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Drop Zone: Adding Fractions with Like and Unlike Denominators Utilizing Strategy

  • Lesson
6-8
2
Number and Operations
Ann Bremner
Location: unknown

In this lesson, students will play card and computer games by adding fractions to make 1. Students will determine how the fractions are related, by first determining what they have and then how much more is needed. Through different interactive games, students will utilize their skills and build upon them to expand their understanding of fractions. Students will be able to determine common denominators and other strategies to add fractions with like and unlike denominators.

Students will develop and show strategy skills for adding fractions through the use of card games and the online math strategy game Drop Zone. The use of a game format will motivate and engage students to learn about adding fractions by using models such as fraction strips to help reinforce skills. Mastery will be fostered through class discussion and playing games. This lesson is most appropriate for students who have a solid understanding of fractions, including the ability to identify equivalency, the facility to find common denominators, and the ability to add fractions with like and unlike denominators.  

To prepare for the lesson, ensure that all computers have an Internet connection and access to Illuminations and Calculation Nation®. Make sure all students have joined Calculation Nation and know their screen name and password.  

In addition, each student should have a copy of the Fraction Cards Activity Sheet (on colored cardstock) and the Fraction Bar Chart (on white cardstock). Copy the Fraction Cards onto several different colors, and give each student with similar abilities a different color. Students will cut out their fraction cards and fraction bars from the cardstock before they begin this lesson.

pdficon Fraction Cards Activity Sheet 

pdficon Fraction Bar Chart 

For reference, students should also be provided with a copy of the Fraction Bar Chart copied onto regular paper. Students should not cut out the bars from this copy, so it is recommended that this sheet be distributed separately from those that need to be cut out. While the cardstock sheets should be distributed at the beginning of the lesson, the Fraction Bar Chart on regular paper can be distributed at the appropriate time in the lesson. 

This lesson consists of three activities for students.  

  1. Students practice adding fractions to make a sum of 1 and use fraction bars to play the game “Make One,” which reinforces the same skill.  
  2. Students play “One and Out,” another card game that requires students to add fraction cards, trying to get as close to 1 as possible. 
  3. Students will use and practice their skills playing Drop Zone, a math strategy game available from Calculation Nation®.  

Tell students that today, they will practice adding fractions to make a sum of 1. Lay down the fraction card \frac{1}{3} on the overhead. Ask students which cards could be added to \frac{1}{3} to make 1. [\frac{2}{3} or \frac{4}{6} since \frac{1}{3} + \frac{2}{3} = 1 and \frac{1}{3} + \frac{4}{6} = 1.] Demonstrate this using the cut-outs from the Fractions Bar Chart. Lay \frac{1}{3} below "1". Then, lay four \frac{1}{6} cards next to the \frac{1}{3}. Line them up to show students that \frac{1}{3} + \frac{4}{6} = 1. Finally, do the same with \frac{1}{3} and \frac{2}{3} to show that their sum is 1.

 DropZoneFractionBar(2) 

You may wish to repeat this warm-up with different fractions until the students seem comfortable with finding fractions that have a sum of 1. After this demonstration, allow students to play “Make One,” as follows:   

  1. Each player places either the Fraction Bar Chart (on regular paper) or the Fraction Bar cut-outs in front of them. Depending on the students' ability, you may want to make it mandatory that students check their answers using the cut-outs. If students are more comfortable adding fractions, then the Fraction Bar sheet can be used as a reference.
  2. Shuffle the deck of Fraction Cards, and place five cards face up on a table or desk. Place the remaining cards face down in a pile.
  3. Player 1 finds two or more cards that add up to 1. Cards are then replaced so there are always five cards face up.
  4. Player 2 then checks Player 1's work using the Fraction Bar Chart or the cut-outs. Player 2 then chooses two or more cards that add up to 1.
  5. Player 1 will check Player 2's work before continuing the game. If there is a disagreement, they should ask the teacher to verify which cards make a sum of 1.
  6. If players are unable to find cards that add up to 1, up to two cards may be replaced. That is, the student can remove one or two of the face up cards and replace them with cards from the face down pile. If a player still cannot find cards that add up to 1, the turn ends, and it becomes the other player’s turn.
  7. Play continues until all cards in the deck have been used and players cannot determine any cards that add up to 1. The winner is the player with the most cards. If necessary, students may also use the Fraction Bar sheet to help them determine equivalent fractions.

Circulate the room while students play the game to assess whether or not students can find fractions that have a sum of 1. If students are having difficulty, you can help identify a fraction card to start with and talk them through which other fraction cards are needed to make 1.  As cards are chosen, discuss the sum of the fraction cards determining how much more is needed or if the chosen cards are incorrect by adding up to more than 1.

When players demonstrate readiness, remove the Fraction Bar Chart or replace the cut-outs with the Fraction Bar Chart (on a regular sheet of paper). Have them play the game again, so that they can make the transition from concrete to abstract.

After playing the game, students should use their cut-outs to create equivalent fractions. Using the cards, demonstrate how to compare fractions with unlike denominators. Show students that \frac{2}{3} is the same as \frac{4}{6}, 1/2 is the same as 6/12, and so on. Have students share other equivalent fractions and verify it using the cut-outs. Then, show students that they can multiply the numerator and denominator by the same number to find an equivalent fraction. For example, we can multiply 2 to the numerator and denominator to show that \frac{2}{3} = \frac{4}{6}. Stress that multiplying the numerator and denominator by 2 is the same as multiplying \frac{2}{3} by 1. Thus, it does not change the fraction. Discuss how these fractions can represent the same values with different denominators.

Then, demonstrate how to add \frac{1}{4} and \frac{1}{{12}}. It will first be necessary to show that \frac{3}{{12}} is equivalent to \frac{1}{4} (by using the Fraction Bars and multiplication), and then the addition can happen: \frac{1}{4} + \frac{1}{{12}} = \frac{3}{{12}} + \frac{1}{{12}} = \frac{4}{{12}}. Explain the importance of finding a common denominator when adding fractions. Give students the problem \frac{1}{3} + \frac{5}{{12}}, and have them find the sum using both Fraction Bars and multiplication.  

Present several more problems to ensure that students are able to solve the problems both using the fraction bars as well as by finding a common denominator (through multiplication). To move from the concrete to the abstract, you can first replace the cut-outs with the Fraction Bar Chart and then the Fraction Cards. A sample question is listed below:

What if I add \frac{2}{3} + \frac{3}{{12}}? What steps are needed to solve the problem?

[Find the common denominator of 12; multiply 3 x 4=12 and 2 x 4=8. So now, \frac{2}{3}\frac{8}{{12}} (have students verify using the cut-outs or the Fraction Bar Chart). Then multiply 12 x 1 = 12 and 3 x1 = 3 so \frac{3}{{12}} = \frac{3}{{12}}. Add \frac{8}{{12}} + \frac{3}{{12}} = \frac{{11}}{{12}}.]

As the second part of this lesson, students can play “One and Out.”. This game may be played in groups of three or more, using two sets of the Fraction Cards. One student is the dealer, and the other students are trying to get as close to 1 as they can, without going over.

The difference between this game and “Make 1” is that getting a sum of exactly 1 is not necessary. For students to be successful at this game, they will need to know how to add fractions with unlike denominators, and it will be important for them to realize that the numerator of the sum must be less than or equal to the denominator.

Play proceeds as follows:

  1. The dealer gives one card to each player, face down. Only the player can look at the card.
  2. The dealer then asks Player 1 if he would like “one more card.” If so, the dealer gives him another card, face up. This continues until all of the face up cards have a sum greater than 1, or when Player 1 decides that he has received enough cards. It then becomes Player 2’s turn.
  3. Play continues in this manner for all players. Players then need to add their fraction cards. If necessary, students may use the Fraction Bar cut-outs or some other manipulative to help them. Students share their total, and the player with the total closest to 1 without going over is the winner.

After the game, bring the class together for a discussion. Use the Questions for Students to lead this discussion, and be sure to emphasize the important elements from the two games that students have played:

  • How to add fractions with like denominators
  • How to find equivalent fractions
  • How to find least common denominator
  • How to add fractions with unlike denominators

appicon Calculation Nation: Drop Zone 

As a final activity, students will play Drop Zone. First, they may challenge themselves by playing against the computer, and then they may challenge others. Look for students who are able to find equivalent fractions and common denominators. What strategies do they use? Guide and assist those students who simply click and drop without understanding. When students have finished playing the game, have them complete a journal entry or exit ticket explaining the strategies they used or any new skills or strategies they’ve learned. Allow time for students to share and explain their thinking as a class. If there are new strategies shared, students may add them to their journal entry or exit ticket.  Look for similarities, differences and connections between strategies and evidence that students conceptually understand the addition of fractions with unlike denominators and how to effectively use strategies to solve.  Discuss how these strategies can be used in solving other problems [mental math, problem solving, subtracting fractions, etc.] Record these on a class chart to post and revisit again.  Allow discussion to continue until you feel the students have a solid grasp of adding fractions with unlike denominators. If necessary, give some examples to the whole class and have them solve.  [\frac{1}{4} + \frac{1}{3}, \frac{2}{6} + \frac{1}{2}].

Assessment Options 

  1. Ask students to determine a common denominator for \frac{1}{3} and \frac{1}{6}. If successful, move to other problems using fractions whose denominators are 3, 4, 6 and 8.Ensure that all students are able to successfully answer these questions. Then, give students fractions to add, such as \frac{1}{3} + \frac{1}{9}. Have them share their strategies.
  2. Have students complete a journal entry or exit ticket explaining the strategies they used to add fractions with unlike denominators.  Strategies used in playing the games can be included as well.
  3. While students are playing the games, take note of which students are able to add fractions with unlike denominators, both with and without manipulatives. Notice which students are using the top row of the Fraction Bar Chart to help them determine equivalent fractions. Take note of the dialogue between students. Listen for explanations/discussions that exhibit good understanding such as clearly explaining the concepts and skills of finding the Least Common Denominators, addition of fractions with like denominators, as well as unlike denominators.

Extensions 

  1. After playing several games of Drop Zone, discuss strategies that students can use to prevent their opponent from successfully adding fractions to make a whole. Discuss ways they can modify the existing fraction to block their move.
  2. Students can use the NCTM Illuminations’ Equivalent Fractions app (now available for iOS and Android, too!) to practice finding equivalent fractions.
    appicon Equivalent Fractions 
  3. Students can play Fraction Game to help reinforce finding the Least Common Denominator in adding fractions to make 1.
    appicon Fraction Game

 

Questions for Students
1. What can you tell me about the fractions \frac{1}{2}, \frac{1}{3}, \frac{1}{4} , \frac{1}{6} , \frac{1}{{12}}?

[They each have a value less than 1. They all have a numerator of 1. Because 2, 3, 4, 6, and 12 are factors of 12, their common denominator is 12.]

2. How can you combine \frac{1}{4} with \frac{1}{3}?

[First, you need to find a common denominator, and then find equivalent fractions with that denominator. For example, since 4 × 3 = 12, then 12 could be used as the common denominator. Then add the fractions. In this case, \frac{1}{4} = \frac{3}{{12}} and \frac{1}{3} = \frac{4}{{12}}, so add \frac{3}{{12}} + \frac{4}{{12}} = \frac{7}{{12}}.] 

3. What is the Least Common Denominator of the following fractions:  \frac{1}{3}, \frac{1}{4}, \frac{2}{6}?

[12; 3x4=12, 4x3=12, 6x2=12.]

4. Why is \frac{3}{4} equivalent to \frac{9}{{12}}?  Can you draw a picture to show this?

[Draw 2 rectangles or circles – one showing \frac{3}{4} and another showing \frac{9}{{12}} – and compare, explaining how they show the same amount shaded.]

Teacher Reflection  

  • Did you find that students were relying too much on the manipulatives? Would they be able to solve the problems without them?
  • How well were the students checking each other's work and taking the time to really think through the addition before taking their turn?
  • Did the games motivate the students and keep them engaged? Which aspects of the games kept them engaged?
  • Were students able to use strategy to prevent their opponents from a winning move?
EquivalentFractions
Number and Operations

Equivalent Fractions

3-5

This applet allows you to create equivalent fractions by dividing and shading squares or circles, and match each fraction to its location on the number line.

FractionModels
Number and Operations

Fraction Models

3-5, 6-8
Explore different representations for fractions including improper fractions, mixed numbers, decimals, and percentages. Additionally, there are length, area, region, and set models.

Learning Objectives

Students will:

  • Add fractions with like and unlike denominators to make a sum of 1.
  • Use estimation to practice combining various numbers

NCTM Standards and Expectations

  • Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results.
  • Use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.
  • Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

Common Core State Standards – Mathematics

Grade 7, The Number System

  • CCSS.Math.Content.7.NS.A.3
    Solve real-world and mathematical problems involving the four operations with rational numbers.

Grade 7, Expression/Equation

  • CCSS.Math.Content.7.EE.B.3
    Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.