# Challenges

# General Information

Challenges are required for all third through fifth graders. Students work on challenges independently or with one other partner of their choice. The focus for the challenges is on communicating problem solving strategies. Students are asked to explain their thinking using words, numbers, and pictures.

Each challenge is explained in an assembly so that students can ask questions if needed, and flyers are sent home each week explaining the challenge. Students have a week to return their solution. At least one winner or pair of winners receives a prize from the prize table for each challenge, and they also present their challenge at the Awards Ceremony on I love Math Day.

Possible sources for challenges include NCTM, NRICH, Gary Lawrence's blog, Math Goodies, and Pierce School.

Below are the challenges that have been used in past years.

# 2019

### Week 1: Book Tower

The two stacks of books on the third floor have exactly the same number of pages. How many pages are in one of the stacks? Don't include the covers.

### Week 2: Left Out

The fifth graders at Sophie Germain Elementary School are taking a grade level picture. The photographer tried lining them up in different length rows and made the following notes:

In two equal rows, one person is left out.

In three equal rows, one person is left out.

In four equal rows, one person is left out.

In five equal rows, one person is left out.

Use the notes above to determine how many fifth graders there are at the school. There are less than 100.

### Week 3: Apartment Doors

Adapted from NCTM

Jordan's job is putting numbers on the apartments in a new building. The apartments are numbered consecutively starting at 1. Consecutively means in order, one right after the other. For example, 2, 3, and 4 are consecutive numbers. So are 15, 16, and 17.

At the hardware store the digits cost $1 apiece. The bill for the digits came to $282. How many apartments will Jordan be numbering?

Extra: On Mondays the store sells 1s for half price. How much could Jordan save by buying the digits on Monday?

# 2018

### Week 1: Lego Sculptures

** **The two Lego sculptures on the third floor are exactly the same. How many Legos are in one of the sculptures?

### Week 2: Triangle Worms

** **In the land of Trianglia, worms are made of triangles, and they grow fast! How many triangles will be needed for a 4-day old worm? 10-day old worm? a 63-day old worm?

Extra: I found a worm that was made of 60 triangles. How old was it? Explain how you know.

Super extra: Can you make a rule that uses a worm's age (in number of days) to find out how many triangles it is made of? You may describe your rule with words or with numbers and symbols.

### Week 3: Card Challenge

How can you arrange a set of 10 cards so that when you alternate turning one face up and putting one facedown on the bottom of the pile, they end up in order from Ace to 10?

Arrange a set of ten cards, numbered Ace to 10, facedown so that the following occurs:

1. When you turn over the top card, it should be a 1. Place it faceup on the table.

2. Move the next card to the bottom of the deck, keeping it facedown.

3. When you turn over the third card, it should be a 2. Place it faceup on the table.

4. Move the next card to the bottom of the deck, keeping it facedown.

5. Continue this way, turning over a card, placing it faceup on the table, and moving the next card to the bottom of the deck.

6. When you're done, all of the cards on the table should be faceup in order from Ace to 10.

Do you need to see it in action? Watch this!

# 2017

### Week 1: Chain Links

How many plastic chain links are hanging on the third floor walls?

### Week 2: Arranging Squares

Nine squares with side lengths 1, 4, 7 , 8, 9, 10, 14, 15 and 18 cm can be fitted together with no gaps and no overlaps, to form a rectangle. What are the dimensions of the rectangle?

A square has 4 sides of equal length. You were given pieces of centimeter grid paper that you can draw on and cut in order to solve this challenge.

### Week 3: 2017

Use only the digits 2, 0, 1, and 7 and the symbols +, -, x, and ÷ to write expressions for the counting numbers 1 through 100.

You may use the digits in any order. (Ex. 1 + 7 + 2 + 0 = _____)

You do not have to use all digits each time. (Ex. 2 + 1 = _____)

Digits may only be used once in an expression. (NOT allowed: 2 + 2 = ___)

You may make two or three digit numbers. (Ex 17 or 127)

Optional: You may use parenthesis, exponents, and decimals.

### Week 4: Number Maze

1. Using the number maze below, can you find a path which adds to exactly 100?

2. What is the smallest sum you can make through the maze?

3. What is the largest sum you can make through the maze?

# 2016

### Week 1: Cheeseballs

How many cheeseballs are in the clear container. We filled a large storage container with cheeseballs. However, inside of the large container was also an empty cheeseball tub.

### Week 2: Popsicle Sticks

With three popsicle sticks you can make one equilateral triangle. How many equilateral triangles can you make with 4 popsicle sticks? How many with 5? How about 6?

An equilateral triangle has 3 equal sides and 3 equal angles. You were given 6 popsicle sticks that you can move around in order to solve this challenge.

### Week 3: Math Book

My History of Mathematics book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit ‘1’ appear in the page numbers? The number 141 would have two 1’s counted.

# 2015

### Week 1: Cheeseballs

How many cheeseballs are in the terrarium?

### Week 2: Hundred Grid

Using only the operations -5 and x2, can you find a way to visit all numbers, 1 to 100?

1. Mark a start number on the hundred grid.

2. Now subtract 5 from the number or multiply the number by 2.

3. Mark off your new number.

4. Now use the new number to subtract 5 or multiply by 2, each time marking off the numbers you visit.

5. You may visit numbers more than once.

Will you be able to visit every number on the grid at least once?

If YES, what was your start number and the steps you followed?

If NO, why not?

### Week 3: Handshakes

Each member of a baseball team and their coaches will shake hands with every coach and player. How many handshakes will there be? The Red Sox team has 10 players and 4 coaches. At the end of the last game the players and coaches all shook hands with each other.

### Week 4: Weights

Determine the weight of each kind of ball using the picture of three scales below. Assume that all soccer balls in the picture weigh the same, all baseballs weigh the same, and all tennis balls weigh the same. Assume that all weights are whole numbers. You will not need decimals.