Multiplication: reSolve Market
View Sequence overviewMultiplying how many by how much gives the whole: $$\text{how many} \times \text{how much} = \text{the whole}$$
Whole class
reSolve Market PowerPoint
Each group
Counters
Each student
Hidden fruit Sheet printed on A3
Blank A3 paper
Task
Show the picture of the reSolve Market on slide 16 of the reSolve Market PowerPoint to continue the context of the reSolve Market.
Revise:
The array is a very helpful representation for multiplication. We can clearly see ‘how many groups’ and ‘how much in each group’. In the reSolve Market we can use how many and how much to look for relationships between the different fruit displays.
But what if we can’t see all the parts of the array?
Show slide 17.
The grocer has boxes of fruit that she is unpacking and putting out on display. Some of the apples, lemons and blueberry punnets are missing.
The box of strawberry punnets is all full, the boxes of oranges and mangoes are full. There are also the right number of kiwi fruit in each bag. We just can’t see them all.
Pose the task: Can you work out how many apples, lemons and blueberry punnets would be in each box if each box was full? Can you work out how many strawberry punnets, oranges, mangoes and kiwifruit there are?
Provide each student with an A3 sheet of paper and Hidden fruit Sheet. Also provide them with access to counters to use if they choose.
Allow students time to work on the task. Ask them to record on their A3 paper the way that they worked out the total number of items that would fit into each box. Students should represent their thinking in any way they choose, including using materials, diagrams, numbers and equations.
This task asks students to apply their understanding of the array, specifically that there are an equal number in each row. They need to work out how many groups/rows and how much in one group/row, then multiply the two quantities to find the product. Students who solve the problem in this way demonstrate a good understanding of the multiplicative structure as represented by the array.
Students may not begin with this strategy; they should be allowed time to explore and work towards this approach for themselves.
It is quite likely that some students will need to draw or model all items to determine how many pieces of fruit (or punnets) are (or can be) in the boxes. This indicates that they do not fully appreciate the equal rows structure of the array.
- Do we need to see everything to be able to know how many there would be?
- We don’t need to see everything if we understand the structure of equal groups and the array.
- What information do we need to work out how many would be in full boxes?
- The information needed is how many groups and how many in each group. In each example this is provided for the students.
Student work samples
Look at these work samples and see how these students have solved the problem.
Discuss with colleagues: What do these students know and what have they shown that they can do? Give evidence from the work samples for your statements.
Look at these work samples and see how these students have solved the problem.
Discuss with colleagues: What do these students know and what have they shown that they can do? Give evidence from the work samples for your statements.
Use reSolve Market PowerPoint to support the class discussion. Slides 18, 19, and 20 present three different students and the strategies that they used to solve the number of fruit to compare and discuss.
Show slide 18.
This is how Tommy worked out the number of apples there would be if the box was full. Look carefully at how Tommy has recorded his solution.
Ask: Can someone explain how Tommy solved the problem?
- Tommy has completed the array and then skip counted the columns to work out the total that can fit in the box.
Show slide 19.
This is how Isla worked out the number of strawberry punnets there are in the full box. Look carefully at how Isla has recorded her solution.
Ask: Can someone explain how Isla solved the problem?
- Isla worked out how many groups and how much in a group. She multiplied the two quantities to find the whole.
Show slide 20.
This is how Anya worked out the number of mangoes in the four full boxes. Look carefully at how Anya has recorded her solution.
Ask: Can someone explain how Anya solved the problem?
- Anya has drawn all and then counted by twos to calculate the total amount.
Show slide 21 which shows all three strategies.
Discuss:
- What is helpful about these different strategies?
- Isla’s strategy is the most efficient and shows that she understands the array structure.
- Students may comment that it is helpful to see all the items to work out how many there are altogether. This is useful for those who have not fully abstracted the array structure.
- Do you need to see everything in an array to know how many there are altogether?
- You only need to know how many and how much. Multiplying these two quantities gives the whole amount.
- Which is the most efficient strategy?
- Isla’s is the most efficient. We want to promote efficiency based on an understanding of the array structure.
Slide 22 shows the lemons, blueberry punnets, oranges and kiwi fruit.
Ask: How could we efficiently work out how many are in each collection if the boxes were full?
This question reinforces the relationship that exists between how many rows/groups (first factor), how much in each row/group (second factor), and the whole (product). The whole can be calculated in the following way:
$$\text{how many} \times \text{how much} = \text{the whole}$$
The array structure
The Hidden fruit task uses partial arrays/groups, so that students use their understanding of the relationship that exists between three quantities:
- how many rows/groups
- how much in each row/group
- the whole
This can be represented as a multiplication fact:
$$\text{how many} \times \text{how much} = \text{the whole}$$
The purpose of this task is to encourage students to begin to abstract the arrays, because they cannot see the complete array. They need to understand the how many and how much structure of multiplication and apply this understanding to solve the problem.
The fruit is arranged in arrays in the different boxes, but not all the fruit is visible, but enough information is visible for students to deduce the number of rows and the number in each row, even though they are unable to see the whole array.
The Hidden fruit task uses partial arrays/groups, so that students use their understanding of the relationship that exists between three quantities:
- how many rows/groups
- how much in each row/group
- the whole
This can be represented as a multiplication fact:
$$\text{how many} \times \text{how much} = \text{the whole}$$
The purpose of this task is to encourage students to begin to abstract the arrays, because they cannot see the complete array. They need to understand the how many and how much structure of multiplication and apply this understanding to solve the problem.
The fruit is arranged in arrays in the different boxes, but not all the fruit is visible, but enough information is visible for students to deduce the number of rows and the number in each row, even though they are unable to see the whole array.
Discuss: What do we need to know to be able to work out how many in the whole array?
Show slide 22.
Explain: We do not need to see or count every part of the array to know how many there are all together. We just need to know how many groups and how much in each group. We can then multiply:
$$\text{how many} \times \text{how much} = \text{the whole}$$
Add this new understanding to the class display along with students’ work samples.
Class display
In the Summarise phase, the following statement is presented to students:
We do not need to see or count every part of the array to know how many there are altogether. We just need to know how many groups and how much in each group. We can then multiply:
$$\text{how many} \times \text{how much} = \text{the whole}$$
This statement reflects the learning goal for the task and is the understanding that we want all in the class to share. As the sequence progresses, this shared understanding is built on.
Presenting everyone’s work with this statement communicates that everyone has contributed to and participated in developing this understanding as shared in the class, regardless of how complex or sophisticated their thinking may or may not be.
In the Summarise phase, the following statement is presented to students:
We do not need to see or count every part of the array to know how many there are altogether. We just need to know how many groups and how much in each group. We can then multiply:
$$\text{how many} \times \text{how much} = \text{the whole}$$
This statement reflects the learning goal for the task and is the understanding that we want all in the class to share. As the sequence progresses, this shared understanding is built on.
Presenting everyone’s work with this statement communicates that everyone has contributed to and participated in developing this understanding as shared in the class, regardless of how complex or sophisticated their thinking may or may not be.