Study Guide
Field 222: MultiSubject: Teachers of Childhood
(Grade 1–Grade 6)
Part Two: Mathematics
Sample ConstructedResponse Item
Competency 0005
Analysis, Synthesis, and Application
Use the data provided to complete the task that follows.
Using the data provided, prepare a response of approximately 400–600 words in which you:
 identify a significant mathematical strength related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment;
 identify a significant area of need related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment; and
 describe an instructional intervention that builds on the student's strengths and that would help the student improve in the identified area of need. Include a strategy for helping the student build a viable argument related to the given standard.
Background Information
Fourthgrade students have been developing an understanding of fractions. The class has reviewed representing numbers on a number line, worked with equivalent fractions in special cases, and compared fractions by reasoning about their size. The class is currently working on the following standard from the New York State P–12 Common Core Learning Standards for Mathematics.
Number & Operations—Fractions (4.NF)
Extend understanding of fraction equivalence and ordering.
5. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
In particular, the teacher has planned a lesson experience in which students will use benchmarks to compare and order fractions. The students have worked with halves, fourths, eighths, thirds, sixths, and twelfths. The teacher has the students work in groups of two.
Description of Class Activity
The teacher gives each pair of students a set of "benchmark cards." Each card is a shaded index card labeled with the number 0, , or 1. The students place the benchmark cards in correct order from least to greatest along a number line drawn on a piece of cardboard.
Each pair of students also has a pack of "fraction cards." Each card is an unshaded index card labeled with a fraction (e.g., , , , , , ). Each student takes a turn selecting a fraction card from the pack. A card that is equal to one of the benchmark cards is placed beneath the benchmark card. The remaining cards are placed between the benchmark cards in the correct order.
While working with their partner, students are encouraged to justify the placement of each card by reasoning about fraction equivalencies and relationships. The student placing the card must explain his or her reasoning while the student observing is encouraged to question and critique the partner's decisions. The teacher has emphasized that both partners will need to be prepared to justify the placement of fractions on the number line.
Excerpt of Interview with Student
As students work, the teacher moves among them and asks questions that require students to explain their reasoning about comparing and ordering fractions. The teacher stops to observe one group's work in progress and asks one of the students several questions. The group's work is shown below, followed by a short excerpt of the discussion between the teacher and the student.
Teacher: How did you and your partner decide where would go on your number line? Student: Well, we put under because we know they are equal. So then we knew that must be less than onehalf because you need one more piece to make it . We also know that is larger than because is the same as . Teacher: How did you decide where to place and on your number line? Student: We know that is more than onehalf because it only takes to equal . And is the same as because you just need one more piece to make them both a whole. Teacher: Can you show me what you mean when you say, "you just need one more piece"? Student: Well, if I think about a pie, I know I only need one more piece to make the whole. Teacher: Can you show me what you are thinking of with a drawing? Student: That's easy! I'll make two pies. One shows and one shows .
Student: See, they both have one piece missing so they are both the same and they must be pretty close to one whole.
Sample Strong Response to the ConstructedResponse Assignment
A strength that the student demonstrates is understanding that fractions with different numerators and denominators can be equivalent. This is shown by the statement, "we put 4/8 under 1/2 because we know they are equal" and later by "1/4 is the same as 2/8." The student could compare fractions that have the same denominators, showing an understanding of the meaning of the numerator. The statement "because you need one more piece to make it 1/2," although lacking in precise terminology, was accurately used to explain that 3/8 was less than 4/8.
A significant area of need is demonstrated in the student's lack of understanding that denominator determines piece size. When comparing fractions with different denominators, the student made no distinction between the size of the pieces that are being considered; he considered only the number of pieces. This is illustrated by the statement "2/3 is the same as 5/6 because you just need one more piece to make them both a whole." Although the student was able to draw both fractions as a pie and shade the correct number of pieces, he apparently did not discern that the missing piece in the pies were different in size, and that, in fact, the pies themselves were not the same size. He noted only that there was one piece missing from each and erroneously concluded that the fractions were equivalent.
Instructional intervention should start with the student using strips of paper to fold and create fraction strips for a variety of fractions (thirds, fourths, sixths, eighths) and to compare 2/3 to 3/4 and to 5/6. The teacher would ask are they all equal? Which of the three is closest to one whole? Compare the amount left to make one whole. Are 1/3, 1/4, and 1/6 equal? Why not? What is the meaning of the denominator in a fraction? What does it tell you?
Using fraction strips, the student could then be asked to compare several fractions that have the same numerator but different denominators, such as 2/6 and 2/4, or 5/8 and 5/6. Because the student appears to have some understanding of equivalent fractions, the next step would be to do work converting 2/3 and 5/6 to fractions with common denominators. The teacher would ask for a comparison of the new fractions, written with common denominators, to each other. Then the student should compare several other pairs of fractions "with one piece missing" in the same manner, finding common denominators. As he works, he should explain his process and his thinking to the teacher.
Students using symbolic notation for fractions may get lost in the symbols and fail to remember that denominators define the size of the fractional part and numerators represent the number of this part. The use of a visual model (the strips) coupled with teacher questioning and student explanations, would help a student to understand key concepts, thus enabling him to build a viable argument regarding equivalent fractions, and allowing him to progress to new ideas.
Performance Characteristics for ConstructedResponse Item
The following characteristics guide the scoring of responses to the constructedresponse assignment.
Completeness  The degree to which the response addresses all parts of the assignment 

Accuracy  The degree to which the response demonstrates the relevant knowledge and skills accurately and effectively 
Depth of Support  The degree to which the response provides appropriate examples and details that demonstrate sound reasoning 
Score Scale for ConstructedResponse Item
A score will be assigned to the response to the constructedresponse item according to the following score scale.
Score Point  Score Point Description 

4  The "4" response reflects a thorough command of the relevant knowledge and skills:

3  The "3" response reflects a general command of the relevant knowledge and skills:

2  The "2" response reflects a partial command of the relevant knowledge and skills:

1  The "1" response reflects little or no command of the relevant knowledge and skills:

U  The response is unscorable because it is unrelated to the assigned topic or offtask, unreadable, written in a language other than English or contains an insufficient amount of original work to score. 
B  No response. 