## INTEGRATING ADDITION AND SUBTRACTION, AND PLACE VALUE

When children first learn about addition and subtraction, it is informal. They focus their learning on the meaning of the concept. Adding means ‘joining groups together’ and subtraction means ‘taking away from a group’.

When adding, children will select 2 or more number cards, make the groups counting by ones, join the groups together, then count the number in the joined group by ones.

When subtracting, children will select 2 number cards, ask themselves ‘which number could I make a group of that will give me enough to take away a group of the other number?’, make 1 group counting by ones, take away a group of the other number counting the number in that group they took away, then count the number in the group they have left.

Once children understand addition and subtraction, they begin to record their understanding using number lines, initially counting by ones, then using place value to bridge to tens numbers.

In a Year 1 classroom recently, we had children with all of these levels of understanding!

The board contain all of the levels, built up during the 10 minutes of Explicit Teaching with the children:

Students then selected the level that challenged them, then use playing card to generate numbers to investigate adding and subtracting.

Walking around the class, stopping to observe and question children, I noticed that some children had chosen the ‘black’ level, but were finding it very challenging. These children had demonstrated their understanding of adding and subtracting counting by ones on a number line, but were not yet ready to add and subtract using place value to bridge 10. What to do?

There was obviously a step in between that the children did not have deep enough understanding of, that was holding them back.

Of course, the step involved Place Value.

We had been investigating Friends of 10, Partitioning and Place Value of Teen Numbers, and I knew these were the concepts that the children now needed to apply to addition and subtraction.

I asked one child, who was adding 8 and 5, if he knew 8’s Friend of 10. He hesitated, thought hard, then said, ‘5?’ Aha! I knew that until he was more fluent in his Friends of 10, he would not be able to add and subtract bridging 10. ‘Ok,’ I said to the child, ‘You’re not going to add today. You’re going to investigate Friends of 10, then you’ll be able to add again.’ I gave the child a 10 frame and some counters. Because we had previously investigated Friends of 10, he quickly learnt to select a number, place the corresponding number of counters on the 10 frame, and record the number and its Friend of 10. The child remained sitting where he was, amongst other children who were still investigating adding and subtracting at just beyond their current level of understanding.

Because every child was investigating at just beyond their current level of understanding, they were all investigating independently (took time to train them but sooo worth it!!), I was free to walk around the entire class, engaging with every child as necessary. I noticed another child, who was adding 7 and 8, had added 3 to make 10, but had partitioned 8 into 4 and 4. I asked her how many she had already added. She told me, ‘7’. I asked again, rephrasing, ‘You started with a group of 7, how many have you added so far?’ This time she told me, ‘3’. I guided her to place 3 under the 8, as the part she had already added. I asked her, ‘so you are supposed to add 8, you’ve already added 3, how many more do you have to add? How did you partition the 8?’. She looked at me blankly. Aha! I knew that until she had developed deeper understanding of partitioning, she would not be able to add and subtract bridging to 10. ‘Ok,’ I said to the child, ‘You’re not going to add today. You’re going to investigate Partitioning, then you’ll be able to add again.’ I gave the child some large connecting blocks. Because we had previously investigated Partitioning, she quickly learnt to select a number, place the corresponding number of blocks in a tower, repeatedly partition the tower, recording the number and the Partitions. The child remained sitting where she was, amongst other children who were still investigating adding and subtracting at just beyond their current level of understanding.

Continuing my walk around the room, I noticed a child who was adding 5 and 8. She had added 5 to make 10, partitioned 8 into 5 and 3, but was counting on her fingers to add the 3 to the 10. I asked her what 10 and 3 was. She again began counting on her fingers. I stopped her, and rephrased, asking, ‘If I have 1 ten and 3 ones, how many do I have?’ She looked at me for a few seconds, then replied, ’10?’. Aha! I knew that until she had developed deeper understanding of Place Value of Teen Numbers, she would not be able to add and subtract bridging to 10. ‘Ok,’ I said to the child, ‘You’re not going to add today. You’re going to investigate Place Value of Teen Numbers, then you’ll be able to add again.’ I gave the child some small connecting blocks. Because we had previously investigated Place Value of Teen Numbers, she quickly learnt to select cards to make a teen number, collect the corresponding number of small blocks, group them into place values of tens and ones, and record using standard place value. Then break 1 ten into 10 ones, and record using non-standard place value. The child remained sitting where she was, amongst other children who were still investigating adding and subtracting at just beyond their current level of understanding.

In the Explicit Teaching segment of the next lesson, we included these place value levels.

We went through the place value levels, and any other levels that the students needed to. The students looked at the levels on the board to find the level that challenged them, thinking, ‘that’s too easy, I’ll look at the next level. That’s too easy too, I’ll look at the next level…’ until they find their level.

Any other students who found moving from adding and subtracting counting by ones, to adding and subtracting bridging 10, a step too far, identified what Place Value understanding they needed to deepen.

All of these levels, and higher levels, are explicitly taught using the Teaching Resources as www.alearningplace.com.au