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