Begin by asking students the following prompts:

Prompts:

What operation was used today to solve this problem?

With student input, land on the following division sentences:

1 260 passengers ÷ 35 passengers per trip = 36 trips

36 trips ÷ 6 trips per hour = 6 hours

Highlight which information was known and the information you were seeking in order to reveal that both contexts were “quotative” division contexts.

Quotative Division:

Dividing a dividend into a measured quantity (or known rate). In this type of division, the size of the groups is known (35 people), the number of parts is unknown (the number of trips).

Highlight the relationship between multiplication and division through this context. Likely, many students relied on repeated addition or partial products in order to determine the number of trips. After determining the number of trips required to be 36, determining the number of hours was likely a known-fact for many students.

There may be an opportunity to explore partial products over addition and over subtraction depending on how students went about determining the number of trips. Did they go past 36 and work back? Did they get close and then work up? How did students take jumps, by one group of 35? By ten?

In the following silent solution animation video, you will see an approach that leverages a partial products strategy through distribution over addition both visually with an area model as well as symbolically highlighting the use of brackets:

Partial products were created with this particular approach by adding “parts” of one factor (which, in this context is actually the quotient we are solving for) and adding the partial products until reaching the product of 1260 (which again, is our dividend for this division problem).

Modelling over subtraction could be done in numerous ways including, but not limited to:

35(10 + 10 + 10 + 10 – 4)

Which is equivalent to the following four partial products:

(35 x 10) + (35 x 10) + (35 x 10) + (35 x 10) – (35 x 4)

Note that a student leveraging this strategy might notice that 35 x 10 is iterated 4 times:

4(35 x 10) – (35 x 4)

Simplifying further:

4(350) – (35 x 4)

If students are forced to evaluate leveraging strategies and models without a calculator (as we always suggest), then you might see some students break this down even further:

4(300 + 50) – (70 x 2)

Notice the decomposing of 350 into two parts; 300 and 50 as well as a doubling and halving strategy applied to double 35 and half 4 resulting in an easier calculation of 70 x 2. Then, students may be able to finalize their computation as something like:

(1 200 + 200) – (140)

While some students will jump on adding 200 to 1 200 for a quick and easy sum of 1 400, some students might see the associative property of addition and subtraction as helpful to quickly subtract 140 from 200.

1 200 + (200 – 140)

Resulting in:

1 200 + 60

= 1 260

Connecting Partial Products to Partial Quotients

Anticipating that some students will leverage partial products as a strategy to “add up” by groups of 35 passengers until reaching the dividend of 1 260, be sure to connect the idea that division works in a related way.

While many students may have used the partial products multiplication strategy as a tool for thinking to reveal the unknown number of trips:

35(x) = 1 260

Then, using known facts to continue growing the product until reaching 1 260:

35(10 + 10 + 10 + 5 + 1) = (350 + 350 + 350 + 175 + 35)

35(10 + 10 + 10 + 5 + 1) = 1 260

A similar approach can be used to solve this problem using division:

1 260 ÷ 35 = x

Although students who used partial products slowly composed one of the factors until reaching the product of 1 260, students who apply this strategy to division will find themselves decomposing the dividend of 1 260 into smaller parts to make quotative division by 35 easier. Almost automatically, students leveraging this strategy might decompose 1 260 into as many 350 parts since it is likely that the division fact of 350 ÷ 35 = 10 is automatic.

(350 + 350 + 350 + 175 + 35) ÷ 35 = x

(350 ÷ 35) + (350 ÷ 35) + (350 ÷ 35) + (175 ÷ 35) + (35 ÷ 35) = x

(10) + (10) + (10) + (5) + (6) = x

36 = x

While some educators feel overwhelmed thinking of division outside of the long division algorithm that many of us learned ourselves as students, it should be noted that long division is built off of the partial quotients strategy. While the “rules” that we were taught tell us that we are looking for how many times the divisor “goes into” the dividend, working our way from the largest place value digit to the smallest, partial quotients offers a more flexible approach to arrive at the same result.

This flexibility makes emerging the strategy of partial quotients much more accessible for all students unlike the very rigid approach of teaching the “rules” of long division.

Look at the partial quotients strategy (on the left) and a more flexible long division strategy on the right:

In both cases, students are the decision makers on what size chunks or “parts” they would like to decompose the dividend into which will also yield more opportunities for all students to practice and extend their known multiplication and division facts.

Facilitator Note: What About Long Division?

Please do not pre-teach the long division algorithm and work “backwards” to partial quotients. If your local curriculum explicitly requires the teaching of the common long division algorithm, then please ensure that students have fluency and flexibility with both types of division and strategies such as partial products before extending the structure to long division.

As an aside, the Common Core State Standards mention using the standard algorithm or long division beginning in grade 6:

CCSS.MATH.CONTENT.6.NS.B.2

Fluently divide multi-digit numbers using the standard algorithm.

CCSS.MATH.CONTENT.6.NS.B.3

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

While we do not advocate teaching division with an end goal being a “standard algorithm”, if your standards require an algorithm be used, ensure that you are not rushing there and that many connections are made along the journey.

If students have already been pre-taught long-division, then it is more challenging for strategies like partial quotients to emerge more naturally. You will likely need to assist in making these connections.

Facilitator Note: What About Factoring The Divisor?

While unlikely students will go this route based on the context and the added complexity given the quantities we are dividing, it is worth noting that decomposing the dividend is not the only way students could go about dividing. As the facilitator, it is important that we understand that we cannot simply decompose the divisor like we did to the dividend in the previous example. 

If we ignore the context, we can leverage partitive division (instead of quotative division as this context calls for) to partition the dividend of 1260 passengers into 35 trips which reveals 36 passengers per part. While the resulting number of 36 is correct, the unit should be 36 trips (not passengers per part). Continuing with this approach, the partitioning of the dividend can be found more easily by factoring the divisor of 35 into 5 and 7. Then, 1260 can be partitioned by 5 and then by 7 (or visa versa) instead of decomposing the dividend as we shared above.

As mentioned, it is helpful for facilitators to understand that when we use partial products, decomposing the dividend lends itself to quotative division, while factoring the divisor lends itself to partitive division.