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5. Egyptian Pizza

  • Page ID
    13601
  • This page is a draft and is under active development. 

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    Pizza Fair & Square

    Grade 5 / Math & Art Integration

    ~ 1.5 Math Blocks

    Big Ideas/Rationale/Essential Understandings/Inquiry

    Number Sense

    Number sense is an intuition about numbers. It develops when students connect numbers to their own real-life experiences and when students use benchmarks and referents. Number sense can be developed by providing rich mathematical tasks that allow students to make connections to their experiences and their previous learning. (Program of Studies: Mathematics)

    Outcomes

    Program of Studies: Mathematics Kindergarten to Grade 12 General Outcome: Develop number sense.

    Specific Outcomes: 8. Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representation to:

    • Name and record fractions for the parts of a whole or a set
    • Compare and order fractions
    • Model and explain that different wholes, two identical fractions may not represent the same quantity
    • Provide examples of where fractions are used
    • [C, CN, PS, R, V]

    Assessment

    What to look for:

    • Procedural Knowledge – Students can identify, extend, and create Egyptian fractions
    • Problem Solving Skills – Students can use different strategies to create and solve Egyptian fraction problems

    Student

    Name

    Conceptual Understanding: Demonstrates and explains:

    Procedural Knowledge:

    Identifies, describes,

    Problem

    Solving Skills: Creates and solves problems

    Communication: Records and explains reasoning and

    -

    -

    Egyptian fraction

    rule Using manipulatives to show understanding

    extends the concept of Egyptian fractions to solve puzzle

    using appropriate strategies - equally dividing parts, following a ‘1/n strategy’

    procedures clearly and completely, including appropriate terminology

    * For the assessment above use approved levels, symbols, or numeric ratings.

    Teacher involvement:

    • Watch an interactive math story about fractions for a recap:
    • https://www.youtube.com/watch?v=Rr9kN2fQYPw
    • The teacher will walk around and visit pairs/groups to check up on progress and discuss their strategies (implementing differentiation).
    • The teacher will use the rubric as a formative assessment.

    Differentiation for Learners

    • For all students, especially ELL (English Language Learners) students, working with partners and group discussions will provide an opportunity for students to learn from each other, and a deeper understanding of appropriate strategies to use.
    • For all students, especially ELL students and visual learners, clear step-by-step oral instructions along with a visual demonstration of the instructions will support their understanding (examples of visual artifacts shown below). Using manipulatives and other resources are excellent ways of representing patterns visually.
    • Manipulatives: Magnetic Fraction Circles, Plastic Fraction Circles, etc.

    Resources/Materials

    Introduction/Hook

    Students will be asked to gather at the carpet area to watch an interactive math story about fractions:

    https://www.youtube.com/watch?v=Rr9kN2fQYPw

    Time

    Teacher Activities

    Learner Activities

    10 min.

    Use the control signal (clapping hand rhythm) to get the students attention. Gather the students to the carpet area.

    Students respond to the control signal. Students are sitting, quiet, and actively listening to instructions.

    Watch the video:

    • Ask questions during the reading: - What do you notice?
      • Have you seen this in real life? - Can you guess the fraction?
    • Ask questions after the reading: - What have you learned?
      • What fractions have you noticed?
      • Did you see any Egyptian fractions? How did you know it was an Egyptian fraction?

    Students are sitting, quiet, and actively listening and participating.

    Development

    Students will be given the instructions. Students will solve a math puzzle by dividing the number of pizzas between friends equally using the concept of Egyptian fractions.

    Math Puzzle:

    Time

    Teacher Activities

    Learner Activities

    50 min

    Questions:

    Suppose you ordered 5 pizzas to share among 8 friends for a party. Pause for a minute and decide with your students how they would solve this problem before carrying on....

    What if there were only 4 pizzas, not 5 to be split amongst 8 friends?

    Explain to students how if that was the case then they would all get at least half a loaf, so you would use 4 of the pizzas to give all 8 of them half a pizza each. But in this question, we have one whole pizza left.

    Now it is easy to divide one pizza into 8, so they get an extra eighth of a pizza each and all the pizzas are divided equally between the 5 friends. On the picture here (Let's assume we have rectangularly shaped pizzas) they each receive one red part (1/2 a pizza) and one green part (1/8 of a pizza):

    and 5/8 = 1/2 + 1/8

    Students work in groups to discuss the math puzzle and brainstorm strategies they would go about to solving the problem.

    Students participate in think, pair, share, to discuss their thinking to the class.

    Students use manipulates to try to solve this puzzle.

    Students first work independently, and then in pairs to solve the puzzle.

    Students that quickly solve the puzzle are given more challenging puzzles to complete using Egyptian fractions:

    • Suppose you had 3 pizzas to share between 4 people, how would you do this?
    • How about 2 pizzas shared amongst 5 people?
    • Or 4 pizzas shared among 5 people?

    Mathematical Method/Background

    Mathematical Method: Fibonacci’s Greedy Algorithm

    1. Using Fibonacci’s Greedy algorithm to find Egyptian fractions with a sum of unit fractions is as follows:

    • Choose the largest unit fraction we can, write it down and subtract it
    • Repeat this on the remainder until we find the remainder is itself a unit fraction not equal to one already written down.
    • At this point, we could stop or else continue splitting the unit fraction into smaller fractions.
    • To use this method to find a set of unit fractions that sum to 1:
    • So we would start with 1/2 as the largest unit fraction less than 1:
    • 1 = 1/2 + (1/2 remaining)
    • so we repeat the process on the remainder: the largest fraction less than 1/2 is 1/3:
    • 1 = 1/2 + 1/3 + (1/6 remaining).
    • We could stop now or else continue with 1/7 as the largest unit fraction less than 1/6 ...
    • 1 = 1/2 + 1/3 + 1/7 + ...

    Mathematical Background:

    1. The Egyptians of 3000 BC had an interesting way to represent fractions. Although they had a notation for 1/2 and 1/3 and 1/4 and so on (these are called reciprocals or unit fractions since they are 1/n for some number n), their notation did not allow them to write 2/5 or 3/4 or 4/7 as we would today.
    2. It turns out that Egyptian fractions are not only a very practical solution to some everyday problems today but are interesting in their own right. They had practical uses in the ancient Egyptian method of multiplying and dividing, and every fraction t/b can always be written as an Egyptian fraction

    1. Remember that

    o t/b<1 and o if t=1 the problem is solved since t/b is already a unit fraction, so o we are interested in those fractions where t>1.

    The method is to find the biggest unit fraction we can and take it from t/b and hence its other name - the greedy algorithm.

    With what is left, we repeat the process. We will show that this series of unit fractions always decreases, never repeats a fraction and eventually will stop. Such processes are now called algorithms and this is an example of a greedy algorithm since we (greedily) take the largest unit fraction we can and then repeat on the remainder.

    Closure

    Total Time: 5 minutes

    Time

    Teacher Activities

    Learner Activities

    5 min.

    Invite students to share their solution and identify the strategies they used to find the solution. Did they use manipulatives? What technique helped them solve the puzzle? What problems were challenging, and why?

    Students are participating in group discussion.

    Source:

    Special Thanks to Noura Ismail.


    This page titled 5. Egyptian Pizza is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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