## Digging Up Improper Fractions: Converting Improper Fractions to Mixed Numbers for “Dig It” Game

• Lesson
3-5
1

This lesson provides a hands-on approach to converting improper fractions to mixed numbers.  In addition, students locate improper fractions on a number line. Students use number cards and counters as manipulatives while exploring the relationship between improper fractions and mixed numbers. Students reinforce their skills while playing a modified version of Calculation Nation’s “Dig It.”

Group students in pairs. Provide each group with number cards, counters, and toothpicks.  Explain to the class that in this lesson they will convert improper fractions to mixed numbers.  Define the terms improper fraction and mixed number.  Explain that an improper fraction is a fraction with a numerator that is greater than its denominator (ex: ), and a mixed number is a number that includes both a whole number and a fraction ().

Have students take out the “9” and the “4” number cards. Ask them to create a fraction with the numbers by placing a toothpick horizontally on the table.  Tell them to place the 9 above the toothpick to represent the numerator, and the 4 below the toothpick to represent the denominator. Write on the board. Ask students, “What kind of fraction is this and how do you know?” [It is an improper fraction; the numerator is greater than the denominator.]

Point to the numerator, and ask what a numerator represents. [The numerator tells how many parts of the whole are in the problem.] Point to the denominator, and ask what the denominator represents. [The denominator tells how many parts are in, or represent, the whole.] Ask students how many parts represent a whole in the fraction . [4.]

Have students take out the number of counters that represent the number of individual parts in (which would be 9 counters). Observe and make an informal assessment of the students who comprehend and the students who don’t comprehend the parts to a fraction.

Model counting out nine counters and explain that these counters represent the numerator, the individual parts, in . Say to students, “I am going to divide these nine parts into groups of four.” Ask students to explain why the nine counters need to be placed into groups of four. [The denominator is four, so a whole is made up of four parts.]

Demonstrate moving the nine counters into groups of four. Have students do the same. Say to students, “Instead of nine individual parts, I now have two groups of four with one counter left over. How many wholes do we have?” [2.] “How do you know?” [There are two groups of counters that each represents a whole, each group has four parts in a set of four, ].

Ask students how many individual pieces are left. [1.] Ask students what part of the whole this one piece represents. [].  On the board write + + .  Say to students, “Since represents one whole, I can simplify the equation.”  On the board write 1 + 1 + . Explain that the expression can be further simplified. The whole numbers can be added together for a sum of two. And the fraction can be added to the whole number for a sum of .

=           + +

=          1 + 1 +

=

Explain that when you ask students to show their work, you want to see each step of how they convert from an improper fraction to its mixed number.

Provide students with the Digging Up Improper Fractions Activity Sheet. Point to the problem you just completed together. Explain that to find an improper fraction on a number line, one strategy you can use is to convert the improper fraction to a mixed number. Demonstrate moving to the two on the number line. Say to students, “The denominator tells us that a whole is divided into four parts.”  Demonstrate dividing the space between the 2 and 3 into fourths.  Say to students, “The numerator tells us how many fourths we move on the number line.  We are at one of the fourths on the number line.  Demonstrate placing a dot on .  Have students find on their number line.

Next, have a student volunteer to draw two cards from the deck. With the two cards, have the student create a new improper fraction. Write the new fraction on the board, and have students write it on their Activity Sheet. Then have each pair convert the improper fraction to a mixed number using counters. Have students show the steps they took to convert the improper fraction to an equivalent mixed number on their Activity Sheet.

As students work, check for understanding.  Ask for a student volunteer to share the steps needed to convert the improper fraction to a mixed number and show how to locate the number on the number line.

Have students work with their partner to create four additional improper fractions using the number cards. Check their work. Students that seem to have an understanding of the objectives can move to the next step in the lesson. Work in small groups with students who seem to be struggling with the concept.

Next, have students log on to Calculation Nation, Dig It. Explain that in the game they are provided with five random numbers. Tell students that you want them to use the numbers to make improper fractions that will allow them to get as many jewels as possible. After they have dragged the numbers to create the fraction, have them write the fraction on their Digging Up Improper Fractions Activity Sheet. Have students show their work to find the equivalent mixed number, and then locate it on the number line. Allow students to use counters as they work. Then, have students locate the improper fraction on the game’s number line to continue their turn.

Have students continue playing Calculation Nation, Dig It converting improper fractions to mixed numbers and locating them on the number line. Explain that when they play this game on their own, they can also make proper fractions, but for the purpose of this lesson they are only making improper fractions.

At the end of the lesson, have students explain how to convert improper fractions to mixed numbers. Listen for terms like numerator, denominator, whole, part, improper fraction, and mixed number. Discuss strategies students used to determine the two numbers they wanted to use to get as many jewels as possible.

Assessment Options

1. As students work through converting improper fractions to mixed numbers, observe if students:

• understand that the numerator represents the number of parts (number of counters)
• understand the denominator represents how many parts are in a set
• correctly divide the number of parts (indicated by the numerator) into the correct number of sets (indicated by the denominator)
• show each step of their process with fractional equivalents ( + = 1 +   = )
• understand how an improper fraction can be equivalent to a mixed number

2. Give each student an index card. Write three improper fractions on the board. Have students convert the improper fractions to mixed numbers. Have counters available for students to use as they work. For example:

• [.]
• [.]
• [.]

3. Watch as students use the Calculation Nation, Dig It game.  Observe if they correctly find the location of the improper fraction.

Extensions

1. Converting from Mixed Number to Improper Fractions
When students seem to have a solid grasp on converting from improper fractions to mixed numbers, write a mixed number on the board. Have students convert it to an improper fraction.  Encourage students to work backward. For example: With the mixed number , ask students how many parts are in a set. [4.] Explain that there are two whole groups, as you place two groups of four counters on the table. Then, place one counter in its own group to represent one of four counters in the fraction. Model counting the counters. Write a nine in the numerator position to represent the nine parts. Write a four in the denominator position to represent that there are four parts in a set.
2. Working with both Proper and Improper Fractions
During the lesson, students were asked only to only create improper fraction in the Calculation Nation, Dig It game. Allow students to play the game as intended, where students are creating and locating both proper and improper fractions.

Questions for Students

1. What is an improper fraction?

[An improper fraction is a fraction where the numerator is greater than the denominator.]

2. What is a mixed number?

[A mixed number includes both a whole number and a fraction. It is a way to represent numbers so the whole and the part are seen separately.]

3. How do you convert an improper fraction to a mixed number?

[You divide the number of parts you have (numerator) by how many parts are in a set (denominator). Dividing allows you to find out how many wholes and parts are equivalent to the improper fraction.]

4. Describe the relationship between an improper fraction and a mixed number.

[Numbers can be represented as both an improper fraction and a mixed number. For example, is equivalent to . In the fraction there are 7 parts that need to be divided into sets of 3. In , there are two whole sets with 3 in each set, and a partial set with one of three parts.]

5. How do you locate an improper fraction on a number line?

[One strategy you can use is to convert the improper fraction to a mixed number. This allows you to calculate how many wholes and parts you have.]

Teacher Reflection

• How did your lesson address auditory, tactile and visual learning styles?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
• What worked with classroom behavior management? What didn't work? How would you change what didn’t work?
• How did including the on-line game affect students’ level of enthusiasm/involvement in the lesson?

### Learning Objectives

Students will:

• Explore the relationship between improper fractions and mixed numbers.
• Convert improper fractions to mixed numbers.
• Explain how an improper fraction and a mixed number are equivalent.
• Locate improper fractions on a number line.

### NCTM Standards and Expectations

• Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.
• Use models, benchmarks, and equivalent forms to judge the size of fractions.
• Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.

### Common Core State Standards – Mathematics

Grade 3, Num & Ops Fractions

• CCSS.Math.Content.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.