Positive Divided By A Positive

In this Math Is Visual video blitheness and visual prompts, we volition explore negative divided by a positive integer division conceptually through concrete and visual representations! We will also tackle this from 2 approaches: quotative division and partitive sectionalization!

Then far in theinteger divisionseries, nosotros have tackled the following ideas:

Whole Number Division and the Ii Types of Division: Quotative and Partitive
Partition of Integers: Negative Divided Past a Negative
Sectionalization of Integers: Negative Divided By a Positive

If yous have yet to view those posts, you should definitely go back prior to moving on to ensure that you are ready to grasp our last step in the Segmentation of Integers Serial: Positive Number Divided By a Negative Number.

In this video, we will address the division of a positive dividend past a negative divisor first using partitive (or fair share) division, then quotative (or measured) division. The viewer volition also be prompted to pause the video and try i their own using concrete manipualtives and/or visuals.

See the instructional guide below aimed to help teachers and/or parents use this activity with their own students/children.

Visual Prompt #1: Partitive Division

In the video above, we start with dividing a negative past a negative through partitive division:

8 ÷ (-2) = ___

in words:

8 divided into -ii groups gives a result of ___ per group

So, let's have the viewer model concretely and/or visually the following:

Integer Division - Positive Divided by Negative partitive 8 divided into -2 groups prompt

We can approach this using a set or linear model where nosotros are spatially splitting the quantity into 2 equal "negatively charged" groups and counting the number of objects in each group. We can call back of this as repeated subtraction if we were to fair share 1 object at a time or ii objects at a time, upwardly to a quantity that a educatee is comfortable to subitize chop-chop for distributing equally to the ii groups.

Integer Division - Positive Divided by Negative partitive 8 divided into -2 groups prompt solution

Visual Prompt #2: Quotative Sectionalization

Now we move on to budgeted the same problem from a partitive partition perspective:

8 ÷ (-2) = ____

or

8 divided into groups of -two gives a result of ____ groups.

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 prompt

The trouble we come across here is that we want to make up one's mind how many groups of -2 nosotros can create from viii.

This is a problem becausewe can't make whatever groups of -ii (or so it may seem).

If we determine first how many groups of +ii we can make (4 in total), nosotros can negate the negative of each group by using negatively charged groups:

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution step 1

And so instead of having four positively charged groups of +two, nosotros can express those groups every bit 4 negatively charged groups of -2.

In other words, we have:

-4 groups of -two

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution step 2

Other possible representations and strategies:

Consider determining how many groups of +2 we can "remove" since division represents repeated subtraction. Once we notice out how many groups of +ii we can remove, we can then express those groupings negatively to create negative values in the groups:

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution approach 2 step 1

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution approach 2 step 2

Nosotros can also think about how many groups of -two we can "add" to go to 0:

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution approach 3 step 1

Since repeated add-on is the reverse performance of repeated subtraction, we can so take those groups and "subtract" them by negating the groupings:

Integer Division - Positive Divided by Negative quotative 8 divided into groups of -2 solution approach 3 step 2

Visual Prompt #three: Your Plough!

Suspension the video and let the viewer apply their knowledge of positive number divided by a negative number using partitive sectionalisation in the following visual prompt:

10 ÷ (-5) = ___

or

10 divided into -v groups gives a result of ___ per group

Integer Division - Positive Divided by Negative partitive 10 divided into -5 groups prompt

Permit time for the viewer to engage in the partitive division before showing the remainder of the video animation or the following still image.

Again, we must note that we want to determine the value of the quantity that will be shared among 5 negatively charged groups or -five groups.

Integer Division - Positive Divided by Negative partitive 10 divided into -5 groups prompt solution step 1

Past off-white sharing 10 beyond the -v groups, we see that each group will comprise two negatives or -two.

Integer Division - Positive Divided by Negative partitive 10 divided into -5 groups prompt solution step 2

How about approaching the aforementioned problem using quotative division:

10 ÷ (-five) = ___

or

10 divided into groups of -5 gives a result of ___ groups

Integer Division - Positive Divided by Negative quotative 10 divided into groups of -5 prompt

By using our conceptual agreement congenital through the previous visual prompts, we tin can now recognize that since there are 0 groups of -five nosotros can create from the +10 given, we must use negatively charged groups.

Integer Division - Positive Divided by Negative quotative 10 divided into groups of -5 prompt solution step 1

Since I tin can create 2 groups of +5, I know I can likewise represent those aforementioned two groups as -2 groups of -5.

Integer Division - Positive Divided by Negative quotative 10 divided into groups of -5 prompt solution step 2

There you accept it! Understanding integer sectionalization conceptually through concrete and visual representations. In my classroom, I'd have students using coloured tiles or linking cubes to physically manipulate the materials to fully conceptualize this very of import thought that frequently is approached using only rules and procedures.

Did you lot use this in your classroom or at home? How'd information technology go? Mail in the comments!

Thanks for sticking around!

Math IS Visual. Let'south teach it that way.