ordered pairs

Desmos Thanksgiving Mystery Dot-to-Dot

/ ivasallay / Leave a comment

Today’s Puzzle:

I wanted to create a Dot-to-Dot in Desmos for my students that wouldn’t require them to type in many ordered pairs. I concluded that if most points could be reflected over the x or y-axis, I could eliminate the need to type in about half the points. With that in mind, I recently created this mystery dot-to-dot you can enjoy over the Thanksgiving weekend.

What will this unfinished dot-to-dot become when the dots are connected, and 90% of the image is reflected over the y-axis?

My sister guessed it was a cat. The image reminds me of a snowman. What did you think it might be?

You can discover what it is by clicking on this pdf and following the instructions: Desmos Mystery Ordered Pair Dot-to-Dot

The instruction will look like this:

Depending on your device, you may be able to click on the lower right-hand corner of the Desmos image below to see how much fun I had transforming it four different ways: I made the image slide along the x-axis,  rotated it 90 degrees, reflected it over the x-axis, and dilated it. (The location of the turkey’s wattle can help you determine if an image is a reflection, a rotation, or a combination of both.) If clicking the lower right-hand corner does not work on your device, click this link. These transformations are all essential concepts for students to learn, and Desmos can make the process quite enjoyable.

Did you guess right? Have a very happy Thanksgiving!

1780 Reflections of a Polygonal Bird

/ ivasallay / Leave a comment

Today’s Puzzle:

What ordered pairs were used to create this bird?

Its eye was formed from an equation of a circle:
(x – 7)²+ (y – 15)² = 3/4.

After creating the polygonal bird using ordered pairs and that circle equation, I wanted to do other things with the bird. Everything I did was like a puzzle for me to figure out.

Could I make it “fly”? Yes!

 

Could I make it reflect itself more than once over the y-axis and the x-axis? Yes! And I could make it do some sliding at the same time!

This next one was the toughest for me to do. I wanted the bird to be in motion rotating counter-clockwise around the origin. I was able to do it, but Desmos wouldn’t save the sliders exactly the way I wanted. I will need your help on this one. Click on this rotating bird link, then push play on slider a. About the time that slider goes to zero, push play on slider b. If you hit the sliders just right, it will look something like this GIF I made, but slower:

Rotating Polygonal Birds

make science GIFs like this at MakeaGif

 

Factors of 1780:

Perhaps our polygonal bird would like to fly to a tree. Here’s a factor tree for 1780 that it can take a rest on.

I knew that 1780 was divisible by 4 because its last two digits are divisible by 4.

  • 1780 is a composite number.
  • Prime factorization: 1780 = 2 × 2 × 5 × 89, which can be written 1780 = 2² × 5 × 89.
  • 1780 has at least one exponent greater than 1 in its prime factorization so √1780 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1780 = (√4)(√445) = 2√445.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1780 has exactly 12 factors.
  • The factors of 1780 are outlined with their factor pair partners in the graphic below.

More About the Number 1780:

1780 is the difference of two squares in two different ways:
446² – 444² = 1780, and
94² – 84² = 1780.

1780 is the sum of two squares in two different ways:
42² + 4² = 1780, and
36² + 22² = 1780.

1780 is the hypotenuse of four Pythagorean triples:
336-1748-1780, calculated from 2(42)(4), 42² – 4², 42² + 4²,
780-1600-1780, which is 20 times (39-80-89)
812-1584-1780, calculated from 36² – 22², 2(36)(22), 36² + 22², and
1068-1424-1780, which is (3-4-5) times 356.

1780 is KK in base 88 because
20(88) + 20(1) = 20(89) = 1780.