Lesson 14
Using Diagrams to Represent Addition and Subtraction
14.1: 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.
- \(34.56000 = 34.56\)
- \(25 = 25.0\)
- \(2.405 = 2.45\)
14.2: Finding Sums in Different Ways
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Here are two ways to calculate the value of \(0.26 + 0.07\). In the diagram, each rectangle represents 0.1 and each square represents 0.01.
Use what you know about base-ten units and addition of base-ten numbers to explain:
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Why ten squares can be “bundled” into a rectangle.
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How this “bundling” is reflected in the computation.
The applet has tools that create each of the base-ten blocks. Select a Block tool, and then click on the screen to place it.
One
Tenth
Hundredth
Click on the Move tool when you are done choosing blocks.
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Find the value of \(0.38 + 0.69\) by drawing a diagram. Can you find the sum without bundling? Would it be useful to bundle some pieces? Explain your reasoning.
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Calculate \(0.38 + 0.69\). Check your calculation against your diagram in the previous question.
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Find each sum. The larger square represents 1, the rectangle represents 0.1, and the smaller square represents 0.01.
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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.
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If you had 500 violet jewels and wanted to trade so that you carried as few jewels as possible, which jewels would you have?
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Suppose you have 1 orange jewel, 2 yellow jewels, and 1 indigo jewel. If you’re given 2 green jewels and 1 yellow jewels, what is the fewest number of jewels that could represent the value of the jewels you have?
14.3: 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 represent collections of base-ten units—tens, ones, tenths, hundredths, etc. We can use them to help us understand sums of decimals.
Suppose we are finding \(0.08 + 0.13\). Here is a diagram where a square represents 0.01 and a rectangle (made up of ten squares) represents 0.1.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/pubrE26H5izVogThEq6PEmYs?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.22a-01%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.22a-01%2520%25281%2529.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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7f85e723a204ffcc80da6542f5812e596b4d9c30e6ce7536748381cef6a7987d)
To find the sum, we can “bundle” (or compose) 10 hundredths as 1 tenth.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/pMSRcX2cWNuCuc6QnBPRthne?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.23a_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.23a_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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6eb11e3a18da59e619625c85acd4179bc2bc1a3e215a75bed8bf1be170fdd08a)
We now have 2 tenths and 1 hundredth, so \(0.08 + 0.13 = 0.21\).
![Base ten diagram. 0 point 21. Two rectangles. 1 small square.](https://cms-im.s3.amazonaws.com/BGfHABxzfXtSWZp3RWwd7Ree?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24a_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24a_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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1ecb5735411392d4791cbda8adea906d078995a3b983f6053b7e6b5224f5ba07)
We can also use vertical calculation to find \(0.08 + 0.13\).
![Vertical addition. First line. 0 point 13. Second line. Plus 0 point 0 8. Horizontal line. Third line. 0 point 21. Above the 1 in the first line is 1.](https://cms-im.s3.amazonaws.com/EHxAD8TAEfBVKMfUebt2kaL5?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24b%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24b%2520%25281%2529.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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dcc50d7ffd99c778ff9c4349fb39a3f880a81a3206bc649b1a23de8f3ee4ce13)
Notice how this representation also shows 10 hundredths are bundled (or composed) as 1 tenth.
This works for any decimal place. Suppose we are finding \(0.008 + 0.013\). Here is a diagram where a small rectangle represents 0.001.
![Base 10 diagram.](https://cms-im.s3.amazonaws.com/PW6d51pwRZXaZ1Dixp2BFftn?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.22a-02%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.22a-02%2520%25281%2529.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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e62b7dedd6c8f7cc27a826943d9f3beeaf76159d10badefe9d5f05332161bcf7)
We can “bundle” (or compose) 10 thousandths as 1 hundredth.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/V8b16XxJDRMY4NueoVFoEie1?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.23a_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.23a_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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d61ad2e65b9878cfb5d45544d567b9ab1687c76b41250797d4076f7dc27685a1)
The sum is 2 hundredths and 1 thousandth.
![Base ten diagram. 0 point 0 2 1. Two small squares. 1 small rectangle.](https://cms-im.s3.amazonaws.com/t8gDJGBMHbCqsoLUihzBnqC7?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24a_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24a_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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e5ecf8f1d2f9252b6cb98b70d360b5d9725b60b2ab50f136f3557a85c098dbdc)
Here is a vertical calculation of \(0.008 + 0.013\).
![Vertical addition.](https://cms-im.s3.amazonaws.com/EMbjxiiKKCWiT8zCHkWZ2BXq?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24b%20%25282%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24b%2520%25282%2529.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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0d59404ae2893337efd2c72e518fea9a87d6019c44ff7a6f02e18c767784b53a)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f2567bb04f1857d9bdffd8d0ebbba03d980f2ecbf252fd721af76d1e8aadb6ba)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c36ebee9659e2aff1145a5aaf4624fea901f767bb1c7923472d3002bbcee7299)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4a6d067f7e52a6a7fe97ab39000cd29895652e365e9b461d4518a38c72c61a36)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1a07b782f542555f85b7943553c9346721ad0b0ec678e973ce263c9c98a15ccb)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da1fa5a9cee4e372024b2c0ea9d2fe423e9f6da1287cc4b54476fb4e90307e7e)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=22e61daf90b449f0b6b22f74e77e280713ca364c717e9136e74a889c80926bc5)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=04ac2cdb0cf4141c13e003552f71852b2ad536702455b6150ef073ef628be15f)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=de29999e51ed98df0c1ce7b330e08a5c9b2a830a8949ad5244531b2005ad7970)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2617d5fda8b16c2632cfdb8fa6d56d85446de5ce9525698edde164d18a0ab635)
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=20240727T002655Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8987a32e0d9ad08241ee1120995757db2d63ea36c1c0f386f94b1a0a13928844)