Division

We are beginning to explore division.

Today’s question (Feb 23):

The students picked out the important information:

Some students divided 128 into 4 groups using a visual representation of 4 groups (4 circles) and then placing an equal distribution of marks in each circle:

Some students used a similar idea, but counted by 5’s or 10’s:

Some students used a combination of repetitive subtraction (counting backwards by 4’s from 128, but when they got to 100 they new to divide that into 4 25’s):

Some students broke 128 into groups (100, 20, 8) and split each number into 4:

Some students used a traditional strategy:

Some students used mental math, realizing that by splitting a number into 1/2 and then into 1/2 again it would be split into quarters (or 4):

Consolidation: The most common strategy used was the strategy that broke the number into easily divisible groups, and then dividing each group by four. In this case it works out to be the place value columns.  Students remarked that it was similar to the Chunky Cheese strategy, only backwards, so they called it the Reverse Chunky Cheese strategy.

 

For homework the students were asked to answer these 4 questions using the Reverse Chunky Cheese strategy:

Today’s work (Feb 24):

The students made some excellent observations after trying their homework.  The problem, they discovered, was that you cannot always choose groupings by place value.

The students realized that the question we did yesterday was split up into its place value numbers, but only because the place value numbers were easily divided by 4:

It looked like this:

So we talked about how we could show the answer differently, so that we don’t focus on place values, but rather on groupings of numbers that easily divide into 4.  This is what we got:

Now it becomes more obvious that the students chose 100 because it divides easily into 4 groups (25 in each).  20 was chosen because it divides easily into 4 groups (5 in each).  And 8 divides into 4 groups (2 in each).

So then the students attacked another question or two.  We decided to rename the strategy, because reverse chunky cheese implies place value.  We call it the Rip Rope strategy because we rip apart tbe big number (into friendly chunks) and then slide those chunks down the ropes (arrows) into their groups (the answer).  Here’s what it looked like:

So the students discovered a math mis-conception and were able to work their through it.  And now we have our first division strategy, the RIP ROPE strategy!

Today’s question (Mar 3):

The students pulled out the important information:

And wrote the question as a math sentence:

Some students answered the question by drawing 24 groups and then evenly placing markers in each group:

Some students used a type of repetitive addition or subtraction:

Some students used the Rip Rope strategy:

Consolidation: Today we had a discussion about how adding or subtraction can be used to do division.  The students connected this to the Adding Attack strategy in multiplication.

It looks like this:

The students named it the Sub-dition strategy, because you can either use subtraction or addition.

Today’s question (Mar 4):

Activation:  Before working on today’s problem we activated our thinking by discussing the question: if the area of a rectangle is 20 (cm square) and one side is 5 cm, can we calculate the length of the other side.  This also lead to a brief connection between division and the Magic Squares strategy for multiplication.

Today’s problem:

The students pulled out the important information:

This is what the front of the class looks like when all the students have solved the problem:

Some of the students divided the number into groups:

Some students used the Sub-dition strategy:

Some students combined the Sub-dition strategy with the Avalanche strategy from multiplication:

Some students used the Rip Rope strategy:

Some students found a way to use the Magic Squares strategy (arrays or area):

Some students used a traditional algorithm:

 

Consolidation: We had a discussion about how area is multiplication, and we made connections between area, division, multiplication, and Magic Squares.  The students named the new strategy D-Squares (Division Squares).  It looks like this: 

If we break it down into steps, it looks like this:

Therefore the answer to 195 divided by 15 is 10 + 3, or 13.

Today’s Question (Mar 7):

To activate our brains, we reviewed the D-Square strategy.  Then we asked the question, “What happens if the number does not divide equally into the groups?”  This opened up a discussion about remainders and possibly fractions.

Today’s question:

The students pulled out the important information:

Some students used the Sub-Dition strategy:

Some students found a way to use the Sub-Dition strategy more efficiently (like the Avalanche strategy):

Some students used the Rip-Rope strategy:

Some students used the D-Square strategy:

Some students found a found a more efficient way to use the D-Square strategy:

Some students used the traditional algorithm:

Consolidation:  Today the students noticed some connections (especially some similarities) between the D-Square strategy and the Traditional algorithm.  We discussed the connection and displayed the two strategies side by side (the D-Square strategy on its side):

So then we looked at the less efficient D-Square strategy and tried to see how we could present it like the traditional algorithm:

Because the students connected this longer strategy to the multiplication strategies (long and short), we decided to call it the “Traditional Long” strategy.  It looks like this:

For homework the students were asked to solve the following questions using both the D-Square strategy and the Traditional Long strategy:

Today’s Question (Mar 8):

Marcus is planning a trip.  It was 904 km to his vacation destination.  He is going to drive.  But along the way he will need to make some pit stops.  He is going to make three stops by dividing the trip up into 4 equal sections.  How far will it be between stops?

The students solved the problem using the Rip-Rope strategy:

Some students solved the problem by doing mental math (similar to Rip-Rope):

Some students solved the problem using D-Squares:

 

Some students solved the problem using the Traditional Long strategy, and some the Traditional Short:

Consolidation: We focused on the short traditional strategy and connected it to the Long strategy.  We discussed similarities and differences.  We were able to display the core ideas by looking at both the long and the short strategy displayed as D-Squares:

Traditional Short:

For homework the students were asked to solve the following problems using the short or long traditional strategy:

Today’s problem (Mar 9):

Today’s problem was very challenging…

The challenge was to see if they could find the amount of cargo (kg) per boat (companies A, B, and C) and compare them to see which company could carry the most cargo per boat.  There was also a second part to the question, asking what would be the cost to move all (1020 kg) of the cargo, at $50 per trip,

There were basically two approaches to the problem in the class.  The first was to compare two companies to the third…

And the other was to find the cost per trip for all three and compare…

It was a fun question and the students showed excellent perseverance.

Today’s Question: we continued to focus on the idea of “cost per unit” today, to build on yesterday’s learning.

The students solved the problem by using the Sub-dition strategy.  The problem, the students noted was that little mistakes can happen when you have to subtract or add that many times…

Some students used the D-Squares strategy:

Some students were using the Rip-Rope strategy:

Some students used the Traditional Long strategy.  On one of the questions they made a mistake which opened a discussion around the question, “Is your answer reasonable?”

Some students used the traditional short strategy:

Consolidation: We had a discussion about how when trying to figure out the “cost per unit” of something, it often implies division.  But one group did not use division at all.  Instead of figuring out the cost per one unit, they tried to calculate the cost for 40 units, avoiding the use of division.  It was a creative idea, and opened a discussion around comparing.  When comparing costs, one must compare “like” amounts (i.e.: how much for 40 tickets from all the ski hills).

Today’s work (Mar 22):
Today we reviewed our division strategies.  We looked at the homework questions and found different ways of solving them.

Here are the homework questions:

And here are the different ways the students solved the questions:
Question #1

Question #2

Question #3

Question #4

Consolidation: We had a discussion comparing the different strategies, but also comparing the work of students who used the same strategy.  We focused on how students could use the same strategy, yet choose different numbers; for example in question #1 (85 divided by 5), using the rip rope strategy, the student (posted) used groups of 12 and 5 to get the answer 17.  But other students in the class who used the rip rope strategy used groups of 10, 5, and 2  (or 5, 5, 5, and 2; or 10 and 7).