As part of our understanding number relationships, we have been practicing reciprocal equations to see and understand how the same 3 numbers in an equation relate to each other symbollically.
For example:
10 + 0 = 10 7 + 3 = 10
0 + 10 = 10 3 + 7 = 10
10 - 10 = 0 10 - 3 = 7
10 - 0 = 10 10 - 7 = 3
We looked at all the "two addend" ways that we could make 10
10, 0
9, 1
8, 2
7, 3
6, 4,
5, 5 We discovered that 5 and 5 does not give 4 reciprocal equations like all the other ways to make 10!
We explored our understandings about the relationship between the 3 numbers that appear in our equations:
For example:
If we are doing 6 and 4 makes 10
- knowing that 10 is the largest number; it is the greatest amount
- that means that for adding, 10 MUST come at the end of the mathematical sentence, because when we add things up we get MORE of something!
- then for subtracting 10 MUST come at the beginning of the mathematical sentence, because when we have the largest amount, we can share it out with others, or take something away
This relationship information is continually explored in the classroom, with an understanding of mathematical vocabulary to say a sentence out loud, the symbolic understanding to know what all the symbols mean and what order or position we write them in and MOST IMPORTANTLY...
We explore constantly with amounts of any kind of material to visually create and move materials from a larger set to subtract to two smaller sets and then take the two smaller sets and put them back together to create the same larger set that we started with!
The concepts of addition and subtraction go quite naturally together for children and help to cement their developing understanding of number relationships.
When we take the time to add or subtract using materials we can see and ALWAYS take time to find the relationship between the numbers we are "moving around", then children see that adding and subtracting isn't just an arbitrary jostling of numbers on a number line or even from a given set of "stuff" - They need to see that no matter how we move that "stuff" around to change the "amount" and therefore the symbolic number that represents it - we can always move everything back again the other direction - addition and subtraction are related that way!
We also constantly explore the concept that having a set amount such as 10 gives us almost limitless possibilities for rearranging and moving that amount of stuff around into visible groups that add up to 10 or divide the 10 into smaller amounts:
While reciprocal equations are quite a specific relationship to help children understand addition and subtraction, and how they connect mathematically, of course children also need to visually understand that...
with 10 again for example:
1 + 1 + 1+ 1 + 1 + 4 + 1 = 10
2 + 4 + 3 + 1 = 10
and so on... that as long as you have any "set amount", you can arrange and rearrange it in any number of ways, and the amount will remain the same. However, if you take something away from your original set, and give it away to someone else, then the original amount changes.
We continue to work on a variety of number relationship and addition strategies such as:
1 more
2 more
counting on from the number I have first
jumping forward on a number line
making doubles
skip counting
making (or noticing if I see) 10 facts
As well as the reciprocal subtraction relationships when we think:
1 less
2 less
counting back from the number I first have
jumping back on a number line
splitting equal between 2 if it's an even number
splitting equal between 2 until only 1 is left over for an odd number
All of this requires constant review and understanding of even and odd, greater than and less than, skip counting by 2's, 5's and 10's, counting forward and backward confidently when we start anywhere!
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