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Lesson 3Adding and Subtracting Decimals with Few Non-Zero Digits

Let’s add and subtract decimals.

Learning Targets:

  • I can tell whether writing or removing a zero in a decimal will change its value.
  • I know how to solve subtraction problems with decimals that require “unbundling” or “decomposing.”

3.1 Do the Zeros Matter?

  1. Evaluate mentally: 1.009+0.391

  2. Decide if each equation is true or false. Be prepared to explain your reasoning.

    a.  34.56000 = 34.56

    b.  25 = 25.0

    c. 2.405 = 2.45

3.2 Calculating Sums

  1. Andre and Jada drew base-ten diagrams to represent  0.007 + 0.004 . Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.

    Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.
    1. If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
    1. Draw or describe a diagram that could represent the sum 0.008 + 0.07 .

  1. Here are two calculations of 0.2 + 0.05 . Which is correct? Explain why one is correct and the other is incorrect.

Two calculations of zero point 2 plus zero point zero five are indicated. The calculation on the left adds zero point 2 and zero point zero 5 by aligning the ones units, tenths unit, and hundredths unit. The sum is zero point 2 5. The calculation on the right adds zero point 2 and zero point zero five by aligning the hundredths unit under the tenths unit. The sum is zero point zero 7.
  1. Compute each sum. If you get stuck, draw base-ten diagrams to help you.
    1. The vertical calculation of zero point 1 1 plus zero point zero zero 5 is indicated by aligning the ones units, tenths units, hundredths units, and thousandths units.
    1. 0.209 + 0.01
    1. 3.02 + 1.14
  • The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
  • Select a Block tool, and then click on the screen to place it.

Click on the Move tool when you are done choosing blocks.

open applet in presentation mode

3.3 Subtracting Decimals of Different Lengths

To represent 0.4 - 0.03 , Diego and Noah drew different diagrams. Each rectangle shown here represents 0.1. Each square represents 0.01.

  • Diego started by drawing 4 rectangles for 0.4. He then replaced 1 rectangle with 10 squares and crossed out 3 squares for the subtraction of 0.03, leaving 3 rectangles and 7 squares in his drawing.

    A base-ten diagram labeled “Diego’s Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 10 squares in that column. The last rectangle is circled with a dashed line and an arrow pointing from the rectangle to the column of squares is labeled “unbundle.” The last three squares are crossed out.

  • Noah started by drawing 4 rectangles for 0.4. He then crossed out 3 of them to represent the subtraction, leaving 1 rectangle in his drawing.

    Noah started by drawing 4 rectangles for 0.4. He then crossed out 3 of them to represent the subtraction, leaving 1 rectangle in his drawing.

  1. Do you agree that either diagram correctly represents 0.4 - 0.03 ? Discuss your reasoning with a partner.

  2. To represent 0.4 - 0.03 , Elena drew another diagram. She also started by drawing 4 rectangles. She then replaced all 4 rectangles with 40 squares and crossed out 3 squares for the subtraction of 0.03, leaving 37 squares in her drawing. Is her diagram correct? Discuss your reasoning with a partner.

    A base-ten diagram labeled “Elena's Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 40 squares in that column. All four rectangles are circled with a dashed line and an arrow pointing from the rectangles to the column of squares is labeled “unbundle.” The last three squares are crossed out.
  3. Find each difference. If you get stuck, you can use the applet to represent each expression and find its value.
    1. 0.3 - 0.05
    2. 2.1 - 0.4
    3. 1.03 - 0.06
    4. 0.02 - 0.007

Be prepared to explain your reasoning.

  • The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
  • Select a Block tool, and then click on the screen to place it.

Click on the Move tool when you are done choosing blocks.

An image of a trash can labeled delete tool.

Subtract by deleting with the delete tool, not crossing out.

open applet in presentation mode

Are you ready for more?

A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.

At the Auld Shoppe, a shopper buys items that are worth 2 yellow jewels, 2 green jewels, 2 blue jewels, and 1 indigo jewel. If they came into the store with 1 red jewel, 1 yellow jewel, 2 green jewels, 1 blue jewel, and 2 violet jewels, what jewels do they leave with? Assume the shopkeeper gives them their change using as few jewels as possible. 

Lesson 3 Summary

Base-ten diagrams can help us understand subtraction as well as addition. Suppose we are finding 0.023 - 0.007 . Here is a diagram showing 0.023, or 2 hundredths and 3 thousandths. 

a diagram showing how to break 0.023 into hundredths and tenths

Subtracting 7 thousandths means removing 7 small squares, but we do not have enough to remove. Because 1 hundredth is equal to 10 thousandths, we can “unbundle” (or decompose) one of the hundredths (1 rectangle) into 10 thousandths (10 small squares).

a diagram showing how to break 0.023 into hundredths and tenths

We now have 1 hundredth and 13 thousandths, from which we can remove 7 thousandths.

a diagram showing how to break 0.023 into hundredths and tenths

We have 1 hundredth and 6 thousandths remaining, so 0.023 - 0.007 = 0.016 .

a diagram showing how to break 0.016 into hundredths and tenths

Here is a vertical calculation of 0.023 - 0.007 .

diagrams showing how to find the differences of numbers

In both calculations, notice that a hundredth is unbundled (or decomposed) into 10 thousandths in order to subtract 7 thousandths. 

Lesson 3 Practice Problems

  1. Here is a base-ten diagram that represents 1.13. Use the diagram to find 1.13 - 0.46 .

    Explain how you found the difference, or label your diagram to show your steps.

    A diagram showing how to break a number into hundreds, tens, and ones

  2. Compute the following sums. If you get stuck, you can draw base-ten diagrams.

    1. 0.027 + 0.004

    1. 0.203 + 0.1

    1. 1.2 + 0.145

  3. A student said we cannot subtract 1.97 from 20 because 1.97 has two decimal digits and 20 has none. Do you agree with his statement? Explain or show your reasoning.

  4. Decide which calculation shows the correct way to find 0.3-0.006 and explain your reasoning.
    diagrams showing how to find the differences of numbers

  5. Complete the calculations so that each shows the correct difference.

    diagrams showing how to find the differences of numbers

  6. The school store sells pencils for $0.30 each, hats for $14.50 each, and binders for $3.20 each. Elena would like to buy 3 pencils, a hat, and 2 binders. She estimated that the cost will be less than $20.

    1. Do you agree with her estimate? Explain your reasoning.
    2. Estimate the number of pencils could she buy with $5. Explain or show your reasoning.

  7. A rectangular prism measures 7\frac{1}{2} cm by 12 cm by 15\frac{1}{2} cm.

    1. Calculate the number of cubes with edge length \frac{1}{2} cm that fit in this prism.
    2. What is the volume of the prism in \text{cm}^3 ? Show your reasoning. If you are stuck, think about how many cubes with \frac12 -cm edge lengths fit into 1\text{ cm}^3 .

  8. At a constant speed, a car travels 75 miles in 60 minutes. How far does the car travel in 18 minutes? If you get stuck, consider using the table.

    minutes distance in miles
    60 75
    6
    18