On this page you will find some brave and hard-working Crane students working on math problems. In the videos, you will have an opportunity to see the more complex problems students are asked to solve today, as well as the ways they can process and attack a problem.
You will see a third grader, a fourth grader, and a fifth grader each working on a fraction problem. The three videos are also meant to give you a sense of the progression of fractions in the Common Core State Standards for Mathematics, and the expectations for student learning in one area of fractions over these three grades.
Grade 3: Paul Works on Showing Equivalence of Fractions Using Visual Models
In third grade, students begin their conceptual understanding of fractions, including the meaning of the numerator and denominator, comparisons of fractions, and the idea of partitioning a whole into equal parts (wholes may include shapes or line segments). The subgroup of third grade Common Core State Standards representative of these goals states "Develop understanding of fractions as numbers."
In the video, Paul works on a problem in which he is given two fractions and asked to compare them to decide whether they are equivalent. In order to show his thinking and prove equivalence, I've asked Paul to compare these fractions through the use of visual models (the rectangles and the number lines).
As Paul works on this problem, he is being asked to incorporate a great deal of what I would term "cognitive work." In other words, there is a lot of important mathematical thinking that connects to bigger math concepts, which Paul uses as he draws visual models to determine equivalence. Some of Paul's cognitive work includes:
- partitioning wholes (such as shapes or line segments) into equal parts.
- focusing on the pieces when partitioning line segments, as opposed to the hash marks that separate the pieces.
- representing fractions of wholes with appropriate shading or labeling.
- incorporating the understanding of possible relationships among fractions (such as fourths and eighths) while drawing, comparing, or considering relative size of a fraction.
- being aware that depending on the denominator, it may not be possible to find an equivalent fraction to one half, and why that is the case; then using that knowledge to draw, compare, or reason about the fraction's relative size.
After completing his visual representations and working through this math task, Paul has developed a deeper understanding of whether the two fractions were equivalent. In addition, he spent quality time with some more challenging cognitive work to complete such a task, which will help him make broader connections and have deeper, longer-lasting conceptual understanding of fractions. This is an example of the kind of work that third graders are doing today to build a foundation for understanding fractions, fraction operations, and beyond.
Grade 4: Ben Works on Multiplying a Fraction by a Whole Number Using a Visual Model
In fourth grade, students continue building conceptual understanding of fractions, through comparison, determining equivalence, and reasoning about relative size. They expand their understanding to consider a set of objects as a whole that can be partitioned into equal groups, in addition to one shape or line segment as a whole that is partitioned into equal parts. Students also make more use of fractions on the number line, consider how fractions are equivalent, find equivalence between improper fractions and mixed numbers, and build understanding of multiplication of fractions (by a whole number only) from the basis of addition of fractions. The subgroup of fourth grade Common Core State Standards representative of these concepts states "Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers."
In the video, Ben works on multiplying a whole number (5) by a fraction (3/4). The number line model highlights a visual representation of the repeated addition of 3/4. Ben carefully partitioned the number line so that he first had equal wholes, and then so that each whole was divided into equal fourths. Then he labeled the number line, and showed five "jumps" of 3/4, landing on 2 3/4. The rectangle model shows generally the same kind of action, although as Ben admitted, he and many other kids prefer that model over the number line, most likely because partitioning the shape equally can be easier than partitioning a number line. Some students might also color each group of 3/4 with a new color or pattern to show each of the five groups.
Grade 5: Libby Works on Solving a Word Problem by Multiplying Two Fractions and Using a Visual Model
In fifth grade, students draw on their conceptual understanding of fractions to learn and fluently perform the four operations of addition, subtraction, multiplication, and division on fractions. Any of these fraction operations can be presented as word problems in which students are expected to determine the correct operation to use to solve the problem (a fairly complex skill), or as simply numeric problems with only numbers and symbols.
Fifth graders must show these operations in action first using a visual strategy that can demonstrate how the operation is working on the fractions, and then they may translate that conceptual understanding from the visual models to a more abstract method for solving. The subgroup of fifth grade Common Core State Standards representative of these goals states "Apply and extend previous understandings of multiplication and division to multiply and divide fractions."
This video shows fifth-grader Libby working on multiplying two fractions after she reads a word problem about finding part of a part of a milkshake. Yes, I meant to say that! In the word problem, one child has half of a milkshake, and the other child drinks one fourth of that half, so how much of the original milkshake did the 2nd child drink? Students in fifth grade are expected to identify the action in the word problem, which in this case is finding part of a part, and then determine which operation(s) should be used to find the solution, which is multiplication in this case. When fractions are multiplied, they can find part of a part. As you may assume, there is a great deal of cognitive work required for students to interpret and solve such a problem. The Common Core State Standards specifically require students to learn these operations conceptually (with the help of visual models) so that the meaning of how and why the operations work is clear to students. Eventually, they are expected to move toward more efficient strategies using only abstract numbers and symbols.
The problem asks the reader to find out how much of the whole milkshake is one fourth of one half. To directly mimic the action in the story, the student would first draw a picture of one half and then find one fourth of the one half. In the video, Libby technically finds one half of one fourth. However, she seems to have determined that the solution would require her to multiply two fractions, and then with the knowledge of the commutative property of multiplication, Libby switches the order of the two factors and appropriately uses a visual strategy to find the correct answer to the problem. [1/4 x 1/2 = 1/2 x 1/4]
This video attempts to show all the previous understanding of fractions that fifth graders must apply to their understanding and use of the four operations for fractions. Again, there is a great deal of cognitive work to incorporate from previous years and bigger math connections to be made in order to be successful with these more challenging concepts of fractions in fifth grade.
