Lesson 19

In this warm-up, students are asked to name the shaded portion of a 10-by-10 grid, which is equal to 1. To discourage students from counting every square, flash the image for a few seconds and then hide it. Flash it once more for students to check their thinking. Ask, “How did you see the shaded portion?” instead of “How did you solve for the shaded portion?” so students can focus on the structure of the fractional pieces and tenths in the image. Encourage students to name the shaded portion in fractions or decimals. Some students may also bring up percentages.

Tell students you will show them a 10 x 10 grid for 3 seconds and that the entire grid represents 1. Their job is to find how much is shaded in the image and explain how they saw it.

Display the image for 3 seconds and then hide it. Do this twice. Give students 15 seconds of quiet think time between each flash of the image. Encourage students who have one way of seeing the grid to consider another way to determine the size of the shaded portion while they wait.

Invite students to share how they visualized the shaded portion of each image. Record and display their explanations for all to see. Solicit from the class alternative ways of quantifying the shaded portion and alternative ways of naming the size of the shaded portion (to elicit names in fractions, decimals, and percentages). To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone solve the shaded portion the same way but would explain it differently?”
• “Did anyone solve the shaded portion in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

Students are given cards, each of which contains an original price and a new price, as shown.

Their job is to sort the cards into two piles: one pile for deals they would take, and another for those they would reject. There are many paths students could use to reason about whether or not to accept a deal. For example, if the original deal was \$3.50 for 10 juice boxes and the new deal is \$2.40 for 6 juice boxes, they could:

• Find and compare the unit rates for both the original pack and the new pack. If the unit rate is the same, the deal is fair. If the unit rate is lower, the clerk is offering a discount. If the unit rate is higher, the clerk is not being fair.

  number of juice boxes	cost in dollars	dollars per box
  10	3.50	0.35
  6	2.40	0.40

• Find the unit rate in the original pack, apply it to the number of items in the new pack, and compare the costs for the same number of items in the original and new pricing schemes. This can be done in two ways, one focused more on column reasoning and the other on row reasoning, as shown.

  

  

  

  

• Use an abbreviated table and bypass calculating the unit rate. Find the multiplier to get from the original to the new number of items, and use the same multiplier to find what the price would be if the deal has not changed. Compare the actual new price to this projected price.

  

  

As students work, attend to how they reason about the deals and make their decisions (deal or no deal).

Show the picture on Card A (or use an actual 4-pack of a beverage with a missing bottle.) Present the following situation:

“You’ve entered a local shop to buy a 4-pack of drinks. You find one last pack of the drink you want on the shelf and, unfortunately, only 3 bottles remain in that pack. You decide to buy it anyway. You take the 3-pack to the check-out counter and ask the clerk to consider a fair price for the incomplete pack. If the cost of a 4-pack was $3.16 and the clerk offers to sell the 3 pack for $2.25, will you take the deal?”

Poll the class for their response and display how many students would and would not take the deal. Then, ask “How could you figure out if the deal is good or not?” Give students a moment of quiet think time to come up with strategies for solving such a problem and then invite a few students to share.

Arrange students in groups of 2. Give each group a set of five cards A–E (or six cards A–F if including the extension problem). Tell students their job is to sort the cards into a ‘Deal’ pile and a ‘No Deal’ pile. Instruct partners to collaborate in finding the answer for card A and divide up the remaining cards between them. Ask students to first work on their cards individually, then share their reasoning with their partner, and lastly, sort the cards together. 

Students who are not fluent in multiplication and division computation work from grade 5 may need some review in order to be successful in this activity.

Select 2–3 students who used different but effective strategies to share their thinking with the class. Encourage students to listen to others’ reasoning. Record the different strategies in one place and display them for all to see. Highlight any similarities and differences (e.g., whether a unit rate was used, whether students compare the original unit rate to the new quantity or the other way around, etc.)

In this activity, students convert between customary and metric units in order to compare lengths. To make some measurements comparable to others, students need to perform multistep conversions and activate arithmetic skills from previous grades. Support students with computations as needed and provide access to calculators as appropriate. Share the following information with students when requested.

• 1 mile = 1,760 yards = 5,280 feet
• 1 yard = 3 feet
• 1 foot = 12 inches
• 1 kilometer = 1,000 meters
• 1 meter = 100 centimeters

Expect students to choose different units of measurements to make comparisons. As students work, identify those who opt for the same unit so that they can partnered or grouped together for discussion.

Give students 1–2 minutes to read the task, and then ask how they think they could compare these lengths. Students are likely to suggest converting all the lengths into the same unit of measurement. Ask students which units might be appropriate in this case and why. (Feet, yards, and meters are better choices than inches or miles.) After discussing some appropriate options, give students quiet think time to complete the activity, and then time to share their explanation with one or more students who have chosen to use the same unit of measurement.

Some students may need to be prompted about the intermediate steps needed to compare units that require several conversions before they can be compared.

Poll the class to see if they agree on who took first, second, third, and last place. If there is widespread agreement, invite two students to share: one student who converted all measurements to feet or yards, and another who converted everything to meters. If there are discrepancies, list the distances run by each animal in each unit of measurement and display them for all to analyze and double check. While the numerical values of the measurements in feet will all be greater than those in meters, the rank order will come out the same.