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Note: This task may be implemented individually, in small groups, or in a whole-group setting. If the task is given in a whole-group setting, the teacher should ask each student to explain his or her thinking and strategy.

The teacher provides the student with the Throwing Footballs worksheet and reads each problem to the student.

Two students were trying to see how far they could throw a football. Sadie threw the football 18 yards and William threw the football 6 yards. How many times farther did Sadie throw the football than William?

Michael has 20 baseball cards and Abed has 5 baseball cards. Michael has how many times as many baseball cards as Abed?

Before the student solves each problem, the teacher asks the student to write an equation with an unknown to model each word problem.

The teacher then asks the student to solve each problem and explain his or her solution strategy.

TASK RUBRIC

Getting Started

Misconception/Error

The student does not understand multiplicative comparisons.

Examples of Student Work at this Level

The student multiplies the two numbers in each problem to solve. The student writes each equation as 18 x 6 = ___ and 20 x 5 = ___.

The student may solve one or both of the problems using addition (18 + 6 = _) or subtraction (18 â€“ 6 =_ ).

Questions Eliciting Thinking

How far did Sadie throw the football? How far did William throw it?

If William had thrown it twice as far, how far would it have gone?

How about Michaelâ€™s cards? How many did he have? Abed?

If Abed had twice (or three times) as many cards, how many cards would he have?

Instructional Implications

Expose the student to other multiplicative comparison situations using manipulatives such as cubes. For example, provide the student with two trains of cubes. The first train has four blue cubes and the second train has 20 cubes total (groups of four in alternating colors). Guide the student to compare in terms of multiplication and ask, â€śHow do the lengths of the two trains compare? Ask the student to write a multiplication equation and to explain his or her reasoning.

Assist the student in making sense of the problem. Then provide similar problems and have the student use manipulatives to model the multiplicative comparison. Guide the student to write a multiplication equation that represents the model.

Consider using the MFAS task Dogs as Pets.

Moving Forward

Misconception/Error

The student struggles to solve multiplicative comparison problems.

Examples of Student Work at this Level

The student is able to solve one of the multiplicative comparison problems yet struggles to solve the second.

The student may or may not write equations correctly.

The student solves one of the problems using addition (or subtraction) providing an incorrect response.

The student needs much prompting to solve the first problem, but is subsequently able to solve the second problem with little prompting.

Questions Eliciting Thinking

If William had thrown the football twice as far, how far would it have gone?

If Abed had twice (or three times) as many cards, how many cards would he have?

Instructional Implications

Provide the student with additional multiplicative comparison word problems. Guide the student in making sense of the problems by first describing the action in the problems in his or her own words. Prompt the student to explain the relationship between the quantities in the problem.

Provide opportunities for the student to write equations that model word problems beginning with the unknown in the result. Then transition the student to writing equations to model multiplicative comparison situations in which the unknown is the number of times larger one quantity is than another.

Have the student solve multiplicative comparison word problems with an unknown product. Consider using the MFAS task Dogs As Pets.

Almost There

Misconception/Error

The student incorrectly writes the equation for one or both problems.

Examples of Student Work at this Level

The student solves each problem correctly and explains his or her reasoning. However, he or she does not write an equation correctly, does not write an equation at all, or writes the equation vertically omitting the equal sign.

Questions Eliciting Thinking

Why did you multiply by six? Was something six times larger?

What are we trying to find in this problem? What symbol should we use and where should it go in our equation?

What are we comparing? How can we write numbers to show that comparison?

Instructional Implications

Model for the student how to write equations that relate to the problems.

Provide opportunities for the student to write equations that model word problems beginning with the unknown in the result. Then transition the student to writing equations to model multiplicative comparison situations in which the unknown is the number of times larger one quantity is than another.

Consider using the MFAS task Writing An Equation to Match A Word Problem.

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 solves each problem correctly and explains his or her reasoning. He or she writes either 6 x ___ = 18 or 18 Ă· 6 = ___ as the equation for the first problem and 5 x ___ = 20 or 20 Ă· 5 = ____ for the second problem.

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Questions Eliciting Thinking

Is there another equation that you could write to model this word problem?

How could you tell another student who was struggling with these types of problems how to solve them?

Instructional Implications

Challenge the student to write an additional equation that matches the word problem.

Have the student share his or her solution strategy and equation with the class.

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