Lesson 3
Adding and Subtracting Decimals with Few Non-Zero Digits
Let’s add and subtract decimals.
3.1: Do the Zeros Matter?
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Evaluate mentally: \(1.009+0.391\)
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Decide if each equation is true or false. Be prepared to explain your reasoning.
- \(34.56000 = 34.56\)
- \(25 = 25.0\)
- \(2.405 = 2.45\)
3.2: Calculating Sums
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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.
- If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
- Draw or describe a diagram that could represent the sum \(0.008 + 0.07\).
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Here are two calculations of \(0.2 + 0.05\). Which is correct? Explain why one is correct and the other is incorrect.
- Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.
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- \(0.209 + 0.01\)
- \(10.2 + 1.1456\)
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- 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.
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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.
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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.
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Do you agree that either diagram correctly represents \(0.4 - 0.03\)? Discuss your reasoning with a partner.
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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.
- Find each difference. If you get stuck, you can use the applet to represent each expression and find its value.
- \(0.3 - 0.05\)
- \(2.1 - 0.4\)
- \(1.03 - 0.06\)
- \(0.02 - 0.007\)
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.
![Base ten diagram. 0 point 23. Two rectangles in the tenths column. 3 small squares in the hundredths column.](https://cms-im.s3.amazonaws.com/41UnLzjaGrfxGPGb48PPWskH?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19aa_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19aa_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2455ae0e4352862c73ff868d322b2ff034d259d79f3abeed2ecc104d2027650d)
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).
![Base ten diagram.](https://cms-im.s3.amazonaws.com/J9avrzBM4cCs2nTq3dK6hkM9?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19bb_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19bb_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7c2f8df220822fc982e5c53b9c8e5f4bf3a5923e9c8692bac0bcdb9f7824c148)
We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/JPvpTK6KU1GH88cd2L7wYCCF?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19cc_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19cc_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1a3f34fa2aa200fb0c33c677448d7d86ef780da4ac7ab28e83cb3498a0e17fd4)
We have 1 tenth and 6 hundredths remaining, so \(0.23 - 0.07 = 0.16\).
![Base ten diagram. 0 point 16. One rectangle in the tenths column. 6 small squares in the hundredths column.](https://cms-im.s3.amazonaws.com/A43uLTTVusNQEEByRhbAJXEN?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19dd_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19dd_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9e9bd084b539cee09eb56f3d89da9a15d97e7828cb47d526fb447dbc7b776200)
Here is a vertical calculation of \(0.23 - 0.07\).
![Vertical subtraction.](https://cms-im.s3.amazonaws.com/G54w5mBLTSGggCQEhyrDR4Zb?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.20a_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.20a_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=bd460464644db2579d9b584c28032b8457ca2dbf53ec288937bccc510aa4b42c)
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.
![Base 10 diagram. 0 point 0 2 3. Two small squares in the hundredths column. Three small rectangles in the thousandths column.](https://cms-im.s3.amazonaws.com/5AxmNJZYnZMjU8Hi93QguVmQ?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19aa_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19aa_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=62cf91dd52dcdd0ae43462ee1e0ff03335c506e2e3655894391e7c0bd813b99b)
We want to remove 7 thousandths (7 small rectangles). We can “unbundle” (or decompose) one of the hundredths into 10 thousandths.
![Base 10 diagram.](https://cms-im.s3.amazonaws.com/vuM4NwwLs6HTdQpDczDZxLz3?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19bb_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19bb_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0c2f4fc8d514cda060bbc9bc43e29eec71ffd746c9397b10c5d638f79c41aab7)
Now we can remove 7 thousandths.
![Base 10 diagram.](https://cms-im.s3.amazonaws.com/mT2keB3z8zxC4Hd1dxSGY4LQ?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19cc_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19cc_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b698af22f62fd1dfaf885787fee2a9083c97e8df27b6e985200531af7b9d3910)
We have 1 hundredth and 6 thousandths remaining, so \(0.023 - 0.007 = 0.016\).
![Base ten diagram. 0 point 0 1 6. One small square in the hundredths column. 6 small rectangles in the thousandths column.](https://cms-im.s3.amazonaws.com/tvA4VipK2PicSc5wNWoeJck1?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.19dd_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.19dd_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=021effb101fbc6bb0a64ceb19454981e173e1dfb46b1a78c6badcd1a244693ab)
Here is a vertical calculation of \(0.023 - 0.007\).
![Vertical subtraction.](https://cms-im.s3.amazonaws.com/yZSqahFfBR1mew66NXJw7eo8?response-content-disposition=inline%3B%20filename%3D%226.5.B2.Image.20a_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B2.Image.20a_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T043009Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4004bf7ee3240e636622853a5e95b339a870c74067cf7575fcc42ffd31a508a6)