NOTICE Lab - Video Club 4: Mangoes

Video Club 4: Mangoes

In this video, a group of students discuss the "Mangoes" problem. There are several clips in the classroom video. In the first clip, two students explain how they did the problem and try to answer the questions of a third student. They evoke the idea of moving backward and relate it to retracing steps in everyday life, like going through your internet search history. In the second and third clip, students explain their thinking about the problem using blocks.

This video club could be completed in one longer session or two shorter sessions. A good place to split the video viewing is at 4:07. We suggest stopping at this point and having a discussion about the first part of the video and then watching the second part of the video and having a discussion.

Video Club Overview

Video Club Goals:

- Teachers will notice how students utilize the idea of “undoing” to think about the problem.
  1. Teachers will notice student’s informal examples that help him to explain undoing.
- Teachers will notice how students utilize the idea of “undoing” to think about the problem.
  1. Teachers will attend to and interpret student’s informal examples that help him to explain undoing, like search history.
  2. Teachers will formulate questions to push student’s thinking further.

Materials:

- Video loaded on Anotemos and shared with all participants
- If watching in person, each participant should have a device on which they can view the video and headphones

Get ready (20 mins). To prepare participants for viewing the video, review the problem. Give participants a few minutes to reason about the problem themselves. Then have a whole group discussion that gives participants the opportunity to share their explanations. The Mangoes problem:

Hummingbird Middle School offers a mid-afternoon “grab and go” snack that their students can get in the cafeteria during their study hall. The best snack day is mango day!

When Terrell got to the cafeteria he found only one bowl full of mangoes left. He was really hungry so he took ⅙ of the mangoes in the bowl. A few minutes later, Hana walked in the cafeteria and took ⅕ of the mangoes that Terrell had left in the bowl. Still later, Gabriela took ¼ of the remaining mangoes. Even later, Sahil took ⅓ of what was left. Finally at the end of study hall, CJ took ½ of the remaining mangoes. Only 3 mangoes remained.

How many mangoes were originally in the bowl?

Exploring the Math

There are a number of ways that students could do this problem:

Guess and check: students might try to guess total number of mangoes and then follow the directions in the problem to check their answer, aiming to end up with three mangoes. Some students might start with a multiple of six, realizing that they first have to divide by six. Students might also revise their guess higher or lower than, based on how many mangoes they end up with at the end. For example, if they start with 24 mangoes, they will end with 4. As a result, they might try 18 next because it is the next smallest multiple of 6.

Draw a picture: Students could also draw a picture and use it to depict what happens in the problem. For example, a student might draw a rectangle or a circle and divide it into sixths, and take one piece away. They subsequently take another piece away, which makes ⅕, ¼, ⅓, and finally ½. They might see that 1 piece is worth three mangoes, meaning that all six pieces are worth 18 mangoes.

Working backwards: Another possibility is that students start with the three mangoes and then work backwards to get to the original answer. With this method, students often ask themselves questions like, “What is three one-half of?” The answer is 6. Students can continue in this way, next asking “What is 6 two-thirds of?” until they get back to 18 original mangoes.
- A common mistake with this method is to work backward by multiplying quantities by the denominators of each fraction in the problem. For example, instead of asking “What is 6 two-thirds of?” students might multiply 6*3, thinking that will give them the previous value. This would be true if ⅔ of the remaining mangoes had been taken.

Using an equation: A student might also use an equation to solve. If x is the original number of mangoes, then Terrell took ⅙x. x-(⅙x)=⅚x.
- Next, Hana took ⅕(⅚x)=(5/30)x=(⅙)x. ⅚x-⅙x=(4/6)x
- Next, Gabriella took (¼)(4/6)=⅙x. (4/6)x-⅙x=(3/6)x
- Next, Sahil took (⅓)(3/6)=⅙x. (3/6)x-⅙x=(2/6)x
- Finally, CJ took (½)(2/6)=⅙x. (2/6)x-⅙x=(1/6)x
- We know ⅙x=3 so x=18 mangoes.

There are also a number of Seeds of Algebraic Thinking that students could activate while solving this problem:

Part to whole relationships- part to whole relationships are a precursor to proportional thinking. In this problem, students think about a “whole” in terms of different quantities of mangoes. For example, if students use a drawing of a rectangle or a circle they might divide that whole shape into six pieces. In this case, each piece of the whole actually represents three mangoes. Students have to figure out how taking parts of that whole thing translates into taking and leaving a certain number of mangoes.

Relational thinking?- Students think about how different quantities in the problem are related to each other and change depending on who takes mangoes. Students might realize that, regardless of the child in the problem who takes mangoes, they will always take three. This happens because as the children in the problem take larger fractions, the number of leftover mangoes they take from gets smaller. As a result, they are always taking ⅙ of the original amount of mangoes.

Undoing- If students work backward in this problem, they will “undo” the end result (3 mangoes) to get back to the original number of mangoes (18 mangoes). Along the way, students need to think about how to get back to the previous number of mangoes in each step and might ask themselves questions like, “What is 9 three-fourths of?” In this case, they might think about adding back mangoes to get to 4/4. They might also turn the problem into an equation like 9=¾x.

Repetition- Depending on how students attempt this problem, it has a number of repeated processes like repeatedly dividing, multiplying, or adding. From a young age, students are familiar with repeated procedures in their own daily lives, like eating three meals a day and brushing their teeth in the morning and at night.

Viewing and Discussing the Video

Have participants watch the approx. 8 minute video clip individually on the Anotemos platform. Participants should be encouraged to draw and comment on the video when they see interesting moments of student thinking.

Discussion: It might be helpful to pause the video after the first video clip (stop at 4:07) and have a discussion about productive everyday student resources, even though the students do not have the right answer to the problem. After the first discussion, the video club group could continue and watch the second and third clips. The second discussion could focus on comparing the student thinking in the second and third clips or the video as a whole. One key part of the second and third clips is focusing on how students use manipulatives to work backward and forward through the problem.

Description of Goal:

Teachers will notice how students utilize the idea of “undoing” to think about the problem.
1. Teachers will notice student’s informal examples that help him to explain undoing.

Teachers will notice how students determine whether different answers to the problem are reasonable.

Table - Video Key Moments

VC4 table

Potential Discussion Questions:

What strategy are Aiden and Felix using to solve the problem?
1. Why does it make sense to them?
2. What questions might you ask to help Aiden and Felix use their strategy more productively?
What does Nadia seem to understand about the problem? How might you support Nadia’s reasoning?
How are the students judging if their answers are reasonable?

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