New Zealand Curriculum: Level 2
Learning Progression Frameworks: Additive thinking, Signpost 4 to Signpost 5
Target students
These activities are intended for students who understand addition to be the joining of sets and subtraction to be the removal of objects from a set. Difference problems require the student to compare the numbers of objects in two different sets and can be solved using either addition or subtraction. Comparison situations are structurally distinct from the part-whole situations students commonly encounter in early instruction about addition and subtraction.
Target students should have already developed a degree of part-whole understanding in addition and subtraction contexts (joining and separating). They should have some number facts to call on, particularly number bonds to ten. Understanding of two-digit place value, including the structure of ‘teen’ and ‘ty’ numbers will support many of the activities described in this intervention.
Allow access to pencils and paper throughout these activities but not to a calculator. You need a set of ice block sticks, bundled in tens and hundreds, masking cards (half A4 size), note paper to record numbers and a marker pen. (show diagnostic questions)
The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using an iceblock stick context but can be changed to other objects that are engaging to your students, such as counters, sticks, shells, or pebbles.
- I am giving you 9 ice block sticks. (Place 9 ice block sticks in the student’s hand and close their hand. Record the number 9 in front of them.)
I have 5 ice block sticks. (Show the student your ice block sticks and close your hand. Record the number 5 in front of you.)
Look at our numbers of iceblock sticks. How many more do you have? (re-word if necessary)
Signs of fluency and understanding:
Using a part-whole strategy that applies basic facts to ten, such as 5 + 4 = 9. “I have four more ice block sticks than you.”
A derived part whole strategy that involves working from a known fact, such as 5 + 5 = 10 so 5 + 4 = 9. This is demonstrated by thinking like “If I had 10 ice block sticks then I would have 5 more than you. I have only 9 ice block sticks, so I have four more ice block sticks than you.”
What to notice if your student does not solve the problem fluently:
Counting on in ones while tracking the counts (possibly on fingers), 5, 6, 7, 8, 9, might indicate that a student has not yet developed part-whole thinking in joining and separating contexts, or that they lack knowledge of number facts to ten.
Supporting activity:
Difference with sets to ten
- I am giving you 17 ice block sticks. (Place bundle of ten and 7 loose ice block sticks in the front of the student and cover the ice block sticks with a piece of card. Record the number 17 in front of them.)
I have 9 ice block sticks. (Show the student your ice block sticks. Mask the materials with a card. Record the number 9 in front of you.)
Look at our numbers of iceblock sticks. How many more do you have?(re-word if necessary)
Signs of fluency and understanding:
Using a part-whole strategy that applies basic facts to ten, such as 9 + 8 = 17. “I have eight more ice block sticks than you.”
A derived part whole strategy that involves working from a known fact, such as 10 + 7 = 17 so 9 + 8 = 17. “If you had 10 ice block sticks then I would have 7 more than you. You have only 9 ice block sticks, so I have eight more ice block sticks than you.”
What to notice if your student does not solve the problem fluently:
Counting on in ones while tracking the counts (possibly on fingers), 9, 10, 11, 12, 13, 14, 15, 16, 17 may indicate that a student has not developed part-whole thinking in joining and separating contexts, or that they lack knowledge of number facts to twenty, such as 10 + 7 = 17.
Supporting activity:
Difference with sets to twenty
- I am giving you 54 ice block sticks. (Place 5 bundles of ten and 4 loose ice block sticks in the front of the student and cover the ice block sticks with a piece of card. Record the number 54 in front of them.)
I have 20 ice block sticks. (Show the student your two bundles of ice block sticks. Mask the materials with a card. Record the number 20 in front of you.)
Look at our numbers of iceblock sticks. How many more do you have? (re-word if necessary)
Signs of fluency and understanding:
Using a place value-based strategy that subtracts two tens, such as 54 - 20 = 34. This indicates that the student understands that difference problems can be solved by subtraction and recognises that subtracting two tens leaves the ones digit unaltered.
Using a place value-based strategy that adds on from 20 to 54, such as 20 + 30 = 50 so 20 + 34 = 54. This indicates that the student understands that difference problems can be solved by ‘adding on’ and recognises that tens and ones can be combined.
If the student uses a written algorithm question them about the meaning of their working to check that they are applying place value knowledge. Lack of understanding shows when students think they are ‘subtracting two’ rather than twenty.
What to notice if your student does not solve the problem fluently:
Counting on, or back, in tens, and ones, may indicate that a student does not understand place value structure, that groups of ten can be treated just like units of one. Counting strategies might include “20, 30, 40, 50, 51, 52, 53, 54” or “54, 53, 52, 51, 50, 40, 30, 20” and might include difficulties in tracking the difference. Student may need paper and pencil to record the separate amounts that make up the difference, 10 + 10 + 10 + 4.
Improvised part-whole strategies can also cause problems for students where there is extra load on working memory. Look for signs like, “I took 4 off the 54 to make 50, then I took off 30 to get 20. I added the 4 back on to get 34.” Students might also compensate the wrong way, such as subtracting 4 for 20 to get 16.
Supporting activity:
Difference when one number is a multiple of ten (decade number)
- I am giving you 85 ice block sticks. (Place 8 bundles of ten and 5 loose ice block sticks in the front of the student and cover the ice block sticks with a piece of card. Record the number 85 in front of them.)
I have 34 ice block sticks. (Show the student your three bundles and four single ice block sticks. Mask the materials with a card. Record the number 34 in front of you.)
Look at our numbers of iceblock sticks. How many more do you have? (re-word if necessary)
Signs of fluency and understanding:
Using a place value-based strategy that subtracts three tens and four ones from 85, such as 80 – 30 = 50, 5 – 4 = 1, 50 + 1 = 51. This indicates that the student understands that difference problems can be solved by subtraction and recognises that tens and ones units can be subtracted separately then recombined. Working might be supported by recording such as an empty number line or equations.
Using a place value-based strategy that adds on from 34 to 85, such as 34 + 50 = 84, 84 + 1 = 85 so 34 + 51 = 85. This indicates that the student understands that difference problems can be solved by ‘adding on’ and recognises that tens and ones can be treated as separate units then be combined. Working might be supported by recording such as an empty number line or equations.
If the student uses a written algorithm question them about the meaning of their working to check that they are applying place value knowledge. Lack of understanding shows when students think they are ‘subtracting three’ rather than thirty.
What to notice if your student does not solve the problem fluently:
Counting on, or back, in tens, and ones, may indicate that a student does not understand place value structure, or that they think that groups of ten can be treated just like units of one. Counting strategies might include “20, 30, 40, 50, 51, 52, 53, 54” or “54, 53, 52, 51, 50, 40, 30, 20” and include difficulty tracking the difference. Students may need paper and pencil to record the separate amounts that make up the difference, 10 + 10 + 10 + 4.
Improvised part-whole strategies can also cause problems for students where there is extra load on working memory. Look for signs like, “I took 4 off the 54 to make 50, then I took off 30 to get 20. I added the 4 back on to get 34.” Students sometimes compensate the wrong way, such as subtracting 4 for 20 to get 16.
Supporting activity:
Differences with two-digit whole numbers with no renaming
- I am giving you 92 ice block sticks. (Place 9 bundles of ten and 2 loose ice block sticks in the front of the student and cover the ice block sticks with a piece of card. Record the number 92 in front of them.)
I have 48 ice block sticks. (Show the student your four bundles and eight single ice block sticks. Mask the materials with a card. Record the number 48 in front of you.)
Look at our numbers of iceblock sticks. How many more do you have? (re-word if necessary)
Signs of fluency and understanding:
Using a place value-based strategy that subtracts four tens and eight ones from 92, such as 92 – 40 = 52, 52 – 8 = 44. This indicates that the student understands that difference problems can be solved by subtraction and fluently subtracts back through decades, such as 52 – 8 can be solved as 52 – 2 – 6 = 44.
Using a place value-based strategy that adds on from 48 to 92, such as 48 + 2 = 50, 50 + 40 = 90, 90 + 2 = 92 so 48 + 44 = 92. This indicates that the student understands that difference problems can be solved by ‘adding on’ and fluently adds up through decades, such as 88 + 4 = 92.
If the student uses a written algorithm question them about the meaning of their working to check that they are applying place value knowledge. Lack of understanding shows when students cannot explain that ‘crossing out the 9 in 92 to make 812” is renaming 92 as “eighty twelve.”
What to notice if your student does not solve the problem fluently:
Improvised part-whole strategies can also cause problems for students where there is extra load on working memory. Look for signs like, “I took 2 off the 92 to make 90, then I took off 40 to get 50. I added the 2 back on to get 52.” While the strategy may work if the student correctly subtracts 8 to get 44, it is inefficient and prone to error.
Taking smaller from larger, especially if connected to performance of an algorithm, is a sign that the student cannot connect their place value knowledge to subtraction. For example, look for 90 – 40 = 50 then 8 – 2 = 6, to get an answer of 56.
Supporting activity:
Differences with two-digit whole numbers using renaming