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.

Still Struggling with Breaking Apart Multiplication?

I had a comment from Richard that some of his kids are having  trouble with the addition and subtraction involved in this process of multiplication and division. Here is something you might try. One of the teachers at my school gave kids a set of array cards with an erasable marker. The used the marker as ' a pair of scissors' to make imaginary cuts in the array. They took one of the larger array cards and drew a line dividing it into two parts.  Then they recorded the equations that would describe the two parts. They erased the line, put the card back in the pile and pulled another card to do it again. This seemed to help kids a lot with notation. Some kids worked slowly and some went very quickly and did many more, but they all understood how to break an array apart. 

When students have a difficult problem, the teacher can help them by drawing the array on the board and thinking with them about ways to make an imaginary cut. Only do this for the kids who are struggling. Give those who are successful a set of multiplication problems with larger numbers and have them solve with break apart and represent as an array.

We discussed giving students homework that was an equation - something like 19 x 6.

Ask them to write a story problem, solve it in two ways and draw the array. For children who are struggling, the problem might be 12  x 6, or 13 x 6. For those that are working quickly, you might chose something like 28 x 9, 0r even 128 x 9.

Hope this helps.

I am adding some pictures of student work, some from daily lessons and some from morning work. 

Division

I have heard it through the grapevine that some of you are having some problems with division. I looked at the problems for division and most of them are accessible by using multiplication to solve them. Some, however, have smaller starting points. If you look at page 66, the problem is 64 divided by 4. If a child knows that ten groups of six make forty, he or she has gotten more than halfway through the problem. You will need to model notation that looks something like this:
10 x 4 = 40
Teacher questions: Do you think that there are more than ten fours in 64? How many more?
Some student will give you a number. If they say something like two groups of four, you write:
2 x 4 = 8.
Teacher: how many groups of four have we found? (12) How much do we have? (48)
Are there more fours hiding in this number?
Kids may choose to go by twos or give you larger numbers. If they give a number that is too large, take it and work through it, asking the students if they are satisfied with the total. Kids need to try things that don't work, and make decisions about the results.
Kids need to use starting points that they know. Sometimes tens will be too large. If they suggest ten and it is too big, and they recognize that, ask if half of ten would work. Use doubles and halves if you can, to add up to a number or to make your strategy more efficient. Here are some possible ways students may do this for 64 divided by four.
5 x 4 = 20
5 x 4 = 20
5 x 4 = 20
1 x 4 = 4

10 x 4 = 40
5 x 4 = 20
1 x 4 = 4

10 x 4 = 20
6 x 4 = 24



Kids need to use their number sense to start to think about how many fours are hiding inside the 64.
Here is an example of a fourth grader who is still counting by ones to solve problems.

Here are some examples of students with appropriate solutions:

I photographed these from students math notebooks. Some are from morning work and some are from daily lessons.

Size, Shape and Symmetry Start Oct. 28

Good luck today with the start of Book 4. Remember that we talked about the  importance of estimating in metric and in standard measurement. Benchmarks are the key to estimation so help your kids find benchmarks that are familiar and useful. Our discussions in the first unit focus a lot on reasonableness and kids need a lot of practice with that idea. They want their estimates to be exact, so remember to discuss the idea of 'close to' and have the students decide if certain estimates are 'close to' the actual measurement. We are stepping back from being the arbiter of right and wrong and giving the students opportunities to decide and justify what is right.
It is a good idea to use the Missing Measures page from the student book because the questions on it are typical of logical thinking questions that you might see on a standardized test. I hope you have a good week with measurement and I will write about Unit 2 later on.

Our Meeting Thursday

Hello all,

Thank you for a good day. I hope you went home with your  head full of new things. Please email me with any questions. Yesterday we were very busy. We looked at some student work on multiplication from the first unit in Book 3: Multiple Towers. Most of the kids are getting good at using arrays to break apart their two-digit multiplication. We talked about strategy in quick image, using doubles and halves to prove equality and get kids to understand the equal sign – not thinking that it means, ‘and the answer is’.

We looked at the Unit test for Size, Shape and Symmetry and found it rather daunting, so we put it aside to do later in the day.

We read and charted The Mathematics in this Unit. We noticed that some of the things taught here are things we learned in high school. Our focus is on measuring with standard units, describing and classifying two-dimensional shapes, describing and measuring angles by relating them to 90 degrees, and finding and understanding area.

Unit 1 focused on measurement which was familiar to everyone. The idea of using centimeters and meters at the same time was new to most of  us, but we recognize that it gives the students an opportunity to compare measures.  We discussed using estimation in measurement and focusing on the phrase ‘close to’. Many kids struggle with estimation and need more practice. We emphasized thinking about benchmarks as a method for estimating.  The discussion in 1.1 at the end of the lesson is about reasonableness and kids need the opportunity to discuss if they come up with different estimates because they have lived in a ‘right answer’ climate previously.

We did Lesson 2.1 and classified polygons by attribute by constructing the Polygons, Not Polygons chart and playing Guess my Rule with the polygons. In lesson 2.2 and 2.3, we made polygons with different numbers of sides and named them by the attribute of size and also looked at concave and convex polygons. We played Guess My Rule with Shape Cards and discussed the attributes by which students may group them. In Lesson 2.4, we sorted quadrilaterals, and made decisions about the characteristics of all quadrilaterals and some quadrilaterals. We also discussed a common confusion about squares and rectangles. Some students erroneously believe that rectangles must have two short sides and two long sides rather that opposite sides equal.  Teachers planned to address this issue at this time in the unit.

In Unit three, we worked extensively with Power Polygons which were new to many of us. We discussed having students use the corner of a paper as their base 90 degree angle because it is difficult to pile the polygons on top of each other. The easiest way to demonstrate is with an overhead, but many schools have gotten rid of those.  Once students have had time to work, maybe a museum walk would be a way to demonstrate and see what others have done. We then used the 90 degree angle to find the value of other power polygons. We practiced building angles and discussed the importance of teacher using the vocabulary frequently and the importance of the discussion at the beginning and end of the task. At the beginning, we set the task, and at the end we cement the learning by having students discuss  the solutions. We anticipate that some students may have difficulty adding and halving the numbers of some angles and we should be ready to ask questions like, ‘Would breaking the number apart help?”.

In Investigation four, we continued to use power polygons to look at symmetry and measure area with triangles and trapezoids. We wanted to look at half the area and then double it, so we see the mathematics in previous books being used here. We spent a considerable amount of time with the geoboards. Some teachers commented that they had them but never knew what to do with them. We used them to decompose shape and find the area in square units. Students will often answer by giving you a single digit as the area. It is important to have the student explain how they figured it out and that the area is ‘six square units’. The idea of framing off triangular portions of irregular shapes and seeing that they are half of a rectangle was surprising to many. We discussed that we had actually derived the formula for area of a triangle.

With ten minute math, we discussed Broken Calculator. Some of the brighter students in several classes seem to struggle with this and want to give up. I talked about Eric Ericson’s theory that children develop industry between six and twelve years old. Those bright and/or gifted students often have not had to struggle with learning, and when they are faced with something difficult, they just give up. It is important to scaffold their learning so they can break down tasks that they would normally give up on.

We spent time on Quick Images. QI changes this month to geometric figures and it is easy to just see if the kids do it correctly. Then there Is no math. QI must be used as an opportunity to teach mathematical language by asking students to describe the shape so that someone who has not seen it can draw it. Yesterday, I asked the teachers to describe the quick image so I could draw it. Initially, their description were imprecise and I made the drawing incorrectly. When they started saying, “Each vertex of the rhombus touches the midpoint of the sides of the rectangle”, the drawing would be correct. Of course, we know some of our students do not have the language that the teachers have, so they may say “It touched the middle of the line”. We would question, “so something touches the rectangle at the midpoint of this line”, giving other kids the opportunity to refer to the midpoint of the other lines. When kids refer to the point of the diamond, the teacher may ask, “Oh, do you mean the vertex of the rhombus?” We discussed that the teacher must use the vocabulary over and over again, and encourage students to use it too, So when the teacher asks, ”Do you mean….” The student must not be allowed just to say ’yes’, but must say “Yes, I mean the midpoint of the sides of the rectangle”.  We talked about the importance of looking carefully at the quick image before we use it,  to decide for ourselves what mathematical language can be drawn from it.

We ended by revisiting the final assessment and teachers felt confident about the content of the test.

I had no complaints and teachers seemed to be happy with what they had learned. Most had never used Power Polygons or Geoboards. At the end of the day, when each teacher reported about what was the most useful thing they learned, those were most mentioned.  One teacher asked about starting a blog and I am considering it.

I enjoy working with this group and look forward to our next meeting on Dec. 2.  Remember, you are welcome to email me with any questions or concerns.

 Marge