Introduction to Division

My focus is on working with students with special needs. Many struggle with rote memorization, including for multiplication facts. I find that skip counting, with scaffolded support in the learning process, provides them access to multiplication and therefore division. To access division, I use an approach of skip counting to find a missing factor and then connect this to division. This post provides details of a handout using this approach.

Overview

This handout focuses on connections to prior knowledge of skip counting and finding a missing factor. The students then make an explicit connection by rewriting division problems as missing factor problems. The handout is linked at the bottom.

Skip Counting

If students are struggling with multiplication, they are likely having trouble with skip counting. I start with a warm up on skip counting with the numbers that are easiest for students to skip count. Note: you can start with 2, 5, 10 only if necessary.

Factor Tree

I have students solve a missing factor problem using a provided skip counting row. Then they are shown that the problem can be rewritten as a division problem which has the missing factor as the answer. That is, division is another way to write a missing factor problem. You can use factor tree handouts and have students practice rewriting the problem as a division problem. Note: I see that most worksheets are used for prime factorization. Use the first two branches as shown in the image below.

Missing Factor

The students are then presented a math sentence only for missing factor. They are to solve for the missing factor. Then they rewrite the math sentence into a division sentence and solve again. I have a separate column to help emphasize that they are lookin to solve a division problem. They have to see the division problem in isolation and then write the quotient.

Division Problems

Finally, the students are presented division problems and rewrite as a missing factor problem. Their mental process can be as follows: “2 times what gives me 10?” and then they skip count by 2s until they reach 10. This can be supported with multiples rows as shown in the factor tree page. A blank page is provided. You can give students a division worksheet and have them copy the problems into the handout.

Access to Handout

Here is a link to the handout.

Partial Quotients Too?!

There is a difference of opinion on what is essential to teach in math. Partial Quotients algorithm (and the Standard Algorithm for division) are topics I believe are worth discussing regarding the need to master these beyond a single digit divisor.

In response to the post on the value of teaching Long Division by muli-digit divisor, there were several responses that cited the value of Partial Quotients. I like that approach much more but the same question still arises. Are students grasping the concept or do they not see the forest among the trees? What is the cost-benefit analysis for this? What do they gain, besides practicing skills?

Long Division: Should it be Long Gone?

I have seen many students struggle with long division (especially the Standard Algorithm but also the Partial Quotients Algorithm). Similarly, I have seen many teachers lament this lack of proficiency. I suggest that it is prudent to conduct a cost-benefit analysis for learning these algorithms for division beyond 1-digit divisors.

If students understand the concept of division and perhaps can do long division with 1 divisor, what is the purpose of teaching long division with a multiple digit divisor (or the partial quotients algorithm).

For a long time square roots were computed using a lengthy algorithm, similar in nature to long division. We don’t teach that any more.

Sharp came out with a scientific calculator in 1978. It had a square root button. No more algorithm.

Long Division – Scaffolded Handout

Many teachers shared that long division is one of the hardest math topics to teach. A major factor is likely related to a lack of mastery of multiplication facts. I have posted about some strategies related to multiplication. Here is a handout my initial attempt at scaffolding long division (WORD, PDF).

There are 4 versions: with and without arrows and with or without all squares. I am interested in feedback and would revise as needed. (Update, if you accessed the files before 8:30AM EST on Dec 4 you will see that I changed the wording in the left columns.)

I did not attempt to include using “R” to identify the remainder as my focus is on the steps.

The handout has a full page of each type. (partial pages shown).

Hack for Multiplication (and division) Facts

A common method to learn multiplication facts is through skip counting. In turn, this is a means of learning division facts (see next paragraph). The challenge for many students is they struggle to learn the skip count routines or cannot engage brute force memorization effectively (e.g., have a working memory deficit).

https://www.homeschoolcreations.net/skip-counting-charts-from-2-through-15-printable/

The challenge with multiplication by skip counting is keeping track of two sets of numbers while memorizing the order of the skip counting. That is another example of the rubbing belly and patting head phenomena in math where one extra task demand undermines the process.

A hack I use to scaffold this process to reduce the task demand during the learning process is to provide rows from a multiplication chart (below) for the facts of focus (3s and 4s in this example). The same approach can be used for division facts, e.g., in the image below right I have the student choose the row of the divisor (3) and then skip count to until reaching the dividend (12). The idea is the student has less task demands while learning the process and seeing the number pattern. This allows for more repetitions or rehearsal.

For students more severely impacted by a disability or who simply struggle with the patting head and rubbing belly of skip counting, the appropriate times table row can be provided for each problem to allow the student to circle (below). This allows for a hands on approach with even less task demand. You could also laminate the rows to make them reusable in lieu of several consumable ones requiring more paper. I like the consumable as I use that for data collection.

Life Skills Math – Not So Easy

As I wrote previously, shopping is dense with math tasks as are grocery stores. Here are some division situations that are sneaky challenging and require a student to know when and why to divide before even reaching for the calculator. I will use these to help illustrate the fact that life skills math is not simply counting money or using a calculator to add up prices. There is a great deal of problem solving and thinking skills that need to be developed.

For example, if a student has $60 to spend on gifts for her 3 teachers the student needs to understand that she can spend up to $20 per teacher (before even talking about taxes).

An entry point for division can involve a dividing situation the students intuitively understand, e.g., sharing food. Start with 2 friends sharing 8 Buffalo wings evenly (below).

This can lead into the 3 teachers sharing the $60 evenly (below). In turn, this can be followed by the online shopping shown above.

This approach can be used to develop an understanding of unit cost (cited in the shopping is dense post). Start with a pack of items to allow the students to see the cost for a single item before getting into unit cost by ounces, for example.

I have had success with teaching these division related concepts using sheer repetition as much of our learning is experiential learning. Using a Google Jamboard as shown in the photos allows for the repetition.

Basic Skills Older Students

A widespread problem at the secondary level is addressing basic skills deficiencies – gaps from elementary school. For example, I often encounter students in algebra 1 or even higher level math who cannot compute problems like 5÷2. Often the challenges arise from learned helplessness developed over time.

How do we address this in the time allotted to teach a full secondary level math course? We cannot devote class instruction time to teach division and decimals. If we simply allow calculator use we continue to reinforce the learned helplessness.

I offer a 2 part suggestion.

  1. Periodically use chunks of class time allocated for differentiation. I provide a manilla folder to each student (below left) with an individualized agenda (below right, which shows 3 s agendas with names redacted at the top). Students identified through assessment as having deficits in basic skills can be provided related instruction, as scheduled in their agenda. Other students can work on identified gaps in the current course or work on SAT problems or other enrichment type of activities.

  • Provide instruction on basic skills that is meaningful and is also provided in a timely fashion. For example, I had an algebra 2 student who had to compute 5÷2 in a problem and immediately reached for his calculator. I stopped him and presented the following on the board (below). In a 30 second conversation he quickly computed 4 ÷ 2 and then 1 ÷ 2. He appeared to understand the answer and this was largely because it was in a context he intuitively understood. This also provided him immediate feedback on how to address his deficit (likely partially a learned behavior). The initial instruction in a differentiation setting would be similar.
  • 2018-12-20 11.20.25

    Long Division for Students who Struggle

    Long division

    Making the concept of long division accessible:

    1. Start with simpler problem
    2. Use base ten blocks (and later with drawings as seen in the photo) and have students model each step of the long division with the base ten blocks.
    3. Do this for each step.
    4. Follow with more complicated problems

    See link  to document with full explanation.