Is it 1 or more than 1? (Improper Fractions and Mixed Numbers)

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Objective

SWBAT identify improper fractions and mixed numbers from a real-world context by drawing models.

Big Idea

Fractions with numerators equal to or larger than the denominator can be written as a whole or mixed number. Models can be used to show improper fractions or mixed numbers.

Opener

10 minutes

In today's lesson, the students learn to identify improper fractions and mixed numbers by drawing models (MP4). This aligns with 4.NF.A1 because this extends the understanding of fraction equivalency.  In previous lessons, the students learned to use fraction strips and number lines to find equivalent fractions.  Today, the students use their understanding of fractions as being equal sized pieces, and use it in real-world scenarios to identify an improper fraction as an equivalent mixed number.

On the Smart board, I show the following scenario:

Tim's mom baked a chocolate cake.  She cuts the cake into 8 pieces.  Tim eats 3 pieces.  What fraction of the cake did Tim eat?  I call on a student and he says 3/8 because the largest number should be at the bottom and the smallest number is at the top.  (I let the students know that  in this lesson we're going to talk about whether the smallest number is always at the top or if the smallest number could be at the bottom.)  On the next slide, the students see that Tim has eaten 3/8 of the cake.  I pointed out to the students that the eighths on the slide are divided into equal size pieces.  When they draw their models in pairs, I want them to remember this.

To extend their understanding, I add to the scenario:

Tim loves chocolate cake.  After a nap, he wanted more cake.  He eats the rest of the cake.  What fraction of the cake did Tim eat in all?  I call on students to respond.  One student says, one whole.  She explains her answer by saying that he ate the remaining cake so it is the whole cake.  Another student disagreed with the answer because she said that he ate 3 pieces, but then there was 5.  I continued questioning the student by reminding her that he ate 3 the first time and 5 the last time, so how many did he eat in all?  The class helps this student out by saying that Tim ate 8 pieces of cake.  I point out the question wants to know how many he ate in all.  The next slide shows that time ate 8/8 of the cake.  I explain to the students that a numerator can be the same as a denominator.  "Is 8/8 equal to 1 or more than 1?" One student responds, it is equal to one because the whole thing is shaded.  To check the students understanding, I have them give me a thumbs up if they think 8/8 is equal to 1.  This is a quick way to assess the students' understanding.  By asking the students this question, it is giving them an understanding that any number over itself in a fraction equals 1 whole.   "Can a numerator be larger than a denominator?" One student tried to explain, but his explanation was not correct.  I told them that this is the topic of our lesson and we will find out in a few minutes.

Whole Class Discussion

15 minutes

I tell the students that we have discovered that a numerator can be the same as the denominator, now let's see if the numerator can be larger than the denominator.  On the Smart board, I have a real-world scenario.

Tim's mom baked 2 loaves of bread.  She cut the bread into 8 slices each.  Tim was so hungry that he ate 10 slices of bread.  What fraction of the bread did Tim eat?  (I point out to the students that the question is asking for a fraction.  They need to give me a numerator and a denominator in their answers.)

I give the students a few seconds to think about the question.  I let them share their answers with their neighbors.  (By allowing the students to share with their neighbors, it gives them different perspectives on the problem.  Also, it gives the students who are shy and afraid to speak out in whole class a voice because they have an opportunity to be heard by someone in the classroom.)  I call on students to share whole class.  As I walked around to listen in on the conversation, I heard several students say 10/16.  I reminded them that they said the denominator is the total number of pieces the whole was cut into.  After much discussion, the majority finally says 10/8.

In the power point, the students can clearly see from the shaded model that Tim has eaten 10/8 of the bread.  I explain to the students that the numerator can be larger than the denominator.  This is called an improper fraction.  An improper fraction has a numerator greater than or equal to its denominator.  I point out that 8/8 was also an improper fraction because the numerator was equal to the denominator. You must keep in mind that we are talking about the same whole.  The 2 loaves of bread must be the same size and same shape (congruent).

We can write 10/8 in another way. This means that Tim ate 8/8 of the first loaf, which is 1 whole.   From the second loaf, Tim ate 2/8.  This tells us that 10/8 is equivalent to 1 2/8.

This is called a mixed number.  A mixed number has a whole number part and a fraction part.  I ask, "Is 10/8 equal to 1 or more than 1?"  From the model, the students see that it is more than 1.  I ask, "If 8/8 is equal to 1 and 10/8 is more than 1, what conclusion can you draw from the two?" One student shares, a fraction equals 1 when there are two numbers that are the same.   Another student adds, the numerator is larger than the denominator in 10/8.  I continue to probe the students:  If a fraction has a numerator that is larger than the denominator will it be equal to 1 or greater than 1?  The students agree that it will be greater than 1.  This type of questioning allows the students to create their own knowledge.  I do not have to tell the students that if the numerator is larger than the denominator, the fraction is greater than one.  They can come up with this understanding on their own if allowed by the teacher.

How to draw it out:

It is important for students to get a conceptual understanding, so I have the students draw out the fractions.  Later, in another lesson, I will show the students how to use calculations to change improper fractions to mixed numbers and mixed numbers to improper fractions.  First, I want them to get a visual of what improper fractions and mixed numbers look like.  I feel that this will better help the students understand the math calculations by connecting it to what they learned by drawing the model.

Improper fractions:  I remind the students that the denominator lets you know how many pieces will be in each whole.  If the improper fraction is 4/3, how many pieces are in each whole?   There are 3 pieces.  On the Smart board, I draw a rectangle and divide it into 3 pieces.  The numerator tells how many equal parts we are describing.  In this problem, the numerator is 4.  This means that we are talking about 4 pieces of something.  If I have 3 pieces here (pointing to the board), how many will I shade out of the 3? Student response: 3.   How many more pieces do we need?  Student response: 1.  I draw another rectangle, divide it into 3 equal sized piecs.  I shade 1 piece.  What is the mixed number for this improper fraction? I point out (in the model) that my whole number is 1 and the fraction is 1/3, therefore 4/3=1 1/3.

Mixed Numbers:  In the mixed number 2 4/5, the whole number is 2 and the fraction is 4/5.  The whole number lets us know that we must draw 2 whole items.  Because we have a fraction, we know that we have to draw 1 additional whole. The denominator in the fraction is 5.  This tells us that the fraction is divided into 5 pieces.

I show 3 rectangles on the board and divide them into fifths.  I get input from the students by questioning them.  (Questioning helps me to check their understanding to see if they are grasping the information being given.)  I ask the following questions:  How many do we shade in the first rectangle?  What about the second rectangle?  So far, how many have we shaded?  How many do we shade in the last rectangle?   What improper fraction is represented in this model?

I tell the students that they will now practice with a parter on drawing models to name improper fractions and mixed numbers.

Skill Building/Exploration

20 minutes

I give the students practice on this skill by letting them work together.  I find that collaborative learning is vital to the success of students.  Students learn from each other by justifying their answers and critiquing the reasoning of others.

For this activity, I put the students in pairs.  I give each pair an Improper Fractions and Mixed Numbers Exploration.docx exploration sheet and crayons. The students must work together to draw models and identify the improper fractions  and mixed numbers.  An example of the Student Work - Skill Building Activity can be found in the resources.

The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and must agree upon the answer to the problem.  Because the students must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill (MP6).  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.

As they work, I monitor and assess their progression of understanding through questioning.

1.  What is the denominator?

2.  How many pieces should you divide the whole into?

3.  How many pieces will you shade?

4.  How many wholes will you need to draw?

5.  Is it equal to 1 or more than 1?

6.  What improper fraction is represented in your model for the mixed number?  What mixed number is represented in your model of the improper fraction?

Early Finishers:  Choose a mixed number.  Draw a model of the mixed number.  Name the improper fraction from your model.

Closure

15 minutes

To close the lesson, I have one or two pairs share their answers.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time.  Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good work samples, as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during the group activity will be addressed whole class.

Each student is given an Exit Ticket Improper Fractions and Mixed Numbers.docx to complete individually.  Group activities are great, but I need to know how well each student is doing on their own. The exit tickets are collected at the end of class.  This gives me further data on how the students are comprehending individually.   All struggling students identified from the data on the exit tickets will receive further instruction in small group.

Results from exit ticket:

After evaluating the exit tickets, I know that we still have work to do on identifying improper fractions and mixed numbers.  Out of 19 students, 11 students could draw the model correctly, 15 knew that the fraction was more than 1 whole, and 8 students got all of the information on the exit ticket correct.  It is encouraging that from their models and what they learned in class, they knew that the number was more than 1.  However, we will continue to work on writing the correct fraction or mixed number for the model.