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”.

ADDING INTEGERS

When using two-colored counters to model addition, build each addend then find the value of the collection. 5 + (-3) = 2 zero pairs

Modeling addition of integers: 8 + (–3) = 5

(Notice that there are no zero pairs.) Here is another example: -4 + (-3) = -7 (Notice that there are no zero pairs.)

Build the following addition problems: -7 + 2 = 2) 8 + -4 = 4 + 5 = -6 + (-3) = -5 4 9 -9

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

Warm Up: Get your folder and a set of integer disks. Add the integers. 4 + (-2) = -5 + -3 = -7 + 2 = 6+2 = -9 + -1 = 3 + -7 =

SUBTRACTING INTEGERS

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: –7 – (–3) 6 – 1 –5 – (–4) 8 – 3 = –4 = 5 = –1 = 5

Can’t do it? Think back to building collections in different ways. 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: Add zero pairs: Subtract –2: 5 – (–2) = 7

Subtract: 8 – 9 = –1

Try building the following: 1) 8 – (–3) –4 – 3 –7 – 1 9 – (–3) = 11 = –7 = –8 = 12

Look at the solutions. What addition problems are modeled?

1) 8 – (–3) = 11 = 8 + 3

–4 – 3 = –7 = –4 + (–3)

3) –7 – 1 = –8 = –7 + (–1)

4) 9 – (–3) = 12 = 9 + 3

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: –8 – 14 2) –24 – (–8) 3) 11 – 15 4) –19 – 3 5) –4 – (–8) 6) 18 – 5 7) 12 – (–4) 8) –5 – (–16)

Warm Up Get folder and counters Solve the following a) 4 + -2 = b) 6 – (-2)= -7 + -8 = d) -5 – (-3) = e) -12 + 6 = f) -24 – 6 =

(-3) + 9 (-5) + (-2) 16 + (-18) 7 – (-4) -5 – 6 -12 – (-3) 16 - 20 16 + (-18) 7 – (-4) -5 – 6 -12 – (-3) 16 - 20 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

MULTIPLYING INTEGERS

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!

means to take away 2 groups of positive 3. 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.

DIVIDING INTEGERS

12÷3 It means take 12 and make 3 equal groups.

-10÷5 What does this mean?

Write a “rule” for dividing integers.

Try these problems using your rule: 20÷-4= -24÷-3 = 36 ÷ -9 = 45 ÷ 5 =

Warm Up: -4 + -6 = 2. -12 + 8 7 – (-4) = 4. -12 – 6 (-5)(8) 6. (-7)(-6) = 7. -56 ÷ - 8 = 8. 42 ÷ - 6 =

Order of Operations PEMDAS 1. Evaluate expressions inside of ___________   2. Evaluate _______ 3. __________and _________from left to right 4. __________ and ________ from left to right

 

Warm Up: Use Order of Operations to solve 3 + 4*2 4. 24 ÷4 * 2 12 – (4 – 5) 5. 10 + 5 – (-3) 30 – (52 – 4) 6. 12 – 5 + 2 - 15

 

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