New Zealand Curriculum: Level 2
Learning Progressions Framework: Multiplicative Thinking, Signpost 3 to 4
Target students
These activities are intended for students who use additive strategies to solve multiplication and division problems. They may have some simple multiplication fact knowledge and be able to skip count in twos, fives, and tens.
The following diagnostic questions indicate students’ ability to use known additive strategies or multiplication facts to solve division problems. In doing so, students apply the commutative property, the distributive property, and the associative property of multiplication to the inverse operation, division. (show diagnostic questions)
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 calculators. The questions should be presented orally and in a written form so that the student can refer to them.
Some of the questions and lessons presented below are in context. To increase motivation you might frame the learning situations in contexts that appeal to the interests of your students. Division applies to situations where a set, the dividend, is either shared into equal sets or is measured in sets of equal size. Students might relate to equal sharing in contexts like allocating money from a job fairly, distributing collections equally among team members, allocating time so all people do their share of the mahi (work), or working out how many vehicles are needed to transport a group to an event.
- Here are ten macaroon biscuits.
Two people share the biscuits equally.
How many macaroons does each person get?
Signs of fluency and understanding:
Anticipates the result of five biscuits each using either doubles, 5 + 5 = 10, or multiplication, 2 x 5 = 10.
What to notice if your student does not solve the problem fluently:
Draws a diagram to support them, such as tallies, as the biscuits are allocated one at a time to each person. Counts the tallies as part of the process of naming the share.
Imaging one by one allocation then counts an imaged share by ones. This may indicate that the student needs experience in using number knowledge to anticipate the results of equal sharing situations.
Supporting activity:
Equal sharing in halves
- Three friends have 12 minutes of skipping time before dinner.
To be fair, how much time should each friend spend in the middle while the others turn the rope?
Explain what you did to solve the problem.
Signs of fluency and understanding:
Uses 3 x 4 = 12 to anticipate that each friend gets 4 minutes.
Explains that 3 x 4 means 3 friends getting 4 minutes each in this context.
Uses additive methods, such as 3 + 3 + 3 = 9 so 4 + 4 + 4 = 12 or 2 + 2 + 2 = 6 so 4 + 4 + 4 = 12. Explains that four added three times makes 12 minutes, so each friend gets 4 minutes of skipping time.
What to notice if your student does not solve the problem fluently:
Allocates the 12 minutes one at a time to each friend, by imaging, or recording tallies or objects to represent minutes. Counts 4 minutes in each friend’s time to confirm the shares are equal. Explains one-by-one sharing. This may indicate that the student needs support to derive from existing addition or multiplication knowledge to solve equal sharing problems.
Supporting activity:
Equal sharing into more than two shares
Each taxi takes four passengers. There are 20 people in the group.
How many taxis do they need to get to the hotel?
Explain how you worked out the answer.
Signs of fluency and understanding:
Uses 5 x 4 = 20 to find that five taxis are needed. Explains that, in this context, 5 refers to the number of taxis and 4 to the number of people in each taxi. The student may use the commutative property, 4 x 5 = 20 so 5 x 4 = 20.
What to notice if your student does not solve the problem fluently:
Uses addition to find the answer, e.g., 4 + 4 = 8, 8 + 4 = 12, 12 + 4 = 16, 16 + 4 = 20, 20 + 4 = 24. Tracks the number of fours added as the number of taxis needed.
Skip counts in fours, 4, 8, 12, 16, 20, tracking the count of five taxis. This may indicate that the student needs experience in using number facts to anticipate the result of division by measurement (How many fours are in 20?).
Supporting activity:
Division by measurement
- Danny picks 40 avocadoes to sell.
If he puts the avocadoes into bags of ten, how many bags can he fill?
If he puts the avocadoes into bags of five, how many bags can he fill?
Explain how you worked out your answer.
Signs of fluency and understanding:
Uses place value knowledge, 4 tens equal forty, to anticipate that 4 bags of ten avocadoes can be made.
Recognises that two bags of five can be made from each bag of ten. Doubles four and explains that twice as many bags of five, 8 bags, can be made.
May work each problem out separately using multiplication knowledge, 4 x 10 = 40 and 8 x 5 = 40, respectively. This may indicate that the student will benefit from exploring how doubling and halving manifests itself in division.
What to notice if your student does not solve the problem fluently:
Uses skip counting or repeated addition to find out the number of tens and fives in 40. Sees the number of bags of ten and five as unrelated. This may indicate that the student needs experience with measurement division in situations where the divisors are related, like ten and five.
Supporting activity:
Related divisors
- Six people are collecting cockles and plan to them equally.
If they find 30 cockles, then each person gets five cockles.
If they find 42 cockles, how many does each person get?
Explain your answer.
Signs of fluency and understanding:
Calculates that 42 is twelve more fish than 30. Since 12 ÷ 6 = 2 that means two more fish per family than if 30 fish were caught. 5 + 2 = 7 which means each family gets 7 fish.
What to notice if your student does not solve the problem fluently:
Makes no connection between 6 x 5 = 30 and 6 x □ = 42. Tries to work out 42 fish shared among six families by skip counting, repeated addition, or working through the “times six” basic facts, if they are known. This may indicate that the student needs experience with applying the distributive property of multiplication to division.
Supporting activity:
Deriving from a known division or multiplication fact