Division

We are beginning our unit on division.  To begin with the students were asked to compare what they think multiplication means and what they think division means:

Then they were asked to look at how to do simple division using a multiplication chart.  Then they were asked how many ways they could show 30 divided by 6:

Sample ideas:

Today’s question (Feb 28):

To activate their brains the students were asked, and came up with different answers:

Today’s question with the Need to Know information pulled out:

Some students used drawings to help answer the question (128 dots put into 4 circles – 32 in each):

Some students used trial and error, starting with 4 groups of 30, then trying four groups of 33 and then trying 4 groups of 32:

Some students put friendly amounts into 4 groups, first 20 into each group (=80) then 10 into each group (=40 total of 120) and then 2 into each group (=8 total 128), showing 32 in each group:

Some students broke up 128 into 100 and 28 and divided each number into 4:

Some groups split chunks of amounts into 4 groups until they reached 128:

Some groups divided 128 into 2 twice (clever!):

Some groups used traditional division:

Consolidation: We focused on the solution that split chunks of amounts into 4 groups.  First 25 was put into each group (total of 100), then 5 into each group (total of 20) and then 2 into each group (total of 8):

Although it looks like the number was split into it’s expanded form, it actually is not, large friendly chunks are simply subtracted from the total.  The students named this strategy “SPLIT DECISION”

What a large number of great strategies that were used on today’s problem.  Great work!

Today’s problem (Feb 29):

To activate our brains we asked and answered, how many ways the students could represent 20 divided by 4:

Today’s problem:

Some students divided 144 into 24 groups:

Some students made and added baskets of 24 baseballs (in groups of 2) until they got to 144:

Some students repetitively subtracted 24 from 144:

Some students repetitively added 24 until they reached 144:

Some students used the traditional strategy to find the answer:

Consolidation: We looked at, and discussed, the idea that dividing 144 by 24 means “how many groups of 24 can 144 be broken up into”.  This can be solved by starting at 0 (or starting at 144) and adding (or subtracting) groups of 24 until you get to 144 (or to 0), and then counting how many times you added (or subtracted) 24 to find your answer.  It looks like this:

We chose to call this strategy:  THE LEANING TOWER OF NUMBERS.

Today’s problem (Mar 2):

To activate our brains we talked about area.  If multiplication can be seen as area, can division be seen as area?   We looked at 20 divided by 5 as a magic square:

Today’s question:

Some students solved the problem using groups:

Some students used the Leaning Tower of Numbers strategy:

Some students used a more efficient Leaning Tower strategy by doubling up numbers:

Some students used the Split Decision strategy:

Some students used a traditional approach:

Some students were able to break up the division question into parts (similar to the magic squares):

Consolidation:  We looked at connection the above solution (and the Spit Decision strategy) to the Magic Square format for solving the problem.  We know that the area has to be 195.  We know that the width is 15.  So if we start with by making the length 10 (a friendly number), then we have an area of 150.  That leaves 45 (195 – 150 = 45).  Then we add 3 to the length which adds 45 to the area (for a total of 195).  Therefore the length is 13.  195 (area) divided by 15 (width) = 13 (length). 

Today’s problem (Mar 5):

To activate our brains by using different strategies to solve this problem.  The new component added was that there was not an exact answer, there were left-overs, which we call remainders.

Today’s problem:

Some students answered using the chocolate chip cookie strategy:

Some students used the Leaning Tower strategy (by adding 17 repetitively), although some students found more efficient ways of doing the Leaning Tower strategy by doubling or even tripling the divisor (17):

Some students used the Spit Decision strategy:

Some students were able to use multiplication to solve the problem:

Some students used the traditional approach:

Consolidation:  We focused on the group that used multiplication to solve the problem and connected it with the Traditional Long strategy.  Below is the traditional strategy done by two different students.  It demonstrates that the traditional long strategy will look different depending on the student.  In the first example the students is trying to divide 387 into groups of 17.  The student makes 10 groups of 17 which equals 170.  The student then subtracts the 170 from the original amount.  They then make 10 more groups of 17 (subtracting another 170).  Then they make 2 more groups of 17 leaving only 13 left over.

Today’s problem (Mar 8):

We activated our brains with a review of some of the division strategies we have used so far.  Then we gave today’s problem.  They were reminded not to fall into the trap of thinking of all our problems as “division” problems, but rather to just think of them as a problem.

Most groups used a variety of strategies that followed the pattern of multiplying 36 (students/class) x 8 (classes) to find out how many students there are (288 students).  Each student wrote a letter, so there were 288 letters, divided equally amongst 4 schools, giving each school 72 letters. 

However one group was really thinking!  They realized that if there are 8 classes of letters, being sent to 4 schools, then each school will get letters from 2 classes.  Therefore the answer is 36 + 36 = 72.  Brilliant.

Consolidation:  We looked at the Traditional Long strategy being done more and more efficiently, that at the end it looks like the traditional short strategy (where you divide the tens, subtract, then divide the ones).

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