We connect dots with toothpicks in a grid as shown below. For example, the grid below has 7 horizontal toothpicks in each row and 5 vertical toothpicks in each column. 
(a) Suppose we instead have a grid of dots that requires 10 horizontal toothpicks in each row and 20 vertical ones in each column. Then, how many total toothpicks will we need? Also, how many total dots are there?
(b) Can you generalize your answer? Suppose we have a grid that requires 
horizontal toothpicks in each row and 
vertical toothpicks in each column. Then, how many total toothpicks will we need? Also, how many total dots are there?
+128474 The number of horizontal dots will be one greater that the number of horizontal toothpicks.
So.....this is 11.
And the number of rows of dots will be one more than the number of vertical toothpicks.
And this is 21
So the number of dots is just 11 x 21 = 231.
The number of horizontal toothicks we need in each row is just one less than the number of dots in that row = 10......and we have 21 rows of these - the same as the number of rows of dots.....so 10 x 21 = 210 horizontal toothpicks.
The number of vertical toothpicks we have in one column is just one less than the number of vertical dots in that column = 20. And we have 11 of these columns......the same as the number of dots in each row. So 20 x 11 = 220 vertical toothpicks
The total number of toothpicks is just 210 + 220= 430.
Let's take the second part of this in relation to the dots. The total number of dots is just the dots in one horizontal row x the number of rows of dots.
Then, the number of horizontal toothpicks is just (the number of horizontal dots in one row - 1) x the number of rows of dots.
And the number of vertical toothicks is just (the number of vertical dots in one column - 1) x the number of columns of dots
I hope that helps.....
+118609 Please anon, follow the rules for reposting.
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