Fractions as Multiples of Unit Fractions: Using Models

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Objective

Students will be able to use unit fractions and multiplication to describe fractions that are multiple of the unit fractions.

Big Idea

Students will develop a deeper understanding of how multiplication is related to fractions by using unit fractions and pictorial representations.

Anticipatory

20 minutes

Fraction Pad

Because my students require additional support, I begin this lesson by using the attached fraction pad and main clue words to review some third grade skills to help them better understand the new information. It is important to meet students where they are in their learning.

To do this I invite students to the carpet to discuss how to apply what they already know to a newer concept. I briefly review mathematical language, and how it is used to help solve real-world problems. I model how to scan through a given word problem and circle key words.  INSTRUCTIONAL TASK.docx

add, addend, sum, subtract, difference, equation, strategies, properties (rules about how numbers work), rectangular arrays, area model, multiply, divide, factor, product, quotient, and check for reasonableness.

Modeling with visuals, and verbal explanations allow students to understand more than just the computation; it brings them deeper into the lesson by allowing them to see the how and why.

 

Model and Inquiry

20 minutes

To generate a deeper understanding, I pose the following scenario:

Teacher:

There are three brown paper bags in front of me. Each bag has two apples in it. I wonder, how many apples I have altogether? What equation could we use to model this situation? (Accept addition, but if it is given ask them if there is another more efficient way?) Is there a drawing that shows this problem?

It is important to represent problems within context, because it creates meaning for the numbers and symbols. Students need many experiences with different ways to solve mathematical problems.  It helps them develop their mental flexibility in working with numbers, and is an essential part of students becoming fluent in solving mathematical equations. 

I model the thinking that is being developed to broaden my students' internal thinking.  I demonstrate how to solve this equation by using a pictorial representation.  Since I have three bags, I point to the bags and say there are two apples in each bag.  If there are three bags with two apples in each bag (counting by twos) 2, 4, 6 ... then 3 bags x 2 apples = 6 apples in all.

Checking for Understanding: 

Keeping in mind that my students require additional support, I stop for a moment to check for understanding.  This is important to review with whole numbers because students generally have develop many gaps with they shift from whole numbers to fractions.  I carefully explain how using models can assist them in their learning.

I ask students, "How many apples are there? What multiplication equation could you use to show the situation? How can I represent this problem using a pictorial representation? How can I represent this problem using multiplication?

It may seem as if I am repeating the same information over again, however, students need as many experiences with problem solving to understand the importance of models and how some skills are repeated in the problem-solving stage.

I ask a couple of volunteers to explain the same concept in their own thinking. This helps to build confidence so students rely on their own understanding, instead of others.


Guided

20 minutes

During this stage of the lesson, I ask students to move into their assigned groups.  I give each group a set of colored pencils, chart paper, and pencils. I tell them they will have 20 minutes to solve the given problem.  I remind them to think about how they solve their equation.  Having students share/discuss their work help promotes independent thinking because students who share their work do not rely on the teacher as much for corrective advice; they usually find ways to solve their problems on their own (MP1).

How much tomato juice is needed for a group of 4 people if each person gets 1/3 cup of juice?  How much tomato juice is needed if they each get 2/3 cup of juice? 

I ask students to discuss different ways to solve the problem with their group. I remind students that using pictures to represent a situation can assist in solving the given equation.  I want my students to be able to explain how multiplication can be used to find how much tomato juice is needed for a group of 4 people, if each person gets 1/3 cup of juice. MP4

As students are working, I circulate to listen in on their thinking and check on the development of their pictorial representations. The Common Core standards requires a significant increase in expectations for my students, and I believe that representations and symbols are essential in developing their understanding of multiplication and fractions. Additionally, asking questions allow me to zone in on students who seem to be having difficulty. MP3 Critique the reasoning of others.

1/3

 

 

1/3

1/3

1/3

 

1/3+ 1/3+ 1/3+ 1/3 = 4/3

4/3 = 4  x  1/3

Quick Check

10 minutes

It is important for students to have time to practice new concepts, so I ask students to move back into their assigned seats.  Then I have students work independently to draw a picture of the  given question.

Can you use multiplication to find how much tomato juice is needed for the 4 people if each person gets 2/3 cup of juice? Show your thinking.

Sample answer:

4 x 2/3; each person gets an equal amount (2/3 cup).

The problem can be represented by a number sentence:

2/3 + 2/3 + 2/3 + 2/3 + = 8/3 = 3/3+ 3/3+ 2/3= 2  2/3 

model 1

model 2

model 3

Multiplication can be used to represent repeated addition, so the sum is equal to 4 x 2/3= 8/3= 2 2/3

 

Independent

20 minutes

Now that students have had multiple experiences modeling a multiplication problem with unit fractions, I want to assess their ability to use their own process and mathematical tools to check for reasonableness when solving a given set of problems. Assessment

I remind students that they can use scratch paper or their math journals to draw their own pictures to model the problem.  As students are working, I begin to take notes of any misconceptions and their current level of understanding. I use the given information to re-teach students and improve my lesson delivery. 

Student Sample