Getting Started |
Misconception/Error The student does not understand that decimal comparisons are only valid when referring to the same sized whole. |
Examples of Student Work at this Level The student says that the reasoning is correct because both models have four shaded in. When prompted, the student says that the two wholes do not need to be the same size.
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Questions Eliciting Thinking What is the same about the models? What is different?
Can we compare decimals using area models if they are not the same size? What would that show us?
How many total squares are in the model? What fraction of the model is shaded?
What does one square represent on a 10 x 10 grid? What does one column represent?
Can you write the decimal (or fraction) for the second model? |
Instructional Implications Demonstrate that comparisons are only valid when the decimals refer to the same size whole using the same size partition (e.g., tenths or hundredths). Provide opportunities for the student to use models (10 x 10 grids or base ten blocks) to compare decimal numbers. Reinforce the relationship between decimals and fractions and guide the student to understand and compare decimals by thinking about them as fractions.
It may be helpful to relate the decimals to money. Relate the place value to the corresponding coin (e.g., the tenths place represents dimes). For example, when comparing 0.4 and 0.04, the student can say that 0.4 is four dimes, or forty cents, and 0.07 is seven pennies, or seven cents. Guide the student to observe that 0.07 is less than 0.4 since seven cents is less than forty cents.
Model reading decimals appropriately (e.g., read 0.4 as â€śfour tenthsâ€ť rather than as â€śzero-point-fourâ€ť).
Consider using the MFAS task Comparing Four Fifths and Three FourthsÂ which assesses this same idea using fractions. |
Making Progress |
Misconception/Error The student is unable to clearly explain the difference in the models. |
Examples of Student Work at this Level The student struggles to clearly explain that the model is flawed because the two wholes are not the same size. After prompting from the teacher, the student may be able to describe that decimal comparisons are only valid when the two wholes are the same size.
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Questions Eliciting Thinking What is different about the two models? Why is that difference important?
Can we compare fractions using area models if they are not the same size? What would that show us?
How many total squares are in the model? What fraction of the model is shaded?
If you write each number as a decimal (or fraction) can you compare them?
How could you represent each of the models as a fraction? |
Instructional Implications Guide the student to interpret 0.04 as four hundredths and to interpret 0.4 in its equivalent form, 40 hundredths. Model explaining that 0.04 is less than 0.4 because 4 hundredths is less than 40 hundredths. Encourage the student to provide place value explanations when comparing decimals.
Partner the student with a â€śGetting Startedâ€ť student. Have the student model for the â€śGetting Startedâ€ť student how to compare decimal numbers using place value understanding.
Provide the student with problems in context to compare decimals with different wholes. For example, Mary had 0.75 of one milliliter of liquid and Zak had 0.5 of a liter of liquid. Explain why Zak has more liquid than Mary. |
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 says that the reasoning is incorrect because the models do not have the same size whole and comparisons of decimals are only valid when they refer to the same size whole. The student may also note that the decimal grid showing 0.4 has a greater fraction of area shaded than the decimal grid showing 0.04. |
Questions Eliciting Thinking How could we make the models comparable in this example?
What would you tell another student to convince them that the models are not equivalent?
Can you represent the value shown by each model on the same number line?
How can you use benchmark fractions (0, , , , and 1) to compare 0.49 and 0.8?
Is always less than ? Why or why not? |
Instructional Implications Ask the student to use strategies other than visual models to compare decimals (e.g., place value and benchmark decimals). Guide the student to reason about the size of the decimal and to see that some decimals can be easily compared using place value or benchmark decimals. Guide the student to be flexible in his or her choice of strategies based on the decimals being compared. |