The “100 Problems Involving the Number 100” PDF that ...
Transcript of The “100 Problems Involving the Number 100” PDF that ...
The “100 Problems Involving the Number 100” PDF that
accompanies this session can be found at:
www.nctm.org/100
The link appears with the May 14, 2020, session.
Other Great Problems…• The Locker Problem
• https://tinyurl.com/UCCS-LockerProblem
• Paper Pool• https://tinyurl.com/NCTM-PaperPool
• Number Pyramids• https://tinyurl.com/NRICH-NumberPyramids
• Orthocenter Travels• https://tinyurl.com/UGA-OrthoPath
Books with Great Problems
• My Best Math and Logic Puzzles, by Martin Gardner
• Visual Mathematics from the Math Learning Center (free)
• Can You Solve My Problems? by Alex Bellos
• MathCounts School Handbook from MathCounts (a new one
free every year)
• Problems from Math Jokes 4 Mathy Folks
Great Places for Great Problems• PlayWithYourMath.com• OpenMiddle.com• NRICH.maths.org• Math Teachers’ Circles
• tinyurl.com/MTC-Resources
• Illuminations.NCTM.org• MathCounts.org• brilliant.org• IllustrativeMathematics.org
Great Places for Great Problems• Mathy People…
• Jim Wilson (tinyurl.com/UGA-JWilson)
• Andrew Stadel (estimation180.com)
• Robert Kaplinsky (robertkaplinsky.com/lessons)
• Mathy Cathy Yenca (mathycathy.com/blog)
• Fawn Nguyen (fawnnguyen.com/)
• Dan Meyer (tinyurl.com/3Acts-DanMeyer)
Patrick Vennebush :: May 14, 2020
Celebrating 100 Years with 100 Problems Involving 100
Warm-Up Problem
7q + 3q
What’s Special about 100?
Use the Chat box…
Why is 100 a special number?
What’s Special about 100?
100° C
13 + 23 + 33 + 43 = 100
What’s Special about 100?
What’s Special about 100?
It’s big enough, but it’s not too big.Eli Vennebush (age 12)
What’s Special about 100?
It’s big enough, but it’s not too big.Eli Vennebush (age 12)
Problem 21: Gauss and Check
What is 1 + 2 + 3 + ⋯ + 10?
What is 1 + 2 + 3 + ⋯ + 1,000?
What is 1 + 2 + 3 + ⋯ + 100?
Problem 21: Gauss and Check
What is 1 + 2 + 3 + ⋯ + 10?
5 × 11 = 55
Problem 21: Gauss and Check
What is 1 + 2 + 3 + ⋯ + 98 + 99 + 100?
(1 + 100) + (2 + 99) + (3 + 98) + ⋯ + (50 + 51)← 50 sets of 101 →
50 × 101 = 5,050
Agenda
1. Why 100?
2. What are Great Problems?
3. Four Problems Involving 100
4. Four Strategies
5. Q & A
Great Problems
The solution of problems is one of the lowest forms of mathematical research… yet its educational value cannot be overestimated. It is the ladder by which the mind ascends into higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem.
Benjamin Franklin Finkel
Great Problems
Solve for x:
3x + 7 = 16
Write an expression for six more
than a number.
What is 40% of 150?
Great Problems
Use the Chat box…
What are the characteristics of a great problem?
Great Problems
• It’s exploratory. That is, the problem requires students to
pose conjectures and test them. In short, students will
need to get messy.
• Students want to get messy with the problem, because
it’s posed in an interesting way.
• Something — and, hopefully, something mathematical —
will be learned by solving the problem.
Great Problems
• A solution strategy isn’t obvious, and there are multiple
solution strategies.
• At least one solution is understandable to every
student.
• The problem is low-floor, high-ceiling, meaning that there
are entry points for every student but challenge for high-
achieving students.
Problem 54: Dog Days
What do you notice?
Problem 54: Dog Days
What do you wonder?
Problem 54: Dog Days
What do you know?Or, what could you
figure out?
Poll
Which of the following could you determine?
A.The height of the large dog
B. The height of the small dog
C.The height of the tableD.The difference in the dog’s heights
E. All of A, B, C, and D
F. None of A, B, C, and D
Problem 54: Dog Days
D = d + T
d + 100 = D + T
Problem 54: Dog Days
On the left:DOG + 100 + dog
On the right:dog + table + DOG + table
Problem 54: Dog Days
What was different?When the large dog is on the ground and the small dog is on the table, the tops of their heads are the same distance off the ground. But when the small dog is on the ground and the large dog is on the table, the top of the small dog’s head is 100 inches lower than the top of the large dog’s head. How tall is the table?
Strategy 1
Notice and Wonder
“Create a safe environment where students share thoughts without pressure to solve the problem.”
NCTM, “I Notice, I Wonder”
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Would you rather have…
100 lbs. of dimes?
100 lbs. of quarters?OR
Poll
I would rather have…
A.100 lbs. of dimes.
B. 100 lbs. of quarters.
C.Neither… I couldn’t carry them.D.Either… they have the same value.
E. Either… I’m broke!
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
Problem 4: It’s Gettin’ Kinda Heavy
https://www.usmint.gov/learn/coin-and-medal-programs/coin-specifications
2.268× 2.55.670
2.500× 2
5.000
Strategy 2
Low Floor,High Ceiling
Low Threshold… • A starting place for everyone
• Allows work at their own level
High Ceiling… • Offers lots of possibilities to do
more challenging mathematics,
too
Problem 3: Letter Product
Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of the letters within the word.
Problem 3: Letter Product
Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of the letters within the word.
Problem 3: Letter Product
Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of the letters within the word.
Use the Chat box…
What problem would you like to solve?
Problem 3: Letter Product
Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of the letters within the word.
How many common English words have a letter product of 100?
Problem 3: Letter Product
Find a common English word that has a letter product of 3,000,000.
John Horton Conway(1937-2020)
Problem 3: Letter Product
http://mathjokes4mathyfolks.com/problem_productforms.html
Strategy 3
Problem Stem
“Take a guess, draw a picture, make a movie… all before ‘solve!’”
Steve Leinwand
Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of the letters within the word.
How many common English words have a letter product of 100?
How many common English words have a letter product of 100?
Problem 100: Covering with Squares
As shown to the right, a square grid with 100 smaller squares can be covered by 100 squares(each measuring 1 × 1).
Problem 100: Covering with Squares
As shown, it can also be covered by 25 squares (each measuring 2 × 2).
Problem 100: Covering with Squares
And, it can be covered by 13 squares (one 6 × 6, two 4 × 4, two 3 × 3, two 2 × 2, and six 1 × 1).
Problem 100: Covering with Squares
Find all values of n for which it is impossible to cover a 10 × 10 grid with n squares of integer side length.
Problem 100: Covering with Squares
Find all values of n for which it is impossible to cover a 10 × 10 grid with n squares of integer side length.
http://bit.ly/cover-squares
Strategy 4
Be Less Helpful
Fighting to get out of its chrysalis helps a butterfly to strengthen its new wings. In fact, were someone to help the butterfly out of its cocoon, bypassing this uncomfortable yet essential struggle, its wings would remain soft and weak, and it would never fly. Similarly, as teachers, we all want to help our students — but, at times, being too helpful is precisely what hurts them most. This is why we must be less helpful.
Joshua Zucker
Problem 20: Farey Tales
Write all fractions from 0 to 1, in order, with every possible denominator less than or equal to 100.
What’s the 100th term of that sequence?
Agenda
1. Why 100?
2. What are Great Problems?
3. Four Problems Involving 100
4. Four Strategies
5. Q & A
Q & A
Questions?
Comments?
Compliments?Complaints?
[email protected] :: @pvennebush
Celebrating 100 Years with 100 Problems Involving 100