Counting Pennies

Charles Loeffler Matt Lohsl Bruce Nance Ann Sebrell

Class: Pre-Algebra/General Math

Materials: Pennies or washers, balance or scales, rulers, plastic cups, graph paper, a Cartesian graph on a transparency.

Goals: To give students practice on measuring skills, graphing, using proportions, organizing data, and modeling problem solving skills.

Time: Four days. One day to introduce the project, form groups, and discuss strategies. One day to review measurements, and collect data. One day for graphing and equation writing. One day for presentations, collating group responses, and summarization.

Background: Knowledge of measuring weight and length, using proportions, graphing, data collection, data recording, and data displaying.

Setting: Hakeem, Barbie, and Punky, the Mattels, are planning a trip to Kings Island. They have been saving pennies in containers of various sizes for a whole year. Without counting all of the pennies, they want to know if they have enough money to go.

Problem: The students are to propose alternate methods of Acounting@ pennies and test them to determine which method is best.

Evaluation: One grade will be given to the entire group. Point values will be assigned for the data sheet, written report, and presentation. The following is a suggested percentage breakdown:

Data Collection - 25%

Written Report - 45%

Math Work - 25%

Communication - 20%

Presentation - 30%

Extension: The minting process has changed over the years. Will the Abest@ method depend on the year of the penny?

Teacher Notes:

1. Have the students bring in pennies with the intent of buying donuts when they are done using them. This will give the class ownership. They are less likely to walk off with the pennies (you can substitute washers).

2. Pass out the data collection sheet to the students, and have them discuss possible ways to count, or estimate, the pennies.

3. Determine and test alternate methods of counting pennies:

a. Weight

b. Length, or height

c. Volume

1. Handfuls

2. Cupfuls

4. Each group is given a method to investigate and a sample of pennies.

5. After each group has determined a model, they would use that model to Acount@ one or more of the Mattels= containers.

6. With the actual value for the container, each group can calculate the relative error.

(May want to use the same terms used by the science department.)

7. Oral reports covering the methods and results of group experimentation and individual summaries of collated data are appropriate.

8. Overhead transparencies can be used to make comparisons between methods, and to collate all group data.

9. Level of guidance can be determined by each teacher, depending on class goals (discovery, maturity, motivation of students, etc.)

One possible device for measuring the height of the pennies is made from two meter sticks.

Possible

Questions: 1. If a weighing method has a relative error of +2% and the true weight is 350 grams, then what was the predicted value?

2. Explain the meaning of a negative relative error.

Counting Pennies

Student Data Collection Sheet

Hakeem, Barbie, and Punky, the Mattels, are planning a trip to Kings Island. They have been saving pennies in containers of various sizes for a whole year. Without counting all of the pennies, they want to know if they have enough money to go.

What are some possible ways for the Mattels to Acount@ their money?

Use the data sheet to test your ideas.

 Method of Measure Trial Number Number of Pennies Measure 1 2 3 4 5 6 7 8 9 10

 Container Number Measure of Container

Counting Pennies

Sample Report Using Weight

Project Description:

Hakeem, Barbie, and Punky, the Mattels, are planning a trip to Kings Island. They have been saving pennies in containers of various sizes for a whole year. Without counting all of the pennies, they want to know if they have enough money to go.

For this project, we decided to weigh six different piles of pennies. We recorded the number of pennies in each pile, and we recorded the weight of each pile. We then calculated the weight for one penny for each trial. After calculating the weight for one penny for each trial, we averaged the six results. This gave us the average weight for one penny. Using this calculation we were able to find the number of pennies in the container based on weight.

Calculations: We measured each stack to three significant digits using a metric balance.

Trial # Number of Pennies Weight Weight per penny

1 15 41.5g 2.77g

2 15 41.6g 2.77g

3 20 55.5g 2.78g

4 5 13.8g 2.76g

5 10 27.4g 2.74g

6 12 32.5g 2.70g

Total 16.50g

Average Weight per Penny 2.75g

The weight of the pennies in our container was 379.0g. This was found by weighing the container while it was empty, then weighing the container filled with the pennies, and subtracting the weight of the container.

Container empty - 5.76g .

Container with pennies - 384.76g.

Pennies only - 379.0g.

The total number of pennies was calculated by dividing the weight of the pennies by the average weight per penny.

Weight of the pennies / Average weight per penny 379.g / 2.75g

Total number of pennies in the container 137.8

This number will round to 138.

Conclusions: By using an average weight for one penny, you can determine the number of pennies in a given container based on its weight.

Counting Pennies

Sample Report Using Height

Project Description:

Hakeem, Barbie, and Punky, the Mattels, are planning a trip to Kings Island. They have been saving pennies in containers of various sizes for a whole year. Without counting all of the pennies, they want to know if they have enough money to go.

For this project, we decided to find the length of six different piles of pennies. We recorded the number of pennies in each pile, and we recorded the height of each pile. We then calculated the height for one penny for each trial. After calculating the height for one penny for each trial, we averaged the six results. This gave us the average height for one penny. Using this calculation we were able to find the number of pennies in the container based on height.

Calculations: We measured each stack to the nearest mm using two meter sticks taped together (with masking tape) to form a Avee@ coin counter.

Trial # Number of Pennies Height Height per penny

1 15 23mm 1.53mm

2 15 22mm 1.47mm

3 20 30mm 1.50mm

4 5 8mm 1.60mm

5 10 15mm 1.50mm

6 12 18mm 1.50mm

Total 9.10mm

Average Height per Penny 1.52mm

The total number of pennies was calculated by dividing the height of the pennies by the average height per penny.

Height of the pennies / Average height per penny

460mm / 1.52mm

Total number of pennies in the container 302.6

This number will round to 303.

Conclusions: By using an average height for one penny, you can determine the number of pennies in a given container based on the total height of the stacked pennies.

Counting Pennies

Teacher Data Collation Sheet

 Method

 Group # Predicted Number Actual Number Relative Error Average Error

Counting Pennies

Teacher Data Collation Sheet

 Method Average Error

 Best Method

Funded in part by the National Science Foundation and Indiana University 1995

Funded in part by the National Science Foundation and Indiana University 1995