Deriving basic multiplication facts
This resource supports teachers to assess and find appropriate activities for students who need acceleration in their understanding and application of deriving basic multiplication 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.
Deriving basic multiplication facts
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.
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 written form (Deriving basic multiplication facts) 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 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.
Required materials
See Materials that come with this resource to download:
- Deriving basic multiplication facts (.pdf)
Activities
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
Here are two small orchards. (Show the 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 that 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, such as 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 indicates that the student needs opportunities to interpret the multiplicative structure in arrays.
- Creates the 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
Here are four sets of five fingers: 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, such as 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, 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 student needs support to see the connection between additive strategies and deriving multiplication facts.
Supporting activity
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 and 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 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
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 and 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
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 the student needs experience 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, they try 4 berries each, then add 4 + 4 = 8, 8 + 4 = 12, to recognise that the kereru get more than 4 berries each. This may indicate that the student needs experience anticipating the result of equal sharing using multiplication facts.
Supporting activity
Teaching activities
Heading
Meaning of multiplication symbols
The purpose of this activity is to support students to decode the meaning of multiplication equations and understanding the role of the multiplier, multiplicand, and product in the equations. Problems are restricted to equal sets situations.
1 of 6Commutative property
The purpose of this activity is to support students to understand and apply the commutative property of multiplication.
2 of 6Deriving by subtracting from tens and fives
The purpose of this activity is to support students to derive new multiplication facts by subtracting from multiples of five and ten. For example, if the student knows 6 x 10 = 60 they can find 6 x 9 = 54 by subtracting six.
3 of 6Equal sharing using multiplication
The purpose of this activity is to support students to use their known multiplication facts to solve equal sharing problems.
4 of 6Deriving by doubling
The purpose of this activity is to support students to derive new multiplication facts from facts they know, using doubling or either the multiplier or the multiplicand. For example, if 2 x 7 = 14 is known then 4 x 7 = ? can be solved by doubling 14.
5 of 6Deriving by adding onto twos fives and tens
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.
6 of 6
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