##
* *Reflection: Developing a Conceptual Understanding
King Fraction - Section 3: Concept Development

Using the activity of the King's Game was an excellent way to build students conceptual development of finding a fraction of a whole number. Students really connected with the idea of the game and referred back to the game as they solved the various problems. Students often need a hands on approach in order to make sense of the mathematics. By physically moving themselves around the room, they were able to own the knowledge for themselves and make sense of the problems. For an interesting read and more information about kids' brains and sitting still, you can check this article out.

My students absolutely loved this lesson and all the components from the game to the partner work. While some educators may argue that finding a fraction of a whole numbers if more of a fifth grade skill, I chose to do this lesson using visual fraction model because of it's real world application. Students often see or hear problems about fractions of groups. Therefore I think this a very real world relevant skill that my students can persist with when allowed to use models.

*Kings Game Good As Gold*

*Developing a Conceptual Understanding: Kings Game Good As Gold*

# King Fraction

Lesson 10 of 14

## Objective: SWBAT find the fraction of a whole number.

## Big Idea: In this lesson, students interact with a story about King Fraction. Through interacting with the story, students gain an understanding for finding a fraction of a whole number.

*50 minutes*

#### Warm Up

*5 min*

For this warm up I ask student to use their whiteboards to solve 3 4/5 + 6 4/5.

When students are finished they should put a thumb up. When most students are finished I ask for a student volunteers for an answer. I then ask the student to briefly explain their strategy for figuring out the answer.

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#### Concept Development

*40 min*

I begin this lesson by reading the story the Fraction King. This is an interactive story in which my students will move around the room at various points. You can access the story here - Fraction_King.pdf. The story also tells the stopping points and what students do during those times.

My students have had previous lessons with multiplying a fraction by a whole number as a skill and within a context. In previous lessons students worked to find how many chocolate bars would they need to buy if they had 8 friends and each friend ate 2/3 of a chocolate bar. In terms of difficulty I believe my students find these types of problems to be easier because they are more familiar with sharing situations and they can "act out" or model the situation as they read it. Though finding a fraction of a whole number requires similar thinking it can not be acted out as easily.

After the story is done and we have several examples of students arranging themselves into various groups sizes in order to find the fraction of a number, I pass out 50 small unifex cubes or small square similarly sized pieces of paper to student partners. I ask students to count out 30 cubes.

*Students work in pairs for this next activity or guided practice portion of this lesson. I give students about 20 seconds to determine which partner will be partner A and which partner will be partner B. Then I instruct partners that they will begin with partner A's blocks. By doing this, it allows one partner to be the "leader" while the other partner takes on more of an observation role. I switch back and forth so both partners have opportunities to lead and observe.*

I ask partners to blocks in 5 equal piles and to ignore the rest of his/her blocks.

Once all the students have their blocks grouped properly I ask them the total number of blocks each person placed in the groups.

I begin asking the students questions like:

What number of blocks is equal to 3/5 of 30? How do you know?

What number of blocks is equal to 1/5 of 30? How did you figure that out?

What number of blocks is equal to 4/5 of 30? What advice do you have for someone who is struggling in how to do this?

Next I instruct to place 32 blocks into 8 equal piles and to ignore the rest.

Once all the students have their blocks grouped properly I ask them the total number of blocks each person placed in the groups.

I begin asking the students questions like:

What number of blocks is equal to 2/4 of 32? How do you know?

What number of blocks is equal to 1/8 of 32? How did you figure that out?

What number of blocks is equal to 5/8 of 32? What advice do you have for someone who is struggling in how to do this?

If students seem to display difficulty, I continue on with the above procedure with other blocks and equal groups. If students seem to be understanding the concept, I display several problems on the smartboard for students to work on.

What is 3/4 of 24?

What is 1/6 of 24?

What is 2/8 of 48?

What is 3/7 of 42?

What is 2/12 of 36?

What is 4/5 of 45?

What is 6/10 of 50?

Students arrange their blocks to calculate the answer to each of the above questions.

As students work I circulate around the class checking for understanding and clearing up misunderstandings.

Note:

3 x 1/5 and 1/5 x 3 are somewhat confusing to educators and experts in terms of where they both fit into the standards.

Understanding that the two are arithmetically equivalent (commutative property and all), they are conceptually very different. The former being multiple copies of a fraction (which could be solved by repeated addition) and the latter being a fraction of a set (which could not be conceptualized as repeated addition). The former supports the fourth grade fraction work on adding (and subtracting) fractions with like denominators… the latter does not.

Lead author Bill McCallum, had this to say:

" I agree with the consensus here 3 x 1/5 in Grade 4, 1/5 x 3 in Grade 5. ... the two are conceptually quite different: the first is what you get by putting 3 segments of length 1/5 together, the second is what you get by dividing a segment of length 3 into 5 equal parts. The fact that these are the same can be shown by some reasoning on the number line; it’s not obvious. ...this doesn’t prevent curriculum writers from developing a pathway that starts talking about simple instances of fraction x whole number before Grade 5, but it does constrain assessment developers."

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#### Wrap Up

*5 min*

For this wrap up I give each student a piece of centimeter graph paper. I then ask each student, independently, to answer this question:

What is 3/4 of 28?

I ask students to use the graph paper to show how they solved this with their unifex cubes.

#### Resources

*expand content*

*Great idea. I used this for the beginning of my observation with a small group of math intervention students. I think they enjoyed moving around and you could see the gears turning as they tried to get themselves into groups. Thanks. | 3 months ago | Reply*

*Responding to mary tyndall*

Thanks Mary. I'm glad they liked it!

| 2 years ago | Reply*expand comments*

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- UNIT 1: Getting to Know You- First Days of School
- UNIT 2: Multiplication with Whole Numbers
- UNIT 3: Place Value
- UNIT 4: Understanding Division and Remainders
- UNIT 5: Operations with Fractions
- UNIT 6: Fraction Equivalents and Ordering Fractions
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- UNIT 10: Measurment
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- LESSON 1: Multiplying Whole Number with a Fraction
- LESSON 2: Multliplying Fractions with Whole Numbers Day 2
- LESSON 3: Inquiry with Mixed Numbers - How far did she run?
- LESSON 4: Fraction CARD GAME
- LESSON 5: All Hands In for Improper Fractions
- LESSON 6: Mixed Number Mystery
- LESSON 7: Adding Mixed Numbers
- LESSON 8: Cats and Strawberries - Inquiry with Subtracting Mixed Numbers
- LESSON 9: Tic Tac Toe - Subtracting Mixed Numbers
- LESSON 10: King Fraction
- LESSON 11: Hi-Tech Fractions
- LESSON 12: Fabulous Fraction Review with Computers
- LESSON 13: Fraction Jeopardy
- LESSON 14: Rocking Operations with Fractions - Assessment