Ellipse and Constant Product

AVERAGE TIME: 1 hour - 1 hour and 30 minutes

SPACES: classroom

MATERIALS: ruler of at least 30 cm, pencil, colours, TALES tables (downloadable in the ATTACHMENTS section)

CONSTANT PRODUCT

After briefly reviewing the product, rectangular-shaped tables are proposed to the class - downloadable at the bottom of the page. The request will be to join pairs of numbers that have the same product.

In the above configuration, the rectangle_2.pdf table was used, choosing as product **24**. The obtained configuration is **half of an ellipse**.

In general, the more divisors the chosen number has, the more the resulting ellipse is defined.

Unlike the parabola, the word ellipse is probably unknown to the class group. This does not mean that it cannot be proposed anyway: when the students encounter new words, an idea may be to let the class play the game of hanging with the new term.

At the end of the activity, we recommend proposing to the class the phrase, "**If you connect numbers on opposite sides with a constant product, you get half an ellipse"**.

TESSELLATION EXPERIENCE WITH THE ELLIPSE

In the configuration created, half an ellipse is obtained. We recommend colouring the ellipse obtained by making the children agree two by two to use similar colours. Once finished the activity, they can join their configurations to obtain a complete ellipse.

THE ELLIPSE IN REALITY

It is advisable to have the children find some ellipse examples in reality, such as those shown in the figure.

*The arena of Verona, the elliptical orbit of the planets.*

FINAL REVIEW OF CONFIGURATIONS

As a final experience, we recommend let the children review all the configurations obtained so far. Among the attachments, you will find ripasso.pdf with a partial verification proposal.

IN-DEPTH ANALYSIS: THE TRANSLATED ELLIPSE

A more complicated experience (which we suggest proposing as a possible in-depth analysis) is to draw a translated ellipse as shown in the figure.

To obtain it, you will always have to combine the numbers that produce 24 but read as the afternoon hours. Therefore, the 3 x 8 product will be represented by the segment that joins **15 on one side with 20 on the other**, as 15 is three in the afternoon and 20 is eight in the evening. The discourse is part of the clock's arithmetic, which consists of looking at numbers as remainders of the division with a given number. The module. 13 modulo 5 for examples is 3, because 13: 5 has the remainder 3.

Sometimes we count in this way unaware, as in the case of time, that we measure it (in the hours) module 24 or 12.

The site of Macchine Matematiche suggested the construction of the ellipse. Among other things, there are tools, also adaptable to primary school, to construct conics on the site.

NATIONAL INDICATIONS OBJECTIVES COHERENT WITH THE ACTIVITY

END OF THIRD GRADE

- Mentally perform simple operations with natural numbers;
- Know with confidence the multiplication tables of numbers up to 10.

END OF FIFTH GRADE

- Carry out the four operations with safety;
- Recognize and describe regularities in a sequence of numbers or figures;
- Recognize rotated, translated and reflected figures;
- Recognize plane representations of three-dimensional objects.

Ellipse and Constant Product

AVERAGE TIME: 1 hour - 1 hour and 30 minutes

SPACES: classroom

MATERIALS: ruler of at least 30 cm, pencil, colours, TALES tables (downloadable in the ATTACHMENTS section)

CONSTANT PRODUCT

After briefly reviewing the product, rectangular-shaped tables are proposed to the class - downloadable at the bottom of the page. The request will be to join pairs of numbers that have the same product.

In the above configuration, the rectangle_2.pdf table was used, choosing as product **24**. The obtained configuration is **half of an ellipse**.

In general, the more divisors the chosen number has, the more the resulting ellipse is defined.

Unlike the parabola, the word ellipse is probably unknown to the class group. This does not mean that it cannot be proposed anyway: when the students encounter new words, an idea may be to let the class play the game of hanging with the new term.

At the end of the activity, we recommend proposing to the class the phrase, "**If you connect numbers on opposite sides with a constant product, you get half an ellipse"**.

TESSELLATION EXPERIENCE WITH THE ELLIPSE

In the configuration created, half an ellipse is obtained. We recommend colouring the ellipse obtained by making the children agree two by two to use similar colours. Once finished the activity, they can join their configurations to obtain a complete ellipse.

THE ELLIPSE IN REALITY

It is advisable to have the children find some ellipse examples in reality, such as those shown in the figure.

*The arena of Verona, the elliptical orbit of the planets.*

FINAL REVIEW OF CONFIGURATIONS

As a final experience, we recommend let the children review all the configurations obtained so far. Among the attachments, you will find ripasso.pdf with a partial verification proposal.

IN-DEPTH ANALYSIS: THE TRANSLATED ELLIPSE

A more complicated experience (which we suggest proposing as a possible in-depth analysis) is to draw a translated ellipse as shown in the figure.

To obtain it, you will always have to combine the numbers that produce 24 but read as the afternoon hours. Therefore, the 3 x 8 product will be represented by the segment that joins **15 on one side with 20 on the other**, as 15 is three in the afternoon and 20 is eight in the evening. The discourse is part of the clock's arithmetic, which consists of looking at numbers as remainders of the division with a given number. The module. 13 modulo 5 for examples is 3, because 13: 5 has the remainder 3.

Sometimes we count in this way unaware, as in the case of time, that we measure it (in the hours) module 24 or 12.

The site of Macchine Matematiche suggested the construction of the ellipse. Among other things, there are tools, also adaptable to primary school, to construct conics on the site.

NATIONAL INDICATIONS OBJECTIVES COHERENT WITH THE ACTIVITY

END OF THIRD GRADE

- Mentally perform simple operations with natural numbers;
- Know with confidence the multiplication tables of numbers up to 10.

END OF FIFTH GRADE

- Carry out the four operations with safety;
- Recognize and describe regularities in a sequence of numbers or figures;
- Recognize rotated, translated and reflected figures;
- Recognize plane representations of three-dimensional objects.