This unit supports students in developing procedural fluency with the multiplication and division of integers, and in developing conceptual understanding of how integers are used in the real world.
Integers are an extension of the whole number system. Therefore, the properties of integers under the four operations should be the same as those for whole numbers. This is described below. Note that the four main properties hold for multiplication but not for division. Furthermore, the set of integers is not closed under division.
The commutative property of multiplication
The order of the addends does not affect the sum. If -3 x 4 = -12, then 4 x -3 = -12. Note that the commutative property does not hold for division. For example, 12 ÷ -3 = -4 but -3 ÷ 12 = -¼.
The distributive property of multiplication
This property, pertaining to the partitioning of factors and recombining those factors, states that a x (b+c) = (a x b) + (a x c). For example, if 5 = -1 + 6, then -3 x 5 = -3 x (-1 + 6) = -3 x -1 + -3 x 6. This property does not hold for division.
The associative property of addition
This property, pertaining to the ‘associating’ of pairs of factors, one pair at a time, states that the grouping of the numbers does not affect the answer. For example, (-4 x 3) x -1 = -4 x (3 x -1). This property does not hold for division.
Inverse operations
Multiplication and division are inverse operations so one operation undoes the other. For example, -2 x -3 = -6 so -6 ÷ -3 = -2.
It is the need for these number laws to hold, that establishes the effect of operations. For example, the multiplying of two negative integers has the same effect as the multiplying of two positive integers.
Lack of closure
The set of integers is not closed under the operation of division. This means that the quotient (answer) for division of an integer by another integer is not always another integer. This lack of closure gives rise to the need for the set of rational numbers.
Many representations of multiplication and division of integers are problematic on two grounds:
Hans Freudenthal (1983) introduced two models for operations on integers, the annihilation and vector models. In the annihilation model positive and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative integer from a pair to equal zero does not alter the quantity being represented. The vector model presents integers as magnitudes with direction. +1 is represented by a vector of length one in a positive direction and -1 as a vector of length one in a negative direction. Both the annihilation and vector models transfer to multiplication and division of integers as repeated addition and subtraction, but will need to be broken down practically in some examples. In 1993 Marcia Cooke introduced a videotape model to represent the multiplication of integers. The factors in the situation were directional speed, in a positive or negative direction, and time, either forward or backward. Cooke’s model adapts to division though creation of ‘missing factor’ problems.
Freudenthal suggested that prompt application of the integer operations was essential for students to appreciate the new possibilities created by enlargement of the number system. In particular, he favoured work with functions in both algebraic and graphical form.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
The unit frames learning in a wider variety of contexts, including money, hills, time, speed, length, area, gears, and pulleys. Consider how you might adapt or replace these contexts with ones that better reflect the interests and cultural backgrounds of your students, and that might make interesting links with learning from other curriculum areas. Students interested in sports might enjoy golf as a context, and those who enjoy computers might find points schemes interesting. Some students may enjoy the context of comparing temperatures from locations around the world. Investigating local places that are above and below sea level could also be an engaging context for students.
Te reo Māori vocabulary terms such as tau tōpū (integer), tau tōraro (negative number), and tau tōrunga (positive number) could be introduced in this unit and used throughout other mathematical learning.
It is important that students have prior experience with addition and subtraction of integers prior to being introduced to this concepts in this unit. Since multiplication is founded on repeated addition, and division on repeated subtraction, knowledge of these operations is essential. Students need to distinguish between integers as quantities (magnitudes with direction) and operations as actions performed on those quantities. For that reason, try to avoid over-simplifications such as ‘two negatives make a positive’, since the negative signs involved relate to different things (direction and operation).
This session introduces multiplication of integers using the annihilation and vector models used in the Level 4 unit Integers. The model represents +1 as a black arrow to the right and -1 as a red arrow to the left (see Copymaster One). Beginning at zero, and combining +1 and -1 in any order, results in a destination of zero. Contexts that can be linked to the addition of +1 and -1 include:
One hill fills one dale One dollar pays one IOU
Look for students to recognise this as a simple equal sets multiplication scenario that can be expressed as 4 x 5 = 20. By changing the unknown, two division problems can be created:
You might model the problems using arrows from Copymaster One or with chalk drawings on concrete.
3 x 8 = _ and 3 x -8 = _ 6 x 7 = _ and 6 x -7 = _ 9 x 2 = _ and 6 x -2 = _
Students may say that when the first factor is negative the model does not make sense. It is not physically possible to have negative sets of anything.
Some might mention that a positive multiplier means that sets are repeatedly added so a negative multiplier must mean the opposite, that sets are repeatedly subtracted. Therefore -3 x 8 = _ means the removal of three sets of eight which results in a ‘deficit’ of -24. The annihilation model of this interpretation relies on making 24 zeros then removing three sets of eight.
Make 24 ‘zeros’
Take away three sets of eight leaving -24.
In this session students explore Marcia Cooke’s model for multiplication of integers.
| Negative Speed | Positive speed |
Negative Time | -2 x -3 = __ | -2 x 3 = __ |
Positive Time | 2 x -3 = -6 | 2 x 3 = 6 |
Problems One and Two
With Problem One:
With Problem Two:
Problem Three
In this session students apply the multiplication of integers to enlargement (dilation) of figures. Students will need access to Geogebra which is a free download. The videos listed below were created using Geogebra and we appreciate their support in allowing us to provide them copyright free.
Unchanged | Changed |
Angles Order (Way parts of figure are connected) Orientation (Direction the figure is facing) Ratios of side lengths | Lengths (x 3) Area (x 9) |
Changes in length and area can be checked using key points. The algebra window gives the co-ordinates of points from the figure and the corresponding points in the image.
Animation Three B shows what happens when a figure is enlarged by a factor of -3. Look again at the variant and invariant features. The change to orientation is particularly interesting. The image is upside down and facing the other way compared to the original.
Students might like to explore what happens with three of four combined enlargements. It is interesting to explore combinations of scale factor -1.
Figure It Out, Number Book Six, Level 4+, page 24 shows how negative enlargement is used in photography and explores the proportional nature of enlargement.
In this session the students explore another application of multiplication of integers, gears and pulleys.
Comment that students are going to see another situation where a rate and direction can be represented using integers.
If the blue gear is attached to a motor and drives the red gear. What will happen to the red gear in each picture as the blue gear turns once? In both situations the red gear will turn twice which is a transmission factor of two. But in the pulley gear (left) the turn will be in the same direction as the drive gear and in the direct gear (right) the turn will be in the opposite direction. The transmission factors are 2 and -2 respectively. How is this like the enlargement situation?
Animation Four A continues and shows the turning of gears. It provides other examples. Stop at each new gear setup and ask the students to name the transmission factors.
You may like to use The Biscuit Factory learning object with your students . This learning object explores gear ratios though it neglects the integer connection. It is a Flash object so is not supported on some devices.
The rest of the lesson is devoted to students working from two e-ako from the Multiplicative thinking pathway (R4.40: Gears as a context for ratios and R5.10: More gears as a context for ratios) These e-ako activities constitute a sequence which culminates in students understanding how the transmission factors in a gear train can be multiplied to find the combined transmission factor. This makes connections to proportional reasoning and the multiplication of fractions. For example, in the train below the blue gear drives the grey gear which in turn drives the red gear.
The grey gear turns 180◦ for each 360◦ turn on the blue gear, in an opposite direction. The transmission factor is -½. The red gear turns 120◦ for each 360◦ turn of the grey gear, in an opposite direction, so the transmission factor is -⅓. The combined effect is the result of -½ x -⅓ = -¹/₆. The red gear makes a 60◦ turn in the same direction for each 360◦ turn of the blue gear.
In this session multiplication and division of integers is generalised, allowing students to investigate number patterns with the operations. Although students do not need to formally refer to the commutative, associative and distributive properties, those properties are implicit in the number patterns they will explore.
x | 2 | -2 |
7 | 14 | |
-7 | 14 |
x | + | - |
+ | + | - |
- | - | + |
Dear families and whānau,
This week we are learning about multiplying and dividing positive and negative numbers. The rules are a bit tricky but we will be working on understanding why things happen and looking at real world applications. Does anyone use integers? It turns out they do!
For example, when we multiply two negative numbers together the product is positive. Why does that happen? Perhaps your student can explain.
Printed from https://nzmaths.co.nz/resource/multiplying-and-dividing-integers at 1:43am on the 27th April 2024