Discrete mathematicsGrade 6
Grade 6
36 weeks
77 pages

FIND THE RULE

GENERALIZE IT. A PATTERN BECOMES A RULE WITH A LETTER, AND THEN THE RULE GETS PROVED.

One letter says what a hundred examples cannot. The year turns patterns into rules written with a variable, then insists the rule be justified: the same count derived two different ways, a formula checked against a case nobody drew.

Download the workbook Teacher guide
36
weekly assignments
two pages each
77
printable pages
proofed in grayscale
21
CA standards touched
named on the page each week
436
answers recomputed
from the problems, on every build

What the student ends the year holding

THE RULE BOOK.

Each rule stated in symbols, with the picture it came from and the argument that it holds for every n, not just the ones that were tried.

It is built one page a week, in the student’s own words, and it is the reason the year holds together rather than reading as thirty-six unrelated puzzles.

Anatomy of a week
01
In class this week
The standard the class is already working on, so the sheet reinforces Tuesday’s lesson.
02
Number Warm-Up
Two minutes of ordinary grade-level fluency. The page starts on familiar ground.
03
Words to keep
The two or three terms this week needs, defined where the student meets them.
04
The main activity
Every map, grid, graph and diagram printed on the page, nothing to prepare.
05
More to try
The practice set, on page two.
06
Add to your Rule Book
The week’s page of the artifact the student builds across the year.
07
Talk About It, then ★ Challenge Zone
One question out loud, a hint, then something genuinely harder.

Every week, in order

THE WHOLE YEAR.

Wk
Strand
Assignment
What the week is
01
Patterns & Rules
A Letter For Any Number
One letter says what a hundred examples cannot.
02
Patterns & Rules
A Pattern That Grows
Draw the shape, count the dots, and let the picture hand you the formula.
03
Patterns & Rules
Triangular Numbers
1, 3, 6, 10, 15. You have met them three times already. Now you get the formula.
04
Ways of Counting
The Handshake Formula, Proved
Grade 2 counted them. Grade 3 found the rule. This week you prove it.
05
Patterns & Rules
Adding One to n
Pair the ends. The whole sum falls out in one line.
06
Patterns & Rules
The Odd Numbers Make a Square
1, then 1 + 3, then 1 + 3 + 5. The answers are square, and the picture says why.
07
Patterns & Rules
Looking Back or Jumping Ahead
Two kinds of rule. One needs the answers before it, and one does not.
08
Logic & Proof
When a Pattern Lies
1, 2, 4, 8, 16. Everybody says 32. Everybody is wrong.
09
Logic & Proof
Proof by Picture
Some arrangements settle a question before a single symbol is written.
10
Logic & Proof
Proof by Cases
Split the question into a few kinds, settle each one, and you have settled them all.
11
Ways of Counting
Counting the Same Thing Twice
Count one thing two ways. The two answers must be equal, so you have a rule for free.
12
Patterns & Rules
Rule Fair
Six patterns. Find each rule, and say how sure you are.
13
Logic & Proof
Case Files
Everything from the first twelve weeks, in one folder.
14
Ratio & Rate
Ratio as a Rule
Three cups of flour to two of water. That is a rule with a letter hiding in it.
15
Ratio & Rate
One of Them
Reduce everything to one, and any two things become comparable.
16
Ratio & Rate
Out of a Hundred
Percent is a ratio wearing a uniform. Everything is out of a hundred.
17
Dots & Lines
Scaling a Network
Every road doubles in price. What happens to the cheapest route?
18
Dots & Lines
Everyone Plays Everyone
A round robin, counted with the handshake formula and read as ratios.
19
Number Structure
Factors, By Primes
Grade 4 found common factors by listing. Primes make it a rule instead.
20
Number Structure
All Four Quadrants
The grid goes left and down as well, and nothing else changes.
21
Shape & Space
Distance on a Grid
A delivery van cannot go through buildings. Its distance is a different distance.
22
Shape & Space
A Square Circle
A circle is every point the same distance from the middle. Change the distance, change
23
Shape & Space
Nets and Solids
Flatten a solid, count its parts, and put it back together.
24
Patterns & Rules
The Formula That Always Holds
Faces, edges and corners. One little sum comes out the same for every solid you try.
25
Logic & Proof
Puzzle Fair
A little of everything from Weeks 14 to 24.
26
Ways of Counting
n Choose k
The counting you have done since Grade 3, finally written as a formula.
27
Ways of Counting
Patterns in the Triangle
Every row, every diagonal and every sum in Pascal's triangle means something.
28
Ways of Counting
At Least One
Three words that always mean the same instruction: count them all, count the ones with
29
Ways of Counting
Three Circles, In Symbols
Grade 5 filled the regions in. This week the picture becomes a formula.
30
Ratio & Rate
How Likely Is It?
Chance is a ratio: the outcomes you want against all the outcomes there are.
31
Data
Data Has a Shape
Nine numbers on a line. The shape says more than any single number can.
32
Data
Center and Spread
One number for where the data sits, one for how far it wanders.
33
Data
Same Average, Different Story
Two classes, one average, and you would not want to swap them.
34
Data
Finding the Middle
The median needs the data sorted, and sorting has a price you already know.
35
Logic & Proof
Pigeonhole, Grown Up
Grade 3 put marbles in boxes. Now the boxes are a city.
36
Patterns & Rules
The Rule Book, Finished
Thirty-three pages of rules. This week they become one book, and the line ends.

This is the book’s own table of contents, read out of the source rather than retyped here. The fourth column is the line printed under the title on the student’s page.

California standards touched

This is enrichment: it reinforces and extends these standards rather than delivering them. No standard here is taught to mastery by this program alone. The list is the set of codes actually named in the “In class this week” line across the 36 weeks, counted from the published book.

6.EE.26.EE.36.EE.46.EE.56.EE.66.EE.96.G.36.G.46.NS.46.NS.66.NS.76.NS.86.RP.16.RP.26.RP.36.RP.3c6.SP.16.SP.26.SP.36.SP.56.SP.5c
Everything, free

Every one of the 436 numeric answers in that guide is recomputed from the problem as the workbook states it, by a script that ships with the source and runs on every build.

The year before
GRADE 5: THE BEST WAY
Cost it.
The end of the line
Grade 6 is the last book in this line. What follows it is not written yet, and the curriculum map says so cell by cell.

No year in this line has been run start to finish in a classroom yet. If you would be the first teacher to do it, we would like to hear from you.