How many cards do we need to display the dates?
From APEC HRDWG Wiki
Teacher Akihiko Takahashi's lesson “How many cards do we need to display the dates of the month on the wall?” was captured on video for the APEC Education Network (EDNET) project called Classroom Innovations through Lesson Study. The lesson is an example of using the Lesson Study process of professional development in the teaching of Mathematics.
Lesson Overview
Understanding the base-ten place-value system is one of the most important objectives for the early grades. This notation system is based on the principles of grouping and place-value. In this system, objects are grouped by ten so that only ten symbols, the digits 0 through 9, are needed to represent any number of the objects. In order to represent any possible numbers in this system, the place that the digit appears in the numeral has a particular size for the group. For example, the ones place represents how many of ones are there, tens place represents how many tens are there, and so on. Thus the 5 in 523 represents five hundreds and the 5 in 675 represents five ones. This means that the same digit can represent different quantities based on its place in the number. Unless students understand that the each place in the numeral holds different value, it is very difficult for students not only to understand or express numbers by using notation system, but also to understand how addition, subtraction, multiplication, and division work.
In order to deepen students’ understanding of the concept of place value notation, this lesson is designed to help students think about how their previous learning of the base-ten place-value system can be used to solve problems in students’ everyday life. Through problem solving, students are expected to deepen their understanding the concept of place value as well as to develop problem-solving skills. It is also expected that the solution to this problem — only six cards are sufficient to display the dates of month everyday throughout a year — could generate students’ interest in using mathematics in their life.
This lesson was taught by Akihiko Takahashi during the Chicago Lesson Study Conference, May 9, 2009.
The goals of the lesson are:
- Students will deepen their understanding of the concept of place value notation through solving a problem related to children’s everyday life.
- Students will understand that organizing their thinking processes and thinking logically are important for problem solving.
[Demonstration Practice  ] (see Resource Types)
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| Lesson Plan | Lesson Video in MPEG 4
(Video Clips and Highlights Available below. Right-click and select "Save Target As" to download the file.) |
Video Clips and Highlights from the Lesson
The clips below are selected from the full list of episodes. The Full Lesson Video may be downloaded for further study.
| Description of Video Clip | Video Links |
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| Students are reminded about the previous lesson in which they were working with 2-digit numbers. |
1. Preparing the Problem
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| Students are shown that day dates can be represented as two-digit numbers. Students are then asked to determine how many cards (each with one digit, 0-9) are needed to represent all of the day dates in a calendar. |
2. Posing and Understanding the Problem
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| Discussions develop about the number of days in a month. The teacher checks on the progress of groups. |
3. Working on the Problem
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| In whole class discussion, we see that the students are not all aware that the maximum number of days that a month can have is 31. This is resolved, and attention returns to the initial problem. |
4. Class Discussion
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| Students are shown that cards can be used more than once (“recycling”). Discussion develops about how fewer than 31 cards can be used to represent 31 days. |
5. Further Discussion
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| Having established that 31 cards are not needed, students discuss how many will be needed. One pair of boys are seen changing their position. Students’ ideas vary on this, with students arguing for 9, 10, and even 22 cards. |
6. Student Work
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| The class explores why 9 cards are not enough, even thought there is on day “0” on the calendar, because the digit 0 is needed to make the two-digit numbers 10, 20, and 30. |
7. Class Discussion
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| If 31 cards are more that are needed, and 9 cards are not enough, how many cards are required? Are 10 cards enough? Students work with sets of cards to explore how the dates can be made. |
8. Working on the problem
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| The class decides that 10 cards are not enough. But how many are needed? Students begin to see the role of the tens place and the need for two cards for “1” (in order to represent 11) and “2” (to represent 22) but a second “3” is not needed. |
9. Class Discussion
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| The class determines that 20 cards are not needed. Instead, only 12 cards are needed. |
10. Converging on a solution
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| Teacher challenges the class to consider using both sides of the cards. Another trick is to use “6” up-side-down to represent a “9”. The teacher shows how it is possible to display dates using two cubes! |
11. Closure
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