Interpreting Remainders Problem Solving for the Rest of Us (Get it? Rest of us? Remainders? Rest of us? That’s what remainders are? Get it now?)

Use the UPSS method whenever you problem solve.

U U PSS U nderstand:  Retell the problem, including the question, in your own words.  Read it twice (or thrice or more).  Take notes.  Study any charts or diagrams.  Look up words and symbols that are new.

U P SS P lan:  Decide what operation(s) to use.  Decide what sequence of steps to use.  Choose an appropriate strategy: o Draw a picture. o Solve a simpler problem. o Act it out. o Find a pattern. o Guess and check. o Work backwards. o Make a chart or organized list. o Use logical reasoning.

UP S S S olve:  Follow the plan and work accurately.  If the first attempt doesn’t work, try another plan.

UPP S S ee if you were right:  Does the answer make sense?  Is it necessary to start over?  Has the question in the problem been answered, and has the answer been labeled?

Here is a problem. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go?

Wow, that’s a lot of information !

Let’s make sure we UNDERSTAND what we are being asked. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go?

That question was... What is the least number of boats that must be rented for the Friday trip so that everyone can go? In my own words: –How many sailboats do they have to rent so that all the people can go sailing on Friday?

Also, let’s cross out information we don’t need. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go? (Or, how many sailboats do they have to rent so that all the people can go sailing on Friday?)

This is what I’m left with. A campground rents sailboats. Thirty- seven campers have signed up for a Friday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go? (Or, how many sailboats do they have to rent so that all the people can go sailing on Friday?)

One more important thing I see. A campground rents sailboats. Thirty- seven campers have signed up for a Friday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go? (Or, how many sailboats do they have to rent so that all the people can go sailing on Friday?)

So, now I think I understand the conditions of this problem. A campground rents sailboats. Thirty- seven campers have signed up for a Friday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. What is the least number of boats that must be rented for the Friday trip so that everyone can go? (Or, how many sailboats do they have to rent so that all the people can go sailing on Friday?)

Now, let’s PLAN. I’m going to use the strategy of drawing a picture. I know there are 37 campers who are going on the sailboats on Friday. I know only four people can go in each boat. I know that a boat has to have at least two people in it. I’m going to draw a picture to see how this will work out.

Here are my 37 campers ☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺ ☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺ But since only four can go in each boat, I will draw them in the boats. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺☺ (36) ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

Now, we’ve got another problem. The problem said: No sailboat can go out with fewer than two people. How are we going to get all 37 campers in boats if the last one can’t go by himself? ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺☺ (36) ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

I’ve got it!

What if I move one of the campers from a full boat to go with the one left over? ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺☺ (36) ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

Do you see what I did? Look again. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺ (36) ☺ ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

This is what I had. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺☺ (36) ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

This is how I changed it. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺ (36) ☺ ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32)

So, how many boats did they need? ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺ (35) ☺ ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32) They needed 10 boats: eight boats with four campers, one with three, and one with two.

Now, let’s SEE if I was right. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺☺ (35) ☺ ☺ (37) ☺☺☺☺ (28) ☺☺☺☺ (32) 4 x 8 = 32 And = 37, so I was right. I am so good, aren’t I?

Let’s try this with a different question. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. At least how many boats should be rented for the Saturday trip?

Let’s make sure we UNDERSTAND what we are being asked. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. At least how many boats should be rented for the Saturday trip?

That question was... At least how many boats should be rented for the Saturday trip? In my own words: –What are the fewest number of boats the campers are going to need for Saturday?

Also, let’s cross out information we don’t need. A campground rents motor boats, sailboats, and canoes. Thirty-seven campers have signed up for a Friday sailing trip, and 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. At least how many boats should be rented for the Saturday trip? (Or, what are the fewest number of boats the campers are going to need for Saturday?

And don’t forget... A campground rents sailboats. 26 campers have signed up for a Saturday sailing trip. A sailboat can hold up to four people. No sailboat can go out with fewer than two people. At least how many boats should be rented for the Saturday trip? (Or, what are the fewest number of boats the campers are going to need for Saturday?

Now, let’s PLAN. I’m going to use the strategy of drawing a picture again. I know there are 26 campers who are going on the sailboats on Saturday. I know only four people can go in each boat. I know that a boat has to have at least two people in it. I’m going to draw a picture to see how this will work out.

Here are my 26 campers ☺☺☺☺☺☺☺☺☺☺☺☺☺ But since only four can go in each boat, I will draw them in the boats. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺ (26)

This time my extra campers are okay because there are two of them. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺ (26)

I’m ready to SOLVE the problem. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺ (26)

They will need 7 boats: six with four campers in each and one for the last two. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺ (26) So, how many boats will they need?

24 ÷ 4 = 6 R2. ☺☺☺☺ (4) ☺☺☺☺ (8) ☺☺☺☺ (12) ☺☺☺☺ (16) ☺☺☺☺ (20) ☺☺☺☺ (24) ☺☺ (26) Now, let’s SEE if I was right. So they would need 6 boats plus one more for the extra two. Boy, I am good!

Next problem: The campground follows strict safety regulations. The rowboats must have exactly three people on board at all times. Boats are sent out every hour, and campers must wait at the dock for the next available boat. At 2:00 PM, 19 people want to go out in rowboats. How many rowboats will go out at 2:00?

Next problem: The campground follows strict safety regulations. The rowboats must have exactly three people on board at all times. Boats are sent out every hour, and campers must wait at the dock for the next available boat. At 2:00 PM, 19 people want to go out in rowboats. How many rowboats will go out at 2:00? How many people can go out in one rowboat?

Next problem: The campground follows strict safety regulations. The rowboats must have exactly three people on board at all times. Boats are sent out every hour, and campers must wait at the dock for the next available boat. At 2:00 PM, 19 people want to go out in rowboats. How many rowboats will go out at 2:00? How many people are waiting to go out in a rowboat?

Next problem: The campground follows strict safety regulations. The rowboats must have exactly three people on board at all times. Boats are sent out every hour, and campers must wait at the dock for the next available boat. At 2:00 PM, 19 people want to go out in rowboats. How many rowboats will go out at 2:00? Solve the problem.

Here’s one solution. 19 ÷ 3 = 6 R1 So 3 boats will go out at 2:00. One person will be left sitting on the dock, picking his toes, and sobbing over his bad luck.

See? ☺☺☺ (3) ☺☺☺ (6) ☺☺☺ (9) ☺☺☺ (12) ☺☺☺ (15) ☺☺☺ (18) I’m the 19 th camper. I’m so sad. I’m going to pick my toes.

So, in summary, solving problems with remainders isn’t so hard. Follow all the UPSS steps. Draw pictures. Check your answers. You’ll do fine.

The End