![]() Instead of looping the numbers around, let’s write them in two rows: 1 2 3 4 5 6 7 8 9 10 The above method works, but you handle odd and even numbers differently. It always bugged me that the same formula worked for both odd and even numbers – won’t you get a fraction? Yep, you get the same formula, but for different reasons. And instead of having exactly n items in 2 rows (for n/2 pairs total), we have n + 1 items in 2 rows (for (n + 1)/2 pairs total). Notice that each column has a sum of n (not n+1, like before), since 0 and 9 are grouped. However, our formula will look a bit different. Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead: 0 1 2 3 4īy counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. Many explanations will just give the explanation above and leave it at that. What if we are adding up the numbers 1 to 9? We don’t have an even number of items to pair up. Wait - what about an odd number of items?Īh, I’m glad you brought it up. And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs. As the top row increases, the bottom row decreases, so the sum stays the same.īecause 1 is paired with 10 (our n), we can say that each column has (n+1). Instead of writing all the numbers in a single column, let’s wrap the numbers around, like this: 1 2 3 4 5Īn interesting pattern emerges: the sum of each column is 11. Pairing numbers is a common approach to this problem. For these examples we’ll add 1 to 10, and then see how it applies for 1 to 100 (or 1 to any number). Let’s share a few explanations of this result and really understand it intuitively. Manual addition was for suckers, and Gauss found a formula to sidestep the problem: So soon? The teacher suspected a cheat, but no. The so-called educator wanted to keep the kids busy so he could take a nap he asked the class to add the numbers 1 to 100. ![]() October 2014 Elem Math 3rd Gradeĥ3 relate decimals to fractions that name tenths and hundredths.ĥ4 determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line.There’s a popular story that Gauss, mathematician extraordinaire, had a lazy teacher. October 2014 Elem Math 3rd Gradeĥ2 compare and order decimals using concrete and visual models and moneyĬompare and order decimals using concrete and visual models and money. ĥ1 represent decimals, including tenths and hundredths, using concrete and visual models and money. October 2014 Elem Math 3rd Gradeĥ0 round whole numbers to a given place value through the hundred thousands place. October 2014 Elem Math 3rd GradeĤ compare and order whole numbers up to 100,000 and represent comparisons using the symbols >,, <, or =. October 2014 Elem Math 3rd Gradeģ represent a number on a number line as being between two consecutive multiples of 10 100 1,000 or ,000 and use words to describe relative size of numbers in order to round whole numbers. October 2014 Elem Math 3rd GradeĢ describe the mathematical relationships found in the base-10 place value system through the hundred thousands place. 1 compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |