How Many Packages? How Many Groups?

We met again last Tuesday and our group is growing. We welcome our new members and hope the day was a profitable one for them. We started with celebrations. Someone reminded me that I had said things would really start to work well after Christmas. And they are!! Many of us are seeing developing fluency because of the ten-minute math - Practicing Place Value. Kids are very familiar with place value and can use it to their advantage. Kids are naming strategies and understanding more about number relationships.

I discussed the reasons for skipping Book 7 at this time. It covers none of the AZ Math Standards and the lessons on volume would confuse the students when it came to nets.

Next we took a detour. I had copied three lessons from the first edition of fourth grade Investigations. There are three lessons that introduce students to the four quadrant grid and graphing points on it. I encouraged the teachers to fit these lessons in wherever they can. I will continue to look for good lessons to cover the rest of the geometry standards.

We started our work by charting out the goals for Unit 6: How Many Packages? How Many Groups?

Then we practiced estimating in multiplication and division. We recognize that there are many more ways to estimate than just rounding. These include: front end estimation, clustering around an average, rounding and adjusting, using friendly numbers, and using benchmarks. We will ask students their estimate and also ask them to gauge whether their estimate will be higher or lower.

We spent a great deal of time practicing breaking numbers apart to multiply. We know that it is important to keep track of all the parts of the problem. Putting the problem in context makes it easier for students to understand.

We made the connection to quadratic equations and the distributive property.

We looked at the types of student responses we might see - from tally marks to elegant equations. We will make an effort to have students who make drawings attach equations to those drawings and eventually leave the drawings behind.

We worked on cluster problems which were designed to lead us to strategies like doubling and halving to solve break apart problems. Many times, we needed to make an array to make sure that we had included all the factors in our solutions. The array is no where near as foreign to us as it was in July  when many of us encountered it for the first time.

We worked on solving a problem by making an easier problem. When we rounded up one factor in the multiplication problem, we had to struggle at first to figure out what we should subtract. Again, we looked at it using arrays and smaller numbers to help us think through what to subtract. I reminded everyone that one of the stated goals of Investigations is that teachers are engaged in ongoing learning about mathematics content, pedagogy and student learning.

We also spent a good deal of time on division and using multiplication with ten to start solving division problems. Again, it was new for most people, so we practiced.  We expect that many students will start division by 'dealing out'. We discussed methods of sharing strategies and moving kids from tally marks to using multiplication to solve division.

The day went by quickly and there were lots of new ideas to take in. We know that we need to practice so that we will become fluent in all the strategies. Our students are not the only ones who are getting better at math.