## Calculator Remainders

• Lesson
6-8
1

In this lesson, students develop a deep conceptual understanding between remainders and the decimal part of quotients. They learn how remainders and group size work together to influence the results that are displayed on a calculator. Students use beans to physically represent quotients that have remainders, and they compare remainders written as fractions of whole groups to the results obtained with a calculator.

Ima's Dilemma Activity Sheet

Ima's Dilemma is an extended activity that students should be allowed to think about for several days. Distribute the Ima's Dilemma Activity Sheet, and read Ima's Dilemma aloud to students. Explain that their job is to help Ima find a solution, but also explain that they do not need to find a solution immediately. In particular, complete the other activities in this lesson to give students the background they will need to attack the problem.

Give students a deadline by which they should complete their work on Ima's Dilemma, and allow at least 2‑3 days for them to think about it. It works well to use Ima's Dilemma as a problem of the week; post the problem on Monday, and require solutions to be submitted by Friday.

For the main part of the day's lesson, allow students to work in pairs. Each group should have at least one calculator and 40–50 beans in a paper cup. Be sure to prepare the cups ahead of time. Distribute the Splitting Beans Activity Sheet and use the Splitting Beans Overhead to complete a sample row. Have students gather 23 beans and put them into groups of 5. Then ask:

• How many whole groups do you have? [4.]
• How many beans are left over? [3.]
Explain that the 3 beans form 3/5 of a group, because they were dividing the beans into groups of 5. Then show students how they can use the calculator to find the same information: 23 ÷ 5 = 4.6. Because 0.6 is equivalent to 3/5, it must be the case that 3 beans are leftover; the other 20 beans were divded into 4 groups of 5.

After completing the example, tell students to complete the activity sheet on their own. It works well to allow students to work on the first few problems individually for several minutes. Then, allow them to form pairs and work together to complete the rest of the activity sheet. Misconceptions are often clarified if students are given an opportunity to discuss the problems with a classmate.

When students finish all questions on the activity sheet, ask, "How does the fraction column relate to the calculator column?" [The calculator column has the decimal form of the fraction.] Discuss Questions 2 and 6. Students should recognize that the number leftover is not the only factor influencing the decimal part of the quotient. Students should realize that the number leftover and the group size together influence the decimal portion. Use the Leftovers Activity Sheet to reinforce this concept. Allow students to use a calculator to determine the number of whole groups and the number leftover.

Randomly select pairs to present each problem, discussing their solution methods. Summarize the solution methods before returning to Ima's Dilemma. As noted above, allow several days for students to find the solution. They should include a thorough description of their solution method. Students usually work this problem by guess and check, systematic lists or a combination. If using guess and check, they try numbers (usually on their calculators) and use the ideas from the Splitting Beans Activity Sheet to determine the remainders. If they use systematic lists, they identify the possibilites for each clue and find one that is on all the lists. If they use a combination of the strategies, they generate their lists by guessing and checking

Assessments

1. Show students division problems (with divisors having at least two digits), one at a time. Have students determine the remainder and reveal their answers so that you can quickly see who understands and who is struggling.
2. Use the Splitting Beans Activity Sheet for assessment. Specifically, check the last row of every table, where students apply their understanding to write a new problem that fits the pattern.
3. Use Splitting Beans Activity Sheet and Ima's Dilemma Activity Sheet to assess students. Their recorded solution method will reflect their understanding.

Extensions

1. Modular arithmetic is a system of arithmetic where numbers repeat once they reach a certain value. It is related to division and remainders. For example, once you go all the way around a clock, the numbers start over again. If you start at midnight and 51 hours pass, the clock is at 3 o'clock, not 51 o'clock. Allow students to complete the Mod Mysteries Activity Sheet.

Questions for Students

1. Why is the decimal part of the quotient on a calculator almost never the same as the remainder?

[The calculator shows remainders as the decimal equivalents of the fraction of a whole group that is leftover.]

2. When is it possible that the numbers to the right of the decimal point in a calculator answer are the remainders of a division problem?

[When you are dividing by powers of 10.]

Teacher Reflection

• Did your students need the hands-on aspect of this lesson, or would they have learned as much without the beans? What makes you think so?
• Were students able to make up their own problems correctly to complete the tables? If not, what was the main reason they weren't able to do it?
• Is there anything you would change about the way you managed the groups? Why?
• Did any of your students use calculators in a way that undermined the goals of the lesson? If so, what can you do to prevent this problem in the future?
• What methods did the students use most to determine the amount of whole groups and leftovers? Does this indicate anything about their current level of readiness for other concepts you teach?
• Will you do this lesson again? If so, what will you change or try?

### Learning Objectives

Students will:

• Recognize that there is a difference between a decimal representation of a quotient on a calculator and the remainder in a division problem
• Develop a method for finding the remainder of a division problem when faced with a decimal answer from a calculator

### Common Core State Standards – Mathematics

• 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.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.