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