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Understanding remainders

Use arrays and practice problems to understand remainders.

Some division problems work out nicely.

start color #01a995, 24, end color #01a995, divided by, start color #aa87ff, 8, end color #aa87ff, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
An array of 24 circles arranged in 3 rows of 8 circles each.

Other division problems don't work out nicely.

start color #01a995, 25, end color #01a995, divided by, start color #aa87ff, 7, end color #aa87ff
An array of 25 circles arranged in 3 rows of 7 circles each and 1 row of 4 circles.
Oh no! We can't split start color #01a995, 25, end color #01a995 circles into perfect groups of start color #aa87ff, 7, end color #aa87ff. We have some circles left over.
How many groups of start color #aa87ff, 7, end color #aa87ff can we make?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

How many circles remain? In other words, how many circles are not in a group?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Key idea: We can use a remainder to describe what remains after dividing.

25, divided by, 7, equals, 3, start text, space, w, i, t, h, space, a, space, r, e, m, a, i, n, d, e, r, space, o, f, space, end text, 4
empty space, equals, 3, start text, space, R, space, end text, 4

Practice problems

Practice problem 1
19, divided by, 9, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
start text, R, end text
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
An array of 19 circles arranged in 2 rows of 9 circles each and 1 row of 1 circle.

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