Lesson 3: Division Review

Day 3

Activity 5: Interpreting Remainders

So far, all the problems you've done in this lesson have had dividends that are fully divisible by the divisor, meaning that there is nothing left over after all the division is finished. You know from previous math levels that this isn't always the case. Often in division, you end up with a number amount left over, even after you've divided all the digits in the dividend. This number amount is called a remainder.

While it may be acceptable in some problems to simply write the remainder after the quotient, it is often better in word problems to interpret the remainder to find the answer. To interpret the remainder means to figure out what needs to be done with the leftover number amount to best solve the problem.

There are four main ways to interpret a division remainder:
  • include the remainder,
  • discard the remainder amount,
  • increase the quotient by 1 to include the remainder amount, or
  • use a decimal point and zeros in the dividend to prevent a remainder.
Let's work through some examples to demonstrate each of these methods.
First, solve the problem using long division. Since 46 is not divisible by 4, you know this problem will have a remainder. Remember that you write a remainder after the whole number in the quotient.
In the quotient, the number 11 shows the number of cookies that Alan will put on each plate for his neighbors. The remainder (2) shows the number of cookies he has left over. Both of these numbers are needed to answer the word problem questions. Each neighbor will receive 11 cookies, and Alan will have 2 left over for himself. The remainder is needed in the answer.

In other problems, the remainder is not directly a part of the answer, so you have to figure out exactly what to do with a left over amount. The following problem provides an example of this.
Using the divisibility rules, you can see that 136 is not divisible by 3, so again, you can expect a remainder.
The number 45 in the quotient shows the number of pots that you can completely fill with potting soil. The remainder (1) shows that after filling those pots, there is still one liter of soil left over.

What should you do with this soil? Ask yourself: Do I have to use all of the potting soil? The answer is no. You cannot completely fill another pot with the remaining soil, so in a problem such as this, you discard the remainder and show the answer as "45 pots."

This won't always be the case. There are other types of problems in which it doesn't work to discard the remainder. For example:
The quotient shows that after completely filling 5 vans, there are still 10 students remaining. Can Mr. Thompson discard those students? Not unless he wants some very unhappy students! In this case, the remainder requires us to add one more to the quotient's main number so that the remainder is included in the answer. Mr. Thompson will need 6 vans for the field trip.

So far you've explored problems that require the remainder to be used, the remainder to be discarded, or the remainder to be rounded. Some problems, however, do not solve correctly if a remainder exists. These problems call for a special technique that involves decimals. Consider the following problem.
Each girl gets $63, but what about that remaining $1? Should they discard that extra money? No way! Can they increase their earned money by $1? Again, the answer is no. They could simply say they have $1 left, but that doesn't really help anything either. If only there was a way to evenly divide that one dollar. Guess what? There is a way, and it involves decimals.

Remember that every whole number has a decimal after the ones place and every decimal has place values to the right of it. Those place values can be written with zeros without changing the actual value of the number. That means that 6.0 and 6.00 have the exact same value as 6. You can use this fact to extend any dividend so that the steps of long division continue to happen until there is no more remainder. Study the following image to see how this works for the previous example:
By adding a decimal and a zero in the tenths place, the division problem could be solved without having a remainder. Iggy and Brianna each earned $63.50. (Even though the last zero wasn't shown in the division problem, you know to include it for answers involving money.)

Practice the skills you've learned by completing the "Interpreting Remainders in Division" activity sheets. Be sure to carefully read any word problems to decide how best to interpret a remainder.
Your child will learn how to interpret remainders in division problems in terms of how they affect the answer that needs to be given. In some problems, having a remainder is not the best solution, so the concept of using a decimal point and zeros is taught, allowing the student to continue using long division until there is no more remainder. If your child would benefit from extra review of the division algorithm, including how it works with decimals, have him watch the video at the following web link.
Web Link

Answer Key:

  1. Hendrick needs to read a 895 page book. If he reads 35 pages a day, how many days will it take Hendrick to finish the book? (26 days) 895 ÷ 35 = 25 R 20; Hendrick must read the remaining pages, which means he needs to add 1 additional day to his answer.
  2. Violet earned $82 last week as a babysitter. If she worked 8 hours, how much did she get paid each hour that she babysat? ($10.25) Violet wants to be paid an exact amount, so a decimal and zeros must be added to $82 to avoid a remainder. $82.00 ÷ 8 = $10.25)
  3. Joey bought a large bag of jellybeans and wants to divide the candy into smaller bags with 24 jellybeans in each. If the large bag has 980 jellybeans, how many small bags can Joey fill? (40 bags)
    980 ÷ 24 = 40 R 20; the remaining jellybeans cannot fill another bag, so they are discarded from the answer.
  4. Mary Elizabeth divided a game's cards evenly among 4 players and then created a bonus pile with the extra cards. If there were 71 game cards, how many cards were in the bonus pile? (3 cards)
    71 ÷ 4 = 17 R 3; the remainder is used in a special way, so it is included in the answer.
  5. Joachim has a rope that is 645.4 inches long. He wants to cut the rope into 7 equal-sized pieces. How long will each piece of rope be? (92.2 inches) 645.4 ÷ 7 = 92.2 inches; the problem does not include a remainder.
  6. A scientist has 79.2 grams of a chemical and needs to put 2.2 grams of the chemical into different test tubes. How many test tubes can she fill? (36 test tubes) 79.2 ÷ 2.2 = 792 ÷ 22 = 36; the problem does not include a remainder.

Activity 6: Basic Skills Review

You will complete the "Basic Skills Review #1" sheet. This review practices skills that you've previously learned. In your answers, be sure to simplify fractions if needed and include unit labels. For example, if a problem is about perimeter, your answer should include a unit of length such as "inches" or "centimeters." Use scratch paper if more room is needed to solve a problem.
Student Activity Page
Your child will complete a Basic Skills Review. These reviews focus on previously learned skills and encourage knowledge retention and increased mathematical fluency.

Answer Key:

  1. Candidate A won 87,356 votes in the election and Candidate B won 145,302 votes. How many votes were cast altogether? (232,658 votes) 145,302 + 87,356 = 232,658
  2. 5.982 + 96.3 = (102.282)
  3. 452,006 − 372,899 = (79,107)
  4. What property of multiplication is shown? 15 × 73 = 73 × 15 (the commutative property of multiplication)
  5. Mortie jogs 8 miles every day. How many miles has he jogged in the past 365 days? (2,920 miles)
    365 × 8 = 2,920
  6. Pie in the Sky Bakery has 3/8 of a coconut pie, 5/8 of a cherry pie, and 1/8 of an apple pie. How much total pie does the bakery have? (1 1/8 pies) 3/8 + 5/8 + 1/8 = 9/8 = 1 1/8
  7. What is the perimeter of the rectangle? (44 centimeters) 14 + 8 + 14 + 8 = 44
  8. Write using digits. Nine million, three hundred fifteen thousand, four hundred eighty-six (9,315,486)
  9. If there are 96 crackers in a package and a serving size is 16 crackers, how many servings are in the package? (6 servings) 96 ÷ 16 = 6
  10. Samantha bought 3 yards of yarn. How many feet of yarn does she have? (9 feet)
    1 yard = 3 feet, 3 yards = 9 feet