Division (deriving from known facts)
This resource supports teachers to assess and find appropriate activities for students who need acceleration in their understanding and application of division (deriving from known facts).
About this resource
New Zealand Curriculum: Level 2
Learning Progressions Framework: Multiplicative thinking, Signpost 3 to 4
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 counting in twos, fives, and tens.
Division (deriving from known facts)
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.
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 Division deriving 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 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.
Required materials
See Materials that come with this resource to download:
- Division deriving (.pdf)
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 one. This may indicate that the student needs experience using number knowledge to anticipate the results of equal sharing situations.
Supporting activity
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
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 to 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 using number facts to anticipate the result of division by measurement (How many fours are in 20?).
Supporting activity
Danny picks 40 avocados to sell.
- If he puts the avocados into bags of ten, how many bags can he fill?
- If he puts the avocados 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, which 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, which may indicate that the student needs experience with measurement division in situations where the divisors are related, like ten and five.
Supporting activity
Six people are collecting cockles and plan to share 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 and 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 applying the distributive property of multiplication to division.
Supporting activity
Teaching activities
Related divisors
The purpose of this activity is to support students to derive new division facts from known facts in a measurement situation. For example, if four sets of six can be made from 24, then eight sets of three can be made from 24.
1 of 5Equal sharing into more than two shares
The purpose of this activity is to support students in applying their knowledge of addition facts and their knowledge of basic multiplication facts to solve division problems. The problems involve equally sharing a set, the dividend, between more than two parties, and the divisor. At this level, we ...
2 of 5Equal sharing in halves
The purpose of this activity is to support students in applying their knowledge of doubles, or “times two” basic multiplication facts, to solve division problems. The problems involve equally sharing a set between two parties. This involves recognising the amount being shared (the dividend) and the ...
3 of 5Division by measurement
The purpose of this activity is to support students to derive from simple multiplication facts. Those facts come from the simplest skip counting patterns: twos, fives, and tens.
4 of 5Deriving from a known division or multiplication fact
The purpose of this activity is to support students deriving new division facts from given multiplication or division facts. Deriving applies the distributive property of multiplication which connects to division by inverse operation. For example, if 6 x 5 = 30 is known then 35 ÷ 5 is worked out as ...
5 of 5
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