6th Grade - Apply A Concept To Other Problems

Problem Solving
Apply a Concept to Other Problems
Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
Identify and explain the method of finding a solution. Show an understanding of how a solution was found by solving similar problems.

Sample Problems


Sammie has 4 ½ cups of flour. She needed to know if she had enough flour to make two cakes. She needs 2 1/8 cups of flour for a lemon cake and 2 ¾ cups of flour for her chocolate cake. Sammie determined that she would need 4 7/8 of flour and went to the store to buy more. How did Sammie determine how much flour she would need?

(She added the flour needed for each recipe; 2 1/8 + 2 ¼.


Sammie has 5 cups of flour. She wants to make 2 cakes. She needs 2 ¼ cups of flour to make a white cake and 2 ½ cups of flour to make a carrot cake. How much flour does Sammie need? Does she have enough?

(Sammie needs 4 ¾ cups of flour. Since she has 5 cups, she has more than enough to make both cakes.)


Amanda bought a pizza for her lunch guests. The pizza had 12 slices. Her guests ate ¾ off the pizza. Amanda determined that there were 3 slices left before looking in the box. How did she find this solution?

(She knew that there was ¼ of the pizza left, because 1/3 + ¾ = 1 pizza. She found ¼ of 12 slices. The word of tells you to multiply. So, Amanda multiplied 12 x ¼ and got 3.)


James bought a large 16-slice pizza for his family. When he left the table to make a phone call, his brothers and sisters ate 7/8 of the pizza. How many slices did they leave for James?

(2 slices)


Tony bought a game that had the original price of $15.00. It was 20% off. Tony determined that the game would cost $12. How did he come to this conclusion? How much would Tony pay for a $10 game that was on sale for 20% off?

(To find the cost of the game minus the discount, Tony could calculate 10% off by moving the decimal point one place to the left to get 1.50 he would then double this to get 20%. So, 20% off is $3. Next, he took the discount away from the original amount, so 15 – 3 = $12. The $10 game is on sale for $8.00.)

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.


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.


A strong understanding of phase 2 is most important to this skill. Your child will have to look problems he/she has completed, as well as problems that others have completed to determine how the problem was solved. This information will then be used to solve similar problems.

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


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.

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, of

Divide: each, divide, quotient, split evenly

These key words can help your child determine which operation was performed in solving a problem. He or she will also be able to look for these words in solving similar problems.


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.

Extra Help Problems


Tyler wanted to combine 2 ½ cups of juice with 3 2/3 cups of lemon soda. He had a punch bowl that holds 6 1/2 cups of liquid. Tyler determined that the bowl was large enough to hold his punch. How did Tyler determine this?

(Tyler added 2 ½ + 3 2/3 and saw that her punch ingredients together would make 6 1/6 cups. This is less than 6 ½, so it will fit in the bowl.)


Steven combined 3/8 a cup of water with ½ cup of flour to make clay for a school project. What were the total cups combined?

(7/8 cups)


Ben had 36 chocolate candies. He ate 2/3 of them. Amanda claims that he must have 12 left. How did she find this out?

(Amanda multiplied 36 x 2/3 to find how many he had eaten; 24. She then subtracted that amount from 36. 36 – 24 = 12.)


Ashley had 150 stickers. She gave 1/3 of them away. How many stickers did Ashley keep?



Misha had 4 ¾ dollars. She spent 1/2 of a dollar on a piece of gum. Her brother determined that she had 4 ½ o 4.25 left in her purse. How did he know this?

(He subtracted; 4 ¾ - ½ = 4 ½.)


Dave had 7 ½ dollars. He spent 4 ¾ dollars on a comic book. How much money did he have left?

(2 ¾ or $2.75)


12 students each brought 6 1/3 cups of water to fill water balloons. Jesse quickly determined that they had a total of 72 cups of water. How did he determine this?

(Jesse multiplied 12 x 6 1/3 cups and got 72.)


If Tia asks 15 friends to bring 2 ½ cups of their favorite juice to mix into a giant punch bowl, how much punch will they make?

(37 ½ cups)


Andy ran 3.4 miles on Wednesday, 4.56 miles on Thursday and 5.62 miles on Friday. Jessica concluded that Andy ran a total of 13.58 miles altogether. How did she figure this?

(She add up all the miles he ran.)


Andy ran 5.47 miles on Monday, 5 miles on Tuesday, 2.3 miles on Wednesday. How many miles did he run altogether?



Jimmy is at a store where all videos are on sale for 10% off. His favorite video is regularly $23.50. He determined that today it would have a discount of $2.35. How did he determine this? If he finds a video with the regular price of 16.50, what will the discount amount be?

(Jimmy found 10% of 23.50 by moving the decimal one space to the left. The discount price of the $16.50 video would be $1.65.)


Red’s is having a sale. All skirt’s are 25% off. Mele found a skirt she’d like that is marked with the original price of $20.00. She decided the skirt would cost her $15 with the discount removed. How did she make this calculation? What would a skirt with the regular price of $30 cost?

(Mele can divide by 4 to calculate the discount price. She then subtracted that from the total. A $30 skirt would cost $22.50 with a 20% discount.)


Greg read the first 110 pages of 330 page book in 2 hours. At this rate, he determined that he could finish the book in 6 hours. How did he find his solution? At this rate, how long would it take him to finish a 1,000 page book?

(One way Greg could find out how many hours it takes to read the whole book is to create a proportion. The first ratio would include the information he knows, 110 pgs/2 hours. The second would include the total pages/the variable. So 110/2 = 330/x. He could then cross multiply to solve. Greg should be able to read a 1,000 page book in about 9 hours.)


A cab ride costs $3.00 for the first mile and $1.25 for each mile after than. Daniel needs a ride to the airport, which is 15 miles away. He expects his ride to cost him $20.50. How did he calculate this? How would the price change if Daniel lived 25 miles from the airport?

(Daniel added $3.00 for the first mile to 1.25 x 14 for the next 14 miles. A cab ride for 25 miles would cost $33.00)


Cecilia makes $4 an hour babysitting. She owes her sister $10. She wants to buy a new shirt for $15.00, but her parents told her she must pay her sister back first. Cecilia decided that she needs to baby-sit for 7 hours. How did she determine this? How many hours would she need to baby-sit to make a $12 donation to the animal shelter and buy a $3 dog toy for her dog?

(Cecilia added up all the money she needed and divided by four. Since there is a remainder, she will have $3 left over. If she uses this $3 for the dog toy, she will only need to baby-sit for 3 hours to get enough money to make her donation.


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