PLACEMAT TASKS

[Solutions are shown after the activities]

TASK 1

TASK 2

TASK 3

TASK 4

TASK 5

TASK 6

SOLUTIONS

TASK 1

What students may see:

Students may notice patterns that deal with:

  • Visual and concrete patterns for consecutive odd, even, and natural numbers.
  • The resulting numeric and algebraic expressions for their sums. For example, the sum of odd numbers 1 + 3 + 5 + 7 = 4×4 and more generally 1 + 3 + 5 + … + N = NxN = N2.

TASK 2

What students may make:

Using link cubes as well as markers, dabbers, and grid paper, students may create their own colourful and appealing arrangements of odd, even, and natural numbers.

The math art may be shared at home, along with practiced dialogues that explain math patterns and concepts:

  • Odd numbers hide in squares. For example, the first 4 odd numbers hide in a 4×4 square.
  • Even numbers hide in rectangles where one side is one unit larger than the other. For example, the first 4 even numbers hide in a 4×5 rectangle.
  • Two sets of natural numbers fit in an even number rectangle.

Sample scaffolding prompts and questions:

  • What math does this art represent?
  • How else might you represent it?
  • How would you explain it to someone at home, who hasn’t seen this before?

TASK 3

How students may respond:

We don’t know how many odd numbers are in the blue square.

We may answer in two parts:

  • First, what is the smallest number of odd numbers that may fit in the square?
    • The answer would be 1.
    • The whole square could represent 1 blue cube or the odd number 1.

Second, what is the largest number of odd numbers that may fit in the square?

  • Let’s imagine that the blue square has a 10 x 10 blue grid that is invisible to us.
    • In this grid, we could fit the first 10 odd numbers.
  • If we then imagine drawing horizontal and vertical lines through each grid square, we can create a 20 x 20 grid.
    • In this grid, we could fit the first 20 odd numbers.
  • We can repeat this and keep increasing the number of odd numbers hiding in the square to infinity.

TASK 4

How students may respond:

From the pattern above:

  • Stage 1 has 1 x 1 = 1 block
  • Stage 2 has 2 x 2 = 4 blocks
  • Stage 3 has 3 x 3 = 9 blocks
  • Stage 4 has 4 x 4 = 16 blocks
  • Stage 5 has 5 x 5 = 25 blocks
  • Stage 10 has 10 x 10 = 100 blocks
  • Stage 100 has 100 x 100 = 10 000 blocks
  • Stage N has N x N = N2 blocks

The number of blocks represent the sum of the odd numbers. For example, 1 + 3 + 5 = 9 = 3×3 = 32.

In younger grades

We have engaged students as young as grade 1 with such activities.

  • They successfully complete the patterns up to stage 10.
    • Instead of using multiplication, they typically skip count by 10.
  • Some extend these patterns to complete the last 2 questions by describing the result:
  • For stage 100, there are 100 rows of 100 blocks.
  • For stage N, there are N rows of N blocks.

NOTE

  • N is a variable.
  • It may represent any stage number.
  • N x N = N2 is an expression for the general rule for finding the sum of the first N odd numbers.

TASK 5

The image above lists the first 10 odd numbers, from 1 to 19, and groups them into 5 pairs, with each pair having a sum of 20.

So, the sum of the first 10 odd numbers 1-19 is 5 x 20 = 100.

We may also use this method to find the sum of the first 10 natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

  • We may create 5 pairs whose sum is 11: 1+10, 2+9, 3+8, 4+7, 5+6.
  • So, the sum of the first 10 natural numbers is: 5 x 11 = 55.

Similarly, we may find the sum of the first 10 even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

  • We may create 5 pairs whose sum is 22: 2+20, 4+18, 6+16, 8+14, 10+12.
  • So, the sum of the first 10 even numbers is: 5 x 22 = 110.

TASK 6

How students may respond:

They grow at a constant rate.

If we join the tops of the bars with a red dotted line, it will be straight.

The sum of odd numbers grows by odd numbers: 1, 3, 5, 7, and 9. This makes sense, as the bars represent sums of odd numbers.

The sums grow at an increasing rate (as the odd numbers added increase in size).

If we join the tops of the bars with a red dotted line, it will be curved.

Next

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