Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Reduce a sum modulo twelve.
- Find an additive inverse shift.
- Identify the group properties of modular addition.
Before you start
Add and subtract integers, divide with remainders, and follow positions around a circle.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A twelve-position classroom dial labels positions 0 through 11. Starting at position 10, advance 5 positions. Where do you finish, and which forward advance returns you to 10?
Why this math matters
Repeating schedules and dial positions wrap around instead of growing without limit. Modular arithmetic captures this behavior. Looking at all possible shifts also introduces a group: a set with an operation that follows specific structural rules.
Set up the model
A useful answer starts with clear assumptions:
- Positions increase around a twelve-position dial and wrap from eleven to zero.
- The operation is addition modulo twelve, not multiplication; zero represents the usual twelve-o'clock position.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Use a clock to meet your first algebraic group
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A twelve-position classroom dial labels positions 0 through 11. Starting at position 10, advance 5 positions. Where do you finish, and which forward advance returns you to 10?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Add and wrap around
10 + 5 = 15; 15 ≡ 3 (mod 12)
Subtract a complete twelve-position turn. Fifteen and three are different integers but represent the same dial position.
Find the undoing shift
5 + 7 ≡ 0 (mod 12); 3 + 7 ≡ 10 (mod 12)
Advancing seven undoes advancing five on this dial. Their total is one full turn, which has the same effect as no movement.
Recognize the structure
set = {0, 1, …, 11}; operation = addition modulo 12
Sums remain in the set, grouping additions does not matter, zero is an identity, and every shift has an inverse. These properties make an additive group.
The result
You finish at position 3; advancing another 7 returns you to position 10.
The group comes from the set together with its operation. The same residues under multiplication do not form a group: for example, no integer multiplied by two gives remainder one modulo twelve.
Common mistakes to catch
- The congruence symbol means matching remainders, not ordinary equality.
- Do not assume division works for every nonzero residue in a modular system.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Starting at position 9, advance 8 on the same dial.
Show a hint
Remove one complete turn from seventeen.
Reveal answer and explanation
Position 5
9 + 8 = 17 and 17 − 12 = 5, so 17 ≡ 5 modulo twelve.
Practice 2
Solve x + 7 ≡ 2 (mod 12), using a representative from 0 through 11.
Show a hint
Subtract seven, then add twelve to choose the stated representative.
Reveal answer and explanation
x = 7
2 − 7 = −5 ≡ 7 modulo twelve. Checking gives 7 + 7 = 14 ≡ 2.
Take the idea with you
For an unfamiliar algebraic structure, specify its objects and operation before testing identity, closure, associativity, and inverses.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.