Division, Order of Operations and an Ugly Fraction Model! Math Coaching at EMES and Beyond

Well... A snow day on Monday, in-service on Friday and Fitness on Thursday gave us only two days to focus on math coaching this week at East Montpelier.  So, we did the best we could with the time we had.

Goals

Fifth grade teachers, Robin and Ellen S. were looking to work on teaching long division using the area of a rectangle model so that they can move onto Divisibility, Least Common Multiple and Greatest Common Factor.  Our math coaching work this fall focused on models for multiplication and using the area of a rectangle to model the partial product method and the standard algorithm for multiplication.


We are looking to develop students who:

• Can understand division as: repeated subtraction, equal shares, and the area of a rectangle model.
• Can divide multi-digit whole numbers using the standard algorithm.
• Can explain the standard algorithm using place value and the distributive property of multiplication terminology and through the area of a rectangle model.
• Can contextualize and decontextualize division problems to apply the concept to both routine and non-routine real world problems.

Sixth grade teachers, Ellen K. and Gary had been working on area and volume, but were looking for help teaching order of operations starting from a concrete model.

We are looking to develop students who:

• Can understand how to model numerical expressions involving multiple operations. (This is essential in developing algebraic thinking.)
• Can simplify expressions involving multiple operations efficiently.
• Can explain why order of operations matters and why the hierarchy is what it is using geometric models.
• Can contextualize and decontextualize problems involving multiple operations (and grouping) to apply the concept to both routine and non-routine real world problems.

Tuesday and Wednesday in the Fifth Grade

We combined the 5th grade classes into one (very large) class both days.  It was a large group (around 40 children), but the students were on their best behavior.  We start out by handing out index cards and having students write "57 divided by 8" and explaining what it could mean and to write a context for the expression if they could.  We saw a range of responses, with most students recognizing the connection between division and multiplication and realizing that "8 didn't go into 57."

Then, we did a counting exercise.  We handed each student a Tool Building Grid.
After clarifying the language: horizontal, vertical, row and column.  I instructed students to write 57 (their "starting number") in row 1, column 4.  Our rule was to "add 8."

We counted up by 8 as a whole group until we had filled in the first five cells of the third column (including 57).

Then we put our next number (97) at the top cell of the fourth column.

I had students fill in the rest of this column independently.  We focused on strategy here: efficient strategies vs. inefficient strategies.  We saw that some students were "counting up by ones" and emphasized that counting up by 10 and subtracting two was a more efficient additive strategy (requiring fewer steps).

Next, I asked students to find the number that would go in the cell that was in row 4, column 7.  I asked them to try to find the most efficient strategy by looking for patterns.

Asking the question, "what do you think the ones digit will be?" got students to recognize that the ones digits across a row were all the same.  Students began to realize that going from one column to the next involved adding 40.

They used this pattern to find the missing number efficiently.

When they saw this pattern, I asked them to think about why.  I called on students randomly, with each student adding to our understanding of the pattern.

Eventually, they were able to tell me that to go from one column to the next we are repeating the process of adding 8 five times.

Then we looked at the cell that was in the fourth row, second column.

Students were able to see that rather than adding 40 twice, they would need to take away 40 twice.

We were able to summarize that starting from 57, if we take away 7 groups of 8, we get to one.   This is the repeated subtraction model for division.  We end up with a remainder of 1.

Next, we followed the area of a rectangle model for division taking students from the concrete model for division to the standard algorithm using this instructional plan:

• Area of A Rectangle Division Instructional Plan. 

At the end of day two, we were disappointed that we didn't have a third day, but Ellen S. and Robin were able to have an idea of what kinds of practice their students still needed.  I left them with these resources:

• Division Area Model Marathon (Practice Activity) (the teacher fills in appropriate practice problems based on formative assessment.
• Division Questioning Cards - so that students can self prompt and get really natural with the language and questioning.
• Division Formative Assessment

Tuesday and Wednesday in the Sixth Grade

We combined Ellen K. and Gary's 6th grade students into one class (which we did twice this fall, too).  We started out with a tiered formative assessment activity.

Students chose an expression to start with.  Most picked purple.  We saw that there was a lot (nearly universal) confusion.  So we knew we had our work cut out for us.

We followed this instructional plan (which will eventually link order of operations back to the two and three dimensional geometry they were studying last week):

• Order of Operations Instructional Plan
• Tiered Order of Operations Formative Assessment

Question on Dividing Fractions

I did get a question on dividing fractions this week so I made this video:

Warning: Don't start with this example!  I have an instructional plan for dividing fractions in my head that I will be trying to find the time to document in the next week.  In the meantime, I have this graphic organizer: Dividing Fractions Graphic Organizer.  It walks you through the order in which I would have students work on creating the concrete models.

You will want to graduate to smaller pieces of paper to represent "square 1" in the name of the Lorax who speaks for the trees.

More to come!  As always, thank you for sending me the question.

Feedback

Please send me your comments, questions and suggestions either by commenting below or by emailing me at edorsey@u32.org.  Thanks for reading!

Next week

I'll be working with U-32 all week on models for operations on fractions.  Hopefully, we will breeze through multiplication and division of fractions so that the seventh grade teachers can get to Ratios and Proportions quickly.  With all of the hard work that our fifth and sixth grade teachers are doing with the area of a rectangle model this year, perhaps next year, we won't need to include this in our Tier 1 instruction.