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?

patrickv@mathlearningcenter.org :: @pvennebush

Celebrating 100 Years with 100 Problems Involving 100