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The Egyptians used a sum of unique unit fractions to represent other fractional values. For example, they could use ½ + ¼ to represent the value ¾. The Egyptians would not have used this representation for whole numbers, but it’s interesting to explore the different ways to make 1 whole with unique unit fractions.

How can unique unit fractions be combined to form 1 whole? 

There are 8 friends sitting at 2 lunch tables. No one is sitting alone. What are some different combinations of friends at tables? 

Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination.

Breanna has a button collection.

• When she puts her buttons in 3 or 7 equal groups, there are 2 buttons left over.
• When she puts the buttons into 2 equal groups, there is 1 button left over.

How many buttons could Breanna have?

Maya counted an odd number of stripes on a zebra’s leg. If the zebra’s leg has an even number of black stripes, how many white stripes might it have?

Kyle wrote a story problem about a garden. The answer is 72. What could his story problem be?

If you could design your own school bus, what would it look like inside? How many rows of seats would there be? How many students would sit in each row? How many students could ride your bus at once?

Kimiko ran a total of 6 miles in 4 days. Each day, she ran ½ mile more than she had run the day before.

• How many miles did Kimiko run on the first day?
• If Kimiko continues this pattern, how many miles will she run on the eighth day?

The purple band is one side of a quadrilateral.

• What types of quadrilaterals can you make?
• How do you know what types of quadrilaterals you have made?
• Are there any types of quadrilaterals that you can't make? How do you know?

Friendly flies like to live together. They live in groups of 3 in square apartments. They enjoy having neighbors in at least 2 apartments they share walls with, but they don't want neighbors on all 4 sides. That's just too much.

Ava made pictures using three pattern shapes. Each pattern shape is worth a different number of points.  She found the total number of points for each picture.

• How many points is each shape worth?
• Can you make a picture worth 20 points?
• Can you make a picture worth 10 points?

Find as many rectangles as you can that have a perimeter that is 4 units more than the number of square units in the area.

Kayla drew 4 two-dimensional shapes. She counted a total of 18 sides. What 4 shapes might she have? How many answers can you find?

Each of these symbols represents a number. Only one of the numbers is a whole number. What could the three numbers be?

Hint: A whole number is a number such as 0, 1, 2, 3....

Use the numbers 1–9 to make the equation equal. You may use each number only once (you won't use them all).
What numbers can you use to make the equation equal?

______  ______  +  ______  =  ______  ______  -  ______

I have 45 cents. I have an odd number of coins. What combination of coins could I have?

Hint: An odd number can be divided into 2 equal groups with 1 left over.

How can you fill the flower with pattern shapes without any of the same shapes sharing a side?

• The first 4 shapes have been placed for you.
• If you use 2 of the same shape, make sure that their sides don’t touch each other. It’s okay if their vertices (corners) touch each other.
• If you get stuck, remove some of the shapes and keep trying.

How many times a day do the hands of a clock form a 90-degree angle?