Showing posts with label Grade 6. Show all posts

Friday, September 4, 2026

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.

Arithmetic patternsGrowing patternsMissing termsQuick Fire + Mastery

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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

Subtract neighboring terms. Are the differences the same?
If not, check whether the differences themselves follow a pattern.
Check simple multiplication or division.
If neither works, test an alternating rule.
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.

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.

4. Alternating operations

5, 10, 12, 24, 26, ?

×2, +2, ×2, +2, so next is ×2.

5. Square numbers

1, 4, 9, 16, 25, ?

These are 1², 2², 3², 4², 5².

6. Missing middle term

8, 13, ?, 23, 28

Add 5 each time.

TRY WITH ME

Use the gaps

Find the next number:

3, 7, 12, 18, 25, ?

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

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 Conversion PSHS NCE Student Hub

Thursday, September 3, 2026

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.

3D diagramsFaces, edges, verticesSurface area + volumeQuick Fire + Mastery

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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?

Space inside means volume.
Use L × W × H.
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 Area PSHS NCE Student Hub

PSHS NCE Geometry and Measurement Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Geometry and Measurement

Use perimeter, area, angle facts, and unit conversions quickly and accurately.

Grade 6–7 friendlyPerimeter + AreaAngles + UnitsQuick Fire + Mastery

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Geometry and Measurement

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Know what the question is measuring

Perimeter is the distance around a shape.

Area is the amount of space inside a flat shape.

Rectangle:
Perimeter = 2L + 2W
Area = L × W

DO IT FAST

Use IDENTIFY → FORMULA → UNIT

IDENTIFY: Is the problem asking for perimeter, area, angle, or measurement?
FORMULA: Use only the formula you need.
UNIT: Check cm vs cm², m vs m², minutes vs hours, and so on.
REASON: Ask whether your answer is the right size.

WORKED EXAMPLES

Six patterns you should recognize

1. Rectangle perimeter

Length 8 cm, width 5 cm.

2(8) + 2(5) = 26 cm

26 cm

2. Rectangle area

Length 9 m, width 4 m.

9 × 4 = 36 m²

36 m²

3. Triangle area

Base 10 cm, height 6 cm.

1/2 × 10 × 6 = 30 cm²

30 cm²

4. Angle fact

Angles on a straight line add to 180°. If one angle is 125°, the other is:

180 − 125 = 55°

55°

5. Unit conversion

3.5 m = ? cm

3.5 × 100 = 350 cm

350 cm

6. Time conversion

2.5 hours = ? minutes

2.5 × 60 = 150 minutes

150 minutes

TRY WITH ME

Perimeter or area?

A rectangular garden is 12 m long and 7 m wide. How much fencing is needed to go around it?

Fencing goes around the garden.
So we need perimeter.
2(12)+2(7)=24+14.

38 m

COMMON TRAPS

Watch out for these

Using area when the question asks “around.”
Around means perimeter.

Forgetting square units.
Area uses cm², m², and so on.

Using a slanted side as triangle height.
The height must be perpendicular to the base.

Converting units in the wrong direction.
1 m = 100 cm, so meters to centimeters gets larger.

REGULAR PRACTICE

Build the skill

These mix perimeter, area, angles, and measurement conversions.

QUICK FIRE

Can you choose the right formula fast?

Five short questions. Decide what is being measured 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?

Geometry formulas getting mixed up?

Send Teacher Abi the problem that confused you and your result. I can help you identify what the question is actually measuring.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Patterns, Sequences, and Number Relationships

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

PSHS NCE Rates, Work, and Distance Problems Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Rates, Work, and Distance Problems

Use the right relationship quickly: distance = rate × time, unit rate, and simple work-rate thinking.

Grade 6–7 friendlyRate × TimeRegular practiceQuick Fire + Mastery

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Rates, Work, and Distance Problems

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Rate tells how much happens in one unit of time

A cyclist travels 60 km in 3 hours.

60 ÷ 3 = 20 km/h

The rate is 20 km per hour.

Three useful relationships:
Distance = Rate × Time
Rate = Distance ÷ Time
Time = Distance ÷ Rate

DO IT FAST

Use WHAT DO I NEED?

Need distance? Multiply rate × time.
Need rate? Divide distance ÷ time.
Need time? Divide distance ÷ rate.
Need work per hour? Think “fraction of the job completed in one hour.”

Fast clue: If the problem gives “per hour,” “per minute,” or “each,” you are probably dealing with a rate.

WORKED EXAMPLES

Six patterns you should recognize

1. Find distance

A car travels at 50 km/h for 4 hours.

50 × 4 = 200 km

200 km

2. Find rate

A runner covers 18 km in 3 hours.

18 ÷ 3 = 6 km/h

6 km/h

3. Find time

A bus travels 180 km at 60 km/h.

180 ÷ 60 = 3 hours

3 hours

4. Unit rate

5 notebooks cost ₱150.

₱150 ÷ 5 = ₱30 each

₱30 per notebook

5. Simple work rate

A machine can finish a job in 4 hours.

In one hour it finishes 1/4 of the job.

Work rate = 1/4 job per hour

6. Combined work

Machine A does 1/4 of a job per hour. Machine B does 1/4 per hour.

1/4 + 1/4 = 1/2 job per hour

Together they finish in 2 hours.

TRY WITH ME

Pick the missing quantity

A van travels 240 km at 60 km/h. How long does the trip take?

We know distance and rate.
We need time.
Time = distance ÷ rate.
240 ÷ 60 = 4.

4 hours

COMMON TRAPS

Watch out for these

Multiplying when you should divide.
Ask first: am I looking for distance, rate, or time?

Ignoring units.
Hours and minutes must match.

Treating work time like ordinary addition.
If one worker takes 4 hours, the rate is 1/4 job per hour.

Missing “each” or “per.”
Those words usually signal unit rate.

REGULAR PRACTICE

Build the skill

These mix distance-rate-time, unit rates, and simple work-rate questions.

QUICK FIRE

Can you choose the right operation fast?

Five short questions. Decide whether to multiply or divide 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?

Rate problems still slowing you down?

Send Teacher Abi the problem that confused you and your result. I can help you identify what the problem is really asking for.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Geometry and Measurement

Previous: Algebraic Expressions and Simple Equations PSHS NCE Student Hub

PSHS NCE Algebraic Expressions and Simple Equations Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Algebraic Expressions and Simple Equations

Translate words into math, understand variables, and solve simple equations by undoing operations.

Grade 6–7 friendlyVariables + equationsRegular practiceQuick Fire + Mastery

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Algebraic Expressions and Simple Equations

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A variable is just a number we do not know yet

If x + 5 = 12, then x is the number that makes the equation true.

x + 5 = 12 → x = 7

Expression: 3x + 4 has no equals sign.
Equation: 3x + 4 = 19 has an equals sign and can be solved.

DO IT FAST

Use TRANSLATE → UNDO → CHECK

TRANSLATE: Turn words into a math expression or equation.
UNDO: Reverse the operation around the variable.
CHECK: Put your answer back into the original equation.

Undo pairs:
+ ↔ −
× ↔ ÷

WORKED EXAMPLES

Six patterns you should recognize

1. Translate words

“Five more than a number n”

n + 5

2. Watch “less than”

“7 less than x”

x − 7

Not 7 − x.

3. Solve addition

x + 8 = 15

Subtract 8 from both sides.

x = 7

4. Solve multiplication

5x = 35

Divide both sides by 5.

x = 7

5. Two-step equation

3x + 4 = 19

Subtract 4: 3x = 15. Divide by 3.

x = 5

6. Story to equation

Four times a number plus 3 equals 27.

4x + 3 = 27

x = 6

TRY WITH ME

Undo one step at a time

2x + 7 = 21

Subtract 7 from both sides.
Now 2x = 14.
Divide both sides by 2.

x = 7
Check: 2(7)+7=21.

COMMON TRAPS

Watch out for these

Changing only one side.
Keep the equation balanced.

Mixing up expressions and equations.
Only equations can be solved.

Reading “less than” in the wrong order.
5 less than x means x−5.

Not checking.
Substitute the answer back in.

REGULAR PRACTICE

Build the skill

These mix translating, evaluating, one-step equations, and simple two-step equations.

QUICK FIRE

Can you undo it fast?

Five short questions. Identify the operation around the variable first.

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?

Equations still feel like a puzzle?

Send Teacher Abi the item that confused you and your result. I can show you what operation to undo first.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Rates, Work, and Distance Problems

Previous: Number Theory and Remainders PSHS NCE Student Hub

PSHS NCE Number Theory and Remainders Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Number Theory and Remainders

Use divisibility, odd-even patterns, and remainders to solve number questions without doing more work than necessary.

Grade 6–7 friendlyFast remainder thinkingRegular practiceQuick Fire + Mastery

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Number Theory and Remainders

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A remainder is what is left after making equal groups

23 ÷ 5 = 4 remainder 3

Five groups of 4 use 20. There are 3 left.

Important: A remainder must always be smaller than the divisor. If you divide by 7, the only possible remainders are 0, 1, 2, 3, 4, 5, or 6.

DO IT FAST

Use MULTIPLE → LEFTOVER → PATTERN

MULTIPLE: Find the nearest multiple below the number.
LEFTOVER: Subtract to get the remainder.
PATTERN: If a number changes, track only the leftover part when possible.
CHECK: The remainder must be smaller than the divisor.

WORKED EXAMPLES

Six patterns you should recognize

1. Find a remainder

What is the remainder when 47 is divided by 6?

42 is the nearest multiple of 6 below 47

Remainder = 5

2. Remainder after adding

A number leaves remainder 4 when divided by 7. What remainder does the number plus 5 leave?

4 + 5 = 9 → 9 leaves remainder 2 when divided by 7

3. Remainder after multiplying

A number leaves remainder 4 when divided by 7. What remainder does 3 times the number leave?

3 × 4 = 12 → 12 leaves remainder 5 when divided by 7

4. Odd and even

Odd + odd = ?

3 + 5 = 8

Even

5. Divisibility shortcut

Is 5,742 divisible by 3?

5 + 7 + 4 + 2 = 18

Yes. 18 is divisible by 3.

6. Possible remainder

Which could be a remainder when dividing by 5?

Possible remainders are only 0, 1, 2, 3, or 4.

4 could be a remainder; 5 or more cannot.

TRY WITH ME

Track only the leftover

A number leaves a remainder of 3 when divided by 5. What remainder will 4 times the number leave when divided by 5?

Ignore the full groups of 5. They will still divide evenly.
Track only the remainder: 3.
Multiply: 4 × 3 = 12.
12 leaves remainder 2 when divided by 5.

Remainder = 2

COMMON TRAPS

Watch out for these

Giving a remainder equal to the divisor.
When dividing by 6, remainder 6 is impossible.

Doing the full multiplication.
If only the remainder matters, track the remainder instead.

Forgetting odd/even patterns.
These can answer some questions instantly.

Using long division when a divisibility rule works.
Check the digits first.

REGULAR PRACTICE

Build the skill

These mix direct remainders, transformed remainders, divisibility, and odd-even reasoning.

QUICK FIRE

Can you spot the remainder fast?

Five short questions. Avoid full calculations when the leftover is enough.

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?

Remainder questions still feel tricky?

Send Teacher Abi the question that confused you and your result. I can show you the fastest leftover method.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Algebraic Expressions and Simple Equations

Previous: Percent PSHS NCE Student Hub