Students will determine the listening time of an audiobook read a 1.5x the speed of the original listening time.

As is true for any task, the intentionality or learning objective can vary depending on what mathematical thinking you are hoping to elicit. The purpose of this task is to engage students in problem solving with rates of change while strengthening their understanding of the connection among fractions, decimals and percent and multiplying and dividing fractions.

Show students the video below of two instances of an audiobook being read at different speeds, side-by-side.

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

For example, students might notice and wonder the following:

• I notice that the book will take 21 hours and 36 minutes to listen to.
• I wonder what this has to do with math class.
• I noticed that there’s pictures of the same book.
• I wonder why we’re seeing two of the same book.
• I notice that one side the book is read faster than the other.
• And many others…

Document what students notice and wonder with you on chart paper, whiteboards, or other visual displays in your classroom.

Then, ask students:

What speed do you think the audiobook was being played during the second half of the video?

Make an estimate and share with a neighbour.

You’ll notice in the Spark tab that there is also a screenshot there for students to reference.

As we typically do when following the Curiosity Path and asking students to first estimate before solving a problem, we want to encourage students to consider quantities that are unreasonably high or unreasonably low.

For example, some students might say that a speed of 1.05x the original reading speed is too low because it would be too hard to notice a difference. Other students might say that a speed of 20x would be way too high because you wouldn’t even be able to understand what they were saying.

Facilitator Note:

Encourage students to get more risky with their range of unreasonable values to try and narrow down the reasonable range they might use to land on an estimate. If students say “1.05x”, ask students to explain what that value means as a percentage. Students who are flexible with fractions, decimals, and percentages will be able to say something like “5% faster” using additive thinking or “105% as fast as” using multiplicative thinking.

Others might be able to say “an additional 20th the speed” recognizing that 5% = 1/20 and using additive thinking.  A multiplicative thinker using a fraction to describe the same increase in speed might say “21 twentieths (or 21/20) the speed of the original”.

Trying to draw on these discussions and even considering modelling using number lines or bar models is a worthwhile way to spend some of this estimation time to leverage and grow number fluency and flexibility.

If you’re looking for more on additive and multiplicative thinking as well as building number sense in your students, consider The Concept Holding Your Students Back course.

Once students have shared their estimates and you’ve recorded them on a numberline on the chalk board, whiteboard, or chart paper, you can share the estimation reveal video to help lead us to our Sense Making Prompt for the day.

Students should now be aware that the first “play” of the audiobook was at normal speed (1x) and the other was at a faster speed, (1.5x).

The prompt we will land on to Fuel Sense Making for this lesson is:

How long will it take to listen to the audiobook at 1.5x the speed of the original?

It should be noted that our prompt is being stated using multiplicative thinking, whereas some students may misinterpret additively.

Have students estimate the length of time it would take to listen to the book at 1.5x speed using friendly numbers. Have students share their estimates with a class elbow partner and then to the whole class.

Provide images highlighting the original length of the book at normal speed (1x) and the unknown listening time at the new faster speed (1.5x).

Give students an opportunity to turn and talk with a partner and discuss what strategies they may need to solve this. Ask students to use this information to update their estimates.

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After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened with this video.

Answer: 14 hours, 24 minutes.

Revisit the student responses. If student approach #3 didn’t emerge amongst your students, lead a discussion to connect listening speed, fraction of listening time, listening time as a percentage, and final listening time.

Return to the questions you recorded from the Notice and Wonder protocol, and answer any questions that have not been answered.

Consider extending this task with one of the following possible extensions or another extension that you pre-plan prior to delivering this lesson:

Extend #1:

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Extend #2:

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