In this activity, students calculate quotients of decimals in the context of length and distance but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

The problems allow students to practice dividing decimals by making use of the structure they observe in the numbers, or by multiplying both the divisor and the dividend by an appropriate power of 10. The questions in Problem Card 1 reiterate the idea of division as a way to answer questions about the size of a group and the number of groups. Those in Problem Card 2 revisit concepts about rate and constant speed. Students may choose to reason with a ratio table, but it would be less efficient than dividing directly, and students would still need to divide decimals somewhere along the way.

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “To solve the problems about the blue rope and red rope, did you multiply the dividend and divisor by powers of 10 before dividing, or did you reason another way?”
• “How did you know what information to ask for? Once you have the information, how did you go about finding the tortoise’s speed in miles per hour?”
• “How did you find the time it would take to travel 4.86 miles?”

Highlight for students that it may not always be necessary to multiply the dividend and divisor by a power of 10 when a division involves one or more decimals. In some cases, we can reason about the relationship between the divisor and dividend directly. For example:

• When finding , we can first think of and then divide the result by 10, instead of calculating .
• When dividing , we can use partial quotients: First find , which is 100, and then find , or how many 5 hundredths are in 75 hundredths, which is 15.

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Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”

• “I picked the norm ‘.’ During the activity, I realized the norm ‘’ would be a better focus because .”

In this activity, students use arithmetic with decimals to study the 110-meter hurdle race. The first question prompts students to draw a diagram to capture and make sense of information about measurements in a hurdle race. The last two questions prompt students to calculate the distance between hurdles and the size of a runner’s stride, both of which involve subtracting and dividing decimals.

Throughout the activity, students need to make sense of problems and persevere in solving them (MP1). As they represent given measurements diagrammatically, use them in calculations, and interpret the results in terms of the situation, students engage in abstract and quantitative reasoning (MP2).

As students work, monitor for a variety of diagrams and reasoning strategies. Select students who organize their understanding of the situation and reasoning in different ways, and ask them to share later.

Students may not realize that there are only 9 spaces—not 10 spaces— between 10 hurdles, leading them to miscalculate the distance between hurdles. Urge students to study the number of spaces in their diagram, or ask them to think about how many spaces are between 2 hurdles, 3 hurdles, 4 hurdles, and so on. and extend the pattern to 10 hurdles.

A calculation error in dividing may lead to a quotient with a non-terminating decimal. Look out for arithmetic errors when students calculate the distance between the first and last hurdles (82.26 meters) and when students perform division. If students end up with a non-terminating decimal for their answer, ask them to revisit each step and see where an error might have occurred.

Select one or more previously identified students to share their diagrams for the first question. Ask other students if theirs are comparable to these, and if not, where differences exist. If not mentioned by students, be sure to highlight proper labeling of the parts of the diagram. Then, discuss the second and third questions. For the second question, discuss:

• How many ‘gaps’ are there between the hurdles? (9)
• What is the distance from the first hurdle to the last hurdle? (82.26 meters)
• What arithmetic operation is applied to the two numbers, 9 and 82.26? Why? (Division, because the 10 equally spaced hurdles divide 82.26 meters into 9 equal groups.)

For the third question, discuss:

• How far does the runner go in three strides? How do you know?
• Is meters (the exact answer) an appropriate answer for the question? Why or why not? (Most likely not, because runners are not going to control their strides and jumps to the nearest centimeter. About 2 meters would be a more appropriate answer.)

Consider asking a general question about hurdle races: Is it important for the runner that the hurdles be placed as closely as possible to the correct location? The answer is yes, because runners train to take a precise number of strides and to hone their jumps to be as regular as possible. Moving a hurdle a few centimeters is unlikely to create a problem, but moving a hurdle by a meter would ruin the runners’ regular rhythm in the race.

In this activity, students apply their understanding of decimals to solve problems in another sporting context. They study the measurements of a tennis court and answer questions about lengths and area.

Visually, it appears as if each half of the tennis court (divided by the net) is a square. Similarly, it appears as if the service line is about halfway between the net and the baseline. Calculations show that in both cases, however, neither half of the tennis court is a square, and that the service line is not half way between the baseline and the net.

Just as with the distances between hurdles in an earlier activity, the measurements of a tennis court are very precisely determined. It is also very important for professional tennis players that the courts on which they play have consistent measurements, as the smallest differences could affect whether the ball is in or out.

There are some subtleties related to measurement in the real world and the idealized version in the task. On a tennis court, the lines have width. For the first two questions of the task, the strips can be taken as dimensionless lines. In the final questions, students deal explicitly with these strips, an opportunity to attend to precision (MP6).

Focus the discussion on how students use calculations involving decimals to answer each question.

Before discussing each question, poll the class on whether they concluded that each half of the court is a square, the service line is halfway between the baseline and the net, and the court is larger or smaller than it should be given a painter’s mistake. Record and display the responses for all to see.

Consider displaying a picture of a tennis court from directly above, as shown, to facilitate discussion. Though it is possible to judge from the picture that the service line is closer to the baseline than it is to the net, it is not as easy to tell, just by looking, whether each half of the court is a square.

Ask students to share their responses and reasoning. Record or display their calculations for all to see. Invite other students to agree or disagree, or to offer alternative ways to reason.

If time permits, consider asking questions such as:

• “Would the painter’s mistake change your answers to the first two questions?” (No, the sides of the quadrilaterals would all increase by 10 cm and they still would not be squares.)
• “Would you notice the painter’s mistake if you were playing on the mis-painted court?” (A professional player may notice, but an unseasoned player may not because 5 cm is very little compared to 11 meters and 24 meters.)