 dollars.

a

Identify the discrepancy between the word-form amount and the spreadsheet amount. State the difference in dollars. [1]

b

Explain one reason why writing large numbers in words increases the risk of errors in financial documents. [2]

c

A council member suggests replacing all word-form amounts in the public budget report with numerals. Analyse whether this change would improve accuracy and transparency in government budget reporting. [2]

Solutions

20QuestionComparing and Ordering DecimalsAssessment Practice

8 marks~12 minCriterion D

A nutritionist compares the sugar content per serving of four cereal brands:

Brand A:

4.5

g
Brand B:

4.4

g
Brand C:

4.6

g
Brand D:

4.5

g

a

Order the four brands from lowest to highest sugar content and identify which brand has the least sugar. [2]

b

The actual sugar content of Brand A is

4.49

g, rounded to

4.5

g. Explain how this rounding affects a comparison between Brand A and Brand D, which also shows

4.5

g. [2]

c

A consumer uses only the rounded values above to decide which cereal is healthiest. Analyse two limitations of this approach. [4]

Solutions

21QuestionApplications of Square Numbers in GeometryAssessment Practice

2 marks~3 minCriterion D

A square kitchen floor has a side length of

4 \, \text{m}

. Each tile covers

1 \, \text{m}^2

.

a

Calculate the number of tiles needed to cover the floor. [1]

b

The side length is measured

0.5 \, \text{m}

too short. Explain how this error affects the number of tiles ordered. [1]

Solutions

22QuestionApplications of Square Numbers in GeometryAssessment Practice

3 marks~5 minCriterion B

A square has a side length of

8

units, split into four regions by dividing each side as

5 + 3

: two smaller squares and two rectangles.

a

Calculate the area of the large square. [1]

b

Calculate the area of each of the four smaller regions and find their total. [1]

c

Explain what the result from part (b) shows about the expression

(5 + 3)^2

. [1]

Solutions

23QuestionComposite Numbers and Their PropertiesAssessment Practice

4 marks~6 minCriterion C

A school plans a rectangular garden with an area of

36 \text{ m}^2

. A student claims that because 36 is composite, both side lengths must also be composite numbers, giving exactly three possible pairs:

4 \times 9

,

6 \times 6

, and

9 \times 4

.

a

List all factor pairs of 36. [1]

b

Identify which factors of 36 are composite numbers. Explain why 1, 2, and 3 are not composite. [2]

c

Analyse whether the student's claim is valid. [1]

Solutions

24QuestionFinding Factors of a NumberAssessment Practice

6 marks~9 minCriterion C

A student designs a rectangular garden with an area of

36 \text{ m}^2

. She lists all whole-number dimension pairs:

Length (m)	1	2	3	4	6	9	12	18	36
Width (m)	36	18	12	9	6	4	3	2	1

She claims there are exactly 9 possible designs.

a

State how many distinct rectangular designs are actually possible, and list them. [2]

b

Explain why the number of possible dimension pairs is no longer limited to whole numbers when measurements have a precision of

\pm 0.1 \text{ m}

. Give one example of a valid non-integer pair. [3]

c

Analyse one real-world constraint, other than measurement precision, that could further limit the possible dimensions of the garden. Justify your answer. [1]

Solutions

25QuestionFinding Factors of a NumberAssessment Practice

6 marks~9 minCriterion D

A school principal needs to organise 48 students into equal-sized groups for a rotation activity. Each group size must be a factor of 48. The table below shows some information about available resources.

Number of teachers available: 6
Maximum room capacity: 20 students
Minimum group size (safety ratio): 4 students

a

List all factors of 48. [2]

b

Explain why group sizes of 1 and 48 are not suitable for this rotation activity. [2]

c

Using the resource information, identify all feasible group sizes and justify which group size you would recommend. [2]

Solutions

26QuestionAbsolute Value of an IntegerAssessment Practice

4 marks~6 minCriterion B

The absolute value of an integer is its distance from zero on the number line.

Integer:

+3,\ -3,\ +7,\ -7,\ +10,\ -10

Absolute value:

3,\ 3,\ 7,\ 7,\ 10,\ 10

a

Describe a general rule for the absolute value of any integer

n

, writing your rule using

n

. [2]

b

Apply your rule to show that

|-15| = 15

, stating which part of your rule you used. [1]

c

A student claims: "Because

|-20|

and

|+20|

give the same result, absolute value tells us nothing useful about the original integer." Justify whether this claim is correct or incorrect. [1]

Solutions

27QuestionAbsolute Value of an IntegerAssessment Practice

6 marks~9 minCriterion D

A city planner checks whether five buildings comply with a height limit of 50 m.

Building 1: 47 m
Building 2: 53 m
Building 3: 49 m
Building 4: 52 m
Building 5: 48 m

The planner measures total deviation by calculating

|\text{height} - 50|

for each building and summing the results.

a

Calculate the total deviation from the height limit. [2]

b

Explain one limitation of using absolute value to measure compliance in this situation. [2]

c

A revised model only penalises buildings that exceed the limit. Analyse whether this revised model is more useful to the city planner, using the data above to support your answer. [2]

Solutions

28QuestionOpposites and Zero on the Number LineAssessment Practice

4 marks~6 minCriterion C

A weather station records daily high temperatures:

Monday:

+8°C

Tuesday:

-3°C

Wednesday:

0°C

Thursday:

+5°C

Friday:

-7°C

A student claims: " $0°C$ means no temperature, so Wednesday had no heat energy."

a

Explain how

0°C

acts as a reference point on the number line, using at least two temperatures from the data to support your answer. [2]

b

Explain one real-world problem that arises from treating

0°C

as "nothing," and refer to

+8°C

and

-3°C

as opposites in your response. [2]

Solutions

Number Concepts & Systems — Free MYP3 Mathematics Practice Questions