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The teacher provides the student with the Are the Equations True? worksheet and says, â€śWithout adding 49 and 23, determine whether the equation 49 + 23 = 47 + 25 is true or false.â€ť

The student is given ample time to respond and the teacher prompts the student to explain his or her thinking.

The teacher then says, â€śWithout adding 75 and 48, determine whether the equation 75 + 48 = 65 + 38 is true or false.â€ť

The student is given ample time to respond and the teacher prompts the student to explain his or her thinking.

TASK RUBRIC

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 49 + 23 does not equal 47 (the number just to the right of the equal sign in the first equation) and 75 + 48 does not equal 65 (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 15 = 15 and 16 = 12 + 4. Ask the student if the equations are true or false.

I see that you did not consider the number 25 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 base ten blocks to show that the sum of the quantities on the left and right sides of the equal sign must have the same value. Then provide the student with addition equations that contain a missing number, n, such as 17 + 9 = n + 14. 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 using base ten blocks.

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 calculate 49 + 23 and 47 + 25 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 49 compare to 47? How does 23 compare to 25?

Instructional Implications

Model for the student how to use comparative relational thinking to determine if the equations are true. Introduce the student to the concept of compensation. For example, show the student the expression 49 + 23. Then decrease 49 by two and ask the student how 23 can be adjusted to compensate for the change (i.e., 49 + 23 = 47 + ?). Make explicit that when 49 is decreased by two, 23 must be increased by two to compensate. Show the student that 49 + 23 = (47 + 2) + 23 = 47 + (2 + 23) = 47 + 25. 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 sums are true.

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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 says, â€śSince 49 is two more than 47, 23 should be two more than 25.â€ť

Questions Eliciting Thinking

If 49 is two more than 47, what needs to be true of the other two addends in order for the equation to be true?

If 65 is 10 less than 75, what needs to be true of the other two addends in order for the equation to be true?

Instructional Implications

Assist the student in identifying the error in his or her reasoning. Introduce the student to the concept of compensation. For example, show the student the expression 49 + 23. Then decrease 49 by two and ask the student how 23 can be adjusted to compensate for the change (i.e., 49 + 23 = 47 + ?). Make explicit that when 49 is decreased by two, 23 must be increased by two to compensate. 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 and explains that since 49 is two more than 47 then the second addend on the right side must be two more than 23, which is the case since it is 25. Therefore, the first equation is true. The student explains that the second equation is false because both numbers on the left side of the equal sign are 10 more than the two numbers on the right of the equal sign.

Questions Eliciting Thinking

In the second equation, how much larger is the expression on the left than the expression on the right?

What symbol can replace the equal sign to make the second equation true?

What number could be changed in order to make the second equation true? Is there another way to change a single number to make the second equation true?

Instructional Implications

Consider using the MFAS task Comparative Relational Thinking in an Addition Equation (4.OA.1.b).

Introduce equations involving multiplication such as 25 x 9 = 5 x 45 or 15 x 6 = 45 x 18. Ask the student to use comparative relational thinking to determine if the equations are true or false.

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