Place value is a polynomial in the base, so changing base changes notation, not quantity.
ThinkCompareExplain
Today’s route
A question becomes an explanation.
Try
Enter through a problem. A partial attempt counts.
Connect
Use base, place value, expansion to name what you notice.
Convince
Build an argument a partner can check.
The goal is a method someone else can use.
The opener
Start with a question.
Opening problem
Convert 101101 in base 2 to base 10.
Think · 2 min
Sketch, test a small case, or make a first guess.
Pair · 3 min
Compare methods. Find one thing you agree on and one question.
Be ready to share your route, even if it is unfinished.
Compare before checking
Put two attempts in conversation.
Explain
Tell a partner how you started. Let them finish before responding.
Question
Which step feels certain? Which step still needs evidence?
Build
Combine one useful idea from each attempt into a new route.
Can you explain your partner’s method in your own words?
Board conversation
Notice before naming.
Look for structure
01Try a small case and record every step.
02Separate what changes from what stays fixed.
03State the pattern in a sentence before naming the theorem.
Make a conjecture
We think ___ because ___. A case that might break this is ___.
What changed? What stayed the same?
Useful language
Now the language has a job.
base
number of digit symbols
place value
digit weight bk
expansion
sum of digit times place value
Use one of these words to explain something you already noticed.
The core idea
Keep the idea. Explain the conditions.
(dk⋯d1d0)b=i=0∑kdibi,0≤di<b
The idea
Place value is a polynomial in the base, so changing base changes notation, not quantity.
What to remember
Digits are coefficients; the base supplies their weights.
What information does this method keep?
Worked example
One route, one decision at a time
The problem
Write 83 in base 5.
Predict each next step
0183 = 3(25) + 8.
028 = 1(5) + 3.
03Read the coefficients 3,1,3.
04Thus 83 = 313 base 5.
Which step needs a reason, not just a calculation?
See the structure
The structure behind base representations
(dk⋯d1d0)b=i=0∑kdibi,0≤di<b
For an integer base b at least 2, digits are coefficients and powers of b are weights.
Read the representation
Place value is a polynomial in the base, so changing base changes notation, not quantity.
Look together
Point to the part that explains the relationship. Change one input and predict what changes.
What can you see here that was harder to see in words?
Two approaches
Compare the reasoning.
Structural method
01Name the objects and conditions.
02Use a reusable relationship.
03Explain why the method must work.
Brute force
01May help with a small test case.
02Can hide the general reason.
03Becomes unreliable when the scale grows.
Which route can you justify? What would you test first?
Reasoning clinic
Find the step that needs repair.
Investigate these concerns
01May help with a small test case.
02Can hide the general reason.
03Becomes unreliable when the scale grows.
Repair together
Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.
A useful correction explains why the original step fails.
Work together
Investigate base representations
Your shared task
Groups invent a small alien numeral system, encode values, and explain its carrying rule.
Explore
4 min - test cases
Compare
5 min - form a rule
Defend
4 min - defend the rule
Everyone should be able to explain the group’s final claim.
Share → connect
What did the room discover?
Bring your evidence
Share a conjecture, a useful example, or a route that changed.
Connections to investigate
01Digits are coefficients; the base supplies their weights.
02A complete solution names the condition that makes the method valid.
03A counterexample is useful when it exposes a missing condition.
Which observation explains the mechanism?
Try → explain → check
Your turn to build the route
The problem
Add 132 base 4 and 23 base 4.
Predict each next step
01Add the units: 2 + 3 = 5 = 11 base 4.
02Write 1 and carry 1.
03Then 3 + 2 + 1 = 6 = 12 base 4.
04The sum is 221 base 4.
Which step needs a reason, not just a calculation?
Convince a skeptic
Turn a pattern into a proof.
The claim
For integer b ≥ 2, every nonnegative integer has a unique base-b expansion with no leading zeros, except the numeral 0 itself.
Build the argument
01Repeated division by b produces valid remainders.
02The quotients eventually reach zero.
03The last digit is forced by the remainder mod b; divide by b and repeat to force every other digit.
What does each step rely on? Where are the conditions used?
Stretch together
Take the idea somewhere new.
Transfer challenge
Find a number whose base-10 and base-2 representations are both palindromes.
Choose a route
Name the objects, the conditions, and a relationship that might help. Start with one test case.
What transfers from the earlier example? What is different?
Peer review
Check the argument together.
Restate
Explain the other person’s route before evaluating it.
Test
Try a small case or boundary case. Does every step still make sense?
Revise
Fix an assumption, add a reason, or find a second method.
Where did your thinking change?
Beyond this problem
Where might this idea help?
A connection
Binary, hexadecimal, checksums, and digital storage all depend on base representation.
Discuss the model
01What objects or quantities are represented?
02Which relationship from today is preserved?
03What assumption would you check before trusting it?
Invent one more situation with the same structure.
Exit ticket
Leave with an explanation.
Question 1
Convert 101101 in base 2 to base 10.
Question 2
Explain why this claim is true: For integer b ≥ 2, every nonnegative integer has a unique base-b expansion with no leading zeros, except the numeral 0 itself.
Question 3
Find a number whose base-10 and base-2 representations are both palindromes.
Choose one. Write an answer with a reason, then compare with a partner.
Reflect together
Make your thinking visible.
I used to think…
Name a claim or approach that changed.
Now I can explain…
Give the reason, not only the result.
I still wonder…
Write a question worth bringing to the next circle.
Share one sentence. Ask one follow-up question.
The idea to keep
Digits are coefficients; the base supplies their weights.
Use the matching practice sets to move from direct examples to proof and transfer.
A good question belongs to everyone in the room.
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