With Every Reason

With Every Reason · For schools

Scope and sequence

The course rests on one idea: a number records a measuring. A unit is laid along a quantity, and the number says how it went. When the old way of measuring stops working, something new is born: a fraction when the unit leaves a bit over, the decimal when the unit is cut by tens, a negative number when a change has a direction, and the rate when two quantities change together. Grades K to 4 build the whole of arithmetic this way with objects. Grades 5 and 6 then give Kiselev’s systematic course, which proves the system again in general.

Below, each grade’s units of edition 2.0 are listed in teaching order, as planned. The Teacher’s Edition will give the day-by-day calendar of the school year, with its practice days, review weeks and circle periods.

Kindergarten

Measuring and comparing quantities where they are, then counting what was measured. Book K leads into Part 0 of Grade 1. Its units will be listed here once Book K is written.

Grade 1

Measuring, comparing and counting; tens and bundles; addition and subtraction with their stories.

  1. Part 0: before the numbers
  2. Counting numbers: tens, tallies, other bundles, the signs
  3. Folding, building, and the clock: halves and fourths
  4. Addition
  5. Subtraction
  6. Stories of putting together and taking apart
  7. Comparing stories; is it fair? the IOU stick

Grade 2

Numbers to 1,000 with the hundred-stick and coins; adding and subtracting to 1,000 with the reason; stories with the unknown anywhere; multiplication and division begun as measuring by a bigger unit; length and the clock.

  1. Counting numbers: the hundred-stick and the ladder of coins
  2. Addition to 1,000, with its reason
  3. Subtraction to 1,000, with its reason
  4. Stories with the unknown anywhere; the bar graph
  5. Comparing stories; two-step strips; the signs
  6. The bigger measure, and equal shares
  7. Measuring length; the clock

Grade 3

Multiplication and division, with the table; equal groups and two-step stories; measuring quantities; fractions as the cut unit; quadrilaterals.

  1. Counting numbers: the nearest mark; adding within 1,000
  2. Multiplication: the big stick named; covering a floor by rows
  3. Division: sharing and measuring; stories of equal groups; two-step stories with a letter
  4. Measuring quantities
  5. Fractions: the cut stick
  6. The chart that fibs; quadrilaterals

Grade 4

Numbers to a million; tenths and hundredths; fractions: every piece cut again, and pieces of one kind added; the column methods with their reasons; long division by one digit; divisibility and primes; angles.

  1. Counting numbers: the ladder to a million
  2. The ladder down: tenths, hundredths, measures converted
  3. Cut every piece again; pieces of one kind
  4. The column methods with their reasons; so many times as many
  5. Long division by one digit; the remainder
  6. Divisibility with bundles; primes; is there a biggest prime?
  7. The turn cut into 360

Grade 5

Kiselev’s systematic course begins: every operation defined and its laws proved, for whole numbers, fractions and decimals; long division; the common unit; multiplying and dividing fractions and decimals; the coordinate grid.

  1. A number is a measure; the ladder both ways
  2. The sum and the difference, for every measure
  3. The product: Kiselev’s definition, its laws, the column method, the box
  4. The quotient defined once; long division of whole numbers
  5. The cut unit
  6. A common smaller unit
  7. Times a fraction: a new meaning, and why it is the right one
  8. Dividing by a unit fraction; decimals multiplied and divided
  9. The grid and the family of shapes

Grade 6

Directed numbers; a fraction by a fraction; ratio, rate and percent; letters for numbers, equations and inequalities; Kiselev’s theorems written in letters; the common measure of two numbers; area, volume and nets.

  1. Directed measures
  2. Measuring a fraction by a fraction
  3. Ratio and rate: two quantities that change together
  4. Ratio at work: shares, percents, and the typical problems
  5. Letters for every number, then the relation written: equations, inequalities, two quantities that change together
  6. Kiselev’s theorems in letters, in Euclid’s order
  7. The common measure of two numbers, at work
  8. A question with many answers
  9. Area, volume and nets: formulas from measuring