Operations with integers can be modeled using two-colored counters. Positive +1 Negative

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) = 2) = 3) = 4) -6 + (-3) =

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: = = = = + 8

Build –6. Now try to subtract +5. Can’t do it? Think back to building collections in different ways.

Remember? +5 = 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) = 12

These examples model an alternative way to solve a subtraction problem.

Subtract: –3 – 5 = –8–8 –3–3 –5 +

Any subtraction problem can be solved by adding the opposite of the number that is being subtracted. 11 – (–4) = = 15 –21 – 5 = –21 + (–5) = –26

Write an addition problem to solve the following: 1) –8 – 142) –24 – (–8) 3) 11 – 154) –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 = × + 2 = ÷ + 2 = ÷ - 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.