Getting Started |
Misconception/Error The student has an operational view of the equal sign. |
Examples of Student Work at this Level The student says both equations are false because 24 Ă· 6 does not equal 12 (the number just to the right of the equal sign in the first equation) and 32 Ă· 12 does not equal 16 (the number just to the right of the equal sign in the second equation). |
Questions Eliciting Thinking What does the equal sign mean?
Show the student the equations 5 = 5 and 6 = 18 Ă· 3. Ask the student if the equations are true or false.
I see that you did not consider the number three to the right of the equal sign. Should we just ignore this number? |
Instructional Implications Consider using the MFAS task True or Not True (1.OA.4.7) which provides insight into the studentâ€™s understanding of the equal sign.
Provide explicit instruction on the meaning of the equal sign. Use a strategy with which the student is comfortable to show that the quotient of the quantities on the left and right sides of the equal sign must have the same value. Then provide the student with division equations that contain a missing number, n, such as 27 Ă· 9 = n Ă· 3. Model for the student how to determine the value of one side of the equation and then use that value to determine the unknown quantity on the other side of the equation. Eventually, model comparative relational thinking to solve equations. |
Moving Forward |
Misconception/Error The student has a relational understanding of the equal sign but is unable to use comparative relational thinking to determine if the equations are true. |
Examples of Student Work at this Level The student understands that the two quantities on each side of the equal sign must have the same value. However, the student needs to divide 24 by six and then 12 by three in order to determine if the equation is true or false. |
Questions Eliciting Thinking You have a good understanding of the meaning of the equal sign. Could you determine if the equations are true by comparing the numbers on each side of the equation?
How does 24 compare to 12? How does six compare to three? |
Instructional Implications Provide experiences that will help the student develop the concept that when both the dividend and divisor are multiplied or divided by the same value, the quotient remains the same. For example, 36 Ă· 12 = 3 and (36 Ă· 4) Ă· (12 Ă· 4) = 9 Ă· 3 = 3. Then model for the student how to use comparative relational thinking to determine if equations relating two quotients are true. Provide the student with additional practice in using comparative relational thinking. Encourage the student to explain his or her thinking to other students.
Provide opportunities for the student to hear the explanations of Got It level students using comparative relational thinking to determine if equations relating two quotients are true. |
Almost There |
Misconception/Error The student attempts to use comparative relational thinking but makes errors. |
Examples of Student Work at this Level The student attempts to use comparative relational thinking but makes an error or becomes confused. For example, the student attempts to compare the numbers multiplicatively but not in a consistent order (e.g., 24 divided by 12 is two but three divided by six is not two) which causes the student to conclude an equation is false. |
Questions Eliciting Thinking If 24 is two times 12, what needs to be true of the two divisors in order for the equation to be true?
If 32 is two times 16, what needs to be true of the two divisors in order for the equation to be true? |
Instructional Implications Assist the student in identifying the error in his or her reasoning. Emphasize that when both the dividend and divisor are multiplied or divided by the same value, the quotient remains the same. Model explaining that the first equation is true since 24 is two times 12 and six is two times three. Provide the student with additional practice in using comparative relational thinking. Encourage the student to explain his or her thinking to other students. |
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 determines that the first equation is true since 24 is two times 12 and six is two times three. The student also says that the second equation is true since 32 is two times 16 and 12 is two times six. |
Questions Eliciting Thinking Can you describe the relationship between the numbers in these equations in terms of division instead of multiplication?
How would you find the missing number that makes this equation true: 48 Ă· 6 = ? Ă· 18? |
Instructional Implications Consider using the MFAS task Comparative Relational Thinking in a Division Equation (4.OA.1.b).
Provide a false equation such as 54 Ă· 18 = 9 Ă· 2. Have the student determine the error and then use comparative relational thinking to rewrite the equation so that it is true. |