Interpreting Remainders

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Interpreting Remainders

• Problem Solving for the Rest of Us
• (Get it? Rest of us? Remainders? Rest of us?
  Thats what remainders are? Get it now?)

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Use the UPSS method whenever you problem solve.
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UPSS

• Understand
• 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.

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UPSS

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

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UPSS

• Solve
• Follow the plan and work accurately.
• If the first attempt doesnt work, try another
  plan.

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UPPS

• See 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?

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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?

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Wow, thats a lot of information!
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Lets 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?

10
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?

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Also, lets cross out information we dont 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?)

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This is what Im 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?)

13
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?)

14
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?)

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Now, lets PLAN. Im 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.
• Im going to draw a picture to see how this will
  work out.

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Here are my 37 campers

• ???????????????????
• ??????????????????
• But since only four can go in each boat, I will
  draw them in the boats.

???? (4)
???? (20)
???? (36)
???? (8)
???? (24)
? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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Now, weve 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 cant go
  by himself?

???? (4)
???? (20)
???? (36)
???? (8)
???? (24)
? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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Ive got it!
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What if I move one of the campers from a full
boat to go with the one left over?
???? (4)
???? (20)
???? (36)
???? (8)
???? (24)
? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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Do you see what I did? Look again.
???? (4)
???? (20)
??? (36)
???? (8)
???? (24)
? ? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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This is what I had.
???? (4)
???? (20)
???? (36)
???? (8)
???? (24)
? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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This is how I changed it.
???? (4)
???? (20)
??? (36)
???? (8)
???? (24)
? ? (37)
???? (12)
???? (28)
???? (16)
???? (32)
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So, how many boats did they need?
???? (4)
???? (20)
??? (35)
???? (8)
???? (24)
? ? (37)
???? (12)
???? (28)
???? (16)
???? (32)
They needed 10 boats eight boats with four
campers, one with three, and one with two.
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Now, lets SEE if I was right.
???? (4)
???? (20)
??? (35)
???? (8)
???? (24)
? ? (37)
???? (12)
???? (28)
???? (16)
???? (32)
4 x 8 32
And 32 3 2 37, so I was right. I am so
good, arent I?
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Lets 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?

26
Lets 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?

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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?

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Also, lets cross out information we dont 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?

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And dont 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?

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Now, lets PLAN. Im 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.
• Im going to draw a picture to see how this will
  work out.

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Here are my 26 campers

• ?????????????
• ?????????????
• But since only four can go in each boat, I will
  draw them in the boats.

???? (4)
???? (20)
???? (8)
???? (24)
???? (12)
?? (26)
???? (16)
32
This time my extra campers are okay because there
are two of them.
???? (4)
???? (20)
???? (8)
???? (24)
???? (12)
?? (26)
???? (16)
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Im ready to SOLVE the problem.
???? (4)
???? (20)
???? (8)
???? (24)
???? (12)
?? (26)
???? (16)
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They will need 7 boats six with four campers in
each and one for the last two.
So, how many boats will they need?
???? (4)
???? (20)
???? (8)
???? (24)
???? (12)
?? (26)
???? (16)
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24 4 6 R2.
Now, lets SEE if I was right.
???? (4)
???? (20)
???? (8)
???? (24)
???? (12)
?? (26)
???? (16)
So they would need 6 boats plus one more for the
extra two. Boy, I am good!
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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 200 PM, 19 people want
  to go out in rowboats. How many rowboats will go
  out at 200?

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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 200 PM, 19 people want
  to go out in rowboats. How many rowboats will go
  out at 200?
• How many people can go out in one rowboat?

38
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 200 PM, 19 people want
  to go out in rowboats. How many rowboats will go
  out at 200?
• How many people are waiting to go out in a
  rowboat?

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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 200 PM, 19 people want
  to go out in rowboats. How many rowboats will go
  out at 200?
• Solve the problem.

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Heres one solution.

• 19 3 6 R1
• So 3 boats will go out at 200.
• One person will be left sitting on the dock,
  picking his toes, and sobbing over his bad luck.

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See?
??? (3)
??? (12)
??? (15)
??? (6)
??? (9)
??? (18)
Im the 19th camper. Im so sad. Im going to
pick my toes.
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So, in summary, solving problems with remainders
isnt so hard. Follow all the UPSS steps. Draw
pictures. Check your answers. Youll do fine.
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The End