6th Grade - Refer Back To The Problem To Check

Problem Solving
Refer Back to the Problem to Check
Evaluate the reasonableness of the solution in the context of the original situation.
Determine if a solution is reasonable in context to the original situation. Explain why a solution is reasonable or unreasonable.

Sample Problems


Jaime read the problem: “A librarian has 129 extra books she wants to divide equally among 2 classrooms. How many books should she give each classroom?” Jaime answered that each class should get 64.5 books. Does his answer make sense in context of the problem?

(No, the problem is about books. You cannot give part of a book to someone. He should have determined that each class would get 64 books and there would be one book left over.)


Talia read the problem, “A pizza shops made 100 free pizzas. They gave away 3 free pizzas to each family that entered the shop. How many families got free pizza?” Her solution was 33 families got free pizza. Is her solution reasonable? Explain.

(No, if they gave 3 pizzas to 33 families, they only gave away 9 pizzas. There were 100 free pizzas given away. This means the 34th family in the door got one free pizza. So, 34 families were given free pizza.)


Four friends are sharing the cost of their teacher’s end-of-year gift. They bought flours for $15.95 and a vase for $5.95. How much is each person’s share? Solve and explain why your result is valid in context to the original problem.

(Two students will pay 5.47 and two will pay 5.48. This will cover the total cost of gift at $21.90. 5.47 + 5.47 + 5.48 + 5.48 = 21.90)


Marielle read the problem, “You save $2 of your allowance in January, $4 in February, and $8 in March. If this savings pattern is continued, how much money will you have saved altogether by April?” Marielle answered, $30. Check the reasonableness of her answer in context to the problem. Explain why she is correct or incorrect. If it is incorrect, explain what needs to be done to fix the problem.

(Marielle’s answer is reasonable. She found the pattern to determine that $16 would be added to saving in April, and the she added all the amounts together to get $30 altogether. She make correct calculations and answered the question.)


Tim read the problem, “You have 30 students in your class. 12 of them are girls. What percent are boys?” Tim answered, 40%. Is his answer reasonable? Explain. If it is incorrect, explain what needs to be done to fix the problem.

(Tim’s answer is not reasonable. If 12/30 students are girls, this means 18/30 are boys. That is more than half. Half is 50%, so 40% cannot be correct. Tim answered the problem for how many girls there are. It will be easy to use this number to find the percent of boys, because together we will have 100% of the class. This means, 60% of the class is boys.)

Learning Tips


Mathematical reasoning is needed for problem solving. In general, sixth grade students should be able to problem solve by using 3 phases of a problem solving process. These phases allow children to think thorough a problem logically to determine a solution. The first phase of problem solving is to be sure to read and understand the problem. The second phase is to plan and solve the problem. The last phase is to look back at the problem and check work. The more children practice working through each of these phases, they better they will become at using mathematical reasoning to problem solve. The third phase is the focus of this skill.


In order to evaluate the reasonableness of an answer, your child will need to be able to determine if phase one and two of the problem solving process have been completed properly. Here’s a quick overview of these two phases.

In phase one, read and understand, sixth graders should be sure to identify and determine each of the following:

What exactly is being asked?

Children should underline the sentence or command in the problem and write a problem goal in their own words.

What do I know from the problem?

Children should highlight important, relevant facts that will help them to meet his/her goal from question one.

What do I know from personal knowledge?

When necessary, children will apply personal knowledge to add important facts to a problem.

In phase two of the problem solving process, 6th graders will use the information obtained in phase one to evaluate relationships and determine how to solve the problem. After choosing a strategy to solve a problem, children will use that strategy to find a solution. There are several strategies that can be used to solve problems. Help your child to use the list of all the options below to determine which could be used to reasonably solve each problem he/she encounters.

Problem Solving Strategies

Choose an operation: + , - , x,

Make an organized list

Make a table

Draw a picture

Make a graph

Look for a pattern

Guess and test

Write an equation

Work Backward

Solve a simpler problem

Act it out or use objects


Be sure that your child understands that phase 3 of the problem solving process is just as important and the first two phases. In fact, it is the focus of this skill. Many children get an answer and stop there. Going back to the original problem, re-evaluating the question and problem facts and comparing them to the solution will allow students to use mathematical reasoning to determine if his/her answer is reasonable. On phase 3, children should:

Compare work to the information in the problem.

Be sure all calculations are correct.

Estimate to see if the answer seems correct.

Make sure the question(s) has/have been answered.

Sample–Check the Validity of the Results from the Context of the Problem

Gina gave Nico ½ of her gummy worms. Nico ate ½ of the gummy worms and gave the rest to Kyle. Kyle kept 5 of the gummy worms and gave the last 7 to Emily. How many gummy worms did Gina keep?

Question: How many gummy worms did Gina Keep?


Gina kept half

Nico took other half

Nico gave half of his half to Kyle

Kyle kept 5 and gave 7 to Emily

Personal Knowledge:

Kyle’s half was 5 + 7 = 12

Strategy(s): Make an organized list and work backward

Gina – ½ of the total = 12

Nico – Kept ½ 12 – gave½ to Kyle = 12 – 12 + 12 =24 (original 1/2)

Kyle – ½ of Nico’s = 12 (Kyle kept 5 and gave 7 to Emily 5 + 7 = 12)

Answer: Gina kept 12, she and Nico had the same amount.

Check the Answer:

  • Compare work to the information in the problem. *There is a problem here. The work shows that Gina kept 12. She and Nico originally had the same amount. Nico had 12 + the 12 he gave away to Kyle. So, Gina actually had 24.

  • Be sure all calculations are correct. Correct.

  • Estimate to see if the answer seems correct.

  • Make sure the question(s) has/have been answered. Fix an answer, if needed. Gina kept 24 gummy worms.


It is important for children to be able to identify relationships within problems and within the real world in order to solve problems. For example, children can use word relationships and mathematical reasoning to determine when to add, subtract, multiply or divide a problem. So words that are often related to each operation are listed below. Relating these terms to the solution of a problem will help your child to evaluate the reasonableness of answers more readily.

Add: altogether, combined, add, both, in all, sum, total

Subtract: difference, fewer, how many more, how much more, left, less, minus, remains

Multiply: product, times, every, at this rate

Divide: each, divide, quotient


When solving division problems with remainders, your child will need to interpret the remainder to determine if the answer should be rounded down, up or written as a fraction or decimal. For example, if you’re asked to find the total number of busses needed to transport students and you get an answer with a remainder, it will be impossible to use this answer, because you can use a part of a bus. For this case, you would need to round up or hire one extra bus so that everyone can fit. However, if you’re dealing with a problem in relation to time you’ll be able to use a decimal or fraction, because time can be written as part of an hour. Logic will need to be used to determine if parts are acceptable or if a remainder needs to be rounded.

Extra Help Problems


Jonah read the problem: “A classroom teacher has 104 extra books she wants to divide equally among her 28 students. How many books should she give each student?” Jonah answered that each student should get 3.71 books. Does his answer make sense in context of the problem? Explain.

(No, you cannot divide books into parts. It only makes sense for each student to be given 3 books each.)


Jameson read the problem, “A bus holds 75 passengers. How many vans are needed to transport 586 people?” She answered, 8 busses. Is Jameson’s answer reasonable? Explain.

(Yes, if you divide 586 by the number of people, you get a decimal answer, 7.81. This needs to be rounded up to 8 busses for the left over people that didn’t fit on the first 7 busses.)


Kyle read the problem, “You are driving at an average rate of 60 mph. How long will it take you to drive 225 miles?” She answered 3.75 hours. Is her answer reasonable? Explain.

(Yes, she has calculated to problem correctly by dividing 225 by 60. She got a decimal answer, which is fine, because 3.75 hours is the same as 3 hours and 45 minutes.)


Jordan read the problem, You ran 2 miles on Monday, 4 miles the following Monday and 6 miles the 3rd Monday. At this rate, what is the total number of miles you will run on the 4th Monday? Jordan answered 20 miles. Is her answer reasonable? Explain.

(No, this answer is not reasonable. Jordan misread the question. The question is asking for the total miles that will be run on the fourth Monday. This means Jordan only needed to complete the pattern. The total miles run on the 4th Monday would be 8.)


The baseball cards Tim wants to buy come in packages of 6 with one autographed bonus card in each. Tim has a collectors book with spots for 125 cards. Tim bought 21 packs of cards to fill his book. Was this a reasonable amount of cards to buy without being wasteful? Explain.

(No, Tim bought too many. He did not count the bonus card in when he did his calculation. Each package actually includes 7 cards total. Tim only needed to buy 18 packs to cards.)


Kylie conducted a survey of 60 sixth graders at her school to find their ages. She found that 48 students are 12, 5 are 11, and 7 are 13. She used this information to conclude that 80% of sixth graders at her school are 12. Is her result reasonable? Explain.

(Yes, 48/60 is 80%.)


Mr. Davis conducted a survey amongst his homeroom. Statistics showed that out of 36 students, 22 had a 3.5 or higher, the rest of the students needed to improve their grades GPA. James used this data to conclude that 61% of Mr. Davis’ homeroom students need to improve their grades. Is James’ answer reasonable? Explain.

(No, James used the data improperly. He actually calculated the percent of students that have grades of 3.5 or higher. He should have used personal knowledge to figure out how many students needed to improve their grade. This is 36 – 22 = 14 students. Since 14/36 need to improve their scores, this can be written as about 39%. )


Dana is baking cupcakes for the school bake sale. One package of cake mix makes 10-12 cupcakes. Dana has committed to bring a total of 8 dozen cupcakes. She bought 9 boxes. Did Dana buy a reasonable amount of cake mix? Explain.

(Yes, there are 12 cupcakes in a dozen. Since the package says that one box makes 10-12 cupcakes, she could have bought 8, but it is probably safer to buy 9, because some batches may only make 10.)


Randy has promised to bake 200 cookies for the school bake sale. His recipe makes about 36 cookies per batch. Randy has made 5 batches. Does he have enough cookies? Explain.

(No, he has about 180 cookies. He will need to make one more batch and keep have extras.)


Jamie is taking a timed math test with 100 problems. She has 1 hour and 20 minutes in which to finish the test. Jamie concludes that she can spend up to 0.012 seconds on each problem. Is her conclusion reasonable? Explain.

(No, if she spent less than a second on a problem, she’d be done in under a minute. If Jamie had made a quick estimate, she’d see that she has almost a minute to solve each problem. This is because 1 hour and 20 minutes is the same as 80 minutes. 80 minutes for 100 questions is .8 minutes per question, which can be translated into 48 seconds.)


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