# The Statistics of Coins

### Summary

Students will use skills of observation, recording, and calculation to provide an overview of years of cents (pennies) in circulation in the local area.

### Coin Type(s)

- Cent

### Coin Program(s)

- Generic

### Objectives

Students will use skills of observation, recording, and calculation to provide an overview of years of cents (pennies) in circulation in the local area.

### Major Subject Area Connections

- Math

### Grades

- Sixth grade
- Seventh grade
- Eighth grade

### Class Time

**Sessions**: One

**Session Length**:
90 minutes

**Total Length**:
46-90 minutes

### Groupings

- Whole group
- Pairs
- Individual work

### Terms and Concepts

- Calculation
- Cent
- Circulation
- Coins
- Penny
- Percent

### Materials

- Pennies
- Lined paper
- Graph paper
- Calculators (if necessary)

- Divide the students into pairs. Supply each pair with a set of ten cents (pennies).
- Have the students record the years marked on each coin.
- Distribute graph paper and have each pair of students create a bar graph showing the number of coins represented by each year.
- Have the students use the data in their own graph to determine the mode's percentage of the total (that is, what percentage of the coins are in the year that occurs most frequently). Model this activity for the students.
- Have the students use their own data to calculate the percentages of the total coins represented by all the years recorded.
- Ask the students to speculate whether they believe their percentages would be close to those represented by the class as a whole.
- Collect the class data, including the number of groups participating and the number of coins represented by each year.
- Have each student create a graph showing the class’ data, then compare the mode on the class graph with the mode on their own graph. Are they the same?
- Looking at just the class graph, have the students calculate the mode's percentage of the total.
- Using the class data, ask the students to calculate the percentages for each year recorded.
- As a class, discuss how the information they discovered compared with what they expected.

### Enrichments/Extensions

Give students hypothetical situations to solve. For example: if they were given a sack with 100 cents in it, determine the probability that they would withdraw a coin with a specific year on it.

Use the students' written work and class participation to assess whether they have met the lesson objectives.

**Discipline**: Math

**Domain**: 4.MD Measurement and Data

**Grade(s)**:
Grade 4

**Cluster**: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit

**Standards**:

**4.MD.1.**Know relative sizes of measurement units within one system of units including km, m, cm, kg, g, lb, oz, l, ml, hr, min and sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table.- For example, know that 1ft is 12 times as long as 1in. Express the length of a 4ft snake as 48in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

**4.MD.2.**Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.**4.MD.3.**Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

**Discipline**: Mathematics

**Domain**: 6-8 Number and Operations

**Cluster**: Compute fluently and make reasonable estimates.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
- develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;
- develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results; and
- develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

**Discipline**: Mathematics

**Domain**: 6-8 Number and Operations

**Cluster**: Understand meanings of operations and how they relate to one another.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
- use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals; and
- understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

**Discipline**: Mathematics

**Domain**: 6-8 Data Analysis and Probability

**Cluster**: Develop and evaluate inferences and predictions that are based on data.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken;
- make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit; and
- use conjectures to formulate new questions and plan new studies to answer them.

**Discipline**: Mathematics

**Domain**: 6-8 Data Analysis and Probability

**Cluster**: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population; and
- select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

**Discipline**: Mathematics

**Domain**: 6-8 Data Analysis and Probability

**Cluster**: Select and use appropriate statistical methods to analyze data.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- find, use, and interpret measures of center and spread, including mean and interquartile range; and
- discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

**Discipline**: Mathematics

**Domain**: 6-8 Number and Operations

**Cluster**: Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- work flexibly with fractions, decimals, and percents to solve problems;
- compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
- develop meaning for percents greater than 100 and less than 1;
- understand and use ratios and proportions to represent quantitative relationships;
- develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
- use factors, multiples, prime factorization, and relatively prime numbers to solve problems; and
- develop meaning for integers and represent and compare quantities with them.