Multiplication: Trays of arrays
View Sequence overviewStudents complete an activity with arrays to build their understanding of the associative property.
Whole class
Trays of arrays PowerPoint
Each group
A small collection of counters
Each student
Transforming arrays Student sheet
Grid paper
Build
Complete the number strings below as a class. We suggest completing the string on two separate days (see the Suggested Implementation for this sequence). Watch the Number strings professional learning embedded in this step to learn how to run a number string with your class.
The first number string is on slide 29 and the second is on slide 30.
| Number string 1 | Number string 2 |
$$3 \times 4 =$$ $$6 \times 4 =$$ $$3 \times 8 =$$ $$6 \times 8 =$$ $$12 \times 4 =$$ $$3 \times 16 =$$ | $$2 \times 6$$ $$2 \times 2 \times 3$$ $$4 \times 3$$ $$6 \times 5$$ $$2 \times 3 \times 5$$ $$2 \times 15$$ |
Associative property
These number strings have been carefully crafted to build students’ understanding of the associative property.
Let’s look closely at the first string.
| Double $3 \times 4$ to get $6 \times 4$ | $$2 \times (3 \times 4) = (2 \times 3) \times 4 = 6 \times 4$$ |
| Double $3 \times 4$ to get $3 \times 8$ | $$(3 \times 4) \times 2 = 3 \times (4 \times 2) = 3 \times 8$$ |
| Double $6 \times 4$ OR $3 \times 8$ to get $6 \times 8$ | $$(6 \times 4) \times 2 = 6 \times (4 \times 2) = 6 \times 8$$ $$2 \times (3 \times 8) = (2 \times 3) \times 8 = 6 \times 8$$ |
| $12 \times 4$ is the same as $6 \times 8$ because it is double 6 and half of 8 | Halve 8 and double 6 $$6 \times (2 \times 4) = (6 \times 2) \times 4 = 12 \times 4$$ |
$3 \times 16$ is the same as $6 \times 8$ because it is half of 6 and double 8 OR $3 \times 16$ is the same as $12 \times 4$ because 3 is a quarter of 12 and 16 is four times 4 | Halve 6 and double 8 $$(3 \times 2) \times 8 = 3 \times (2 \times 8) = 3 \times 16$$ Quarter 12 and 4 fours $$(3 \times 4) \times 4 = 3 \times (4 \times 4) = 3 \times 16$$ |
Use an array and equations to represent students’ strategies. The Number strings professional learning video embedded in this step illustrates how you might do this.
Discuss with colleagues: What relationships do you notice between the numbers in the second string?
These number strings have been carefully crafted to build students’ understanding of the associative property.
Let’s look closely at the first string.
| Double $3 \times 4$ to get $6 \times 4$ | $$2 \times (3 \times 4) = (2 \times 3) \times 4 = 6 \times 4$$ |
| Double $3 \times 4$ to get $3 \times 8$ | $$(3 \times 4) \times 2 = 3 \times (4 \times 2) = 3 \times 8$$ |
| Double $6 \times 4$ OR $3 \times 8$ to get $6 \times 8$ | $$(6 \times 4) \times 2 = 6 \times (4 \times 2) = 6 \times 8$$ $$2 \times (3 \times 8) = (2 \times 3) \times 8 = 6 \times 8$$ |
| $12 \times 4$ is the same as $6 \times 8$ because it is double 6 and half of 8 | Halve 8 and double 6 $$6 \times (2 \times 4) = (6 \times 2) \times 4 = 12 \times 4$$ |
$3 \times 16$ is the same as $6 \times 8$ because it is half of 6 and double 8 OR $3 \times 16$ is the same as $12 \times 4$ because 3 is a quarter of 12 and 16 is four times 4 | Halve 6 and double 8 $$(3 \times 2) \times 8 = 3 \times (2 \times 8) = 3 \times 16$$ Quarter 12 and 4 fours $$(3 \times 4) \times 4 = 3 \times (4 \times 4) = 3 \times 16$$ |
Use an array and equations to represent students’ strategies. The Number strings professional learning video embedded in this step illustrates how you might do this.
Discuss with colleagues: What relationships do you notice between the numbers in the second string?
Number strings
Number strings are a short (15-20 minutes) routine, using a sequence of number problems. The sequence of number problems is carefully crafted to highlight number relationships and to build specific mental strategies. This string highlights the use of distributive property for multiplication based on place value parts.
Watch this video to learn how to run a number string in your classroom.
Number strings are a short (15-20 minutes) routine, using a sequence of number problems. The sequence of number problems is carefully crafted to highlight number relationships and to build specific mental strategies. This string highlights the use of distributive property for multiplication based on place value parts.
Watch this video to learn how to run a number string in your classroom.
Conversation and number strings
Conversation is a critical element of a number string activity, as students are held accountable for defending their own strategies and making sense of others’. Students have a greater chance of making sense of and using the mathematical ideas that arise from other students’ work when they are made public and explicit.
Focus the class conversation on how students used the mathematical relationships between the multiplication problems: how students used one problem to solve another. This structured practice supports students to develop their mental calculation strategies and deepens their understanding of the mathematical models that are used.
Conversation is a critical element of a number string activity, as students are held accountable for defending their own strategies and making sense of others’. Students have a greater chance of making sense of and using the mathematical ideas that arise from other students’ work when they are made public and explicit.
Focus the class conversation on how students used the mathematical relationships between the multiplication problems: how students used one problem to solve another. This structured practice supports students to develop their mental calculation strategies and deepens their understanding of the mathematical models that are used.
Use Trays of arrays PowerPoint to review the learning from the previous two tasks.
Show slide 31.
To find how many bags of bread rolls there would be we divided 12 into 2 groups by representing it as $2 \times 6$. We doubled 5 to make 10 by grouping the 2 and 5.
Show slide 32.
To find how many boxes of cakes there would be we divided 24 into 4 groups by representing it as $4 \times 6$. We quadrupled 3 to make 12 by grouping the 3 and 4.
Pose the activity: Illustrate why these equations are equivalent. You can use arrays and/or numbers.
Provide students with Transforming arrays Student sheet and a sheet of grid paper to create arrays. Also provide students access to counters if they would prefer to represent their arrays in this way. Allow students time to work through the activity.
Conduct a class discussion to look at the arrays and numbers that they used to illustrate the equivalence between equations.
Associative property of multiplication
This task provides a context for students to continue to explore the associative property of multiplication. The associative property is a fundamental property of our number system, that applies to addition and multiplication.
The addition or multiplication of a set of numbers gives the same output no matter how the numbers are grouped. This is known as the associative property.
In multiplication, this can be expressed algebraically as:
$$(a \times b) \times c = a \times (b \times c)$$
Let’s consider the associative property applied to $5 \times 2 \times 4$:
This can be solved as $(5 \times 2) \times 4$ or $10 \times 4$. It can also be solved as $5 \times (2 \times 4)$ or $5 \times 8$. We can represent this in the following way:
$$(5 \times 2) \times 4 = 5 (2 \times 4)$$
$$10 \times 4 = 5 \times 8$$
This task provides a context for students to continue to explore the associative property of multiplication. The associative property is a fundamental property of our number system, that applies to addition and multiplication.
The addition or multiplication of a set of numbers gives the same output no matter how the numbers are grouped. This is known as the associative property.
In multiplication, this can be expressed algebraically as:
$$(a \times b) \times c = a \times (b \times c)$$
Let’s consider the associative property applied to $5 \times 2 \times 4$:
This can be solved as $(5 \times 2) \times 4$ or $10 \times 4$. It can also be solved as $5 \times (2 \times 4)$ or $5 \times 8$. We can represent this in the following way:
$$(5 \times 2) \times 4 = 5 (2 \times 4)$$
$$10 \times 4 = 5 \times 8$$
Work samples
Look at how these students have transformed arrays to show how the two facts are equivalent.
Discuss with colleagues: What do these students know and what have they shown that they can do?
Look at how these students have transformed arrays to show how the two facts are equivalent.
Discuss with colleagues: What do these students know and what have they shown that they can do?