Architecture of Grace

Architecture of Grace

Mathematics · The Interior · Grades 6–8

The Interior Mathematics

Fourteen activities that walk beside a 7th-grade long-term plan, September through December. Put a link on the board; students type it and work. No accounts, no logins, no installs — and nothing a student does leaves their computer unless the student sends it.

Jump to an activity

Read this first

Built for the student who has stopped trying.

Every activity keeps the same promises, and they are the reason a checked-out kid will touch it:

Nothing leaves unless they send it.No account, no login, no score, no ranking. Work saves to that one machine and the student can erase it, and the page works with the Wi-Fi off. When a class link carries a destination, one Send button appears at the end — it sends a single summary row, only when the student taps it, and never their taps or their writing.

Watch one first.Each activity opens with a fully worked example that belongs to nobody — the student who freezes on problem 1 can see the moves before touching their own.

Hints are the lesson.Four rungs on every problem — cue, strategy, picture, worked example — and the feedback names the actual error (walked the wrong way, flipped the division, forgot to flip a sign) instead of saying “wrong.”

You still get evidence.A “For the teacher” panel on each device builds one probe row — independent vs. supported, hints used, which half of the skill slipped. Read it over the student's shoulder, copy it, or print it. Three equivalent problem sets let you re-probe for weeks without measuring memory.

September – December

Your plan, with a door for every unit

Arranged in the order of the 7th-grade long-term plan. Tap “Show on the board” and the link fills the screen, big enough for the back row.

September · Integers

Review week · before integers

Where It Goes

Locating numbers on the line — integers, tenths, halves, and opposites. The floor every later unit stands on.

How Far From Zero

Distance from zero, walked and counted — including the answer to “why can't it be −7?”, said out loud.

Adding & subtracting integers

Walk the Line

The same Walk the Line as the textbook lesson (pgs 203–204) — but the line is walkable, the arrow draws itself, and “right distance, wrong direction” gets its own sentence.

Subtracting integers · enrichment & zero pairs

Land On It

Backwards: the answer is given, the student builds expressions that land on it — three ways each, from both sides. Zero pairs get discovered instead of announced.

Add/subtract with decimals & fractions

Smaller Steps

The same walk on finer lines — tenths, halves, quarters. For the student who panics at fractions, the whole lesson is: nothing new happened, the steps just got smaller.

When the test comes

Test Prep · Integers

This month’s skills in real test format — study cards you tap to reveal, a hints sheet of the classic traps, and a practice quiz at three levels.

October · Multiply, divide, properties

Multiplying & dividing integers

Sign and Size

Two questions wearing one coat, asked separately: which sign, then how big. Sign slips and times-table slips get counted apart, because they are different instruction.

Rewrite fractions as decimals

Two Names

3/4 and 0.75 are one number with two names, and the bar is a division sign. Every classic error — 0.34, dividing upside-down, cutting a repeat short — is on the tray, waiting to be named.

Distributive property & factoring

Both Ways

One area rectangle read in two directions — expand it, then factor it back. The two real errors (forgetting the x, adding instead of multiplying) sit on the tray.

Name the property

Name the Move

First say what moved — order, grouping, a special number, a spread — then match it to the word wall. Commute means travel; the vocabulary explains itself.

Order of operations

What Goes First

The page never asks for the final answer — only “which one goes next.” Wrong moves are refused with the rule that refused them, and the left-to-right traps are all here.

When the test comes

Test Prep · Multiply, Divide & Properties

This month’s skills in real test format — study cards you tap to reveal, a hints sheet of the classic traps, and a practice quiz at three levels.

November · Expressions

Combining like terms · adding & subtracting linear expressions

Like and Unlike

Three stages per problem: flip every sign behind the minus, gather the x's, then combine. Four of the ten problems are subtraction — the shape the November test actually takes.

When the test comes

Test Prep · Expressions

This month’s skills in real test format — study cards you tap to reveal, a hints sheet of the classic traps, and a practice quiz at three levels.

December · Equations

Writing expressions & equations from words

Say It in Math

Clue word first, then write it — with “5 less than n” sitting right next to “5 − n” every time, because that reversal is where this skill lives or dies.

One-step equations · all four operations, decimals & fractions included

One-Step Equations

A balance where no move is refused — every move is legal, only one isolates x. The equation is held as exact fractions, so even a strange road shows true numbers.

Two-step equations

Last On, First Off

The same balance, wrapped twice. Dividing first is allowed — the fractions that appear make the argument for subtracting first better than any rule could.

When the test comes

Test Prep · Equations

This month’s skills in real test format — study cards you tap to reveal, a hints sheet of the classic traps, and a practice quiz at three levels.

A class period, start to finish

How it runs

1

Put the link on the board.Tap “Show on the board” above — or the same button inside any activity's teacher panel. The address fills the screen.

2

Students type it and work.Ten problems, hints on demand, feedback that names the error. English and Spanish, one button apart. A student who was absent can open the same link at home.

3

Walk the rows and read the probe.“For the teacher” at the bottom of each student's screen shows independent vs. supported, hints used, and which half of the skill slipped. Copy the row or write it down — nothing travels on its own.

4

No devices? Print it.Every activity prints as a clean worksheet of the same ten problems — and the page itself still works with JavaScript off or the network down. Sets 2 and 3 are the same skills with new numbers, for re-probing weeks later.