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How many _______ are in. . . ?


Alignments to Content Standards: 6.NS.A.1

Task

Solve each problem using pictures and using a number sentence involving division.

  1. How many fives are in 15?
  2. How many halves are in 3?
  3. How many sixths are in 4?
  4. How many two-thirds are in 2?
  5. How many three-fourths are in 2?
  6. How many $\frac16$’s are in $\frac13$?
  7. How many $\frac16$’s are in $\frac23$?
  8. How many $\frac14$’s are in $\frac23$?
  9. How many $\frac{5}{12}$’s are in $\frac12$?

IM Commentary

This instructional task requires that the students model each problem with some type of fractions manipulatives or drawings. This could be pattern blocks, student or teacher-made fraction strips, or commercially produced fraction pieces. At a minimum, students should draw pictures of each. The above problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them. If the task is used to help students see the connections to the invert-and-multiply rule for fraction division (as described in the solution) then they should already be familiar with and comfortable solving Number of Groups Unknown (a.k.a. “How many groups?”) division problems with visual models.

By helping students see how division is modeled through the use of visual models, students can develop a sense for how the fraction division algorithm is derived. The visual model for each problem can be connected to the algorithm for fraction division, and the carefully chosen sequence of problems shows the evolution of the algorithm (note that the types of pictures shown require different kinds of interpretations to see the connection to the algorithm). It also provides opportunities for the students to talk about the specific phenomena that fraction division results in a larger quotient when you divide by a number less than 1.

This task was submitted by Victoria Peacock to the fifth Illustrative Mathematics task writing contest.

Solution

  1. Opening Question: How many 5’s are in 15?

    Sol_1_1b2c44db62659c0ade19b941000903aa

    There are three sets of 5 in 15.

    Write this as a division problem: $$ 15\div 5 = 3$$

    You may want to do a couple of examples like this with whole numbers to familiarize students with thinking about division as how many of a number are in another number.

  2. How many halves are in 3?

    Sol_2_fe658e3347ea3166f2748e36ccafbbe3

    There are 6 half-sized pieces in 3 wholes.

    Write this as a division problem: $$3 \div \frac12 = 6$$

    We can connect this to the algorithm for dividing fractions: $$3 \div \frac12 = 3 \times \frac21 = 3\times2$$ and we can see that there are 3 wholes with 2 halves in each whole, so there are $3\times 2 = 6$ halves in $3$.

  3. How many sixths are in 4?

    Sol_3_2a3ddd4f6d791e6ce888da00979ac22c

    There are 24 sixth-sized pieces in 4.

    Write this as a division problem: $$4 \div \frac16 = 24$$

    We can connect this to the algorithm for dividing fractions: $$4 \div \frac16 = 4 \times \frac61 = 4\times 6 $$ and we can see that there are 4 wholes with 6 sixths in each whole, so there are $4\times 6 = 24$ sixths in $4$.

  4. How many two-thirds are in 2?

    Sol_4_f0ff852a59df481b4c3b8ed3d301bc9b

    Each whole yields a two-thirds and one half of another two-thirds, therefore 3 sets of two-thirds can be made.

    Write this as a division problem: $$2 \div \frac23 = 3$$

    We can connect this to the algorithm for dividing fractions: $$2 \div \frac23 = 2 \times \frac32 = \frac{2 \times 3}{2}$$ and we can see that there are 2 wholes with 3 thirds in each whole, so there are $2\times 3$ thirds in $2$. Because we want to know how many two-thirds there are, we have to make groups of $2$ thirds, or divide the number of thirds we have by $2$. So there are $(2\times 3)\div 2 = 6\div 2 = 3$ two-thirds in 2.

  5. How many three-fourths are in 2?

    Sol_5_a517c1201b98e2e956b1de6df111f3e5

    2 complete sets of three-fourths can be made and 2 of the 3 pieces need to make $\frac34$ are left over, so we have another $\frac23$ of a three-fourths.

    Write this as a division problem: $$2 \div \frac34 = 2 \frac23$$

    We can connect this to the algorithm for dividing fractions: $$2 \div \frac34 = 2 \times \frac43 = \frac{2 \times 4}{3}$$ and we can see that there are 2 wholes with 4 fourths in each whole, so there are $2\times 4$ fourths in 2. Because we want to know how many three-fourths there are, we have to make groups of $3$ fourths, or divide the number of fourths we have by $3$. So there are $(2\times 4)\div 3 = 8\div 3 = 2 \frac23$ three-fourths in 2.

  6. How many $\frac16$’s are in $\frac13$?

    Sol_6_7026fe1f62cd9bf2972837ec9ca95424

    It takes two one-sixths to make a third.

    Write this as a division problem. $\frac13 \div \frac16 = 2$

    We can connect this to the algorithm for dividing fractions: $$\frac13 \div \frac16 = \frac13 \times \frac61$$ and we can see that there are $1 \times 6 = 6$ sixths in a whole. The picture shows $\frac13$ of those 6 pieces. There are $\frac13 \times 6 = 2$ sixths in one-third.

  7. How many $\frac16$’s are in $\frac23$?

    Sol_7_eb0a9438030f63f26eca174a1e4e606a

    It takes 4 one-sixth sized pieces to make the two-thirds.

    Write this as a division problem:$$\frac23 \div \frac16 = 4$$

    We can connect this to the algorithm for dividing fractions: $$\frac23 \div \frac16 = \frac23 \times \frac61 $$ and we can see that there $1 \times 6 = 6$ sixths in a whole. The picture shows $\frac23$ of those 6 pieces. There are $\frac23 \times 6 = 4 $ sixths in two-thirds.

  8. How many $\frac14$’s are in $\frac23$?

    Thw_thirds_1_e99cf1d94f5d5e5dc22b17b43779da1e

    If the whole is divided into twelfths, four of them represent $\frac13$ and three of them represent $\frac14$ as shown in the picture above.

    Two_thirds_2_30dce908c553afe10c24b32b8288af8f

    In left picture, we have $\frac23$, and in the right we have highlighted 2 twelfths that we are going to move to make it easier two see how many $\frac14$ there are in $\frac23$.

    Two_thirds_3_54dab303066a02f6f29e7fa85b7f29ae

    On the left we can see where those 2 twelfths were moved to, and on the right we have colored them normally so we can focus on how many fourths there are. There are 2 fourths, and $\frac23$ of another fourth. So all together, there are $2\frac23$ fourths in $\frac23$.

    Write this as a division problem: $$\frac23 \div \frac14 = 2 \frac23$$

    We can connect this to the algorithm for dividing fractions: $$\frac23 \div \frac14 = \frac23 \times \frac41 = \frac{2\times4}{3\times1}$$ and we can see that there $2 \times 4 = 8$ twelfths in two-thirds. To find how many fourths there are, we need to see how many groups of 3 twelfths there are; in other words, there are $8\div 3 = 2\frac23 $ fourths in two-thirds.

  9. How many $\frac{5}{12}$’s are in $\frac12$?

    Sol_14_d1347f430ecdcbaaf445d936f4a38c8d

    One complete set of five-twelfths will fit into a one-half but there is still a one-twelfth left over which is one of the five needed to make another set of five-twelfths.

    Write this as a division problem: $$\frac12 \div \frac{5}{12} = 1 \frac15$$

    We can connect this to the algorithm for dividing fractions, although with this picture it is a bit tricky: $$\frac12 \div \frac{5}{12} = \frac12 \times \frac{12}{5} =(\frac12 \times 12)\div 5 = \frac65 = 1\frac15$$ We can see in the picture that the whole is divided into 12 equal pieces and we are interested in dividing half of those pieces by 5.

    In other words, ($\frac12$ of a whole $\times$ 12 pieces in the whole) $\div$ 5 of the twelve pieces in the set.