6th Grade - Calculate And Check Your Answers

 
     
 
     
 
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6th
Problem Solving
Calculate and Check Your Answers
Make precise calculations and check the validity of the results from the context of the problem.
Find the exact answer to a problem, by using precise calculations. Re-read problems to check answers to make sure they make sense in relation to the problem.
 

Sample Problems

(1)

Jaime read the problem: “A librarian has 329 extra books she wants to divide equally among 4 classrooms. How many books should she give each classroom?” Jaime answered that each class should get 82.25 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 82 books and there would be books left over.)

(2)

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?

Solve and explain why your result is valid in context to the original problem.

(34 families got free pizza. Even though 3 does not go into 100 evenly, the answer is correct, because the question asked how many families got free pizza, not 3 free pizzas. The first 33 families got 3 pizzas and the 34th family only got one, the last one.)

(3)

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)

(4)

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 June?” Marielle answered, $64. 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 is incorrect. The actual question asked, How much money will you save altogether by June. Marielle wrote only the amount that will be saved in June. She needs to add up all the amounts that have been put in savings from January through June.)

(5)

Danny saved $3 of his allowance in June, $6 in July, and $9 in August. If this savings pattern is continued, how much of his allowance will he save in December? Solve and explain why your result is valid in context to the original problem.

(Danny will save $21 in December. The savings pattern is adding $3 each month. The question asked, how much of his allowance will he save in December. December is 7 months away from when he started saving in June. $ 3 x 7 = 21. The calculations are correct and the problem has been reasonably answered based on the context of the problem.)

Learning Tips

(1)

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.

(2)

Work with your child to help him/her successfully complete phase one of the problem solving process. Be sure that your child reads the problem a few times before attempting to solve it. 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.


Sample–Phase 1: Identify the Question/Facts/Personal Knowledge

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

Facts:

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

(3)

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

Sample–Phase 2: Choose a Strategy and Solve

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

Facts:

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 = 24

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)


(4)

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?

Facts:

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 answer if needed. Gina kept 24 gummy worms.


(5)

Many students experience anxiety when asked to use mathematical reasoning to problem solve. Often times they become overwhelmed with the amount of information in the problem and the inability to use a standard algorithm to solve a math problem. If this is the case for your child, you can help him/her to overcome these fears by breaking the problem down into smaller parts. One way to do this is to come up with a system of color-coding or note taking for the solution of these sorts of problems. For example, you may want to have your child highlight important facts from the text in yellow and then underline the question being asked in green. This will greatly benefit the visual learner. In addition, the visual learner may need to transfer the information highlighted to a table, such as the one show below. Help your child to come up with a system that he/she will be able to remember and use independently.

Sample Problem Solving Table

My Goal

(Here, I restate the question in my own words)

Problem Facts

(Use this space to write in the important numbers and information the problem gives me)

Facts I Know

(This is where I tell any information I know from personal knowledge that needs to be added to the problem. I won’t always use this space.)

Solve It

(I will use this area to show my work on the strategy I used to solve the problem.)




Check It

  • ?

  • Facts

  • Calculations

  • Estimate

(Check off each box to make sure I check my work.)

(6)

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

(1)

Jonah read the problem: “A librarian has 127 extra books she wants to divide equally among 3 classrooms. How many books should she give each classroom?” Jonah answered that each class should get 42 books. Does his answer make sense in context of the problem? Explain.

(Yes, three will not go into 127 evenly, so there will be one book left over. However, this book can not be split evenly, so it must be left over.)

(2)

Jameson read the problem, “A van holds 15 passengers. How many vans are needed to transport 52 people?” She answered, 3.47 vans. Does his answer make sense in context of the problem? Explain.

(No, you cannot have a fraction of a van .47, you need a whole number to represent each van.)

(3)

Jameson read the problem, “A van holds 15 passengers. How many vans are needed to transport 52 people?” She answered, 3.47 vans. Explain how Jameson can fix this problem to make the results valid, based on the context of the problem.

(Each person needs a seat in a van. If the vans hold 15 passengers each, 4 vans will be needed to transport all 52 people. 3 vans will only hold 45 people, but 4 vans will hold 60 people.)

(4)

Kyle read the problem, “You are driving at an average rate of 50 mph. How long will it take you to drive 125 miles?” She answered 2.5 hours. Does his answer make sense in context of the problem? Explain.

(Yes, she has calculated to problem correctly by dividing 125 by 50. She got a decimal answer which is fine, because 2.5 hours is the same as 2 ½ hours.)

(5)

Jordan ran 2 miles on Monday, 4 miles the following Monday and 8 miles the 3rd Monday. At this rate, what is the total number of miles Jordan will run by the end of the 4th Monday? Explain why your answer is reasonable.

(30 miles. This is reasonable, because the pattern shows that Jordan is doubling his time each Monday. This would mean that he would run 16 miles on the fourth Monday. The question asks to find the total miles run, so all the miles must be added up 2 + 4 + 8 + 16 = 30.)

(6)

The baseball cards Tim wants to buy come in packages of 8. He has a frame he’d like to put them in that has 12 spaces. How many packages of cards should Tim buy? Explain why you answer is valid.

(Tim needs to buy two packs. If he buys two packs, he will have 16 cards, which is more than enough to fill his frame.)

(7)

Mrs. Stearns bought 75 cupcakes for the 6th grade. How many dozens was that? Explain why your result is valid.

(6.25 dozen; This is valid because she can buy a part of a dozen. The calculations are correct, because there are 12 in a dozen and 75 divided by 12 is 6.25 or 6 ¼ dozen.)

(8)

Juan’s mom bought him a large bag of rainbow colored candies. He counted out ¼ and gave them to his sister. His sister ate half of the fourth and then gave the rest away: 20 to her best friend and 10 to her younger brother. How many candies did Juan keep? Explain why your answer is valid.

(Juan kept 180 candies. We know that his sister had ¼ of his candies. She ate ½ of them and gave ½ away. The ½ she gave away was 20 + 10 = 30. This means she also ate 30. So, we know that ¼ of the total is 60 (30 given + 30 eaten). If there are 60 in ¼, we can multiply by this by 3 to find out how many Juan kept, because if the gave away ¼, he had to keep ¾. ¾ + ¼ = one whole pack. 60x 3 = 180.)

(9)

Juan’s mom bought him a large bag of rainbow colored candies. He counted out ¼ and gave them to his sister. His sister ate half of the fourth and then gave the rest away; 20 to her best friend and 10 to her younger brother. How many candies came in the package? Explain why your answer is valid.

(240 candies came in the package. Since we know that 60 is ¼ of the package, we can multiply that by 4 to find the total amount of candies in the package. This is because there are ¼ x 4 = 1. So, 60 x 4 = 240.)

(10)

Tyler is renting buses to send the 6th graders on a field trip. Each bus holds 75 students. There are 172 students going on the field trip. Tyler ordered 4 buses. Did Tyler order the correct number of buses? Explain.

(No, he ordered too many buses. He should have ordered 3 buses. He is wasting money.)

(11)

Addison is making lemonade for her class picnic. Her teacher asked her bring enough for 40 cups. She brought 3 gallons. Will she have enough lemonade? Explain.

(Yes, there are 16 cups in one gallon. So, 16 x 3 = 48 cups. She will have more than enough lemonade.)

(12)

Eight tour guides split 290 students into groups. Mr. Wallace said that every tour guide should take 36 students. Is he correct?

(No, if the do this, two students will be left without a tour group.)

 

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