Teachable Moments is a glimpse into the CPS classroom experience. Each month, we collect snapshots from around the city that provide insight into the quality of curriculum and the talent of the teaching staff found in all our schools. CAMBRIDGE, MA [04/02/09] A message from CPS Math Coordinator Mark Healy: Hello Cambridge! The spring months are an exciting time for the CPS Math Community! In March, schools across Cambridge celebrated two math days. The first day, Square Root Day, happened on March 3. The date, written fully, 3/3/09, can be thought of as 3 x 3 = 9, which means the square root of 9 is 3. The next time this will occur is April 4, 2016! The day allowed teachers to discuss with students across our district the idea of squaring and square roots. The second day, Pi Day, happened on March 14. This day is celebrated every year on March 14. As Pi has an approximate value of 3.141592, this is translated into 3/14; some people take this even further into celebrating Pi at 1:59 pm. Pi Day brought some exciting celebrations of the number thought to be discovered by Pythagoras in our schools. This April brings several math conferences held in Washington to improve the instruction of mathematics across the country. Several CPS math teachers will be attending these meetings, which include the National Council of Supervisors of Mathematics and the National Council of Teachers of Mathematics. Our community holds professional development as an extremely important value in meeting the needs of our students. Our teachers are excited for the opportunity to learn new strategies and bring them back to our classrooms! This coming May brings the MCAS, the Massachusetts Comprehensive Assessment Systems. This is a state-wide test that assesses student understanding of important math skills. While standardized testing isn’t the only measure of academic success, we are still excited to see how well our students will perform. Our math classrooms have experienced some exciting and innovative new approaches to learning concepts and we are ecstatic to be able to showcase them this month. Regards, Mark. Healy Mark Healy can be reached at mhealy@cpsd.us.
At Cambridge Rindge & Latin School, students want to know “How Do You Find The Space A Bundt Cake Takes Up?”
The class thinks about ways they could slice up this ring of deliciousness, and students notice that one cross-section of it looks like a parabola, the usual way one would slice cake to serve it... but if the knife slices the cake along a horizontal plane and then slices again about 1 cm below that, they get a big spongy shape like the metal washers that go around bolts. “I slide this washer of cake onto a plate for them to look at,” Mr. Patterson said. “It's easy to find out how much volume this washer takes up (we use the formula for the areas of circles and subtract the inner empty circle from the outer circle to get the actual cake washer volume). “ And now, students see that a bundt cake is just a stack of washers, circles with a hole in the center. If they add those layers of washers up, from the wide bottom one to the narrow top layer, they have the volume of a cake. “And that's the basic idea of calculus: break things up into small or thin pieces, find the volume of a small piece, then add up all the pieces to answer the big question,” Mr. Patterson said. “And then we eat the cake.”
Children brainstormed what five meant to them and then shared their idea along with a dance move. Some ideas were: going to Dad’s exercise club, moving to a new apartment, growing up, going to karate class, and running faster than ever!” Brooke wrote in her weekly newsletter to parents. Ms. Barrett and Donna Douglas, assistant teacher, read “It's Hard to Be Five” by Jamie Lee Curtis. Children lined up in groups of five, counted up to five and back down to zero, and made a snack of 5 crackers (2 of one and 3 of another). “We enjoyed celebrating FIVE in JK and will continue to reflect on this powerful number in growing up and in mathematics!” When you walk into this classroom you see students busily playing house, building with blocks, working in the art center and doing a great deal of math! This year the Jr. Kindergarten classrooms implemented a math curriculum called SRA Real Math Building Blocks for Pre-K. The activities are introduced in an engaging way by the teacher and reinforced with games, small group activities and a well-designed set of computer activities. Building Blocks supports our skillful Jr. Kindergarten teachers in infusing the school day with exciting mathematics. Fletcher Maynard Academy Junior Kindergarten Teacher Rosalind Aldrich O’Sullivan says she has a fun lesson that inspires children to love to count. Most junior kindergartners can say the words one, two, three, four etc. But is that counting or chanting? Chanting means the children know the right sounds in the right order. Counting is about quantity. There are many activities that take advantage of natural counting opportunities. One counting experience that my scholars enjoyed was to be explorers at home with their parents. This project, included below, not only gives counting experience to the students in a fun way but it encouraged parents to participate in their children’s learning.
The number of questions asked can be based upon your children. I do not change the wording of the questions because I want my student to be familiar with the words, “How many.” It is fun for the children to compare their answers in class. Fourth grade students at the Baldwin School are participating in a research initiative from MIT and TERC math. For four days, students had the opportunity to pilot the use of a wireless network of tablet computers in many different subject areas.
Student participation and focus has been exemplary! All students are enjoying having their work projected and discussed. “As teachers we are enjoying the instant feedback, accountability and ability to anonymously discuss answers. The software allows us to instantly give feedback to individual students directly on their screen. We are also finding that students are using the technology to ask for help in a low risk way by sending a message directly to the teacher's computer,” said teacher Michelle Frazier. “The use of this First grade teachers Amy Moylan and Maria Castro say “The Ultimate Math Club” is popular with their students at the Amigos School.
Parents are encouraged to submit ideas about how their child makes connections at home. Here is an example: Lucas was curious about his grandfather’s coin that he found at home. He wanted to know how old it is. He brought it to school and shared his interest in learning about how old the coin is. The coin had a date of 1936. Lucas said that his grandfather was 73 years old, so maybe that’s how old the coin was. We then asked Lucas to prove his theory using math skills. At the Cambridgeport School, teachers Gina Roughton, Jill Fousek, and Nancy Horowitz, use “LogoPaths” software to help students learn 2-D geometry.
Ms. Roughton, Technology Integration Specialist and Ms. Fousek, third grade math teacher, have been collaborating for weeks to provide students with multiple opportunities to practice, reinforce, and extend their understanding of geometry and measurement through the use of technology. Using LogoPaths’ “Free Explore,” program, students draw rectangles with given perimeters. Through this activity, they discover that opposite sides of rectangles are equal, that the first two sides of a rectangle always equal half the total perimeter, and that rectangles have four 90 degree angles. The third graders continue to explore 90 degree angles with “Get the Toys,” as they move the turtle through a maze by giving commands to move forward, backward, and turn 90 degrees. The future activity, “Feed the Turtle,” will require students to identify and make turns greater than or less than 90 degrees. Students say they look forward to each LogoPaths session in the Computer Lab. The program is also available on computers in the classroom so that students can continue to practice and apply their math understandings. At the Graham and Parks Alternative School, teachers and students find creative ways to use the Investigations curriculum. In Alexandra Seherr-Thoss' Kindergarten, the students worked on a unit on Shapes. In order to identify shapes in a context, Ms. Seherr-Thoss assembled several instruments, placed them in the middle of the rug, and had small groups of students come up to choose an instrument with a particular corresponding shape; circle, triangle, trapezoid, rectangle, sphere, and so on. Once each small group had chosen their appropriately shaped instrument, they performed a little band concert!
In Judith Richard's third grade class, the students were doing their unit on multiplication. They were asked to think of reasons why people multiply and then applied the ideas of combinations, rate, area, and scale to a few projects which needed multiplication. For example, students designed ice cream shops with choices of toppings, containers, and flavors. They made advertisements for their goods, with claims about how many different combinations of ice creams choices each store had to offer. In order to determine if their claims were truthful, the students had to multiply their various choices to see if they had truth in advertising. Next they set up stores, and had to use multiplication to see what kinds of foods they could purchase, depending on their rates of cost. In addition, the students designed area rugs for their stores, and had to use multiplication to determine the correct areas for their model stores.
Eileen Hanson's fifth graders were studying decimals and reviewing powers of ten, which seemed a prime opportunity to show the documentary film, The Powers of Ten, by Ray and Charles Eames. The film depicts the relative scale of the universe in factors of ten. It begins with the camera on a picnicking couple, and then zooms out at a rate of one power of ten per 10 seconds. The zoom-out continues until we have a view of 1024 , the size of the observable universe. The camera then zooms back, at a rate of a power of ten per 2 seconds, until we are at negative powers of ten, in the man's hand. We end at 10-16 meter, until we are looking at a proton in a carbon atom. The film offered a context with which students could have a different application and understanding of the powers of ten, as well as augment their work with decimal numbers. * Submitted by Toby Caplin, Grade 6 Teacher/Math Coach High Dive
A new twist on the meaning of "high stakes" mathematics? Indeed. But it's not just the life-or-death nature of the problem that makes it engaging. Students respond to the idea that they are learning about situations in which the math can be used. In this case, they need to understand angular speed and trigonometry in order to determine the position of the diver on the wheel as a function of time. They need to understand how to apply quadratics to develop expressions for the time it takes an object to fall a certain distance. They need to know about components of velocity so they can take the motion of the wheel into account. This year, at the end of the unit in late February, a remarkable thing happened. Motivated by pride in their work and an enthusiasm to show others what it's like to "learn math this way," my students decided to create an exhibit for the hall outside our classroom. They produced a poster that is 20 feet long to show the complex equation they had to solve for the unit problem. They used colors and highlighting methods, along with a legend, to illustrate the different concepts embedded in their equation; sort of a road map of their understanding. They created supporting displays and even made parts of the exhibit interactive, to encourage visitors to participate actively in the ideas. Soon after their display went up, a colleague, after admiring their work, invited the students to present to her nearby pre-calculus class. Following a day of "lesson planning" they visited the other class, presented the problem and the ideas they developed to approach a solution. They then brought the pre-calc students to their display in the hall, and gave a description of their big equation. My students asked questions they had prepared and invited questions from the visiting students. When they were done they received a spontaneous round of applause that gave me the kind of chill you get when you know you are witnessing an extraordinary moment. I observed their understanding deepen in each stage of this process. As they prepared their exhibit, and then planned their presentation, they were taking different perspectives and reframing their thinking. Their understandings grew more flexible and more confident. Much of this experience happened by design. The IMP curriculum encourages inquiry, engagement, and collaboration. But the students also brought something special to this learning experience. They aspired to share the story of their learning with the school community. With a rich and challenging problem like this one from IMP, and a willingness to let the students really express themselves, this turned out to be one of the most powerful teaching experiences I have had in a long time. Pi Day Celebration
Each year, on or about March 14th, classrooms across the city celebrate pi-day by doing a variety of activities stimulating student thinking and providing students with opportunities to think about mathematics flexibly. These activities support the learning of both NCTM and Massachusetts State Frameworks. These activities also include a literacy component and I am always struck by the number of middle school students who enjoy teachers reading aloud to them. Julia Lau’s grade 6, 7 and 8th grade classrooms were bubbling with excitement while engaged in mathematics and literature focused on pi! Students were enthusiastic as they challenged one another to remember the greatest number of digits in the irrational number “pi” often remembered and approximated as 3.14. One eighth grade student (Aayushmaan) was able to recite 154 digits of pi to win the prize of an “apple pi”. The six and seventh grade winners, Tasha and Brianna, recited 61 and 42 digits of pi respectively and both students also were awarded an apple Pi. Students shared strategies of how they were able to remember long strings of numbers including the chunking of numbers in strings of 3 and also associating the number with a word of letter. I could feel the support among students and teachers and the positive classroom culture as students cheered each other on to success. The lesson plan also included problems that required students to apply the algorithm to calculate the circumference and area of different circles. Ms. Lau used skillful questioning techniques to uncover the relationship among the variables. Students were allowed to use calculators to find the measures; however, Ms. Lau pushed student thinking to articulate the meanings of pi and the measures of circumference and area. Students had previously studied development of the formulas in the CMP2 unit “Covering and Surrounding.” The day also included a great Pi day “Rap” version of the song “Lose Yourself” by Eminem, all of the lyrics related to the digits of pi and the irrational classification of pi in the number system. Students were so excited and struck by the familiar tune and its application to mathematics. Students concluded by singing songs such “Happy Pi Day” and “Pi Day” to the Jingle Bells tune to wrap up the day. |
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Math teacher Joel Patterson places a bundt cake on the table in front of his students. “How much space (volume) does it fill?*”
What happens when every child in Brooke Barrett’s Jr. Kindergarten class at the 
The name of the club was voted on by the first graders. This club provides challenging activities that support the Investigations curriculum and first graders inquiry about how math relates to the real world. We are all excited about this club and hope that all children will take new challenges that are interesting to heighten their understanding of math concepts. 
Once a week for the last month, Room 104’s math class has been taking place in the Computer Lab where students use the LogoPaths software from the Investigations curriculum. Third grade students are commanding turtles to move forward, backward, and make right and left turns - all while using their understanding of properties of rectangles and transferring that understanding to the computer. 
Judy Lazrus and Anne Smith's combined 1-2 class are enjoying a unit on Giants in order to work on scale, measurement, multiples, patterns, and proportions; all the content of the curriculum. On her way to school, Ms. Lazrus found a giant footprint and needed her students to help determine the size of the creature who belonged to the footprint. In their journey, they measured a student and made their own class giant two times this girl's dimensions. The 8 foot giant is in their classroom now! The class offered their giant hand made presents, which were doubled in size, as well as made up the culture, chores, belief systems, and celebrations for their class giant, and read giant stories from all around the world. They also worked on scale by making their own drawings, and increasing and decreasing them proportionally 3 or even 4 times. Using giant insects, the class measured to see how much more, or how much less they are than a standard foot. They were able to graph their findings about giant animals, choosing the appropriate scale for their Y axis and answering questions using comparison words and measurement. Finally, using a scale of 1 foot = 1 inch, the students are making a mural of whales and large sea animals. If you look, you will see little G&P first and second graders scaled to swim along with their giant animal friends.
One of my favorite units in IMP (Interactive Mathematics Program) has always been the first unit in Year 4, called "High Dive." It involves calculating when (and thus from what position) a diver must be dropped from a moving Ferris wheel so that he will land in a tub of water. And if that's not tough enough, the tub of water also happens to be moving. Clearly, precision is the key to success in this situation. One can easily make the case that the diver's life depends on doing the math.