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, multiplication and division to situations that involve equal sharing. The questions are given 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. When asking these questions allow access to pencil and paper but not to a calculator. (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 cubes context but can be changed to other contexts that are engaging to your students, such as Lego™, smarties, novelty counters, or toy vehicles.
- Here is a pile of 10 cubes. If you share the cubes equally between two people, how many cubes will each person get?
(Ask for a prediction then let the student equally share the cubes into two sets of five. Physically modelling the situation is not needed if the student responds quickly.)
What fraction of the pile of cubes does each person get?
Signs of fluency and understanding:
Anticipates that 10 cubes will be equally shared into two sets of five, using factual knowledge, such as 5 + 5 = 10 (additive) or 2 x 5 = 10.
What to notice if your student does not solve the problem fluently:
Shares cubes out one by one to establish equal shares. This may indicate that the student does not yet anticipate the result. Lack of anticipation may be due to inadequate number fact knowledge or not recognising the opportunity to use that knowledge. Equal sharing should meet these criteria:
- All the objects are shared (exhausted)
- The correct number of parts are created (two parts)
- The parts are equal (five cubes)
Watch for students to recount the cubes and check that the parts are equal. Recounting indicates a belief that validating equality is an integral part of the equal sharing process.
Supporting activities:
Anticipating the result of equal sharing
- Here are 15 cubes. You share the cubes equally among three people. How many cubes does each person get?
Write an equation for what you have done. Write a division equation for what you have done.
Write a fraction equation for what you have done.
Signs of fluency and understanding:
Anticipates the equal shares of five fluently without physical modelling. Uses multiplication or division fact knowledge, such as 3 x 5 = 15 or 15 ÷ 3 = 5. May or may not know that the operation can be represented as 1/3 x 15 = 5.
What to notice if your student does not solve the problem fluently:
One by one dealing, either physical or imaged. This may indicate that the student does not understand that the result can be predicted. They may also lack the number facts required to make the prediction.
Additive methods including progressively trying addends, such as 4 + 4 + 4 = 12 then 5 + 5 + 5 = 15, or a combination of addition and equal sharing, such as 3 + 3 + 3 = 9, one more cube to each part uses 12 cubes in total (10, 11, 12) and one more uses all 15 cubes. These methods may indicate the student has yet to connect multiplication with equal addition, and/or lacks multiplication facts to apply. Students might also use multiplication or division to share the cubes, but not recognise how to represent this using fraction language and symbols.
Supporting activities:
Using multiplication facts to anticipate equal sharing
- Here is a pile of 40 cubes. You are going to share the cubes equally among five people. How many cubes will each person get?
(Ask for a prediction then let the student equally share physically if needed. Do not physically share if the response is fluent and correct)
Write a division equation for what you have done.
Write a fraction equation for what you have done.
Signs of fluency and understanding:
Anticipates the equal shares of eight fluently without physical modelling. Uses multiplication or division fact knowledge, such as 5 x 8 = 40 or 40 ÷ 5 = 8. Represents the operation as 40 ÷ 5 = 8 and 15 x 40 = 8.
What to notice if your student does not solve the problem fluently:
Composite dealing, whether physical or imaged, signals that the student lacks belief that the result can be predicted using number facts. For example, the student may give each share two cubes, recognise that ten cubes are used, repeat the action four times, then count the shares by twos.
Additive methods including:
- progressively trying addends, such as 10 + 10 + 10 + 10 = 40 then removing the same number from each part to create the fifth share,
- a combination of addition and equal sharing, such as 5 + 5 + 5 + 5 + 5 = 25, recognise that adding one more cube to each part uses 5 cubes in total (10, 11, 12) and skip counts 25, 30, 35, 40 (tracking the extra cubes added to each part).
These methods may indicate the student has yet to fully connect multiplication with equal addition, and/or lacks multiplication facts to apply, in this case multiples of five.
Inability to write division and fraction equations indicates that the student needs exposure to use materials in equal sharing situations.
Supporting activity:
Using multiplication and division facts to equally share - Here is a pile of 36 cubes. You are going to share the cubes equally among four people. How many cubes will each person get?
(Ask for a prediction then let the student equally share physically if needed. Do not physically share if the response is fluent and correct)
Write a division equation for what you have done.
Write a fraction equation for what you have done.
Signs of fluency and understanding:
Anticipates the equal shares of nine fluently without physical modelling. Uses multiplication or division fact knowledge, such as 4 x 9 = 36 or 36 ÷ 4 = 9. Represents the operation as 36 ÷ 4 = 9 and 14 x 36 = 9.
Anticipates the equal shares of nine fluently by deriving from a known multiplication or division fact, such as 4 x 10 = 40 so 4 x 9 = 36.
What to notice if your student does not solve the problem fluently:
Additive methods including:
- progressively trying addends, such as 10 + 10 + 10 + 10 = 40 then removing one from each part to create shares of nine
- a combination of addition and equal sharing, such as 5 + 5 + 5 + 5 = 20, recognising that adding three more cubes to each part uses 12 more cubes (32 in total) then adding one more to each part.
These methods may indicate the student relies on addition fact knowledge and/or lacks connection between multiplication and equal addition.
Partial multiplication, including trying to halve 36 then halve again to create quarters, or combining simpler known facts, such as 4 x 5 =20 and 4 x 2 = 12 to ‘build up’ the answer. The heavy load on working memory makes these strategies prohibitive.
Inability to write division and fraction equations indicates that the student needs exposure to use of the signs to model equal sharing situations.
Supporting activity
Creating division facts and fraction facts to equally share
- Here is a pile of 54 cubes. You are going to share the cubes equally among three people. How many cubes will each person get?
Write a division equation for what you have done.
Write a fraction equation for what you have done.
Signs of fluency and understanding:
Uses an efficient place value based, multiplicative strategy to anticipate the equal shares of 18 fluently. Strategies might include:
- Apply the distributive property for multiplication, such as 3 x 10 = 30, 3 x 8 = 24, therefore 3 x 18 = 54.
- Apply the distributive property for multiplication combined with rounding, such as 3 x 20 = 60, 3 x 2 = 6, therefore 3 x 18 = 60 - 6.
- Apply proportional adjustment, such as 6 x 9 = 54 so 3 x 18 = 54 (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 ‘two’ came from and what it means.
What to notice if your student does not solve the problem fluently:
Additive methods, including progressively trying addends, such as 10 + 10 + 10 = 30 then progressively adding to each part, 5 + 5 + 5 = 15 so 15 + 15 + 15 = 45, etc. These methods indicate the student relies of addition fact knowledge and/or does not yet see the connection between multiplication and equal addition. The heavy load on working memory makes these strategies prohibitive without use of recording.
Inability to write division and fraction equations indicates that the student needs exposure to use of the signs to model equal sharing situations.
Supporting activity:
Applying place value to equal sharing