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

Landmarks and Large Numbers - Dec. 7

We had a good meeting on Tuesday. We started with celebrations. Many people reported that their students are beginning to love math. While we were doing the unit on geometry, we found that many of our students who struggled with multiplication had much more success with geometry. They were able to share their strategies and become leaders in the math discussion. There was agreement that anticipating potential  problems and deciding what to do with them has been helpful. We will continue to do this at each session.

We then looked at the unit test and decided that we could do it in the traditional way, but no other. That helped set the goal for the day which was learning new ways to solve addition and subtraction problems.

We read and reviewed the mathematical goals in this unit.

In Investigation 1, we worked on place value and adding and subtracting multiples of 10, 100, and 100. Teachers made 1000 books and we talked about out strategies in naming the pages. We expect the students to experience some confusion with this because the numbers that start with 100, such as 134, are on the second page, also called the 200 page. The emphasis is on the students developing and refining strategies for finding numbers efficiently using benchmarks. We played the game Changing Places which adds and subtracts multiples of 10 and 100 from a number students choose. When we worked on  the session How Many Miles, we practiced using the number line ourselves. This is usually introduced and used in second and  third grade, but we are new to Investigations so the students are unfamiliar with it. We discussed the problems students have with thte US standard algorithm, and generally agreed that the open number line was a useful tool for the kids to have in their repetoire. We then planned Investigation 1 to include: math content, routine, focus question(s), questions to guide student learning, potential problems, and ongoing assesment.

Before we went on to Investigation 2, we spent time looking at the different types of addition strategies. Our goal is that students are able to solve a problem in two different ways using these strategies. These included: breaking the numbers apart, and changing the numbers. We practiced these strategies because we need to be proficient so that our kids may become proficient. We looked at the extended algorithm as a way of fostering understanding of the US Standard Algorithm. Many students want to use the US Algorithm even though they are continually wrong. We will promote use of the new strategies by requiring the students to solve problems  two ways. When students are consistently using the US Algorithm incorrectly, we will encourage them to use two other strategies until they understand what is happening in addition and subtraction. We read the Teacher Note: Addition Strategies. Teachers found this a helpful reference that they could go back to when the reached Investigation 2. We also practiced using starter problems in addition. This was an unfamiliar strategy for the teachers, but they  understood it quickly after we did a few problems.  We played Close to 1000, and looked at strategies to improve students in mental math compution. We discovered that starting from the hundreds place was more helpful than trying to do the algorithm in our heads. We also looked for combinations of nine and ten to further proficiency. I proposed that they get the directions to Close to 100 to do first. Close to 100 could be a good intervention for struggling students.We planned Investigation 2. Session 2.4 reintroduced the US Algorithm with the intent that students learn it using the values of the numbers rather than a procedure. However, we still want students to solve problems two ways.

We spent a rather short amount of time on Investigation 3. Many of the sessions were similar to those in Investigation 1, but used numbers in the thousands. We had a discussion about how to construct the 10,000s chart and where to put it in the room. It needs to be displayed as it is used in other lessons.

We then spent a significant amount of time looking at, dissecting and practicing strategies for subtraction. These included subtracting in parts, adding up and subtracting back, (often using the number line) changing the numbers and subtracting by place. In our experience, the US Algorithm for subtraction is very confusing for students. We talked about typical errors that students might make and what questions we could ask, and what strategies we could suggest instead so that students could understand what was happening in subtraction. We read Teacher Note: Subtraction Strategies.

Subtracting back

We looked at our Ten Minute Math piece. We are familiar with Broken Calculator and many of the teachers reported that it is improving student mathematical fluency. Practicing Place Value is a new routine. In it, students practice reading, writing and saying numbers. Then they add multiples of ten, one hundred and one thousand to a given number. We recognize that this routine prepares students for math that comes later in the unit.

We finished with a quick look at three types of subtraction problems:
Removal:
There were six ice cream bars in the refrigerator. My brother ate four. How many are left for me?
Comparison:
My snake is 16  inches long. Michael's snake is 10 inches long. How much longer is my snake?
Missing part:
There were six ice cream bars  in the refrigerator. My brother ate some. There are two left. How many did he eat?

The consensus was that the most valuable part of the day was learning and using the new addition and subtraction strategies. I hope you do well implementing in your class and please post any questions to the blog.

Thank you for your participation on Tuesday and I look forward to our next meeting.

Why Are We Changing the Way We Teach Math

http://www.nytimes.com/2010/12/07/education/07education.html?_r=1&src=me&ref=homepage


This article comes from the New York Times and is a good argument for the need to change the way we teach mathematics.

Electric Shock?

http://bodyodd.msnbc.msn.com/_news/2010/11/04/5404646-want-to-be-a-math-whiz-try-a-touch-of-electric-shock


I thought you might enjoy this article, but remember, do not try this at home, or at school.

What Classroom Math Posters Look Like

 I love the notation on the one on the left: 23 x 9 = (20 x9) + (3 = 9).

I have attached some multiplication posters where students were examing mathematical strategies for different sized numbers. Students did the work on regular paper and the teacher had them explain the process as she wrote what they did. She did this for two reasons: 1) other students need to be in the habit of listening and learning from others and 2) posters written by the teacher can be see more easily than those written by students.