Integers with Manipulatives

Operations with integers can be modeled using two-colored

counters.

Positive

+1

Negative

-1

The following collections of counters have a value of +5.

Build a different collection that has a value of +5.

What is the smallest collection of counters with a value of +5?

As you build collections of two-colored counters, use the smallest collection,

but remember that there are other ways to build a collection.

The collections shown here are “zero pairs”.

They have a value of zero.

Describe a “zero pair”.

Now let’s look at models for operations

with integers.

What is addition?

Addition is combining one or more addends (collections of

counters).

When using two-colored counters to model addition, build each

addend then find the value of the collection.

5 + (-3)zero pairs

= 2

Modeling addition of integers:

8 + (–3) = 5

Here is another example:

-4 + (-3)

(Notice that there are no zero pairs.)

= -7

Build the following addition problems:

1) -7 + 2 =

2) 8 + -4 =

3) 4 + 5 =

4) -6 + (-3) =

-5

9

4

-9

Write a “rule”, in your own words, for adding

integers.

What is subtraction?

There are different models for subtraction, but when using the

two-colored counters you will be using the “take-away” model.

When using two-colored counters to model subtraction, build a collection then take away the

value to be subtracted.

For example: 9 – 3 = 6

take away

Here is another example:

–8 – (–2) = –6

take away

Subtract : –11 – (–5) = –6

Build the following:

1) –7 – (–3)

2) 6 – 1

3) –5 – (–4)

4) 8 – 3

= –4

= 5

= –1

= 5

We can also use fact family with integers.

Use your red and yellow tiles to verify this fact family:

-3 + +8 = +5+8 + -3 = +5+5 - + 8 = -3+5 - - 3 = +8

Build –6.

Now try to subtract +5.

Can’t do it? Think back to building collections in

different ways.

Remember?

+5 =or

or

Now build –6, then add 5 zero pairs.

It should look like this:

This collection still has a value of –6. Now subtract 5.

–6 – 5 = –11

Another example:

5 – (–2)

Build 5:

5 – (–2) = 7

Add zero pairs:Subtract –2:

Subtract: 8 – 9 = –1

Try building the following:

1) 8 – (–3)

2) –4 – 3

3) –7 – 1

4) 9 – (–3)

= 11

= –7

= –8

= 12

Look at the solutions. What addition problems

are modeled?

1) 8 – (–3) = 11 = 8 + 3

2) –4 – 3 = –7 = –4 + (–3)

3) –7 – 1 = –8 = –7 + (–1)

= 9 + 3 4) 9 – (–3) = 12

These examples model an alternative way to solve a subtraction

problem.

Subtract: –3 – 5 = –8

–3 –5+

Any subtraction problem can be solved by adding the opposite of

the number that is being subtracted.

11 – (–4) = 11 + 4 = 15

–21 – 5 = –21 + (–5) = –26

Write an addition problem to solve the following:

1) –8 – 14 2) –24 – (–8)

3) 11 – 15 4) –19 – 3

5) –4 – (–8) 6) 18 – 5

7) 12 – (–4) 8) –5 – (–16)

What is multiplication?

Repeated addition!

3 × 4 means 3 groups of 4:

3 × 4 = 12

++

3 × (–2) means 3 groups of –2:

3 × (–2) = –6

+ +

If multiplying by a positive means to add groups, what

doe it mean to multiply by a negative?

Subtract groups!

Example:

–2 × 3 means to take away 2 groups of

positive 3.

But, you need a collection to subtract from, so build a collection of zero pairs.

What is the value of this collection?

Take away 2 groups of 3. What is the value of the remaining collection?

–2 × 3 = –6

Try this:

(–4) × (–2)

(–4) × (–2) = 8

Solve the following:

1) 5 × 6

2) –8 × 3

3) –7 × (–4)

4) 6 × (–2)

= 30

= –24

= 28

= –12

Write a “rule” for multiplying integers.

Division cannot be modeled easily using two-colored counters, but since division is the inverse of

multiplication you can apply what you learned about multiplying to

division.

Since 2 × 3 = 6 and 3 × 2 = 6, does it make sense that -3 × 2 = -6 ?

Yes

+2 × -3 = -6 and -3 × +2 = -6 belong to a fact family:

+2 × -3 = -6-3 × +2 = -6-6 ÷ +2 = -3-6 ÷ -3 = +2

If 3 × (–5) = –15, then –15 ÷ –5 = ?and –15 ÷ 3 = ?

If –2 × –4 = 8, then 8 ÷ (–4) = ? and 8 ÷ (–2) = ?

3–5

–2–4

Write a “rule” for dividing integers.