Repeated addition as it relates to multiplication.

Part 3

I continued the next day with a lesson on repeated addition.  I posed this question to the students:

Making groups of three, what addition sentence could you write for 12 cubes?

This part of the lesson is the YOU part in which each student has to find the solution on his/her own.  Of course as the teacher, if I did see a student not even attempting I tried to go over and encourage him/her and have the student think of some possible approaches but I never did any of the problem solving for the child.  The learning is in the struggle!   Each student received a bag of cubes and a blank sheet of paper which they divided into fourths to record their solutions.  We reread the question and I made sure they understood what was being asked.  Then I let them start while I walked the room looking for students who had recorded their solutions.  When I saw the first student who had drawn and recorded 3 + 3 + 3 + 3 = 12, I had that student go up and record that on the white board.

The student explained to the class what they did and then I proceeded to give the next instruction. 

What other ways can you add equal groups to get 12? 

Find some more ways I instructed them.  Again, I went around and noticed how the students were quickly grouping by 4s, 6s, and 2s.  I had some come up and record their solutions.  Those same students explained to the class their addition sentences. 

Students come to the board to record their solutions.
Now came the YOU ALL part.  The students now worked with a table partner to combine their 12 cubes to make 24.  The students now had to work together to see how many addition sentences they could come up with for 24 cubes.  Students were very motivated to get the first solution because they really wanted to share the solution on the board.  But at the same time I was amazed at the rich discussions that were going on about how to proceed, which groups would they use, and of course the skip counting.  They easily constructed repeated addition sentences for the different ways to make 24.  I had partners come up and record their solutions and explain how they did it. 

Next, is the WE part.  In this part, I take over more and now show the students how this relates to multiplication.  I asked: in this addition sentence (3 + 3  + 3 + 3 = 12), how many “times” did we write 3?  Of course they responded 4 times.  So I asked:  does that mean 4 “times” 3 is 12?  I was practically blinded by all the light bulbs going off over all the students heads!  They could now see the connection of “times” to repeated addition.  From there, I modeled how to turn another repeated addition sentence into a multiplication sentence.  At this point I also introduced the vocabulary of factors and products.  They then had more practice by turning all the repeated addition sentences that they had recorded on their papers into multiplication sentences.  Once that was done, it was time to open the Go Math book and do the independent work in the book (which matched perfectly what we had just learned).  
 I’m just amazed at how easier and more productive this “reverse gradual release of responsibility” model is working for my students.  They are more engaged, eager to try and find solutions, better at explaining their work and are far less frustrated than the beginning of the year when I relied solely on what Go Math provided. 

That is the end of the series on implementing the Japanese style of teaching math.  By the way, if you care to Google it, it is called Bansho in Japanese.
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