Rich tasks
Jigsaw multiplication
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1. Jigsaw multiplication
- Ensure that students are familiar with how a multiplication grid works and that they realise the tables on this grid are not ordered 1-10 as they usually are.
- Encourage students to look for facts that they recognise first and then derive those they don’t know, to check that each number is placed correctly.
Using multiplying by twos and doubling page 106
- Use x 2 multiplication facts to solve x 4 multiplication facts.
- Use x 4 multiplication facts to solve x 8 multiplication facts.
- Use an array to demonstrate doubling strategies and solve multiplication problems.
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Teacher notes
- Arrays are helpful for developing understanding of the commutative (3 x 5 = 5 x 3)
and the distributive (3 x 5 = 2 x 5 + 1 x 5) property of multiplication.
- This series adopts New Zealand conventions for interpretation of multiplication written as equations. The first number (multiplier) indicates the number of groups, and the second number (multiplicand) indicates the size of each group. So, 3 x 5 should be read as “3 groups of 5”, with an answer (product) of 15.
- It is helpful if students in the same class are encouraged to draw and interpret arrays in the same way. In the student book, each group is drawn horizontally with subsequent groups added underneath. Many students find it easier to align rows and columns when using grid paper. This ensures that groups contain the correct number of items.
- Encourage students to combine groups using addition facts, for example, repeated addition, doubling or multiplication facts (x 2 to find x 4) to find the product, rather than skip counting at this level.
- This link takes you to a PM with a 10 x 10 array.
1/A4 page: http://www.caxed.co.nz/assets/3B-PM-Happy-hundreds-array_45063_1.pdf
4/A4 page: http://www.caxed.co.nz/assets/3B-PM-Happy-hundreds-array-x4_45062_1.pdf
You could use Teacher Resource Sheet 14 - Using a 10 x 10 array to show students how they can be used for exploring or solving multiplication equations with facts students do not yet know.
Multiplying by tens and fives page 109
- Use x 10 multiplication facts to solve x 5 multiplication facts.
- Use a Slavonic abacus or an array to demonstrate halving strategies and solve multiplication problems.
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Teacher notes
- Students were first introduced to the relationship between the 5 and 10 times tables in book 2A. It is helpful if they understand that the pattern involves doubling the size of the group and therefore halving the number of groups, for example, 2 groups of 5 = 1 group of 10.
- Using materials such as the Slavonic abacus, $5 and $10 notes or block towers are good ways of supporting students’ understanding of this concept, which can also be represented on arrays or number lines.
Multiplying by threes page 112
- Solve x 3 multiplication facts by adding one more to each group of 2.
- Use an array to demonstrate “adding more” strategies and solve multiplication problems.
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Teacher notes
- Arrays support students’ understanding of an unknown fact, for example 6 x 3 can be solved by splitting 3 into 2 and 1.
So (6 x 2) + (6 x 1) = 12 + 6 = 18.
Teacher Resource Sheet 17 - On or Off?
Multiplying by adding some on or taking some off page 114
- Solve multiplication problems by adding or subtracting groups.
- Use an array to show how multiplication problems can be solved by adding or subtracting groups.
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Teacher notes
- This section extends the strategy used through “Multiplying by threes” to other multiplication facts. Continued use of arrays will support students’ understanding of which facts could be helpful.
- If students have confident recall of facts to 10 x 10 they are ready to move on and use strategies with multiplication problems with factors outside this range.
x 2, x 3, x 5 and x 10 multiplication practice page 116
- Practice x 2, x 3, x 5, x 10 multiplication facts
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Teacher notes
- When students have learned the rules for x 0 and x 1, and learned their x 2, x 5 and 1 x 10 multiplication facts they have all the facts they need to derive facts to 10 x 10. This is why these facts are learned first.
- Deriving facts that are unknown is a helpful way to learn about the properties of multiplication, but students still need to practice so that recalling facts to 10 x 10 becomes automatic.
- The strategies initially used to derive these facts can then be used to work out more complex multiplication problems.
- If students already have recall of their facts to 10 x 10 they should be encouraged to use this knowledge. It is important to extend the range of factors beyond 10 x 10 at this point to ensure students understand the properties of multiplication and can apply these strategies, for example doubling x 6 to find x 12, or finding x 11 by adding 10 x and 1 x etc.
