3.1: Do the Zeros Matter?

Evaluate mentally: $1.009+0.391$

Decide if each equation is true or false. Be prepared to explain your reasoning.
a. $34.56000 = 34.56$
b. $25 = 25.0$
c. $2.405 = 2.45$
Let’s add and subtract decimals.
Evaluate mentally: $1.009+0.391$
Decide if each equation is true or false. Be prepared to explain your reasoning.
a. $34.56000 = 34.56$
b. $25 = 25.0$
c. $2.405 = 2.45$
Andre and Jada drew baseten diagrams to represent $0.007 + 0.004$. Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.
Here are two calculations of $0.2 + 0.05$. Which is correct? Explain why one is correct and the other is incorrect.
Click on the Move tool when you are done choosing blocks.
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.
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.
Do you agree that either diagram correctly represents $0.4  0.03$? Discuss your reasoning with a partner.
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.
Be prepared to explain your reasoning.
Click on the Move tool when you are done choosing blocks.
Subtract by deleting with the delete tool, not 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.
Baseten 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.
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).
We now have 1 hundredth and 13 thousandths, from which 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 $0.023  0.007$.
In both calculations, notice that a hundredth is unbundled (or decomposed) into 10 thousandths in order to subtract 7 thousandths.