ILLINOIS TEACH & TALK

Interpreting Remainders and Estimation Strategies

 

Remainder- the amount left after some calculation.  

 

In problem situations, students must interpret and use remainders with respect to context (4.OA.3)  For example, what is the smallest number of buses that can carry 250 students, if each bus holds 36 students?  The whole number quotient in this case is 6 and the remainder is 34; the equation 250=6 x 36 + 34 expresses this result and corresponds to a picture in which 6 buses are completely filled while a seventh bus carries 34 students.  Notice the answer to the stated question (7) differs from the whole number quotient.

 

On the other hand, suppose 250 pencils were distributed among 36 students, with each student receiving the same number of pencils.  What is the largets number of pencils each student could have received?  In this case, the answer to the stated question (6) is the same as the whole number quotient.  If the problem had said that the teacher got the remaining pencils and asked how many pencils the teacher got, then the remainder would have been the answer to the problem.  

 

Common Core Standards Writing Team. (2013, September 19). 

Progressions for the Common Core State Standards in Mathematics(draft).  K-5 Counting and Cardinality and Operations and Algebraic Thinking. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.

 

Rounding- replacing a value with an approximately equal value. 

 

Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies. Estimation strategies include, but are not limited to:

 

• Front-end estimation with adjusting (using the highest place value and estimating from the front end, making  

  adjustments to the estimate by taking into account the remaining amounts),

• Clustering around an average (when the values are close together an average value is selected and multiplied by  

   the number of values to determine an estimate),

• Rounding and adjusting (students round down or round up and then adjust their estimate depending on how

  much the rounding affected the original values),

• Using friendly or compatible numbers such as factors (students seek to fit numbers together -e.g., rounding to

  factors and grouping numbers together that have round sums like 100 or 1000),

• Using benchmark numbers that are easy to compute (students select close whole numbers for fractions or

  decimals to determine an estimate).

 

Example 1:

On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. How many miles did they travel total? Some typical estimation strategies for this problem:

 

Student 1

I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200 together, I get 500.

 

Student 2

I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34. When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had, I end up with 500.

 

Student 3

I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530.

 

The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range (between 500 and 550). Problems will be structured so that all acceptable estimation strategies will arrive at a reasonable answer.

 

Example 2:

Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of water. On the first day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in 6 packs with 6 bottles in each container. About how many bottles of water still need to be collected?

 

Student 1

First, I multiplied 3 and 6 which equals 18. Then I multiplied 6 and 6 which is 36. I know 18 plus 36 is about 50. I’m trying to get to 300. 50 plus another 50 is 100. Then I need 2 more hundreds. So we still need 250 bottles.

 

Student 2

First, I multiplied 3 and 6 which equals 18. Then I multiplied 6 and 6 which is 36.

I know 18 is about 20 and 36 is about 40. 40+20 = 60.

300 - 60 = 240, so we need about 240 more bottles.

 

This standard references interpreting remainders. Remainders should be put into context for interpretation. Ways to address remainders:

• Remain as a left over

• Partitioned into fractions or decimals

• Discarded leaving only the whole number answer

• Increase the whole number answer up one

• Round to the nearest whole number for an approximate result

 

Example 3:

Write different word problems involving 44 ÷ 6 = ? where the answers are best represented as:

Problem A: 7

Problem B: 7 r 2

Problem C: 8

Problem D: 7 or 8

Problem E: 7 2/6

 

Possible solutions:

 

Problem A: 7.

Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill?

44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.

 

Problem B: 7 r 2.

Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left?

44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches and have 2 left over.

 

Problem C: 8.

Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would the fewest number of pouches she would need in order to hold all of her pencils?

44 ÷ 6 = p; p = 7 r 2; Mary can needs 8 pouches to hold all of the pencils.

 

Problem D: 7 or 8.

Mary had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received?

44 ÷ 6 = p; p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.

 

Problem E: 7 2/6.

Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of pouches that Mary filled?

44 ÷ 6 = p; p = 7 2/6