New Zealand Curriculum: Level 3
Learning Progression Frameworks: Multiplicative thinking, Signpost 4 to Signpost 5
Target students
These activities are intended for students who understand multiplication as the repeated addition of equals sets, and who know some of the basic multiplication facts.
The following diagnostic questions indicate students’ understanding of, and ability to apply division to situations that involve repeated equal subtraction. The questions are in order of complexity. If the student answers a question confidently and with understanding proceed to the next question. If not, then use the supporting activities to build and strengthen their fluency and understanding. Allow access to pencil and paper but not to a calculator unless stated. (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 a people counters context but can be changed to other contexts that are engaging to your students, such as Lego™, smarties, novelty counters, or toy vehicles.

Here are 12 boys and girls. They need to get themselves into pairs. How many pairs will there be?
(Ask for a prediction then let the student equally share the people counters into six sets of two. Physically modelling the situation is not needed if the student responds quickly.)
Signs of fluency and understanding:
Anticipates that 12 people can form six pairs, using factual knowledge, such as 6 + 6 = 12 (additive) or 6 x 2 = 12 (Multiplicative).
What to notice if your student does not solve the problem fluently:
Forming pairs, using two people counters each time, to establish the number of pairs is a sign that the student does not yet anticipate the result. Lack of anticipation may be due to inadequate number fact knowledge or not recognizing the opportunity to use that knowledge. Equal subtraction should meet these criteria:
 All the objects are distributed (exhausted)
 The correct size of parts is used (Two people in each pair)
 The number of parts is correct (Six pairs)
Watch for students recounting or subitising the people to check the pairs are equal. Recounting indicates a belief that validating equality is an integral part of the equal subtraction process.
Supporting activities:
Anticipating the result of equal subtraction (quotative division)

Here are 20 people. They get into teams of four people. How many teams do they make?
Write an equation for what you have done. Can you write a division equation?
Signs of fluency and understanding:
Anticipates the number of teams, five, without physical modelling. Uses multiplication or division fact knowledge, such as 5 x 4 = 20 or 20 ÷ 4 = 5.
What to notice if your student does not solve the problem fluently:
One by one forming of teams, whether physical or imaged, signals that the student lacks belief that the result can be predicted, and/or lacks knowledge of number facts to make the prediction.
Additive methods might include progressively trying addends, such as 4 + 4 + 4 = 12 then 12 + 4 = 16, then 16 + 4 = 20, keeping track of the fours that are created. These methods indicate the student has yet to connect multiplication with equal addition, and/or lacks multiplication facts to apply. Due to issues of memory load students may have difficulty tracking the number of fours that are created.
Supporting activities:
Using multiplication facts to anticipate equal subtraction (quotative division)

Here are 18 people. A car can take 3 people. How many cars do you need to take all the people to the show?
(Ask for a prediction then let the student form sets of three, if needed. Do not physically share if the response is fluent and correct)
Write a division equation for what you have done.
Signs of fluency and understanding:
Anticipates the number of equal sets of three fluently without physical modelling. Uses multiplication or division fact knowledge, such as 6 x 3 = 18 or 18 ÷ 3 = 6. Represents the operation as 18 ÷ 3 = 6.
What to notice if your student does not solve the problem fluently:
Repeatedly creating sets of three, whether physical or imaged, signals that the student lacks belief that the result can be predicted using their available number facts.
Additive methods might include progressively adding threes and may include a trusted fact, such as 3 + 3 + 3 = 9, then building up until all 18 people are used. These methods indicate the student has yet to fully connect multiplication with equal addition, and/or lacks multiplication facts to apply, in this case multiples of three.
Inability to write a division fact may indicate that the student does not connect repeated equal subtraction with division.
Supporting activities:
Using division facts to anticipate the result of repeated subtraction

There are 54 players. They get into teams of six for a volleyball tournament. How many teams do they make?
(Do not model the problem physically. Look for mental or written strategies)
Write a division equation for this problem.
Signs of fluency and understanding:
Anticipates the number of equal teams using mental or written strategies. Uses multiplication or division fact knowledge, such as 9 x 6 = 54 or 54 ÷ 6 = 9. Represents the operation as 54 ÷ 6 = 9.
Anticipates the number of teams fluently by deriving from a known multiplication or division fact, such as 10 x 6 = 60 so 9 x 6 = 54.
What to notice if your student does not solve the problem fluently:
Additive methods might include progressively adding sixes until the target of 54 is reached, such as 6 + 6 = 12 (2 teams), 12 + 12 = 24 (4 teams), 24 + 24 = 48 (8 teams), 48 + 6 = 54 (9 teams). Additive methods indicate the student relies of addition fact knowledge and/or lacks connection between multiplication and equal addition.
Partial multiplication strategies might include using a known fact, such as 5 x 6 = 30 then building on additively. The heavy load on working memory makes these strategies prohibitive unless written recording is used.
Inability to write a division equation indicates that the student needs exposure to use of the signs to model repeated subtraction (quotative division) situations.
Supporting activities:
Creating facts to solve quotative division problems

There are 120 people to take to the kapa haka festival. Each minibus takes 8 people. How many minibuses are needed?
(Do not model the problem physically. Look for mental or written strategies.)
Write a division equation for this problem.
Signs of fluency and understanding:
Uses an efficient place value based, multiplicative strategy to anticipate the number of minibuses needed (15). Strategies might include:
 Apply the distributive property for multiplication, such as 8 x 10 = 80, 5 x 8 = 40, therefore 15 x 8 = 120.
 Apply proportional adjustment, such as 30 x 4 = 120 so 15 x 8 = 120 (less likely strategy).
If the student elects to perform a written algorithm, ask place value related questions to find out if they understand the meaning of the symbols, such as “Tell me where the ‘four’ came from and what it means.”
What to notice if your student does not solve the problem fluently:
Additive methods might include progressively adding eights such as 8 + 8 = 16 then 16 + 8 = 24, …etc. until 120 is reached. These methods indicate the student relies of addition fact knowledge and/or lacks connection between multiplication and equal addition. The heavy load on working memory makes these strategies prohibitive without use of recording.
Partial multiplicative strategies usually include use of a known multiplication fact, in this case 10 x 8 = 80, followed by repeated addition until the target is reached. Likewise, these strategies need to be supported by recording.
Inability to write a division equation indicates that the student needs exposure to use of the signs to model repeated subtraction (quotative division) situations.
Supporting activities:
Apply place value to quotative division