# Lesson 4

Explore Place Value Relationships (optional)

## Warm-up: Notice and Wonder: Maintain Your Balance (10 minutes)

### Narrative

The purpose of this warm-up is for students to recognize that a balance shows when two weights are equal. This will be useful when students look at weights representing the different decimal place values, providing students with a concrete way to think about the meaning of the digits in a decimal number and the multiplicative relationships between the place values (MP7). The weights will also be used to connect the word form of a decimal number to thousandths and the decimal form.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Each of the smaller weights weigh 0.001 of an ounce. The larger weight weighs one hundredth of an ounce. If there were 20 weights that each weighed 0.001 of an ounce on one side, what would need to be on the other side to stay balanced?” (2 hundredths, 1 hundredth and 10 thousandths, 20 thousandths)

## Activity 1: Balance the Weight (15 minutes)

### Narrative

The purpose of this activity is for students to examine relationships between the different decimal place values. In earlier lessons, students represented decimal numbers using words, fractions, diagrams, and symbols. The diagrams help to reveal that a thousandth is 1 tenth of a hundredth and a hundredth is 1 tenth of a tenth. In this activity, students systematically examine these relationships. For example, there are many different ways to represent 2 tenths. It is also 20 hundredths or 200 thousandths or 1 tenth and 10 hundredths. Through the idea of weights, students investigate these different equivalences. The weights give students a visual and physical way to reason about the different place values and their relationships (MP2, MP7).

The activity synthesis focuses on two key ways to balance a weight or represent a decimal number:

• use the least number of weights to represent a three-digit decimal weight, which is the expanded form students studied in the previous lesson (for example 3 tenths, 8 hundredths, and 5 thousandths).
• use only the smallest weights, thousandths, to represent a three-digit weight, which is the way the decimal number is usually said in words or written as an equivalent fraction (385 thousandths).
MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “What sets of weights will balance the nuggets?”. Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

### Launch

• Display the image of balance.
• “Balances are used to weigh things. On one side you put the object you want to weigh and on the other side you put weights. When the two sides balance, they have the same weight.”

### Activity

• 8–10 minutes: independent work time
• Monitor for students who explicitly use the values of the different decimal places in their reasoning by:
• thinking about the digits in the decimals separately
• understanding how to express tenths in terms of hundredths or thousandths, and hundredths in terms of thousandths

### Student Facing

For each problem, you have a balance and weights of 0.1 ounce, 0.01 ounce, and 0.001 ounce.

1. A gold nugget weighs 0.2 ounces.
1. What is one set of weights you could use to balance the nugget? Explain or show your reasoning.
2. What is another set of weights you could use to balance the nugget? Explain or show your reasoning.
3. How many 0.01 ounce weights would you need to balance the nugget? What about 0.001 ounce weights?
2. Another nugget weighs 0.385 ounce.
1. What is one set of weights you could use to balance the nugget? Explain or show your reasoning.
2. What is the smallest number of weights you can use to balance the nugget? Explain or show your reasoning.
3. What is the largest number of weights you can use to balance the nugget? Explain or show your reasoning.
3. Write a decimal number for the weight of the gold nuggets that balanced with:
1. 266 of the 0.001 ounce weights
2. 150 of the 0.01 ounce weights
3. 27 of the 0.1 ounce weights

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses for the weights they use to balance the 0.385 ounce gold nugget.
• “How do you know that 3 of the 0.1 ounce weights, 8 of the 0.01 ounce weights, and 5 of the 0.001 ounce weights will work?” (The 3 tenth ounce weights give the 3 from the decimal, the 8 hundredth ounce weights give the 8, and the 5 thousandth ounce weights give the 5.)
• “We can represent this with expanded form.”
• Display equation: $$0.385 = 3 \times 0.1 + 8 \times 0.01 + 5 \times 0.001$$
• “How do you know that 385 of the 0.001 ounce weights will also work?” (Because that is the same as 3 tenth ounce weights, 8 hundredth ounce weights, and 5 thousandth ounce weights.)
• Refer to the decimal, 0.385
• “How can you say or name this decimal number?” (Three hundred eighty five thousandths or 3 tenths, 8 hundredths, and 5 thousandths)

## Activity 2: Weights and Place Values (10 minutes)

### Narrative

The purpose of this activity is to provide further practice relating the different forms of decimals. This includes expanded form, word form, and decimal form. Using the balance and weight of gold nuggets as a context, students go back and forth between different ways of representing these weights (MP2). The synthesis highlights the meaning of the digits in a decimal and how that relates to the expanded form of the decimal.

### Launch

• Display the image.
• “What decimal can I write for the weight of the gold nugget?” (0.124)
• Write equation: $$0.124 = \left(1 \times 0.1\right) + \left(2 \times 0.01\right) + \left(4 \times 0.001\right)$$
• “How does the balance show the expanded form of 0.124?” (There is 1 tenth of an ounce weight, 2 hundredth of an ounce weights, and 4 thousandth of an ounce weights.)

### Activity

• 8–10 minutes: independent work time
• Monitor for students who see, in the first problem, that the digits in the decimal for the gold weights are the same as the number of weights for that decimal place value.

### Student Facing

1. Weights are used to balance some gold nuggets. Write the weight of each gold nugget in expanded form.
1. three 0.1 ounce weights, five 0.01 ounce weights, and eight 0.001 ounce weights
2. six 0.1 ounce weights and two 0.001 ounce weights
3. two 0.01 ounce weights and six 0.1 ounce weights
2. Here are the weights of some gold nuggets in word form. Write the weights in expanded form.
1. two hundred eighty three thousandths of an ounce
2. four hundred nine thousandths of an ounce
3. A gold nugget weighs 0.527 ounces.
1. What is the value of each of the digits in the decimal 0.527?
2. How does the expanded form of 0.527 show the value of each digit in the decimal?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share the expanded form of the decimal 0.527.
• Display the expression: $$\left(5 \times 0.1 \right) +\left(2 \times 0.01\right) +\left( 7 \times 0.001\right)$$
• “What is the value of the 5 in 0.527?” (5 tenths)
• “How does the expanded form show this?” (It shows the 5 is $$5 \times 0.1$$ or 5 tenths.)
• “What is the value of the 7 in 0.527?” (7 thousandths)
• “How does the expanded form show this?” (It shows the 7 is $$7 \times 0.001$$ or 7 thousandths.)
• “How is going from word form to expanded form different than going from decimal form to expanded form?” (The decimal form shows the place values. With the word form, everything is given in terms of thousandths, so I need to figure out what the individual place values of the number are.)

## Activity 3: Comparing Place Values with Weights (10 minutes)

### Narrative

The purpose of this activity is for students to use the weights from the previous activity to support place value understanding, specifically to see the multiplicative relationships between different decimal place values (MP7). These relationships will be taken up in greater detail in the next unit but the weights provide a convenient way to see these relationships which complements the diagrams students used in earlier lessons.

Students first compare weights of two gold nuggets, one weighed using 0.1 ounce weights and the other using 0.01 ounce weights. The two nuggets have the same weight because ten 0.01 ounce weights are equivalent to one 0.1 ounce weight, as students saw in the warm-up. Students move from here to making multiplicative comparisons between place values. They can use the weights to help visualize or calculate or they might use a diagram like those in the previous lesson.

Representation: Internalize Comprehension. Synthesis: Provide students with a partially completed graphic organizer, such as a two column table, to record the relationship between tenths and hundredths, hundredths and thousandths, and tenths and thousandths.
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 2 minutes: independent work time
• 8 minutes: partner work time
• Monitor for students who use different strategies to compare the values of the 6s in the gold nugget weights:
• using the value of each place in the decimal
• thinking about fractions or representing the decimals as fractions
• using a diagram such as a hundredths grid

### Student Facing

1. How many 0.01 ounce weights will balance one 0.1 ounce weight? Explain or show your reasoning.
2. How many 0.001 ounce weights will balance a 0.1 ounce weight? Explain or show your reasoning.
3. The table shows the weights of 3 of the gold nuggets Diego and his friends found panning for gold.

Fill in the blanks. Explain or show your reasoning.

gold weight
(grams)
nugget A 0.6
nugget B 0.06
nugget C 0.006
1. Nugget A weighs $$\underline{\hspace{0.9cm}}$$ times as much as Nugget B.
2. Nugget A weighs $$\underline{\hspace{0.9cm}}$$ times as much as Nugget C.
3. Nugget C weighs $$\underline{\hspace{0.9cm}}$$ times as much as Nugget B.
4. Nugget C weighs $$\underline{\hspace{0.9cm}}$$ times as much as Nugget A.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How are the weights of the nuggets the same? How are they different?” (They all have a 6 in them. They each balance with 6 of one of the unit weights. The value of the 6 is different for each nugget.)
• “How do hundredths, the weights for B, compare to tenths, the weights for A?” (There are 10 hundredths in each tenth or a tenth of a tenth is a hundredth.)
• “Nugget B weighs how many times as much as Nugget A? How do you know?” ($$\frac{1}{10}$$ because Nugget A is 6 tenths and a tenth of that is 6 hundredths since a tenth of a tenth is a hundredth.)
• “How do thousandths, the weights for C, compare to tenths, the weights for A?” (There are 100 thousandths in a tenth and one hundredth of a tenth is a thousandth.)
• “Nugget A weighs how many times as much as Nugget C?” (100, there are 100 thousandths in a tenth.)

## Lesson Synthesis

### Lesson Synthesis

“Today we investigated different ways to write decimal numbers by thinking about a balance and the different ways we can balance a given object. Here are the weights that balance two gold nuggets.”

Display the table:

gold weight
nugget 1 two 0.1 ounce weights
nugget 2 twenty 0.01 ounce weights

“How many ounces do the two gold nuggets weigh?” (2 tenths and 20 hundredths)

“How do you know those two weights are equivalent?” (1 tenth is 10 hundredths so 2 tenths is 20 hundredths)

“How many thousandth ounce weights would you need to balance each of these nuggets?” (I would need 200 thousandth weights because 10 of them is a hundredth and so 20 hundredths is 200 thousandths.)

Display the decimals: 0.2, 0.20, 0.200

“What do you notice? What do you wonder?”(I notice that they all have the same value, 2 tenths, 20 hundredths, 200 thousandths. I wonder why there are so many different ways to express a decimal number.)

## Cool-down: Worth its Weight in Gold (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.