Getting Started |
Misconception/Error The student is unable to generate an equivalent fraction for . |
Examples of Student Work at this Level The student may understand the meaning of however, the student guesses a fraction that is equivalent or does not know how to generate an equivalent fraction. |
Questions Eliciting Thinking What do you know about the fraction ? What does the four in the denominator mean? What does the one in the numerator mean?
Can you think of another way to represent ? Can you use these fraction bars to show me?
Can you draw a picture to show ? Can you draw another picture that shows eighths? How many eighths would need to be shaded in to show ? |
Instructional Implications Use an area model to show that is equivalent to . Also use a number line model to show that is located at the same point as .
Provide clear instruction to the student on the meaning of the word equivalent, telling the student that equivalent means, “equal, or the same value as.” Next, using the available manipulatives, show the student how different fractions can represent the same amount beginning with those fractions that are equivalent to . |
Moving Forward |
Misconception/Error The student needs much prompting to find a fraction equivalent to . |
Examples of Student Work at this Level The student initially does not know an equivalent fraction. However, after working with the fraction bars or another manipulative, he or she is able to determine that is equivalent to . |
Questions Eliciting Thinking I see that you are showing that both of these fraction bars are the same length. What does that mean?
In both pictures, is the same as . What does that mean? Can you say more about how you know they are equal?
What does the numerator mean? The denominator? |
Instructional Implications Encourage the student to use what he or she has learned from this task to generate other equivalent fractions. Consider beginning with .
As the student finds other pairs of equivalent fractions, encourage him or her to justify both orally and in writing why the pairs are equivalent. Have the student begin by formulating an oral justification. Then, have the student transition to a written justification. |
Almost There |
Misconception/Error The student struggles to use a manipulative or model to explain why the fraction he or she generated is equivalent to . |
Examples of Student Work at this Level The student states that (or another equivalent fraction) is equivalent to and begins to use a model or manipulative to prove the fractions are equivalent. However, the student’s response is lacking clarity or is not fully accurate. |
Questions Eliciting Thinking Can you say more about how you know they are equal?
Can you use the number line to explain how these two fractions are equivalent?
When both of your pictures have the same amount shaded, what does that say about the two fractions?
I noticed that both two and eight are greater than one and four, so how can both fractions be equal? |
Instructional Implications Encourage the student to use representations that are clear to him or her. The teacher may guide the student to develop an area model first. After the student becomes proficient using area models to represent equivalent fractions, the teacher should introduce representing equivalent fractions on the number line.
As the student finds other pairs of equivalent fractions, encourage him or her to justify both orally and in writing why the pairs are equivalent. Have the student begin by formulating an oral justification. Then, have the student transition to a written justification. |
Got It |
Misconception/Error The student provides complete and correct responses to all components of the task. |
Examples of Student Work at this Level The student generates a fraction equivalent to and is able to use a model or manipulative to explain how he or she knows that the two fractions are equivalent. |
Questions Eliciting Thinking Think about the fraction . Can you list some other fractions that are equivalent to it?
What do you notice about the fractions and ? Do you see a relationship between the numerators? Do you see a relationship between the denominators? Do you think that will always work when finding equivalent fractions?
How much larger is the numerator than the denominator in the fraction ? What about the numerator and denominator in ? Do you notice a relationship? |
Instructional Implications Provide additional examples of fractions that are equivalent to , and encourage the student to look for patterns in the relationship between the numerators and the denominators. Encourage the student to notice that the numerators and denominators change proportionally when they are equivalent. Once the student sees this relationship, encourage him or her to explain it. |