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.

Fraction Cards and Decimal Squares

We met on January 25th and started with celebrations. Everyone agreed that students are more enthusiastic about math. Mathematical fluency has improved with the consistent use of ten minute math.
We congratulated Michelle on her high Galileo scores.

We looked at the goals of the unit and made posters of the key ideas.

We then looked at the Unit test and recognized that we would solve the problems by using the traditional algorithm. It came as a bit of a shock to realize that we would teach it an entirely different way that focused on the meaning of the fraction. We discussed the importance of giving kids lots of opportunities to think about "How much of the whole does this cover?". We read 'Why Are Fractions Difficult' and 'Developing Meaning for Fractions'.

We did the lessons in Investigation 1 and used 4 x 6 arrays to find meaning for fourths, eighths, thirds and sixths. We discussed relationships among fractions and how students could use that to add fractions on a grid. We found fractions of groups of things and halves of different wholes. This reminded us that we could use doubles and halves in fractions as we did in multiplication. It was interesting to see that we could add fractions using relationships on the grid. When we worked on combinations that equaled one using the grids, it was sometimes confusing. We recognized that we need to practice this skill if we are going to teach it effectively. We recognize that there are many opportunities in this Investigation for students to recognize equivalent fractions and fractions that are equal to or more than one whole.

 We used 5 x 12 grids to find fractional parts of a different whole. When using these grids, there are many opportunities for students to explain and justify.

 The fractions on the poster below are exampled of equations that students could make and justify using the visualization on the grid.

In Investigation 2, the initial task is making fraction cards. Students do this in groups of four.  Students are making a variety of cards, including fractions that are less than one, equal to one and more than one. It is critical that the representations that the students make on their fraction cards are correct, because they will be used in other lessons and in math workshop. One of the important discussion during this time is comparing two fractions with a missing piece.

We want students to recognize the value of the missing piece in relation to one whole. In the photo above the first fraction is 1/3 away from a whole. The second fractions is 1/6 away from the whole. Since 1/6 is smaller than 1/3, the second fraction is larger than the first. Students need to understand this concept in order to play Capture Fractions where the largest fraction wins. Students spend a lot of time identifying equivalent fractions and comparing fractions to the landmarks of 0, 1/2, 1 and 2. There is frequent use of number lines to order fractions and justify the order through reasoning about fraction equivalencies and relationships.

In Investigation 3,  we work on representing and comparing decimals. We use a variety of grids that show one whole, tenths, hundredths and thousandths so that students can visualize the value of decimals in relation to one whole. We discussed that students who read a decimal as point 7 rather that seven tenths will have enormous difficulties with representation. Students MUST read decimals correctly.  We played Fill Two, a game that reinforces decimal representation.

These types of opportunities to represent decimals will prevent students from the typical mistake of thinking that a decimal with more numbers is larger than a decimal with fewer numbers.

There are two types of Ten Minute Math in this Unit: Practicing Place Value which includes reading, writing and saying numbers up to 10,000, adding and subtracting multiples of ten to larger numbers, reading and writing decimals, and adding and subtracting tenths and hundredths to decimals; and Quick Survey which includes describing the features of days and interpreting and posing questions about the data.

This unit was a significant departure from what we learned as children - to memorize a procedure. It was sometimes difficult for us to represent and name parts of a whole without relying on the old memorized procedure. This unit is a good example of one of the underlying goal of Investigations: Teachers are engaged in ongoing learning about mathematics.

Response to a question

Richard asked me what program Massachusetts uses. Each city decides for itself - but the Boston School District uses Investigations and many of the trainings and workshops are held in Massachusetts. Massachusetts is one of the high scoring states, and Arizona is perpetually near or at the bottom.
Marge