How each demonstration works
Every demonstration follows the same path the book does, so you always know what comes next.
Predict first
Read the question and the equation from the chapter, then make a guess. A wrong guess is useful.
Change one value
Pick from a short menu. The picture and the numbers update together, starting from the book's own example.
Read the work
Every state shows the calculation behind it, small enough to redo with a pencil.
Check yourself
Try a new question, then open the answer. Each demonstration names the chapter section it came from.
Reentry
The first task is not calculation. It is finding something on the page that can be named and checked.
You Already Do Mathematics
Sorting becomes mathematics the moment the rule is public enough for someone else to test.
- 1More than one number can be the odd one out
- 2A list records, a rule decides
- 3Two rules on six cards
- 4Every number gets one location
Why Your Mind Goes Blank
The arithmetic stays the same. What changes is where the middle of the work can live.
- 1Same arithmetic, different workspace
- 2A tens-and-ones record
- 3Two routes to the same sum
- 4Leave a route back
The Most Useful Wrong Answer
A counterexample points at the sentence that promised too much, never at the person who wrote it.
- 1One allowed case is enough
- 2Same calculation, different domain
- 3Odd numbers that build squares
- 4Two leftovers make a pair
Patterns Are the Point
Name the change, name what counts as the same, then check every feature that standard needs.
- 1Carry every link through the pairing
- 2One failed link is a counterexample
- 3What a reflection keeps and what it moves
- 4An invariant is not yet a symmetry
Quantity
You can now make a rule visible, test an edge case, and ask what survives a change. Quantity gives those habits numbers to work on.
How Big Is It?
Read size from place first, then state what an estimate changed before you check it exactly.
- 1Compare from the left
- 2A grouping changes the question
- 3The example theater
- 4Which way does the estimate miss?
Addition Moves, Subtraction Measures
A sign belongs to a number, a move has a direction, and every subtraction can be checked by addition.
- 1Start, change, end
- 2Subtract by adding the opposite
- 3Change or distance?
- 4Crossing zero, checked by addition
Multiplication Scales, Division Asks
Every product has two roles in it, and every division asks which role is missing.
- 1An array shows the roles
- 2Signs in a product
- 3Sharing or grouping?
- 4When something remains
Whole Numbers Have Architecture
Ask what a number is made of, and its factors, powers and square roots answer.
- 1Factor pairs as rectangles
- 2Different trees, same primes
- 3A negative base or a minus outside
- 4A square root reverses a square
Units and fractions
Whole numbers gave you scale, operations, and structure. Fractions keep those ideas but change the unit being counted.
Fractions Are Places
A fraction names a place on the number line: count equal steps from zero.
- 1Count steps, land on a place
- 2Seven fourths passes one
- 3One ribbon, two correct names
- 4One place, more than one name
Common Units Explain the Rules
Add counts only when they count the same size of part.
- 1The denominator names the unit
- 2Thirds and fourths become twelfths
- 3A sum that passes one whole
- 4Decimal places are common units too
When Multiplication Shrinks
A factor below 1 keeps only part of the amount, and dividing by a small group counts many groups.
- 1The factor decides grow or shrink
- 2A share of a share is an overlap
- 3Count the groups that fit
- 4A group count can contain a fraction
Ratios Run the World
Name what is compared, what counts as one, and which base a percent is of.
- 1The order states the question
- 2A share of a whole, per hundred
- 3Equal percentages, unequal bases
- 4Price per one ounce
Relationships
The quantities from earlier chapters are about to receive names and begin relating to one another. Variables, equations, graphs, and functions compress those relationships, but they do not erase the meaning underneath.
A Variable Is a Name
A letter saves a place for any value the context allows, and the grouping tells you what to do with it.
- 1Add two, then triple
- 2Which terms can combine
- 3Four regions in one rectangle
- 4A fixed fee and a rate
Equality Is a Promise
An equals sign says two sides have the same value. Solving keeps that promise one equal change at a time.
- 1Test a candidate
- 2Keep the pans balanced
- 3Unknowns on both sides
- 4The one reversal
A Graph Is a Story
A graph reports two quantities at once, and its labels, domain and scale are part of the story.
- 1Between the tick marks
- 2Slope from two points
- 3Intercepts and the domain
- 4Where the lines cross
Functions Are Relationships
A function is a dependable assignment: each allowed input gets exactly one output.
- 1Exactly one output
- 2Root first, then power
- 3Second differences
- 4Three families
Shape and motion
Relationships now acquire dimensions, angles, distances, and motion. A diagram can make those features visible, but it must still name its units and assumptions.
Measurement Is a Model
A number without a unit is not wrong. It is unfinished.
- 1Find the whole-number product, then place the point
- 2Convert a side first, then multiply
- 3The boundary and the interior have different jobs
- 4An exact result and its rounded display
Triangles Know the Distance
Find the right angle first, then let the squares of the legs tell you the diagonal.
- 1Legs, hypotenuse and the quick check
- 2Four triangles explain the squares
- 3Trap a square root that is not whole
- 4When the right angle is missing
Why Circles Keep Saying Pi
Every ideal circle answers the same question: how many diameters fit around the boundary?
- 1The ratio that never changes
- 2Recover the radius first
- 3Double the radius
- 4Measure an angle by its arc
How a Circle Becomes a Wave
A wave can begin as a point going around in circles.
- 1The angle decides the ratio
- 2Where the turning point lands
- 3Unroll one coordinate into a wave
- 4Midline first, then amplitude
Uncertainty and data
The next questions involve outcomes that are not known in advance and data that can be summarized in more than one honest way. The rebuilding habit remains the same: name the collection, denominator, assumption, and representation before accepting the conclusion.
Random Is Not the Same as Equal
Random does not mean every pattern takes turns.
- 1Zero and one are the walls
- 2Eleven sums, thirty-six pairs
- 3Take a token out of the bag
- 4Eight paths of three flips
The Base Rate You Forgot
A percentage is only half a sentence until you say which group it is a percentage of.
- 1Read the word among
- 2Split the population first
- 3Three rates, three denominators
- 4The positive column is a mixture
Build an Unfair Game
A game can smile at you while its expected value does the accounting.
- 1Weigh each outcome by its chance
- 2An average no play can produce
- 3Solve for the fair loss
- 4Same center, different spread
The Average Is Hiding Something
An average is a useful witness. It is not the whole story.
- 1Every repeat is a person
- 2Same mean, different data
- 3Sort first, then find the middle
- 4One high value pulls the mean
Nearness and infinity
Finite lists and exact endpoints have served us well. This part asks what happens when a rule continues without a last step or when values approach a target without arriving there.
Patterns That Grow
A pattern is not a prophecy until someone says how it continues.
- 1Count the additions
- 2Add 10 or multiply by 1.10
- 3Where does counting start?
- 4Ten percent keeps the whole
A Hotel With No Vacancy
A full hotel can still make room, provided its rooms are labels rather than walls.
- 1One new guest
- 2Any finite number of new guests
- 3A whole second list of guests
- 4Why 0.999... is 1
Getting Close Counts
Getting close is not the same as giving up before you arrive.
- 1Partial sums record finite progress
- 2The multiplier decides
- 3A tolerance you can meet and keep
- 4Continuity is a three-part check
Change and accumulation
You have already read functions, graphs, units, limits, and changing patterns. Calculus connects them.
What Is Happening Right Now?
An average can tell what happened over an interval. It cannot tell what was happening at one moment.
- 1An average rate needs two times
- 2Shrink the interval without making it zero
- 3Nearby secants approach a direction
- 4A corner has no single rate
Rules of Change, Unmasked
A derivative rule is a receipt for work someone has already checked.
- 1Constants have no change to report
- 2The power pattern from the quotients
- 3A product is not two isolated rates
- 4Check the chain rule at one point
Where Is the Best Point?
The best point is never best until the question says best for what, and over which inputs.
- 1One constraint, one variable
- 2The derivative names a candidate
- 3The completed square sets a ceiling
- 4When the turning point is off limits
How Much Has Accumulated?
The gauge reports the rate. The running total keeps the receipts.
- 1Rate times duration, block by block
- 2Net change or total magnitude?
- 3Where you sample matters
- 4Unequal widths, each in its own place
The Speedometer Meets the Odometer
One accumulated quantity keeps direction. The other records every bit of movement.
- 1Two ledgers for one trip
- 2Accumulation stores the rate in its slope
- 3Check, then subtract endpoints
- 4A velocity that crosses zero