IB MYP Year 9 Maths Surds – Criterion B Investigating Patterns | MYP 3
IB MYP Year 9 Maths Surds – Criterion B Investigating Patterns
Discover the Pattern. Build the Rule. Test It. Then Prove Why It Works.
Criterion B – Investigating Patterns
46 Marks · Suggested Time: 70 Minutes
Mathematics becomes much more powerful when students stop seeing it simply as a collection of questions with answers—and begin asking:
- What do I notice?
- Does this always happen?
- Can I predict what comes next?
- Can I describe the pattern mathematically?
- Can I create a rule that works for every case?
- Can I prove why my rule must work?
That is the thinking at the heart of IB MYP Mathematics Criterion B – Investigating Patterns.
The LittleEdventure IB MYP Year 9 Maths Surds – Criterion B Higher Level Practice Workbook uses surds as a setting for mathematical investigation. Students organise results, observe relationships, make conjectures, construct general rules, verify those rules using new cases and use algebraic reasoning to justify or prove their discoveries.
The workbook is designed around the IB MYP Mathematics Year 9 / MYP 3 Number and Algebra curriculum and focuses specifically on Criterion B – Investigating Patterns. The worksheet itself identifies Criterion B as having a maximum level of 8.
More Than a Surds Worksheet
Many surds worksheets focus mainly on simplifying expressions, rationalising denominators and completing calculations. This resource takes a different approach.
Systematically examine mathematical results to discover relationships.
Identify what changes, what stays constant and how quantities are related.
Turn observations into a precise mathematical rule.
Test the proposed rule using a new case.
Use algebraic reasoning to explain why the rule works generally.
The worksheet explicitly teaches this investigation cycle: organise → observe → conjecture → verify → prove.
Created Specifically for IB MYP Criterion B
What Is Criterion B – Investigating Patterns?
Criterion B asks students to use mathematics to discover patterns rather than simply apply a method that has already been demonstrated.
The workbook develops the three Criterion B strands:
Select and apply mathematical problem-solving techniques to discover complex patterns.
Describe patterns as general rules that are consistent with the findings.
Prove, or verify and justify, the general rules discovered.
These three strands are stated directly in the worksheet and form the structure of the investigations.
Understand What Criterion B Levels 1–8 Look Like
The workbook includes student-friendly Criterion B band descriptors showing how expectations develop from identifying simple patterns to constructing and proving general mathematical rules.
| Level | What Students Work Towards |
|---|---|
| 1–2 | Find simple patterns and make predictions. |
| 3–4 | Describe patterns and suggest general rules. |
| 5–6 | Find complex patterns, use variables and verify rules with new cases. |
| 7–8 | Develop correct general rules and prove, verify or justify them using mathematical reasoning. |
For the higher levels, students are encouraged to produce an algebraic rule, test it with new cases and provide a general proof that works across the appropriate values.
Learn the Complete Mathematical Investigation Cycle
Organise → Observe → Conjecture → Verify → Prove
Work out several cases and record the results systematically in a table using exact surd form.
Look for mathematical structure. Identify what stays the same, what changes and how it changes.
State the observed relationship in words and then express it algebraically using a variable.
Test the proposed rule using a new case that was not used to discover the rule.
Use algebraic reasoning to show why the rule works for every appropriate value.
Verification Is Not the Same as Proof
This distinction is particularly important for students working towards higher Criterion B achievement.
Tests whether a proposed rule works for particular examples. It provides evidence, but it cannot cover every possible case.
Uses general mathematical reasoning, typically with algebra, to demonstrate why the relationship works across all appropriate cases.
The worksheet explicitly warns students that testing an additional value is not the same as proving a general rule.
Learn the Language of Mathematical Investigation
Systematically examine something to find patterns.
Give a detailed account of the mathematical pattern.
Formulate a rule that works across cases using a variable.
Provide evidence by testing the proposed rule.
Give valid mathematical reasons or evidence to support a conclusion.
Use logical mathematical steps to establish that a rule is true generally.
These Criterion B command terms are explicitly included in the workbook to help students understand what different mathematical instructions require.
Four Deep Mathematical Investigations
The main practice section contains four substantial investigations. Together they account for 46 marks.
| Investigation | Focus | Marks |
|---|---|---|
| B1 | Conjugates & Telescoping Sums | 12 |
| B2 | Powers of (1 + √2) | 14 |
| B3 | Nested Surds / Un-nesting Surds | 12 |
| B4 | Rational Products of Surds | 8 |
| Total | Criterion B Investigations | 46 |
The worksheet and answer key both confirm these four investigations and their mark allocation.
Investigation 1: Conjugates That Cancel
From Rationalising a Denominator to Discovering a Pattern
Students investigate fractions of the form:
1 ÷ (√(n + 1) + √n)
They rationalise successive examples, organise the results and identify the relationship between the resulting expressions.
Students then write a general rule and prove why it works for every positive integer n.
The investigation develops further into a sum of consecutive expressions:
S(N) = 1/(√2 + √1) + 1/(√3 + √2) + … + 1/(√(N+1) + √N)
Students investigate how the middle terms cancel, establish a general rule for S(N), solve for a specified value of the sum and determine when the sum is an integer.
Investigation 2: Powers of (1 + √2)
A Pattern Investigation Involving Recurrence, Algebra and Approximation
Every power of (1 + √2) can be written in the form:
aₙ + bₙ√2
Students calculate successive powers and investigate how the coefficients aₙ and bₙ change.
They then derive rules for generating the next pair of coefficients and prove those rules algebraically.
The investigation continues with the expression:
aₙ² − 2bₙ²
Students identify its alternating pattern and use their recurrence rule to justify why the sign changes from one value to the next.
Finally, they examine the ratio aₙ ÷ bₙ and investigate the number that the sequence approaches. The answer key identifies this limiting value as approximately √2 ≈ 1.4142.
Investigation 3: Un-Nesting Surds
Discover the Rule Instead of Simply Memorising It
Students investigate expressions involving square roots inside square roots by squaring sums of neighbouring roots.
They discover the structure behind:
√((2n + 1) + 2√(n(n + 1)))
The investigation then generalises the relationship to positive numbers a and b, followed by verification with a new case.
Students also investigate the corresponding subtraction form and consider why the order of the roots matters when simplifying a square root expression.
This develops an important higher-level habit: a mathematical rule should include the conditions or limitations under which it is valid.
Investigation 4: When Is a Product of Surds Rational?
From Examples to a Mathematical Conjecture
Students investigate when multiplying two square roots can produce a rational result.
They calculate examples, classify results as rational or irrational and examine the product of the numbers beneath the roots.
From these observations, students construct a conjecture about when:
√a × √b
is rational.
They then verify the conjecture using new examples and justify why the rule must be true.
The final task challenges a mathematical claim about rational products and rational sums, giving students an opportunity to use a counterexample to determine whether a universal statement is false.
Skills Developed
Systematically explore unfamiliar mathematical relationships.
Identify structure in exact mathematical results.
Turn observations into precise mathematical statements.
Express patterns as rules using variables.
Test general rules using independent cases.
Use algebra to justify why a rule works generally.
Use a valid counterexample to disprove a universal claim.
Communicate patterns, conditions and conclusions precisely.
Maintain exact surd form so mathematical relationships remain visible.
Identify where a generalisation applies and where it does not.
Surds Knowledge Used Throughout
The workbook also includes a toolkit covering the algebra required for the investigations and proofs, including products and quotients, difference of two squares, conjugates, perfect-square relationships, equal surd expressions and index form.
What’s Included?
Who Is This Workbook For?
Designed specifically for students studying IB MYP Mathematics Year 9 / MYP 3 Higher Level.
Useful for students preparing for Investigating Patterns assessments and class investigations.
Particularly suitable for students developing Level 7–8 skills in generalisation, verification and proof.
Includes guidance, a toolkit, model investigation, four investigations and a self-assessment checklist.
The individual investigations can also be used separately for classroom teaching, tutoring or extension work.
Helps students understand not just the answer, but how to investigate, generalise and justify mathematical relationships.
Help Your Child Learn to Think Like a Mathematician
Sometimes a student can solve every question in a textbook exercise and still feel uncertain when faced with an investigation.
There may be no formula to apply and no example that looks exactly the same. Instead, there may simply be a table of mathematical results and a question:
What do you notice?
This workbook gives students a structured way to approach that uncertainty.
Organise what you know.
Look carefully.
Find a relationship.
Express it mathematically.
Test it.
Then ask: Why must this be true?
The aim is not simply to help students complete one Criterion B task. It is to develop greater confidence when mathematics asks them to discover something independently.
Self-Assessment & Revision Checklist
The final page allows students to reflect on the skills developed throughout the four investigations.
- Organise results clearly using exact values.
- Describe mathematical patterns precisely.
- Write a general rule using a variable and define the variable.
- Verify a rule using a new case.
- Prove a rule using algebra.
- State limitations or conditions of a rule.
- Use a counterexample to show that a claim is false.
The worksheet's final checklist directly covers these investigation skills and divides the resource into B1, B2, B3 and B4 for a total of 46 marks.
Workbook Details
| Curriculum | IB MYP Mathematics |
|---|---|
| Year | Year 9 |
| MYP Stage | MYP 3 |
| Subject Area | Number / Algebra |
| Criterion | Criterion B – Investigating Patterns |
| Difficulty | Higher Level |
| Maximum Criterion Level | 8 |
| Total Marks | 46 |
| Suggested Time | 70 minutes |
| Equipment | Pen and pencil; calculator only for decimal checks |
| Format | Digital printable worksheet + separate answer key |
The worksheet specifies 46 marks, a suggested time of 70 minutes and calculator use only for decimal checks. The accompanying answer key provides completed solutions and marking guidance.
Criterion B Level Guide
| Score | Typical Best-Fit Level | Typical Profile |
|---|---|---|
| 0 | 0 | No pattern identified. |
| 1–10 | 1–2 | Some table values correct; simple predictions. |
| 11–22 | 3–4 | Tables correct; rules suggested in words. |
| 23–34 | 5–6 | Rules written with variables and verified with new cases. |
| 35–46 | 7–8 | Correct general rules with algebraic proofs and stated limitations. |
These score bands are provided as a guide in the answer key; the answer key also notes that Criterion B should be judged on the quality of reasoning rather than score alone.
Frequently Asked Questions
Important: This is an IB MYP-aligned practice and revision resource created by Little Edventure. It is not an official IB assessment paper or official IB publication.