# Lesson 10

Solve Problems with Decimals

## Warm-up: Notice and Wonder: The Luge (10 minutes)

### Narrative

The purpose of this Notice and Wonder is for students to consider the sport of luge and give them some numerical data that they will work with later in the lesson. The times and top speeds have been created and do not represent actual times from an event. The table is not labeled in order to encourage students to think about the meaning of the numbers.

### 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?

A B
48.532 82.13
48.561 82.75
48.626 82.81
48.634 83.07
48.708 82.80

### Activity Synthesis

• “The person in the picture is performing a sporting event called the luge. Athletes go down a steep ice track on a sled.”
• “The numbers on the left are the times, in seconds, it took different athletes to complete the course. The numbers on the right are the maximum speed, in mph.” Consider labeling the columns of the table.
• Invite students to share what they notice about the numbers, including that in the first column, they get larger from top to bottom and they all have 3 decimal places. In the second column, there are only two decimal places and the numbers are not in increasing or decreasing order.

## Activity 1: How Accurate Is It? (20 minutes)

### Narrative

The purpose of this activity is for students to investigate a situation in which knowing a value to the thousandth place is important. Many high speed athletic events such as sprinting, cycling, downhill skiing, and the luge (studied here), are measured to the thousandth of a second in order to distinguish athletes whose finish times are very close to one another. Students examine the finishing times for the luge athletes, introduced in the warm-up, and what would happen if the times were only measured to the nearest hundredth of a second, tenth of a second, or second.

Students may use number lines to help answer the questions, but as in the previous lesson, will need to think carefully about how to label the number line.

MLR5 Co-Craft Questions. Keep books or devices closed. Display only the table, without revealing the questions. Give students 2–3 minutes to write a list of mathematical questions that could be asked about this situation, before comparing their questions with a partner. Invite each group to contribute one written question to a whole-class display. Ask the class to make comparisons among the shared questions and their own. Reveal the intended questions for this task and invite additional connections.
Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Invite students to share activities that they have competed in, participated in, or watched in which athletes’ speed determined their victory.
Supports accessibility for: Memory, Attention

### Launch

• Groups of 2
• Display a stopwatch that records time to the hundredth of a second.
• “What can you do in one second?” (stand up, wave my hand, say my name)
• “What can you do in one tenth of a second?” (blink, type one letter)

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• use place value understanding to round the numbers
• plot the numbers on a number line to round them

### Student Facing

athlete time (seconds) speed (mph)
Athlete 1 48.532 82.13
Athlete 2 48.561 82.75
Athlete 3 48.626 82.81
Athlete 4 48.634 83.07
Athlete 5 48.708 82.80
1. How would the results of the race change if the times were recorded to the nearest second?
2. How would the results of the race change if the times were recorded to the nearest tenth of a second?
3. How would the results of the race change if the times were recorded to the nearest hundredth of a second?
4. An athlete recorded a time of 48.85 seconds to the nearest hundredth of a second. What are the possible times of this athlete recorded to the thousandth of a second?
5. An athlete recorded a time of 48.615 seconds to the nearest thousandth of a second. What are the possible times that this athlete recorded to the nearest hundredth of a second?

### Activity Synthesis

• Ask previously identified students to share their responses.
• “How does rounding the times to the nearest second impact each of the athletes?” (It makes all of the times greater and impossible to distinguish. It impacts the fastest athletes the most as their times are shifted up the most.)
• “How does rounding the times to the nearest tenth of a second impact each of the athletes?” (It makes the times of the 1st, 3rd, and 4th athletes faster and the times of the 2nd and 5th athletes slower. It makes the second athlete tie for second place instead of winning second place.)
• Display the image from the solution or a student generated image.
• “How can you use the number line to find the times to the thousandth of a second that round to 48.85 seconds?” (I can label the tick marks and then take the ones that are closest to 48.85 and the one halfway between 48.84 and 48.85.)

## Activity 2: Compare Speeds (15 minutes)

### Narrative

The purpose of this activity is for students to order decimals and examine the effect of rounding on numbers continuing to use the luge context. In this activity, students investigate the top speeds of the athletes. In this case, the numbers are not listed in decreasing order because the top speeds do not correspond to the fastest times. Students order the top speeds before and after they have been rounded. Then they find a speed between two given speeds when the thousandths place is added. Since the speeds of the riders are not given to the thousandth, students will need to create values for the riders. There is one set of values students could pick, namely 82.804 and 82.805, where there is no value in between to the thousandth. If students choose these values, ask “Are there different possible top speeds for these athletes?”

### Launch

• Groups of 2
• “We will now look at the top speeds that the different athletes recorded.”
• Highlight how fast these speeds are: most speed limits on freeways are between 65 and 75 mph and the athletes are only inches from the ice.

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time

### Student Facing

The table shows the top speeds, in miles per hour, of 5 luge athletes:

athlete speed (miles per hour)
Athlete 1 82.13
Athlete 2 82.75
Athlete 3 82.81
Athlete 4 83.07
Athlete 5 82.80

1. List the top speeds of the athletes in decreasing order.
2. Do any of the athletes have the same top speed rounded to the nearest tenth of a mile per hour? What about rounded to the nearest mile per hour?
3. There was a sixth athlete who was faster than the rider at 82.80 mph, but slower than the rider at 82.81 mph. What could the speeds of the 3 athletes be if all measured to the nearest thousandth of a mile per hour?

### Activity Synthesis

• “Are there different speeds the athlete at 82.80 mph could have, measured to the nearest thousandth of a mile per hour?” (yes)
• “What is the greatest? The least?” (82.804, 82.795) “What about for the athlete at 82.81? What are their fastest and slowest speeds to the thousandth of a mile per hour?” (82.814, 82.805)
• Invite students to give a possible set of top speeds, to the thousandth of a mile per hour, for athletes 3, 5, and 6.

## Lesson Synthesis

### Lesson Synthesis

“Today we studied numbers that represented times and top speeds of luge riders and how they are affected when rounded to different places.”

“What are some reasons to round numbers?” (It gives a general idea of the size of a number. It’s easier to understand how big a number is when it is a round number.)

“What are some reasons to keep numbers unrounded?” (If we need to know the exact size of the number then it can be important not to round it. If we want to compare two numbers, then we may need more digits to decide which is larger.)

“How is rounding decimals the same as rounding whole numbers?” (I need to think about place value and then find the closest hundredth or tenth or one just like I would look for the nearest ten, hundred, or thousand for whole numbers.)

## Student Section Summary

### Student Facing

In this section, we represented decimals to the thousandths place.

The shaded region of the diagram represents 0.542. The 5 shaded rows are each a tenth or 0.1, the 4 shaded small squares are each a hundredth or 0.01, and the 2 shaded tiny rectangles are each a thousandth or 0.001. The decimal 0.542 can be represented in other ways

• $$\frac{542}{1,000}$$
• five hundred forty-two thousandths
• $$(5 \times 0.1) + (4 \times 0.01) + (2 \times 0.001)$$

We can also locate 0.542 on a number line.

The number line shows that 0.542 is closer to 0.54 than to 0.55 so 0.542 rounded to the nearest hundredth is 0.54.