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.

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PSHS NCE REVIEW PATH

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Next competency: Counting, Data, and Probability

Previous: Measurement and Unit ConversionPSHS NCE Student Hub

PSHS NCE Measurement and Unit Conversion Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Measurement and Unit Conversion

Convert length, mass, capacity, time, area, and volume units without getting lost in the zeros.

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Measurement and Unit Conversion

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Convert only after you identify the unit

Length: mm, cm, m, km
Mass: mg, g, kg
Capacity: mL, L
Time: seconds, minutes, hours
Area: cm², m²
Volume: cm³, m³

Important: Area and volume conversions are not the same as length conversions. Squared units and cubic units change faster.

CONVERSION GUIDE

Use the relationship, not memorized guessing

Length

1 km=1,000 m

1 m=100 cm

1 cm=10 mm

Mass

1 kg=1,000 g

1 g=1,000 mg

Capacity

1 L=1,000 mL

Time

1 hr=60 min

1 min=60 sec

Why area is different

Since 1 m = 100 cm:

1 m² = 100 cm × 100 cm = 10,000 cm²

Why volume is different

Since 1 m = 100 cm:

1 m³ = 100 × 100 × 100 = 1,000,000 cm³

WORKED EXAMPLES

Seven patterns you should recognize

1. Meters to centimeters

3.6 m × 100 = 360 cm

2. Meters to kilometers

2,500 m ÷ 1,000 = 2.5 km

3. Kilograms to grams

4.2 kg × 1,000 = 4,200 g

4. Liters to milliliters

1.75 L × 1,000 = 1,750 mL

5. Hours to minutes

2.5 hr × 60 = 150 min

6. Square meters to square centimeters

2 m² × 10,000 = 20,000 cm²

7. Elapsed time

Start: 2:35 PM. End: 4:10 PM.

1 hour 35 minutes

TRY WITH ME

Which way should the number move?

Convert 4.8 km to meters.

1. 1 km = 1,000 m.
2. Kilometers are larger units than meters.
3. Going to the smaller unit means the number gets larger.
4. 4.8 × 1,000 = 4,800.

4,800 m

COMMON TRAPS

Watch out for these

Moving the decimal without knowing why.
Use the unit relationship first.

Treating hours like base 100.
Time uses 60 minutes per hour, not 100.

Using 100 for m² to cm².
It is 10,000 because both dimensions are converted.

Forgetting the unit in the answer.
A number without a unit may not fully answer the question.

REGULAR PRACTICE

Build the skill

QUICK FIRE

Can you convert without hesitation?

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency.

NEED HELP?

Too many zeros?

Send Teacher Abi the exact conversion that confused you. I can help you decide whether to multiply or divide before you calculate.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Patterns, Sequences, and Number Relationships

Previous: Solid Figures, Surface Area, and VolumePSHS 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

PSHS NCE Perimeter and Area Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Perimeter and Area

Know whether a problem is asking for the boundary or the space inside, then use the shortest correct formula.

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Perimeter and Area

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Around or inside?

Perimeter = distance around the outside.
Area = amount of flat space inside.

Fast clue: fence, border, frame, edge, around → think perimeter.
floor, field, tile, cover, inside → think area.

VISUAL GUIDE

See what each formula measures

RECTANGLE

Rectangle: P = 2L + 2W, A = L × W

TRIANGLE

Triangle area: A = 1/2 × base × height

PARALLELOGRAM

Parallelogram area: A = base × perpendicular height

TRAPEZOID

Trapezoid area: A = 1/2 × (b₁ + b₂) × h

WORKED EXAMPLES

Six patterns you should recognize

1. Rectangle perimeter

Length 9 cm, width 4 cm.

2(9) + 2(4) = 26 cm

26 cm

2. Rectangle area

Length 9 cm, width 4 cm.

9 × 4 = 36 cm²

36 cm²

3. Square

A square has side 7 m.

P = 4(7) = 28 m
A = 7 × 7 = 49 m²

4. Triangle area

Base 12 cm, height 5 cm.

1/2 × 12 × 5 = 30 cm²

30 cm²

5. Parallelogram area

Base 10 m, perpendicular height 6 m.

10 × 6 = 60 m²

60 m²

6. Missing dimension

A rectangle has area 48 cm² and width 6 cm. Find the length.

48 ÷ 6 = 8 cm

8 cm

TRY WITH ME

Choose perimeter or area first

A rectangular playground is 15 m long and 8 m wide. A fence will go all the way around it.

1. Fence means around the outside.
2. So use perimeter.
3. 2(15) + 2(8) = 30 + 16.

46 m of fencing

COMMON TRAPS

Watch out for these

Using perimeter for flooring.
Flooring covers inside space, so use area.

Forgetting the 1/2 in triangle area.
A triangle is half of a matching parallelogram or rectangle.

Using the slanted side as height.
Height must be perpendicular to the base.

Wrong units.
Perimeter uses linear units. Area uses square units.

REGULAR PRACTICE

Build the skill

These mix rectangle, square, triangle, parallelogram, trapezoid, and missing-dimension problems.

QUICK FIRE

Around or inside?

Five fast questions. Identify the quantity before calculating.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Still mixing up perimeter and area?

Send Teacher Abi the exact item that confused you. I can help you identify what the problem is measuring before you calculate.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Solid Figures, Surface Area, and Volume

Previous: Angles and Basic GeometryPSHS NCE Student Hub

PSHS NCE Angles and Basic Geometry Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Angles and Basic Geometry

See the shapes, not just the words. This reviewer uses simple diagrams to teach angle relationships and basic plane-geometry ideas in a Grade 6–7 friendly way.

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Angles and Basic Geometry

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Basic ideas to keep in mind

Point = exact location
Line segment = has two endpoints
Ray = starts at one point and goes on in one direction
Angle = formed by two rays with the same endpoint

Quick facts:
Right angle = 90°
Straight angle = 180°
Full turn = 360°
Angles in a triangle = 180°

VISUAL GUIDE

See the relationships

COMPLEMENTARY ANGLES

Complementary angles add up to 90°.

SUPPLEMENTARY ANGLES

Supplementary angles add up to 180°.

VERTICAL ANGLES

Vertical angles are equal.

TRIANGLE ANGLE SUM

The interior angles of a triangle total 180°.

WORKED EXAMPLES

Six patterns you should recognize

1. Identify angle types

An angle measuring 35° is acute. An angle measuring 90° is a right angle. An angle measuring 135° is obtuse.

2. Complementary angles

x + 37 = 90

Subtract 37 from 90.

x = 53°

3. Supplementary angles

x + 125 = 180

Subtract 125 from 180.

x = 55°

4. Vertical angles

If one angle in an X-shaped intersection is 72°, the vertical angle across from it is also 72°.

Vertical angles are equal.

5. Angles in a triangle

48 + 67 + x = 180

Add 48 and 67 to get 115. Then 180 − 115 = 65.

x = 65°

6. Parallel and perpendicular lines

Parallel lines never meet. Perpendicular lines meet to form a right angle.

Shortcut: If the corner is 90°, the lines are perpendicular.

TRY WITH ME

Which relationship is it?

Two angles form a straight line. One angle is 146°. What is the other angle?

1. A straight line means the two angles are supplementary.
2. Supplementary angles add to 180°.
3. 180 − 146 = 34.

34°

COMMON TRAPS

Watch out for these

Mixing up complementary and supplementary.
Complementary = 90°. Supplementary = 180°.

Forgetting triangle angle sum.
Angles inside a triangle always total 180°.

Not recognizing vertical angles.
Opposite angles in an X-shape are equal.

Using a picture only by appearance.
Use the given measurements and facts, not just how “big” the drawing looks.

REGULAR PRACTICE

Build the skill

These mix angle types, complementary, supplementary, vertical angles, triangle angles, and line relationships.

QUICK FIRE

Can you spot the angle fact fast?

Five quick questions. Use the shortest valid idea.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Still mixing up angle relationships?

Send Teacher Abi your result or the exact item that confused you. I can help you see which geometry fact to use first.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Perimeter and Area

Previous: Rates, Work, and Distance PSHS NCE Student Hub