The Penny Drop

Eric Imbrescia & Amanda Sawyer

Researchers believe that students develop a better understanding of fractional concepts when multiple meaning of fractions are emphasized in the classroom (Clarke, Roche & Mitchell, 2008; Lamond, 2012). In particular, researchers found that contextual problems help develop and build fractional understanding (Cramer & Whitney, 2010; Lewis, Gibbons, Kazemi, & Lind, 2015).  Therefore, we created the Penny Drop as a problem-solving activity that helps students engage with the measurement meanings of fractions. Wilkins and Norton (2018) and Lamon (2007) state that instruction based on the measurement meaning of fractions seems to promote the strongest understandings of fractions. The measurement meaning of fraction is defined as the iteration of the unit fraction to determine the length of an object. Fifth grade teachers are required to have their students compare and order fractions, and the Penny Drop could provide a simple realistic problem scenario that supports fraction reasoning using a data collection process.

Problem scenario.

Teachers begin the Penny Drop by presenting the following problem scenario:

You are a contestant in the Penny Olympic Event: The Penny Drop!  You are going to try to drop 20 pennies into a jar. You will record the number of pennies that successfully landed into the jar. The numerator is the number of pennies you dropped into the jar. The denominator is the total number of pennies you dropped. Your goal is to find out at which point as you dropped the 20 pennies were you performing your “best.”  (See the student activity sheet.)

Classroom setup.

In order to complete the activity, you will need the following materials:

· mason jars or plastic cups,

· 20 pennies,

· Fraction Strip Model (Figure 1), and

· The Penny Drop recording sheets (Figure 3).

The students can work in pairs or small groups. Each student will place their jar or cup on the floor between their feet while standing straight and tall, no kneeling or bending over.  The students will drop their pennies one at a time attempting to have them land into the jar. One student is the dropper and one student is the recorder, or students can drop and record their own data.

The teacher starts the competition by saying, “One, two, three, drop.” On the Penny Drop recording sheet, the students will record their fraction with the numerator being the number of pennies that landed into the jar and the denominator being the total number of pennies dropped. Then, they fill out the fraction strip model by first identifying which fraction strip represents the number of shots taken, denominator, and they color in the number of shots made, numerator (see Figure 1). Students will begin at the whole, the top of the page, then progress sequentially down the page coloring in each fractional part from left to right. When the first student completes twenty penny drops, they switch roles with their partner and repeat the activity so that each student has a data set.

An alternative implementation has each group drop a penny at the same time using the “One, two, three, drop” phrase. This allows time for discussion about outcomes. In either case, for the first drop, have students record the first fraction as 0/1 or 1/1. After the students have dropped their fifth penny, have them share their results and name their fraction. You can focus your discussion on conceptual understanding while promoting the measurement language within the activity. For example, part of a whole is 3 out of 5 pennies landed in the jar. Relate this to the measurement language, three units of 1/5 landed into the jar.

You can discuss whether or not they are doing better, worse, or the same after the first penny drop as students share their reasoning. You can stop after the 10th, 15th, and 20th penny drops to have the same discussion. This allows for conversations about equivalent fractions and proportional reasoning as the students reason through creating equivalent fractions to make comparisons.

When the students dropped all of their pennies, you can ask them to place specific fractions on a number line on the Penny Drop activity sheet. These selected fractions are chosen because of their relationship. For example, halves, thirds, and fourths can be used as a basic starting point for everyone to approximate their locations on a number line. You can explore equivalence using outcomes such as 1/2 and 2/4 or 2/2 and 3/3. For example, eighths connect to fourths by doubling or by halving. Ninths allow students to evaluate if their fraction is larger or smaller than the location of eighths on the number line if the ninth penny landed inside or outside of the jar. Twelfths connect to halves, thirds, or fourths. Fifteenths connect to fifths, and eighteenths offer a frame of reference to ninths. Students work together and discuss where to place their fractions within the context of the problem using the fraction strip model for support.

Next, on the Penny Drop activity sheet, the students will grapple with identifying their “best” performance within the first ten shots and again during their last ten shots. For this part of the activity “best” is an intentionally ambiguous term. For example in Figure 2, some students might think that 6/10 is their best because they made six drops, even though earlier they made 5/7 which would be a larger proportion. This can be an opportunity for the class to discuss the idea of “best” and whether we can define it by just looking at the numerator, or whether the proportion of drops made is a better indicator of “best”.

Students who focus only on the numerator can reason through whether 6/8 is the same as 6/9 and 6/10. The students can also visually understand this “best” comparison using the fraction strip model. This allows students to see that the fraction strip that is closest to representing a whole value could be the “best.” A student’s notion of “best” in this situation is often related to whether students are relative, additive, or absolute, multiplicative, thinkers (Lamon, 2012). This thinking is developmental, and students who have not developed relative thinking often struggle with fractions because they do not understand the relationship between the numbers (Lamon, 2012). Therefore, it is important to help students become relative thinkers by emphasizing the multiplicative relationships between fractions, as shown with the measurement language.

Figure 1. Fraction strips.

Where’s the math?

Many students struggle with comparing fractions and rely on finding common denominators or converting fractions to decimals or percents. In this activity, students are comparing and explaining strategies for two different meanings of fractions: part of a whole and measurement. The students create a part of a whole meaning of fractions by comparing the total number of shots made to the total number of shots possible. The activity supports the development of the measurement meaning of fractions by iterating the unit fraction which is constructed through each shot made. The fraction strip model supports students to visually understand the iteration when they color the number of pennies in the jar. From example in Figure 2, when a student colors the fraction 3/5, the student must color 1/5 three times.

This activity emphasizes common numerators when students miss dropping a penny into the jar consecutively. Using 20 total number of penny drops leads students to proportional reasoning when they, for example, double and halve fractions to make comparisons.

Figure 2. Fraction story.

Modifications and extensions.

This activity can be altered to focus on comparing fractions with smaller denominators by reducing the number of pennies students drop. You will ask students to compare specific fractions such as halves, fourths, and eighths. Students can focus on their successful proportions after 2, 4, and 8 pennies dropped, or 3, 6, and 12 pennies dropped. Conversations about a student who has one penny in the jar after several consecutive attempts (i.e. 1/2, 1/3, and 1/4) opens the door for discussions about comparing unit fractions.

To increase the challenge, students can drop pennies one at a time within a given time period, such as 30 seconds. They can keep track of the number of pennies that landed in the jar, the number of pennies that missed, and the total number of pennies dropped. You can ask them to compare their proportional scores with other students in the class. Conversations again focus on the best performance. For example, did someone who had 13 pennies land in the jar out of 30 pennies dropped perform better than someone who had 11 pennies land in the jar out of 15 pennies dropped? This modification produces fractions with larger denominators that requires a more in-depth conversation about a method to compare the fractions. This can lead to students to calculate their best by converting the fraction to a decimal using long division.

Teachers can use the fraction strip model as an extension for the activity. After the fraction strips are completed, the students can share their performances or the performances of another student using the model. They can analyze when they increased their shooting performance when pennies landed in the jar in consecutive drops. The class can discuss the impact between having a penny land in the jar early in the game or having a penny land in the jar later in the process. The chart can also be used to find streaks of when the pennies landed into the jar or when the pennies did not land in the jar, noting the starting point and the ending point of the streaks can be compared.

Figure 3. Directions.

Conclusion.

The Penny Drop activity provides students with the opportunity to compare, order, and interpret fractions using their own data. We found when we completed this activity with a fifth-grade class, for some students the activity provided:

1. a clear picture of how to visually represent fractional concepts, and

2. a model to move beyond using common denominators to compare fractions.

Connecting the mathematical meaning to the concrete representation of fractions, the Penny Drop activity offers the opportunity to engage students in meaningful learning while promoting the student’s ability to make sense of fractions.

References

Clarke, D., Roche, A. & Mitchell, A. (2008). 10 practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 373-380.

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D. V. Lambdin & F. K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 15-22). Reston, VA: NCTM.

Lamon, S. (2012). Teaching fractions and ratios for understanding; Essential content knowledge and instructional strategies (3rd ed.). New York, NY: Taylor & Francis Group.

Lamon, S (2007). Rational numbers and proportional reasoning: towards a theoretical framework for research. In F Lester (Ed.), Second Handbook of research on mathematics teaching and learning, (pp. 629–667). Reston: NCTM.

Lewis, R., Gibbons, L., Kazemi, E., & Lind T. (2015) Unwrapping students’ ideas about fractions. Teaching Children Mathematics, 22(3), 158-168.

Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5:27. https://stemeducationjournal.springeropen.com/articles/10.1186/s40594-018-0119-2

Eric Imbrescia

imbresea@jmu.edu

Amanda Sawyer