PLACEMAT TASKS

[Solutions are shown after the activities]

TASK 1

TASK 2

TASK 3

TASK 4

SOLUTIONS

TASK 1

WHAT STUDENTS MIGHT SEE

1.  Shrinking patterns

Students may notice the shrinking patterns of pink, blue, yellow, green, and so forth. They may also notice that these patterns represent the fractions 1/2, 1/4, 1/8, 1/16, and so forth.

They may wonder how many of the fractions in this pattern would fit in a square, and what the smallest fraction might be.

2.  Infinity in your hand

Students may propose that the infinite number of the fractions 1/2, 1/4, 1/8, 1/16, and so forth, would fit in a single square.

This implies that 1/2 + 1/4 + 1/8 + 1/16 + … = 1.

It also implies that they can hold infinity in their hand, which is further elaborated in Activity 2.

TASK 2

HANDS ON

Working in pairs, students use five 8×8 grids to shade and represent the fractions 1/2, 1/4, 1/8, 1/16, and 1/32. They then cut out the shaded parts and join them to form a new shape (as shown above).

We have worked in several classrooms, in grades 2 and up, where students completed this activity. Students quickly notice that these fractions will fit in a single square.

Many students are convinced that the sum 1/2 + 1/4 + 1/8 + 1/16 + … is 1. Others, who view the square as never completely filling, are convinced that the sum would not be more than 1.

WALKING TO THE DOOR

Referring to the image on the right, you may ask students if it is possible to walk to the door, and beyond, if they first walk halfway to the door, then half the remaining distance, then the remaining distance, and so on. One teacher candidate noted: Of course it is possible. Even if I take a single step and then look back, I can imagine that I travelled the infinite fractions 1/2, 1/4, 1/8, 1/16, and so forth. Infinity is everywhere.

TASK 3

INFINITY ART

The images above are an opportunity for students to engage aesthetically with the concepts of infinity and area representations of fractions.

MATH ART For more inspiration, view the Math Art video at https://youtu.be/yLte3H9mqmo

TASK 4

SOLUTION

In the 2 images on the right, we see that:

  • the pink rectangle is twice the size of the blue square;
  • the yellow rectangle is twice the size of the green square;
  • the purple rectangle is twice the size of the orange square;
  • and so forth.

So, the rectangular pieces make up 2/3 of the square below, and the remaining square pieces make up the remaining 1/3.

That is, 1/2 + 1/8 + 1/32 + … = 2/3.

And 1/4 + 1/16 + 1/64 + … = 1/3.

0.999… = 1

Does 0.999… equal 1?

Yes. Here is a proof.

  • 1/3       = 0.333…
  • 1/3       = 0.333…
  • 1/3       = 0.333…

SUM =3/3 = 1 = 0.999…

Or, using the fractions in the previous solution above:

  • 2/3      = 0.666…
  • 1/3      = 0.333…

SUM = 3/3 = 1 = 0.999…

Next

Coding puzzles … https://learnx.ca/mt/infinity-coding