Study Guide

Field 222: Multi-Subject: Teachers of Childhood
(Grade 1–Grade 6)
Part Two: Mathematics

Sample Constructed-Response 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:

Background Information

Fourth-grade 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 one-half 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 one-half 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 Constructed-Response 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 Constructed-Response Item

The following characteristics guide the scoring of responses to the constructed-response 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 Constructed-Response Item

A score will be assigned to the response to the constructed-response 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:
  • The response thoroughly addresses all parts of the assignment.
  • The response demonstrates the relevant knowledge and skills with thorough accuracy and effectiveness.
  • The response is well supported by relevant examples and details and thoroughly demonstrates sound reasoning.
3 The "3" response reflects a general command of the relevant knowledge and skills:
  • The response generally addresses all parts of the assignment.
  • The response demonstrates the relevant knowledge and skills with general accuracy and effectiveness.
  • The response is generally supported by some examples and/or details and generally demonstrates sound reasoning.
2 The "2" response reflects a partial command of the relevant knowledge and skills:
  • The response addresses all parts of the assignment, but most only partially; or some parts are not addressed at all.
  • The response demonstrates the relevant knowledge and skills with partial accuracy and effectiveness.
  • The response is partially supported by some examples and/or details or demonstrates flawed reasoning.
1 The "1" response reflects little or no command of the relevant knowledge and skills:
  • The response minimally addresses the assignment.
  • The response demonstrates the relevant knowledge and skills with minimum accuracy and effectiveness.
  • The response is minimally supported or demonstrates significantly flawed reasoning.
UThe response is unscorable because it is unrelated to the assigned topic or off-task, unreadable, written in a language other than English or contains an insufficient amount of original work to score.
BNo response.