First we used toothpicks to create a pattern of squares increasing in size. Then we asked a question: How many toothpicks are required to create figure number 125 in the sequence?

Next we created a table to record information we knew.

The first column is a counter for the position in the sequence; 0, 1, 2, 3, 4. The second column contains the number of small squares in the figure; 0, 1, 4, 9, 16. The third column shows the number of toothpicks used in the figure; 0, 4, 12, 24, 40.

For us the pattern was not obvious, so we took a hint from the video and added more columns to help figure out how many toothpicks would be required for step 125. The forth column shows the number of horizontal toothpicks in each figure and the fifth column shows the number of vertical toothpicks in each figure. They are the same; 0, 2, 6, 20.

From here my son noticed a number pattern; The number of squares in each figure plus the figure's position in the sequence always add to the number of vertical or horizontal toothpicks used in the figure. From previous math, he knew that the number of squares in the figure was equal to the sequence number squared. In addition he noticed that the number of vertical toothpicks plus the number of horizontal toothpicks equals the total number of toothpicks.

For example, the third square in the sequence; 3 + 9 = 12 ----------- the 9 came from 3^2 --------- and 12 + 12 = 24 ----------- the number of toothpicks in the figure.

Replacing the sequence number with an n, I helped him to rewrite what he discovered algebraically. (n + n^2) + (n+n^2) equals the number of toothpicks in the figure.

My eleven year old found a slightly different pattern. The number of vertical or horizontal toothpicks always equals the counter number times the counter number plus one.

For example, the third square in the sequence; 3 x (3+1) = 12 and 12 + 12 = 24. Again replacing the counter with an n we wrote her solution algebraically. n x (n + 1) + n x(n + 1) equals the number of toothpicks in the figure.

I created a sixth column in the table entitled counting with multiplication, to help my six year old. We noticed that the total number of vertical or horizontal toothpicks used in each figure was equal to the counter times the counter plus 1; 0, 1x2, 2x3, 3x4. She was able to use this information to predict the number of toothpicks used in the next few squares in the sequence.

I was baffled that the two older kids both found different solutions which seemed to work, so while they built bigger figures to check their predictions, I did a little algebra to verify.

(n + n^2) + (n+n^2) = n x (n + 1) + n x(n + 1)

2(n + n^2) = 2n(n + 1)

2n^2+2n = 2n^2+2n

They were both correct! By they way, it would take 31, 500 to build a 125 x 125 figure. We didn't check this one.

Next my eleven year old and I found an equation to go with a different pattern. This pattern was created placing small circles in a triangle. In each new row there is one additional circle.

Again a table was created to record the known information. Column one is the counter; 0, 1, 2, 3, 4, 5, 6, 7. Column two shows the figure. Column 3 shows the total number of circles required to create the figure; 1, 3, 6, 10, 15, 21, 28, 36.

Again we needed a little help leading us to the solution, so I suggested doubling the number of circles required (a little bird gave me that hint:)) -, 6, 12, 20, 30, 42, 56, 72. Looking at the bigger numbers we noticed that they were answers to common single digit multiplication problems; 0, 1x2, 2x3, 3x4, 4x5, 5x6, 6x7, 7x8, 8x9.

From there we determined the number of circles required to create the figure was equal to the the figure's position in the sequence, times the position in the sequence plus 1, divided by two. Written algebraically it looks like this; n(n + 1)/2

Remember the little bird that gave me the hint? Well it actually came from the Visual Patterns website. Posted on the website are over 100 similar problems and a solution manual can be obtained after sending a quick email to the site's author.

This activity is so good for building number sense, exercising those brain math muscles, and building thinking and problem solving skills. We plan to try working similar problems at least once per week to improve these skills.

This post is linked to:

Montessori Monday

Trivium Tuesday

Hip Homeschool Hop

Thanks for setting this out in so much detail. When I did Jo Boaler's online course on teaching maths last summer she talked a lot about using algebra in this kind of way but I hadn't really fully processed it yet. You've done a great job of demonstrating how algebraic thinking can be learned at several different levels.

ReplyDeleteWow! This looks really neat! Thank for sharing at Trivium Tuesdays

ReplyDelete