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 multiplication facts to find other facts that they do not know. In doing so, students apply the commutative property, the distributive property, and the associative property. They also learn to use multiplication as the inverse operation to solve division problems. (show diagnostic questions)
The questions are presented 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 embedded in context. To increase motivation you might frame all learning situations in contexts that appeal to the interests of your students. Multiplication applies to situations where equal sets are combined. Students might relate to forming equal sets in contexts like minutes taken for routine tasks, like a brushing their teeth or taking a shower, or other measurement contexts like packets of items, lengths of materials such as mats in foot lengths, cupfuls needed to fill a bottle, or apples needed to make 1 kilogram.
- Draw a picture that shows 4 x 5.
Explain where 4 and 5 are in your picture.
Signs of fluency and understanding:
Draws a representation showing four sets with five objects in each set. (This is the standard convention in New Zealand)
Draws a representation showing five sets with four objects in each set. (This is the convention in some other countries)
Clearly explains the role of the multiplier as “how many sets of.”
Clearly explains the role of the multiplicand as “the number of items in each set.”
What to notice if your student does not solve the problem fluently:
Unable to draw a picture of 4 x 5.
Draws 20 objects but is unable to explain the role of 4 and 5 in the multiplication fact. This may indicate that the student rote learnt some multiplication facts without developing an understanding of what the facts represent.
Records the symbols 4 x 5 = 20 but is unable to explain the meaning of the numbers in the equations. This also suggests the student may have rote learnt facts without understanding.
Supporting activity:
Meaning of multiplication symbols
- Here are two small orchards. (Show picture)
Write a multiplication equation for the number of trees in each orchard.
Why do both orchards have the same number of apple trees?
Signs of fluency and understanding:
Records 3 x 4 = 12 (Maggie’s Orchard) and 4 x 3 = 12 (Arohia’s Orchard). Note the factor order might be reversed depending on whether the number of rows or columns is used for the multiplicand.
Explains that the orchards have the same number of trees using the commutative property, i.e., 4 x 3 = 3 x 4
What to notice if your student does not solve the problem fluently:
Unable to create multiplication equations for the numbers of trees. This indicates that the student needs opportunities to interpret the multiplicative structure in arrays.
Creates correct equations for the arrays and uses counting methods to find the total number of trees in each orchard. Counting might be by ones or by skip counting, possibly by twos. This may indicate that the student needs to learn some basic multiplication facts, particularly for twos, fives, and tens.
Records the equations and notices the products are equal. Does not refer to the reversal of factors (commutative property) in explaining why the products are equal.
Supporting activity:
Commutative property
- Here are four sets of five fingers. That shows 4 x 5 = 20.
What does 6 x 5 equal? Explain your answer.
Signs of fluency and understanding:
Fluently recognises that 6 x 5 is another set of two hands and gives the product as 6 x 5 = 30. The students may use place value knowledge that 3 x 10 = 30.
What to notice if your student does not solve the problem fluently:
Unable to decode 6 x 5 as “six hands of five fingers” in this context. This indicates the student needs more support to connect multiplication expressions and equations to physical representations.
Counts on by ones, 20, 21, 22, 23, …, 30, or skip counts in fives, 20, 25, 30. This may indicate that the students needs support to see the connection between additive strategies and deriving multiplication facts.
Supporting activity:
Deriving by adding onto twos, fives, and tens
- Here are five sets of ten. 5 x 10 = 50.
What does 5 x 9 equal? Explain your answer.
Signs of fluency and understanding:
Subtracts 5 from 50 to get 45. Explains that one dot must be taken from each tens frame.
What to notice if your student does not solve the problem fluently:
Unable to see how 5 x 9 can be found using 5 x 10. This may indicate that the student needs support in modelling and connecting multiplication facts.
Counts back by ones to find the product of 45. This may indicate that the student understands how to connect the multiplication facts but does not see the opportunity to use their additive strategies.
Supporting activity:
Deriving by subtracting from tens and fives
- Here are two sets of six cans. 2 x 6 = 12.
What does 4 x 6 equal? Explain your answer.
Signs of fluency and understanding:
Doubles 12 to get 24 as the product. Explains that 4 x 6 is twice 2 x 6.
What to notice if your student does not solve the problem fluently:
Does not make a connection between 2 x 6 and 4 x 6. Calculates 4 x 6 using repeated addition, 6 + 6 = 12, 12 + 6 = 18, 18 + 6 = 24. May attempt skip counting, 6, 12, 18, 24 but counting on and double tracking the counts of six is more likely, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, …, 24. This may indicate that the student needs support connecting the structure of the 2 x and 4 x facts.
Supporting activity:
Deriving by doubling
- Three kererū share 18 karaka berries equally.
How many berries does each kererū get?
Signs of fluency and understanding:
Divides 18 ÷ 3 = 6 or uses 3 x 6 =18 to work out that each kereru gets six berries.
Finds a useful multiplication fact, such as 3 x 5 = 15, and derives from that fact to establish 3 x 6 = 18 so each kereru gets 6 berries.
What to notice if your student does not solve the problem fluently:
Shares the berries using one by one dealing, either physically or by imaging, then counts the shares to establish that each kereru gets 6 berries. This may indicate that that the student needs experience with anticipating the result of equal sharing using number facts.
Uses trial and error by predicting the number of berries for each kereru then uses addition to check if the prediction works. For example, tries 4 berries each then adds 4 + 4 = 8, 8 + 4 = 12, to recognise that the kereru get more than 4 berries each. This may indicate that that the student needs experience with anticipating the result of equal sharing using multiplication facts.
Supporting activity:
Equal sharing using multiplication