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.
Warm Up Copy this Into your Notebook On your Warm up Sheet, Write: Notes on Modeling Integers
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
SUBTRACTING 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: –7 – (–3) 6 – 1 –5 – (–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
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)
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
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.
does it make sense that -3 × 2 = -6 ? 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 = ? 3 and –15 ÷ 3 = ? –5 If –2 × –4 = 8, then 8 ÷ (–4) = ? and 8 ÷ (–2) = ? –2 –4
Write a “rule” for dividing integers.