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Building the Foundation to Algebra. Rational Numbers. Goals. Develop a conceptual understanding of fractions as parts of regions and parts of sets. Determine fractional parts when the whole varies. Determine the size of the whole from fractional parts.
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Building the Foundation to Algebra Rational Numbers
Goals • Develop a conceptual understanding of fractions as parts of regions and parts of sets. • Determine fractional parts when the whole varies. • Determine the size of the whole from fractional parts. • Connect mixed number and improper fraction representations.
Hexagon Fractions • Use a hexagon as a base. • Cover the hexagon with other pattern block pieces. • Take another hexagon and build a different design on top of it. • Make as many designs as you can that cover the hexagon. • How many different designs can you build? How do you know you found all of them? • Make fraction number sentences to describe each of your designs, e.g., 1 = ½ + ½.
Hexagon Fractions • Make a triple hexagon shape. • Use that shape as the whole. (The ONE) • Determine what fractional part each pattern block shape represents: • Hexagon • Trapezoid • Rhombus • Triangle
The Large Hexagon • Use the large hexagon shape as the whole. (The ONE) • Determine what fractional part each pattern block shape represents: • Hexagon • Trapezoid • Rhombus • Triangle
How is this Possible? From her work with pattern blocks in third grade, Lynn always thought that the trapezoid was called ½. But when she made her triple hexagon, the trapezoid wasn’t called ½ anymore! What happened? How is this possible?
How is this Possible? Lynn was trying to figure out which was larger, 1/3 or 1/2. “My third grade teacher said that in fractions, larger is smaller and smaller is larger, so 1/2 is larger than 1/3.” But then she looked at the three pattern block problems she just did. “The hexagon is 1/3 and the trapezoid is 1/2. The hexagon is bigger than the trapezoid. So, 1/3 IS larger than 1/2. I knew larger couldn’t be smaller!” What happened? How is this possible?
Making Connections • Why do the same pattern blocks have different values for the hexagon, triple hexagon, and large hexagon? • What is the relationship between the size of the whole shapes and the fractional value of the pattern block pieces?
Looking through Teacher Lenses • How would you characterize the level of this task: High or low cognitive demand? • What mathematical ideas are embedded in the task? • What makes this worthwhile mathematics?
Flight Problem On the flight from Pittsburgh to San Francisco, I fell asleep after traveling half the trip. When I awoke, I still had to travel half the distance that I traveled while sleeping. For what part of the entire trip did I sleep?
From Parts to Wholes What is the whole if… • the rhombus is 1/2? • the rhombus is 1/3? • the rhombus is 1/4? • the trapezoid is 3/4? • the hexagon is 2/3? • the hexagon is 3/5? • the rhombus is 2/9?
Pattern Block Puzzles Use your pattern block pieces to build the following shapes. Sketch your shape on the recording paper: • A triangle that is 1/3 green and 2/3 red. • A triangle that is 2/3 red, 1/9 green, and 2/9 blue. • A parallelogram that is 3/4 blue and 1/4 green. • A parallelogram that is 2/3 blue and 1/3 green. • A trapezoid that is 1/2 red and 1/2 blue. Build larger versions of your solutions with the same fractional parts.
Pattern Block Puzzles • What strategies did you use to solve the puzzles? • What happened when you tried to build a larger version of your puzzle? What patterns did you notice?
Disappearing Cookies Patty Peterson put out a plate of freshly baked cookies. As her family came home from a hard day at school, they helped themselves: • Peggy took 1/5 of the cookies. • Paula took 3/8 of the cookies left on the plate. • Porter took 1/3 of the remaining cookies. • Pansy took 2/5 of the remaining cookies. • Polly took 1/2 of the remaining cookies. • Payton took 2/3 of what was left. • When Penny got there, there was only one cookie left! How many cookies did Patty Peterson bake?
Make a set of 12 candies 1/4 = candies 2/4 = candies 3/4 = candies Make a set of 20 candies 1/5 =candies 3/5 =candies 5/5 =candies Chocolate Fractions How does knowing the number of candies in a unit fraction help you figure out the number of candies in other fractions?
2/3 of 15? 5/7 of 21? 4/9 of 27? 1/2 of 27? 1/3 of 19? 2/3 of 19? Chocolate Fractions Describe a method for finding the fractional part of any set.
Chocolate Fractions 5 9 of 36 = 20
Name It! • Materials: 1 small pack of candies per player • Number of Players: 2 – 6 • Object of the game: To score the most points by writing fraction sentences to describe your set of candies.
Name It! Directions: • Close your eyes and count out 24 candies. The 24 candies are your “Whole”. • Write as many fraction statements as you can to describe your Whole. • Compute your score. • 1 point for each statement that contains 24ths. • 3 points for each statement that contains unit fractions (fractions with numerator of 1) with denominators less than 24. • 5 points for each statement that contains non-unit fractions (i.e., fractions with numerators of 2 or more) with denominators less than 24.
De-Briefing the Game • How did you figure out the different fractional parts? • What strategies did you use to increase your number of points?
How much is the whole? • Vonnie: “5 blue candies are 1/3 of my whole.” • Colleen: “6 brown ones are 1/4 of my whole.” • Marilee: “8 red ones are 2/5 of my whole.” • Judy: “12 yellow candies are 3/4 of my whole.” • Ken: “15 orange candies are 3/5 of my whole.” How can you find the number of items in the whole given any fractional part?
