Using Models to Compare Decimals

Getting Started

The student does not have an effective strategy for comparing decimals.

The student does not draw a visual decimal model to compare the decimals and only refers to the digits. For example, the student says that 0.08 is greater than 0.7 because eight is larger than seven.

The student does not use the visual model he or she drew to compare the decimals. The student says that he or she lined up the decimals in his or her head to compare. Although the student is able to correctly compare 0.08 and 0.7 using his or her method, the visual model does not match and the student still has misconceptions about tenths and hundredths. The student says 0.30 is greater than 0.3.

The student uses the decimal grids to represent and compare the decimals. The student has a misconception with the tenths and hundredths place values and shades in seven squares for 0.7, eight squares for 0.08, three squares for 0.3 and thirty squares for 0.30. As a result, the student is unable to compare the decimals correctly.

What does the eight in 0.08 mean? What is the place value of the eight? What is the place value of the seven?

How would you show these decimals using 10 x 10 grids?

What does the tenths grid represent? What does the hundredths grid represent?

How can you show seven-tenths on the tenths grid? How could you show seven-tenths on the hundredths grid?

Is eight greater than 70? So even though eight is greater than seven, will a number that contains a digit eight always be greater than a number that contains a digit seven?

Provide direct instruction on the meaning of decimal representations emphasizing place value. Guide the student to visually represent decimals using 10 x 10 grids. Relate the numbers of shaded squares to the way the decimals are written. Then use the less than, greater than, or equal to symbol to compare the decimals. Provide the student with additional pairs of decimals and have the student compare the decimals by first representing each decimal on a 10 x 10 grid.

Review the relationship between decimals and their fraction equivalents. Assist the student in relating the denominator of the equivalent fraction to the number of digits in the decimal representation. Guide the student to scale a number line appropriately in order to graph decimals. Once the student is able to correctly scale a number line, have the student use the number line to graph and compare decimals. Consider using the MFAS task Using Benchmark Fractions On a Number Line.

Moving Forward

The student attempts an effective strategy but makes errors.

The student draws a model correctly for 0.3 and 0.30 and records the comparison with the equal symbol. However, the student represents 0.08 as eighty-hundredths and is unable to correct with prompting.

The student draws decimal grids to compare the decimals but draws one model larger than the other. The student is unable to compare the decimals correctly because the wholes are not the same size and the student compares the decimals based on the size of the parts shaded.

What does the eight in 0.08 mean? What is the place value of the eight?

How many parts should the whole be to represent tenths? What about hundredths?

Can I use the hundredths grid to represent tenths? How many hundredths would I need to color in to equal one tenth?

If I want to compare these two decimals by drawing a model, should the whole in each drawing be the same size? Why or why not?

Demonstrate that comparisons are only valid when the decimals refer to the same size whole (e.g. 0.4 refers to tenths while 0.07 refers to hundredths)f. 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. Consider using the MFAS task Comparing Four Tenths.

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 “four tenths” rather than as “zero-point-four”).

Almost There

The student is unable to justify decimal comparisons appropriately using place value or uses inequality symbols incorrectly.

The student correctly determines that 0.08 is less than 0.7 and explains his or her thinking with a visual decimal model. However, the student is unable to correctly compare the decimals using place value. The student refers to the seven in 0.7 as being in the tens place and the eight in 0.08 as being in the ones place.

The student is able to compare the decimals correctly by using the models and place value, but the student does not understand that the models must be the same size in order to compare the decimals.

The student is able to use the decimal grids to compare the decimals. When recording the comparison using the less than or greater than symbol, the student writes the symbol backwards. When asked to read the comparison aloud, the student is able to read the comparison correctly. For example, the student says, “Eight hundredths is less than seven tenths.”

You are correct, 0.08 is less than 0.7. However, look at the symbol you used to compare the decimals. Do you think you used the correct symbol? Why or why not?

How is the place value of the eight in 0.08 different from the eight in 800? How is the place value of the seven in 0.7 different from the seven in 70?

How can you use the model you drew and shaded to help you explain the comparison using place value? What is the difference in the two models you drew? One model only has ten parts, while the other has 100 parts. What do the parts in each represent?

Model reading decimals appropriately (e.g., read 0.7 as “seven tenths” rather than as “zero-point-seven”). Provide opportunities for the student to read and write decimals.

Provide clear instruction on using the less than and greater than symbols correctly. Then provide additional decimals and have the student compare the decimals and record the comparisons with the less than, greater than, or equal to symbol.

Remind the student that comparisons are only valid when the decimals refer to the same size whole (e.g., 0.4 refers to tenths while 0.07 refers to hundredths). Provide opportunities for the student to draw models to compare decimal numbers. Encourage the student to be consistent when partitioning rectangular area models so comparisons are valid.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly writes 0.08 < 0.7 and 0.3 = 0.30 and can justify his or her thinking using a correct visual decimal model as well as place value reasoning.

How could you use place value to compare the two decimals without having to draw a model?

How can you use benchmark fractions (0, , , , and 1) to compare 0.49 and 0.8?

How would you compare 0.45 and 0.451? What about 0.045 and 0.0451?

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.

Consider using the MFAS Task Compare Decimals, which provides opportunities for the student to compare decimals in a variety of ways.

Have the student partner with another student. Provide the students with index cards with decimals written on them. Have the students play War., Both students flip over a card with a decimal written on it. The student with the card with the greatest decimal value keeps both cards. The game ends when one student runs out of cards.