Lesson 3

Adding and Subtracting Decimals with Few Non-Zero Digits

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.

1. $$34.56000 = 34.56$$
2. $$25 = 25.0$$
3. $$2.405 = 2.45$$

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.

1. If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
2. Draw or describe a diagram that could represent the sum $$0.008 + 0.07$$.
2. Here are two calculations of $$0.2 + 0.05$$. Which is correct? Explain why one is correct and the other is incorrect.

3. Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.
1. $$0.209 + 0.01$$
2. $$10.2 + 1.1456$$
• 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 (the arrow) when you are done choosing blocks.
• Subtract by deleting with the delete tool (the trash can), not by crossing out.

3.3: Subtracting Decimals of Different Lengths

To represent $$0.4 - 0.03$$, Diego and Noah drew different diagrams. Each rectangle represented 0.1. Each square represented 0.01.

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

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

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 to represent subtraction of 0.03, leaving 37 squares in her diagram. Is her diagram correct? Discuss your reasoning with a partner.

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 (the arrow) when you are done choosing blocks.
• Subtract by deleting with the delete tool (the trash can), not by crossing out.

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.

Summary

Base-ten diagrams can help us understand subtraction as well. Suppose we are finding $$0.23 - 0.07$$. Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.

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

We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.

We have 1 tenth and 6 hundredths remaining, so $$0.23 - 0.07 = 0.16$$.

Here is a vertical calculation of $$0.23 - 0.07$$.

Notice how this representation also shows a tenth is unbundled (or decomposed) into 10 hundredths in order to subtract 7 hundredths.

This works for any decimal place. Suppose we are finding $$0.023 - 0.007$$. Here is a diagram showing 0.023.

We want to remove 7 thousandths (7 small rectangles). We can “unbundle” (or decompose) one of the hundredths into 10 thousandths.

Now we can remove 7 thousandths.

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

Here is a vertical calculation of $$0.023 - 0.007$$.