🧠 The 7 Thinking Lenses

These are the mental tools you use to solve problems, understand ideas, and think deeply. When you can name how you're thinking, you become a stronger thinker!

🔍

Comparing

"I compared things"

When you use the Comparing lens, you look at two or more things and find what's the same and what's different. This helps you group things, understand relationships, and make smart choices.

Questions You Ask:

What's the same?
What's different?
Which one is bigger/smaller/faster?
How do these belong together?

Examples:

Math: "Is 5 a big number? Well, compared to what?"

Real life: "A square and rectangle both have 4 sides, but a square's sides are all equal."

Math skill: Classification, sets, greater than/less than, equivalence

🎯 Practice Questions (12)

Q1. Which is bigger: 100 pennies or 1 dollar?

100 pennies

They're equal! 100 pennies = 100 cents = 1 dollar. This uses the Comparing lens to see that different forms can have the same value.

1 dollar

They're equal! 100 pennies = 100 cents = 1 dollar. This uses the Comparing lens to see that different forms can have the same value.

They're equal

They're equal! 100 pennies = 100 cents = 1 dollar. This uses the Comparing lens to see that different forms can have the same value.

Q2. A square has 4 equal sides. A rectangle has 4 sides. What makes them different?

Number of sides

A square has ALL sides equal, but a rectangle only has opposite sides equal. Comparing helps us see what's same (4 sides) and different (side lengths).

All sides equal vs only opposite sides equal

A square has ALL sides equal, but a rectangle only has opposite sides equal. Comparing helps us see what's same (4 sides) and different (side lengths).

Shape color

A square has ALL sides equal, but a rectangle only has opposite sides equal. Comparing helps us see what's same (4 sides) and different (side lengths).

Q3. Is a whale a fish?

Yes, it lives in water

Comparing whales to fish: both swim, but whales breathe air and feed milk to babies. Classification requires comparing multiple features.

No, it's a mammal

Comparing whales to fish: both swim, but whales breathe air and feed milk to babies. Classification requires comparing multiple features.

Sometimes

Comparing whales to fish: both swim, but whales breathe air and feed milk to babies. Classification requires comparing multiple features.

Q4. Which is heavier: 1 kg of feathers or 1 kg of rocks?

Feathers

They weigh the same! Comparing mass vs size helps us see that 1 kg = 1 kg regardless of what it's made of.

Rocks

They weigh the same! Comparing mass vs size helps us see that 1 kg = 1 kg regardless of what it's made of.

Same weight

They weigh the same! Comparing mass vs size helps us see that 1 kg = 1 kg regardless of what it's made of.

Q5. Triangle has 3 sides. Pentagon has 5 sides. How many sides does a hexagon have?

Hexagon has 6 sides. Comparing shapes by counting sides is a key classification tool.

Q6. Are all rectangles squares?

Yes

No! All squares are rectangles (4 sides, right angles), but not all rectangles are squares (need equal sides). This is subset thinking.

Sometimes

No! All squares are rectangles (4 sides, right angles), but not all rectangles are squares (need equal sides). This is subset thinking.

Q7. Which fraction is larger: 1/2 or 1/4?

1/2

1/2 is larger. Comparing fractions: when the top number is the same, the smaller bottom number = bigger fraction.

1/4

1/2 is larger. Comparing fractions: when the top number is the same, the smaller bottom number = bigger fraction.

Equal

1/2 is larger. Comparing fractions: when the top number is the same, the smaller bottom number = bigger fraction.

Q8. 5 is bigger than 3. 3 is bigger than 1. So 5 is bigger than 1. This uses:

Comparing

This is Comparing with transitivity: if A > B and B > C, then A > C. A fundamental comparison property.

Testing

This is Comparing with transitivity: if A > B and B > C, then A > C. A fundamental comparison property.

Breaking apart

This is Comparing with transitivity: if A > B and B > C, then A > C. A fundamental comparison property.

Q9. Which costs more: 3 apples at $2 each, or 2 oranges at $3 each?

Apples ($6)

Both cost $6! Comparing requires calculating total cost, not just counting items or unit price.

Oranges ($6)

Both cost $6! Comparing requires calculating total cost, not just counting items or unit price.

Same ($6)

Both cost $6! Comparing requires calculating total cost, not just counting items or unit price.

Q10. Hot and cold are:

Opposites

Opposites! Comparing helps us identify opposite concepts like hot/cold, big/small, fast/slow.

Same thing

Opposites! Comparing helps us identify opposite concepts like hot/cold, big/small, fast/slow.

Unrelated

Opposites! Comparing helps us identify opposite concepts like hot/cold, big/small, fast/slow.

Q11. Which group is bigger: {1, 2, 3} or {a, b, c, d}?

Numbers

Letters has 4 elements vs 3 numbers. Comparing sets means counting members, not what type they are.

Letters

Letters has 4 elements vs 3 numbers. Comparing sets means counting members, not what type they are.

Same size

Letters has 4 elements vs 3 numbers. Comparing sets means counting members, not what type they are.

Q12. You have 2 quarters and 5 dimes. Your friend has 10 nickels. Who has more money?

You (100¢)

You: 50¢ + 50¢ = 100¢. Friend: 10 × 5¢ = 50¢. Comparing requires converting to same units!

Friend (50¢)

You: 50¢ + 50¢ = 100¢. Friend: 10 × 5¢ = 50¢. Comparing requires converting to same units!

Same

You: 50¢ + 50¢ = 100¢. Friend: 10 × 5¢ = 50¢. Comparing requires converting to same units!

🔢

Ordering

"I thought about order"

When you use the Ordering lens, you arrange things in a sequence. This could be by size, time, importance, or any other rule. Order helps you see patterns and understand steps.

Questions You Ask:

What comes first?
What's in the middle?
What's the biggest or smallest?
What happens before and after?

Examples:

Math: "Put these in order from smallest to biggest: puddle, ocean, pond, lake."

Real life: "The butterfly life cycle: egg → caterpillar → chrysalis → butterfly."

Math skill: Number lines, sequences, measurement, transitivity (if A > B and B > C, then A > C)

🎯 Practice Questions (12)

Q1. Put in order from smallest to largest: pond, ocean, puddle, lake

puddle → pond → lake → ocean

Puddle (tiny) → Pond (small) → Lake (big) → Ocean (huge). Ordering by size means finding the smallest and largest first!

ocean → lake → pond → puddle

Puddle (tiny) → Pond (small) → Lake (big) → Ocean (huge). Ordering by size means finding the smallest and largest first!

pond → puddle → ocean → lake

Puddle (tiny) → Pond (small) → Lake (big) → Ocean (huge). Ordering by size means finding the smallest and largest first!

Q2. What comes after 10, 20, 30, ...?

40! The pattern adds 10 each time. Ordering helps us see the sequence rule.

Q3. The months are ordered by:

Alphabetically

Time sequence! January comes before February because that's the order in which they happen during the year.

Number of days

Time sequence! January comes before February because that's the order in which they happen during the year.

Time sequence

Time sequence! January comes before February because that's the order in which they happen during the year.

Q4. If A is taller than B, and B is taller than C, who is shortest?

C is shortest! This is transitivity: ordering lets us compare even if we didn't directly measure A vs C.

Q5. Which happens first: eating breakfast, going to bed, eating lunch?

Breakfast

Breakfast comes first in a typical day. Ordering by time of day is a sequence.

Lunch

Breakfast comes first in a typical day. Ordering by time of day is a sequence.

Bed

Breakfast comes first in a typical day. Ordering by time of day is a sequence.

Q6. Number line: 0, __, 2, __, 4. What goes in the blanks?

1 and 2

1 and 3! The sequence goes 0, 1, 2, 3, 4. Ordering numbers on a line shows their sequence.

1 and 3

1 and 3! The sequence goes 0, 1, 2, 3, 4. Ordering numbers on a line shows their sequence.

0.5 and 2.5

1 and 3! The sequence goes 0, 1, 2, 3, 4. Ordering numbers on a line shows their sequence.

Q7. Life stages: Baby → Child → Adult → ___?

Teenager

Elder! This is the life cycle sequence. Ordering by age/stage of life.

Elder

Elder! This is the life cycle sequence. Ordering by age/stage of life.

Infant

Elder! This is the life cycle sequence. Ordering by age/stage of life.

Q8. In a race, if you're 2nd and you pass the person in 1st, what position are you?

1st place

1st place! Passing 1st means you become 1st. Ordering changes when positions swap.

2nd place

1st place! Passing 1st means you become 1st. Ordering changes when positions swap.

3rd place

1st place! Passing 1st means you become 1st. Ordering changes when positions swap.

Q9. Arrange by size: ant, elephant, mouse, whale

ant → mouse → elephant → whale

Ant (smallest) → Mouse → Elephant → Whale (largest). Always order from one extreme to the other!

ant → elephant → mouse → whale

Ant (smallest) → Mouse → Elephant → Whale (largest). Always order from one extreme to the other!

whale → elephant → mouse → ant

Ant (smallest) → Mouse → Elephant → Whale (largest). Always order from one extreme to the other!

Q10. What's in the middle of 1, 2, 3, 4, 5?

3 is in the middle! Ordering helps us find the median (middle value) of a sequence.

Q11. If winter comes after fall, and spring comes after winter, what comes after spring?

Fall

Summer! The seasons follow a cycle: Spring → Summer → Fall → Winter → (repeat). Ordering can be circular!

Summer

Summer! The seasons follow a cycle: Spring → Summer → Fall → Winter → (repeat). Ordering can be circular!

Winter

Summer! The seasons follow a cycle: Spring → Summer → Fall → Winter → (repeat). Ordering can be circular!

Q12. Steps to make tea: 1) Boil water 2) __ 3) Add milk. What's step 2?

Drink tea

Add tea leaves! Ordering shows the sequence of steps. You can't have tea flavor without adding tea leaves first.

Add tea leaves

Add tea leaves! Ordering shows the sequence of steps. You can't have tea flavor without adding tea leaves first.

Clean cup

Add tea leaves! Ordering shows the sequence of steps. You can't have tea flavor without adding tea leaves first.

💭

Imagining

"I imagined a story"

When you use the Imagining lens, you picture things in your mind that aren't right in front of you. You think about "what if" and create mental models. This is how you solve word problems and understand abstract ideas.

Questions You Ask:

Can I picture this in my mind?
What if this was different?
How would this work in real life?
Can this idea exist by itself?

Examples:

Math: "What is 'half' without anything to cut? Can the idea of 'half' exist by itself?"

Real life: "If I fold this paper into an origami crane, what shape will it be?"

Math skill: Word problems, modeling, abstraction, visualization, geometric thinking

🎯 Practice Questions (12)

Q1. If I cut a pizza into 8 slices and eat 4, what fraction is left?

1/8

1/2 is left! You need to imagine the pizza: 4 slices out of 8 = 4/8 = 1/2. Imagining helps visualize fractions.

1/2

1/2 is left! You need to imagine the pizza: 4 slices out of 8 = 4/8 = 1/2. Imagining helps visualize fractions.

4/8

1/2 is left! You need to imagine the pizza: 4 slices out of 8 = 4/8 = 1/2. Imagining helps visualize fractions.

Q2. A square has 4 sides. If you remove one corner, how many sides does the new shape have?

3 sides

5 sides! Imagine cutting the corner: you create 2 new sides where you cut. Imagining geometric transformations.

4 sides

5 sides! Imagine cutting the corner: you create 2 new sides where you cut. Imagining geometric transformations.

5 sides

5 sides! Imagine cutting the corner: you create 2 new sides where you cut. Imagining geometric transformations.

Q3. If you fold a paper in half, then in half again, how many sections do you have?

4 sections! Imagine: first fold = 2 parts, second fold = 2 × 2 = 4 parts. Mental visualization of folding.

Q4. Can 'half' exist without something to cut in half?

Yes, as an idea

Yes! 'Half' is an abstract concept → the idea exists even without a physical object. Imagining helps us work with abstractions.

No, must have an object

Yes! 'Half' is an abstract concept → the idea exists even without a physical object. Imagining helps us work with abstractions.

Only in math

Yes! 'Half' is an abstract concept → the idea exists even without a physical object. Imagining helps us work with abstractions.

Q5. Picture a triangle. Now flip it upside down. Does it still have 3 sides?

Yes

Yes! Flipping doesn't change the number of sides, just the orientation. Imagining transformations while preserving properties.

Depends

Yes! Flipping doesn't change the number of sides, just the orientation. Imagining transformations while preserving properties.

Q6. If all birds can fly, and penguins are birds, can penguins fly?

Yes

Trick question! The premise 'all birds can fly' is FALSE (penguins can't fly). Imagining helps us test if statements match reality.

This is a logic trick!

Trick question! The premise 'all birds can fly' is FALSE (penguins can't fly). Imagining helps us test if statements match reality.

Q7. What happens if you keep adding 1 to a number forever?

You reach the biggest number

You keep going infinitely! Imagining infinity requires thinking beyond physical limits → pure abstract thought.

You keep going infinitely

You keep going infinitely! Imagining infinity requires thinking beyond physical limits → pure abstract thought.

The number explodes

You keep going infinitely! Imagining infinity requires thinking beyond physical limits → pure abstract thought.

Q8. A 2D square lives on paper. If it became 3D, what would it be?

Cube

Cube! Imagining dimensions: add depth to a square → you get a cube. This is spatial visualization.

Rectangle

Cube! Imagining dimensions: add depth to a square → you get a cube. This is spatial visualization.

Sphere

Cube! Imagining dimensions: add depth to a square → you get a cube. This is spatial visualization.

Q9. Word problem: 'Sara has 3 apples. She gets 2 more.' To solve, you imagine:

3 apples, then adding 2

Imagine 3 apples, then 2 more appear → 5 total. Imagining creates a mental model to solve word problems.

The word 'more'

Imagine 3 apples, then 2 more appear → 5 total. Imagining creates a mental model to solve word problems.

Sara as a person

Imagine 3 apples, then 2 more appear → 5 total. Imagining creates a mental model to solve word problems.

Q10. Can you imagine a color that doesn't exist?

No, impossible

Yes! Your mind can imagine things that don't exist in reality. Imagining goes beyond what you've seen.

Yes, in your mind

Yes! Your mind can imagine things that don't exist in reality. Imagining goes beyond what you've seen.

Only artists can

Yes! Your mind can imagine things that don't exist in reality. Imagining goes beyond what you've seen.

Q11. If time went backwards, what would happen to your age?

Get younger

Get younger! Imagining 'what if' scenarios—even impossible ones → helps develop flexible thinking.

Stay same

Get younger! Imagining 'what if' scenarios—even impossible ones → helps develop flexible thinking.

Get older

Get younger! Imagining 'what if' scenarios—even impossible ones → helps develop flexible thinking.

Q12. 'Fairness' is a concept you can:

Touch and hold

Only imagine! 'Fairness' is abstract → it exists as an idea, not a physical thing. Imagining abstractions is key to philosophy and math.

Only imagine and discuss

Only imagine! 'Fairness' is abstract → it exists as an idea, not a physical thing. Imagining abstractions is key to philosophy and math.

See with your eyes

Only imagine! 'Fairness' is abstract → it exists as an idea, not a physical thing. Imagining abstractions is key to philosophy and math.

🧪

Testing

"I checked if it always works"

When you use the Testing lens, you check whether something is always true, sometimes true, or never true. You look for exceptions and think about when rules work and when they don't.

Questions You Ask:

Does this always happen?
Can I find an example where it doesn't work?
When is this true? When is it false?
What does it depend on?

Examples:

Math: "Is the answer always wrong if you did it a different way? No! Different methods can give the same answer."

Real life: "If I keep cutting a cake in half forever, will I ever have nothing left?"

Math skill: Conditional logic, proof, counterexamples, always/sometimes/never reasoning

🎯 Practice Questions (12)

Q1. Is 5 always a big number?

Always

Sometimes! 5 fingers is normal, but 5 stars is tiny. Testing means checking: does this ALWAYS work, or does it depend?

Sometimes

Sometimes! 5 fingers is normal, but 5 stars is tiny. Testing means checking: does this ALWAYS work, or does it depend?

Never

Sometimes! 5 fingers is normal, but 5 stars is tiny. Testing means checking: does this ALWAYS work, or does it depend?

Q2. If all cats have tails, and Fluffy is a cat, does Fluffy have a tail?

Always yes

Always yes! Testing logic: IF the rule is true (all cats have tails), THEN the conclusion follows. This is deductive reasoning.

Maybe

Always yes! Testing logic: IF the rule is true (all cats have tails), THEN the conclusion follows. This is deductive reasoning.

Need more info

Always yes! Testing logic: IF the rule is true (all cats have tails), THEN the conclusion follows. This is deductive reasoning.

Q3. Can you find a number where 5 is NOT bigger than 3?

No, 5 > 3 always

No! 5 > 3 is ALWAYS true (in standard numbers). Testing means trying to find counterexamples → if you can't, the rule always works.

Yes, in negative numbers

No! 5 > 3 is ALWAYS true (in standard numbers). Testing means trying to find counterexamples → if you can't, the rule always works.

Depends on language

No! 5 > 3 is ALWAYS true (in standard numbers). Testing means trying to find counterexamples → if you can't, the rule always works.

Q4. Is cutting a pizza into equal slices 'fair sharing'?

Always fair

Depends! Equal slices = equal SIZE, but what if people have different hunger levels? Testing fairness requires checking the context.

Depends on context

Depends! Equal slices = equal SIZE, but what if people have different hunger levels? Testing fairness requires checking the context.

Never fair

Depends! Equal slices = equal SIZE, but what if people have different hunger levels? Testing fairness requires checking the context.

Q5. Does 2+2 always equal 4?

Always

Only in base-10! In base-3, 2+2=11. Testing mathematical rules means knowing the SYSTEM they apply to.

Only in base-10

Only in base-10! In base-3, 2+2=11. Testing mathematical rules means knowing the SYSTEM they apply to.

Sometimes

Only in base-10! In base-3, 2+2=11. Testing mathematical rules means knowing the SYSTEM they apply to.

Q6. If you flip a coin 10 times and get heads every time, will the next flip be tails?

Yes, it's due

Still 50/50! Testing probability: past results DON'T affect future flips. This is testing the gambler's fallacy.

No, still 50/50

Still 50/50! Testing probability: past results DON'T affect future flips. This is testing the gambler's fallacy.

Probably tails

Still 50/50! Testing probability: past results DON'T affect future flips. This is testing the gambler's fallacy.

Q7. Is the middle of {1,2,3,4} the same as the middle of {1,2,3,4,5}?

Yes

No! First has middle 2.5 (even count), second has middle 3 (odd count). Testing 'middle' depends on how many items!

Depends

No! First has middle 2.5 (even count), second has middle 3 (odd count). Testing 'middle' depends on how many items!

Q8. Can a rule be right for some cases but wrong for others?

No, rules are absolute

Yes! 'Fast is better' works for races but not homework. Testing means checking if a rule works in ALL situations or just SOME.

Yes, context matters

Yes! 'Fast is better' works for races but not homework. Testing means checking if a rule works in ALL situations or just SOME.

Only in math

Yes! 'Fast is better' works for races but not homework. Testing means checking if a rule works in ALL situations or just SOME.

Q9. If a pattern is Red-Blue-Red-Blue, what comes next?

Always Red

Could be anything! Testing patterns: we ASSUME it continues, but patterns can break. Only MORE data confirms the rule.

Probably Red

Could be anything! Testing patterns: we ASSUME it continues, but patterns can break. Only MORE data confirms the rule.

Could be anything

Could be anything! Testing patterns: we ASSUME it continues, but patterns can break. Only MORE data confirms the rule.

Q10. Is zero 'nothing'?

Always nothing

Depends! Zero apples = nothing to eat, but 0°C isn't 'no temperature'. Testing concepts reveals they change by context.

Depends on context

Depends! Zero apples = nothing to eat, but 0°C isn't 'no temperature'. Testing concepts reveals they change by context.

It's a number

Depends! Zero apples = nothing to eat, but 0°C isn't 'no temperature'. Testing concepts reveals they change by context.

Q11. Can you prove 'all swans are white' by seeing 100 white swans?

Yes, that's enough

No! You'd need to check EVERY swan ever. Testing universal claims (all/none) requires checking all cases, not just examples.

No, you'd need to see ALL swans

No! You'd need to check EVERY swan ever. Testing universal claims (all/none) requires checking all cases, not just examples.

Depends

No! You'd need to check EVERY swan ever. Testing universal claims (all/none) requires checking all cases, not just examples.

Q12. If Method A gives answer 42, and Method B gives 42, are both correct?

Yes, same answer

Maybe! Same answer doesn't prove correctness. Testing means checking the METHOD, not just the result. Both could be wrong but lucky!

Maybe, depends on the problem

Maybe! Same answer doesn't prove correctness. Testing means checking the METHOD, not just the result. Both could be wrong but lucky!

No, only one way is right

Maybe! Same answer doesn't prove correctness. Testing means checking the METHOD, not just the result. Both could be wrong but lucky!

🔨

Breaking Apart

"I split it into pieces"

When you use the Breaking Apart lens, you take something whole and divide it into smaller parts. This helps you understand how things are built and makes big problems easier to solve.

Questions You Ask:

What are the parts?
How can I split this up?
What's inside this whole?
If I break it down, does it get easier?

Examples:

Math: "If I cut a pizza into 8 slices, each slice is 1/8 of the whole pizza."

Real life: "A sentence can be broken into words. A word can be broken into syllables."

Math skill: Fractions, decomposition, factors, place value, part-whole relationships

🎯 Practice Questions (12)

Q1. A pizza is cut into 8 slices. If you eat 3 slices, what fraction did you eat?

3 slices

3/8! Breaking the whole pizza into 8 parts, you took 3 parts. This is part-whole thinking.

3/8 of the pizza

3/8! Breaking the whole pizza into 8 parts, you took 3 parts. This is part-whole thinking.

5/8 left

3/8! Breaking the whole pizza into 8 parts, you took 3 parts. This is part-whole thinking.

Q2. The number 15 can be broken into:

10 + 5

Both! Breaking apart numbers: 15 = 10+5 (decomposition) = 3 × 5 (factors). Different ways to split the same number.

3 × 5

Both! Breaking apart numbers: 15 = 10+5 (decomposition) = 3 × 5 (factors). Different ways to split the same number.

Both

Both! Breaking apart numbers: 15 = 10+5 (decomposition) = 3 × 5 (factors). Different ways to split the same number.

Q3. If you break 'CAT' into sounds, you get:

/c/ /a/ /t/

/c/ /a/ /t/! Breaking words into phonemes (sounds) is how we learn to read. Breaking apart language!

C and AT

/c/ /a/ /t/! Breaking words into phonemes (sounds) is how we learn to read. Breaking apart language!

CA and T

/c/ /a/ /t/! Breaking words into phonemes (sounds) is how we learn to read. Breaking apart language!

Q4. What's inside a fraction like 3/4?

3 parts out of 4 equal parts

Both! 3/4 means '3 parts out of 4' AND '3 ÷ 4'. Breaking apart the fraction shows two meanings.

3 divided by 4

Both! 3/4 means '3 parts out of 4' AND '3 ÷ 4'. Breaking apart the fraction shows two meanings.

Both

Both! 3/4 means '3 parts out of 4' AND '3 ÷ 4'. Breaking apart the fraction shows two meanings.

Q5. A sentence has words. Words have:

Letters

Both! Breaking language: Sentence → Words → Syllables → Letters. Multiple levels of decomposition.

Syllables

Both! Breaking language: Sentence → Words → Syllables → Letters. Multiple levels of decomposition.

Both

Both! Breaking language: Sentence → Words → Syllables → Letters. Multiple levels of decomposition.

Q6. The number 100 broken into place value is:

1 hundred + 0 tens + 0 ones

1 hundred + 0 tens + 0 ones! Place value breaks numbers by position: hundreds, tens, ones.

50 + 50

1 hundred + 0 tens + 0 ones! Place value breaks numbers by position: hundreds, tens, ones.

10 × 10

1 hundred + 0 tens + 0 ones! Place value breaks numbers by position: hundreds, tens, ones.

Q7. If you split 12 cookies equally among 3 friends, each gets:

3 cookies

4 cookies! Breaking 12 into 3 equal groups: 12 ÷ 3 = 4. Division is breaking into equal parts.

4 cookies

4 cookies! Breaking 12 into 3 equal groups: 12 ÷ 3 = 4. Division is breaking into equal parts.

6 cookies

4 cookies! Breaking 12 into 3 equal groups: 12 ÷ 3 = 4. Division is breaking into equal parts.

Q8. What are the factors of 12?

1,2,3,4,6,12

1,2,3,4,6,12! Factors break a number into pieces that multiply back: 1 × 12, 2 × 6, 3 × 4 all = 12.

2,4,6,8,10,12

1,2,3,4,6,12! Factors break a number into pieces that multiply back: 1 × 12, 2 × 6, 3 × 4 all = 12.

Only 1 and 12

1,2,3,4,6,12! Factors break a number into pieces that multiply back: 1 × 12, 2 × 6, 3 × 4 all = 12.

Q9. A rectangle can be split into:

2 triangles

Both! Breaking shapes: diagonal cut = 2 triangles, grid = 4 squares (if it's a special rectangle). Same whole, different parts.

4 squares

Both! Breaking shapes: diagonal cut = 2 triangles, grid = 4 squares (if it's a special rectangle). Same whole, different parts.

Both possible

Both! Breaking shapes: diagonal cut = 2 triangles, grid = 4 squares (if it's a special rectangle). Same whole, different parts.

Q10. The number 48 broken into prime factors is:

2 × 24

2 × 2 × 2 × 2 × 3! Breaking into PRIME factors means splitting until you can't split anymore (only prime numbers left).

2 × 2 × 2 × 2 × 3

2 × 2 × 2 × 2 × 3! Breaking into PRIME factors means splitting until you can't split anymore (only prime numbers left).

6 × 8

2 × 2 × 2 × 2 × 3! Breaking into PRIME factors means splitting until you can't split anymore (only prime numbers left).

Q11. If a task takes 60 minutes, you can break it into:

6 parts of 10 min each

Both! Breaking time: 60 = 6 × 10 = 4 × 15 = 3 × 20. Different ways to split the same duration.

4 parts of 15 min each

Both! Breaking time: 60 = 6 × 10 = 4 × 15 = 3 × 20. Different ways to split the same duration.

Both

Both! Breaking time: 60 = 6 × 10 = 4 × 15 = 3 × 20. Different ways to split the same duration.

Q12. A story has a beginning, middle, and end. This breaks the story by:

Characters

Sequence/structure! Breaking a story into parts by WHEN things happen (narrative arc). This is structural decomposition.

Sequence/structure

Sequence/structure! Breaking a story into parts by WHEN things happen (narrative arc). This is structural decomposition.

Page numbers

Sequence/structure! Breaking a story into parts by WHEN things happen (narrative arc). This is structural decomposition.

🔮

Finding Patterns

"I found a pattern"

When you use the Finding Patterns lens, you notice what repeats or follows a rule. Patterns help you predict what comes next and see connections between different things.

Questions You Ask:

What repeats?
What's the rule?
What comes next?
Does this same pattern appear somewhere else?

Examples:

Math: "A bead pattern (red-blue-blue) is like a music rhythm (loud-soft-soft)!"

Real life: "The number pattern 2, 4, 6, 8... follows the rule 'add 2 each time.'"

Math skill: Sequences, algebraic thinking, functions, generalization, skip counting

🎯 Practice Questions (12)

Q1. 2, 4, 6, 8, ... What's the pattern rule?

Add 2 each time

Add 2 each time! Finding patterns means discovering the RULE that generates the sequence.

Multiply by 2

Add 2 each time! Finding patterns means discovering the RULE that generates the sequence.

Even numbers

Add 2 each time! Finding patterns means discovering the RULE that generates the sequence.

Q2. Red-Blue-Red-Blue is the SAME pattern as:

A-B-A-B

Both! Patterns are about STRUCTURE, not content. A-B-A-B and 1-2-1-2 follow the same alternating pattern.

1-2-1-2

Both! Patterns are about STRUCTURE, not content. A-B-A-B and 1-2-1-2 follow the same alternating pattern.

Both

Both! Patterns are about STRUCTURE, not content. A-B-A-B and 1-2-1-2 follow the same alternating pattern.

Q3. If Monday-Coding, Tuesday-Coding, Wednesday-__, what's the pattern?

Coding (daily)

Can't tell yet! You need MORE data points to confirm a pattern. 2 examples aren't enough to be sure.

Rest (alternating)

Can't tell yet! You need MORE data points to confirm a pattern. 2 examples aren't enough to be sure.

Can't tell yet

Can't tell yet! You need MORE data points to confirm a pattern. 2 examples aren't enough to be sure.

Q4. Which shape doesn't fit: Circle, Square, Triangle, Car?

Car (not a basic shape)

Car! Finding patterns includes finding what DOESN'T belong. All others are geometric shapes.

Circle (only curved)

Car! Finding patterns includes finding what DOESN'T belong. All others are geometric shapes.

Triangle (3 sides)

Car! Finding patterns includes finding what DOESN'T belong. All others are geometric shapes.

Q5. Skip counting by 5s: 5, 10, 15, __, 25

20! Skip counting is a pattern: add the same number repeatedly. Patterns help predict what's next.

Q6. Is 'A-B-C' the same pattern as 'Do-Re-Mi'?

Yes, both are sequences

Yes! Both are ordered sequences (alphabetic/musical scale). Patterns transcend specific content → it's about the structure!

No, different content

Yes! Both are ordered sequences (alphabetic/musical scale). Patterns transcend specific content → it's about the structure!

Only in music

Yes! Both are ordered sequences (alphabetic/musical scale). Patterns transcend specific content → it's about the structure!

Q7. Even numbers (2,4,6...) follow the pattern:

Divisible by 2

Both! Patterns can be described by MULTIPLE rules: divisibility AND last digit. Different views of same pattern.

End in 0,2,4,6,8

Both! Patterns can be described by MULTIPLE rules: divisibility AND last digit. Different views of same pattern.

Both

Both! Patterns can be described by MULTIPLE rules: divisibility AND last digit. Different views of same pattern.

Q8. Fibonacci: 1,1,2,3,5,8,... What's the pattern?

Add 1

Add previous two! Each number = sum of the two before it. Patterns can build on previous terms (recursive).

Add previous 2 numbers

Add previous two! Each number = sum of the two before it. Patterns can build on previous terms (recursive).

Double each time

Add previous two! Each number = sum of the two before it. Patterns can build on previous terms (recursive).

Q9. Day-Night-Day-Night is like:

On-Off-On-Off

On-Off! Both are binary alternating patterns. Finding patterns helps you see connections across different domains.

Summer-Winter

On-Off! Both are binary alternating patterns. Finding patterns helps you see connections across different domains.

Both

On-Off! Both are binary alternating patterns. Finding patterns helps you see connections across different domains.

Q10. Which continues the pattern: Square, Cube, __?

Rectangle

Tesseract! Pattern: 2D square → 3D cube → 4D tesseract. Patterns can extend into dimensions you can't even see!

Tesseract (4D)

Tesseract! Pattern: 2D square → 3D cube → 4D tesseract. Patterns can extend into dimensions you can't even see!

Circle

Tesseract! Pattern: 2D square → 3D cube → 4D tesseract. Patterns can extend into dimensions you can't even see!

Q11. All squares are rectangles. This pattern is:

Subset relationship

Subset! Pattern: every square fits the rectangle pattern (4 sides, right angles), but not vice versa. This is hierarchical pattern.

Size pattern

Subset! Pattern: every square fits the rectangle pattern (4 sides, right angles), but not vice versa. This is hierarchical pattern.

Color pattern

Subset! Pattern: every square fits the rectangle pattern (4 sides, right angles), but not vice versa. This is hierarchical pattern.

Q12. Pattern: Every other letter starting at A: A, C, E, __

G! Skipping one letter each time: A(skip B)C(skip D)E(skip F)G. Patterns show what to SKIP too!

❓

Asking Why

"I asked why"

When you use the Asking Why lens, you look for reasons and explanations. You build chains of cause and effect to understand how things work and why they happen.

Questions You Ask:

Why does this happen?
What's the reason?
How does this cause that?
Can I explain this to someone else?

Examples:

Math: "Why does ice float on water? Because ice is less dense than liquid water, so it's lighter."

Real life: "Why do we need to sleep? Because our brains need rest to organize memories and recharge."

Math skill: Causal reasoning, proof, justification, explanation, logical chains

🎯 Practice Questions (12)

Q1. Why does ice float on water?

Ice is magic

Ice is less dense! Asking why reveals the CAUSE: when water freezes, it expands, making it less dense, so it floats.

Ice is less dense than water

Ice is less dense! Asking why reveals the CAUSE: when water freezes, it expands, making it less dense, so it floats.

Ice is colder

Ice is less dense! Asking why reveals the CAUSE: when water freezes, it expands, making it less dense, so it floats.

Q2. Why do we sleep?

Because we're tired

Brain needs rest! Asking why goes beyond the obvious → sleep lets your brain process the day and recharge. That's the deeper reason.

Brain needs rest to organize memories

Brain needs rest! Asking why goes beyond the obvious → sleep lets your brain process the day and recharge. That's the deeper reason.

To pass time

Brain needs rest! Asking why goes beyond the obvious → sleep lets your brain process the day and recharge. That's the deeper reason.

Q3. Why is the sky blue?

Because it reflects the ocean

Light scatters! Sunlight hits air molecules, blue light scatters more. Asking why explains the MECHANISM, not just observation.

Light scatters in atmosphere

Light scatters! Sunlight hits air molecules, blue light scatters more. Asking why explains the MECHANISM, not just observation.

Paint from clouds

Light scatters! Sunlight hits air molecules, blue light scatters more. Asking why explains the MECHANISM, not just observation.

Q4. Why does 5+3 = 8?

Because we count

Definition of addition! 'Why' in math traces back to DEFINITIONS and axioms. 5+3=8 because that's what adding means.

Definition of addition

Definition of addition! 'Why' in math traces back to DEFINITIONS and axioms. 5+3=8 because that's what adding means.

Math rules

Definition of addition! 'Why' in math traces back to DEFINITIONS and axioms. 5+3=8 because that's what adding means.

Q5. A plant grows toward the sun. Why?

Plants like sun

Photosynthesis needs light! Asking why reveals PURPOSE: plants need sun's energy to make food. This is FUNCTION.

Photosynthesis needs light

Photosynthesis needs light! Asking why reveals PURPOSE: plants need sun's energy to make food. This is FUNCTION.

Plants are smart

Photosynthesis needs light! Asking why reveals PURPOSE: plants need sun's energy to make food. This is FUNCTION.

Q6. Why do magnets attract metal?

Magic force

Magnetic field! Asking why explores invisible FORCES: magnets create fields that pull on certain metals (iron, nickel).

Magnetic field interaction

Magnetic field! Asking why explores invisible FORCES: magnets create fields that pull on certain metals (iron, nickel).

Magnets are sticky

Magnetic field! Asking why explores invisible FORCES: magnets create fields that pull on certain metals (iron, nickel).

Q7. Why does 2 × 3 = 3 × 2?

Lucky coincidence

Commutative property! Asking why reveals mathematical LAW: order doesn't matter in multiplication. This is a proven property, not luck.

Commutative property

Commutative property! Asking why reveals mathematical LAW: order doesn't matter in multiplication. This is a proven property, not luck.

Both equal 6

Commutative property! Asking why reveals mathematical LAW: order doesn't matter in multiplication. This is a proven property, not luck.

Q8. Why do seasons change?

Earth's orbit shape

Earth's tilt! Asking why traces the ROOT CAUSE: Earth tilts 23.5°, so different areas get more sun at different times of year.

Earth's tilt

Earth's tilt! Asking why traces the ROOT CAUSE: Earth tilts 23.5°, so different areas get more sun at different times of year.

Sun gets tired

Earth's tilt! Asking why traces the ROOT CAUSE: Earth tilts 23.5°, so different areas get more sun at different times of year.

Q9. Why is zero important?

Placeholder in numbers

Both! Zero is a placeholder (makes 10 different from 1) AND represents absence. Asking why reveals multiple roles.

Represents nothing

Both! Zero is a placeholder (makes 10 different from 1) AND represents absence. Asking why reveals multiple roles.

Both

Both! Zero is a placeholder (makes 10 different from 1) AND represents absence. Asking why reveals multiple roles.

Q10. Why does the moon change shape?

Moon rotates

Different lit portions! The moon doesn't change → we see different parts lit by the sun as it orbits Earth. Asking why = understanding perspective.

We see different lit portions

Different lit portions! The moon doesn't change → we see different parts lit by the sun as it orbits Earth. Asking why = understanding perspective.

Moon grows/shrinks

Different lit portions! The moon doesn't change → we see different parts lit by the sun as it orbits Earth. Asking why = understanding perspective.

Q11. Why do we use base-10 numbers?

10 is magic

We have 10 fingers! Asking why about human choices: base-10 is cultural/historical, tied to how we count on our hands.

We have 10 fingers

We have 10 fingers! Asking why about human choices: base-10 is cultural/historical, tied to how we count on our hands.

Math rules

We have 10 fingers! Asking why about human choices: base-10 is cultural/historical, tied to how we count on our hands.

Q12. Why are bubbles round?

Soap makes circles

Surface tension! Asking why reveals physics: bubbles form the smallest surface area possible for the volume → that's a sphere. Nature optimizes!

Surface tension minimizes area

Surface tension! Asking why reveals physics: bubbles form the smallest surface area possible for the volume → that's a sphere. Nature optimizes!

Bubbles copy balls

Surface tension! Asking why reveals physics: bubbles form the smallest surface area possible for the volume → that's a sphere. Nature optimizes!

💡 The Power of Naming Your Thinking

When you can say "I'm using the Comparing lens" or "This needs the Testing lens," you become aware of how you think. This meta-cognitive skill helps you choose the right thinking strategy for any problem → in math, science, reading, or life!