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I DON’T UNDERSTAND...

Writing a Strong Paragraph

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PSHS NCE Patterns, Sequences, and Number Relationships Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Patterns, Sequences, and Number Relationships

Spot how numbers change, find the rule, and predict what comes next without checking every possible answer.

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Patterns, Sequences, and Number Relationships

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Ask: what changes each time?

Sequence = numbers arranged in an order that follows a rule.

Same difference:
5, 9, 13, 17
Add 4 each time.

Same factor:
3, 6, 12, 24
Multiply by 2 each time.

Changing difference:
2, 5, 9, 14
Add 3, then 4, then 5.

Alternating rule:
4, 7, 14, 17, 34
+3, ×2, +3, ×2.

FAST METHOD

Check the gaps first

1. Subtract neighboring terms. Are the differences the same?
2. If not, check whether the differences themselves follow a pattern.
3. Check simple multiplication or division.
4. If neither works, test an alternating rule.
5. Use the simplest rule that explains every term.

WORKED EXAMPLES

Six common pattern types

1. Same difference

12, 17, 22, 27, ?

Add 5 each time.

32

2. Same factor

2, 6, 18, 54, ?

Multiply by 3 each time.

162

3. Growing differences

4, 7, 11, 16, 22, ?

Differences are +3, +4, +5, +6. Next is +7.

29

4. Alternating operations

5, 10, 12, 24, 26, ?

×2, +2, ×2, +2, so next is ×2.

52

5. Square numbers

1, 4, 9, 16, 25, ?

These are 1², 2², 3², 4², 5².

36

6. Missing middle term

8, 13, ?, 23, 28

Add 5 each time.

18

TRY WITH ME

Use the gaps

Find the next number:

3, 7, 12, 18, 25, ?

1. Find the differences: +4, +5, +6, +7.
2. The difference increases by 1 each time.
3. Next difference is +8.

33

COMMON TRAPS

Watch out for these

Guessing from only the first two terms.
The rule must fit the whole sequence.

Missing an alternating pattern.
Try checking odd-position and even-position steps separately.

Forcing a complicated rule.
Entrance tests usually reward the simplest consistent pattern.

Ignoring changing differences.
If the first differences are not equal, inspect how they change.

REGULAR PRACTICE

Build the skill

QUICK FIRE

Can you spot the rule fast?

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency.

NEED HELP?

Can't see the pattern yet?

Send Teacher Abi the sequence. I can help you test the differences, factors, and alternating rules without giving away the method too early.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Counting, Data, and Probability

Previous: Measurement and Unit ConversionPSHS NCE Student Hub

PSHS NCE Solid Figures, Surface Area, and Volume Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Solid Figures, Surface Area, and Volume

Recognize 3D shapes, count faces, edges, and vertices, and solve surface-area and volume problems without mixing them up.

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Solid Figures, Surface Area, and Volume

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Surface area is outside. Volume is inside.

Surface area = total area of all outside faces.
Volume = space inside a solid.

Cube: 6 square faces, 12 edges, 8 vertices.
Rectangular prism: 6 rectangular faces, 12 edges, 8 vertices.

VISUAL GUIDE

See the solid first, then the net

First look at a simple 3D drawing with only the outside edges. Then look at the net. If the solid feels hard to picture, the net makes the faces easier to count and is often easier for surface area.

CUBE

3D view

Net view

If the cube sketch feels tricky, the net shows the 6 square faces more clearly.

Cube formulas: Volume = s³ · Surface area = 6s²

RECTANGULAR PRISM

3D view

Net view

The prism’s net helps you see all 6 rectangular faces, which is useful when finding surface area.

Rectangular prism formulas: Volume = L × W × H · Surface area = 2(LW + LH + WH)

TRIANGULAR PRISM

3D view

Net view

In the 3D view, it can be hard to keep track of the side faces. The net makes it easier: 3 rectangles and 2 triangles.

Triangular prism volume: area of triangular base × prism length

SQUARE PYRAMID

3D view

Net view

The pyramid sketch shows the shape, but the net makes the faces easiest to count: 1 square and 4 triangles.

Square pyramid volume: 1/3 × base area × height

WORKED EXAMPLES

Six patterns you should recognize

1. Count parts of a cube

A cube has 6 faces, 12 edges, and 8 vertices.

2. Rectangular prism volume

L=8 cm, W=5 cm, H=3 cm.

8 × 5 × 3 = 120 cm³

120 cm³

3. Cube surface area

Side length = 4 cm.

6 × 4² = 6 × 16 = 96 cm²

96 cm²

4. Rectangular prism surface area

L=6, W=4, H=2.

2(24 + 12 + 8) = 88

88 square units

5. Triangular prism volume

Triangular base: b=6 cm, h=4 cm. Prism length=10 cm.

(1/2 × 6 × 4) × 10 = 120 cm³

120 cm³

6. Square pyramid volume

Square base side=6 cm, height=9 cm.

1/3 × 36 × 9 = 108 cm³

108 cm³

TRY WITH ME

Outside or inside?

A gift box is 10 cm long, 6 cm wide, and 4 cm high. How much space is inside the box?

1. Space inside means volume.
2. Use L × W × H.
3. 10 × 6 × 4 = 240.

240 cm³

COMMON TRAPS

Watch out for these

Using square units for volume.
Volume uses cubic units, such as cm³.

Using cubic units for surface area.
Surface area uses square units, such as cm².

Counting hidden edges twice.
Count each edge once, even if it is dashed or hidden.

Forgetting the 1/3 for pyramids.
A pyramid with the same base and height as a prism has one third the volume.

REGULAR PRACTICE

Build the skill

QUICK FIRE

Can you tell the solid fact fast?

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency.

NEED HELP?

Still mixing up surface area and volume?

Send Teacher Abi the item that confused you. I can help you identify whether the problem is asking about the outside or the inside.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Measurement and Unit Conversion

Previous: Perimeter and AreaPSHS NCE Student Hub