Friday, March 14, 2014

Discovering Pi (π), 3.14159265359…

By Sharri Zachary, RME Mathematics Research Coordinator and RME Collaborator Patti Hebert, Garland ISD

As presented in the opening session of our RME conference, there are three key components that we, as educators, should maintain as we transition into the new math TEKS: (1) balance, where the emphasis is on students’ conceptual understanding and procedural knowledge (2) focus, where we centralize instruction around the “big” ideas, and (3) coherence, where the instruction is aligned within and across grade levels.

Consider this standard from the revised math TEKS for grade 7:
The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships.

The student is expected to: 
5(B) –describe π as the ratio of the circumference of a circle to its diameter

In honor of Pi Day, I would like to share an activity that you may want to consider for use with your students, Sidewalk Circles. At the end of this activity, the students should be able to explain that:
  1. The distance from the center to the edge of a circle is "1" /"2" the distance from one side of the circle to the other side of the circle through the center (OR the distance from one side of the circle to the other side of the circle through the center is about 2 times the distance from the center out to the edge of the circle).
  2. The distance all the way around the outside of the circle is about 3 times the distance from one side of the circle to the other side of the circle through the center (discovery of Pi).
You will need the following materials for each team:
  • One (1) center tool (we used small funnels with shoelaces strung through the end of the funnel to keep the shoelace from coming through)
  • Chalk
  • One (1) pre-marked ribbon piece (with indicated measures, 10 cm, 23 cm, and 36 cm) 
  • One (1) 2.5 m piece of string 
  • One (1) tape measure
    Take the class outside to an unused pavement area. (If raining, let students use large pieces of butcher paper to complete this activity.)

    Student teams directions:
    1.  Pick a center point and mark it with a clear mark so that you will know where it needs to be every time you are creating a circle.
    2. Stretch your pre-marked ribbon out tight and wrap it around the piece of chalk so that the first mark on your ribbon is at the edge of the chalk.

    3. HOLD THE RIBBON TIGHT as you move the chalk around the center drawing a circle on the sidewalk. Work as a team and do not let the center move.
    4. You must take 3 measurements for EACH circle. Use the 2.5 m string. Lay it out, then take it to a measuring tape to find the actual measurement: a) From the center to the edge of the circle b) From one side of the circle to the other side of the circle THROUGH the center c) Around the outside of the circle.
    5. Repeat this process for the other 2 tape marks on your ribbon.
    The general premise is that each group of students will create sidewalk circles using the pre-marked lengths of the ribbon piece (one each: 10 cm, 23 cm, 36cm), a center point, and chalk. They will use string and a tape measure to find the distance from the center to the edge of the circle (radius) and the distance around the entire circle (circumference). They will repeat these processes for all three measurements until they have drawn one circle for each measurement. The group should discuss their measurements and use reasoning skills to analyze the relationships among the measurements.

    Wednesday, March 12, 2014

    Developing Numerical Magnitude Understanding in Young Children

    By Dawn Woods, RME Elementary Mathematics Coordinator

    Research shows general math achievement is closely related to children’s understanding of numerical magnitudes, or the amount of a quantity (Gersten, Jordan, & Flojo, 2005; Siegler & Booth, 2004). Children develop the ability to quantify and order numbers through subitizing and counting (Clements and Sarama, 2009) and many children can answer the question “which is more, 5 or 3?” by five years of age. However, some children may be unable to tell which of two numbers is bigger or which number is closer to another number and may not have developed the “mental number line” representation of numbers (Gersten, Jordan, & Flojo, 2005; Griffin, Case, & Sigler, 1994; Clements and Sarama, 2009).

    This concept of numerical magnitude is a core component of number sense, which is widely viewed as crucial to success in mathematics (National Council of Teachers of Mathematics, 2006). Furthermore, existing data on the relationship between mathematical proficiency and understanding of magnitudes are consistent with the view that helping young children develop a better understanding of numerical magnitudes may lead to improved performance on mathematics tasks (Laski & Siegler, 2007).

    So with this research in mind, how can teachers and parents help young children develop a better understanding of numerical magnitudes? One way is to use a clothesline as number line in order to build understanding of numerical relationships (Suh, 2014). The list of activities below can help young children develop flexible thinking with numbers.

    • Encourage young children to equally space and hang number cards on the number line, using benchmark numbers such as 0, 5, and 10 as a guide. As the child masters this range of numbers, expand or change the range. Encourage the child, as he/she hangs the number cards, to reason and talk about mathematical ideas such as: 
      •  Is your number card closer to 0 or 5? How do you know? 
      •  Is your number card closer to 5 or 10? How do you know? 
      •  How far is 4 from 10? How do you know? 
    •  Support children’s reasoning about comparing and ordering numbers by having them justify solutions. For example, 
      •  Which number is bigger, 4 or 5? Why? 
      •  Why is 225 smaller than 250? 
    • Discussing placement of fractions and decimals highlights equivalency concepts. As children work with these number cards ask questions such as, 
      •  Which fraction is equivalent to 1/2? 
      •  Are 0.09 and 0.90 the same or different number? How do you know?
    Clements, D. & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routlege. 

    Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293 – 304. 

    Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24–49). Cambridge, MA: MIT Press. 

    Laski, E., & Siegler, R. (2007). Is 27 a big number? Correlational and casual connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development 78(6), 1723-1743. 

    National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Washington, DC: National Council of Teachers of Mathematics. 

    Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75, 428 – 444. Suh, J. (2014). 

    Line ‘em up! Teaching Children Mathematics, 20(5), 336.