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Operations with integers can be modeled using two-colored
counters.
Positive
+1
Negative
-1
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The following collections of counters have a value of +5.
Build a different collection that has a value of +5.
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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.
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The collections shown here are “zero pairs”.
They have a value of zero.
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Describe a “zero pair”.
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Now let’s look at models for operations
with integers.
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What is addition?
Addition is combining one or more addends (collections of
counters).
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When using two-colored counters to model addition, build each
addend then find the value of the collection.
5 + (-3)zero pairs
= 2
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Modeling addition of integers:
8 + (–3) = 5
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Here is another example:
-4 + (-3)
(Notice that there are no zero pairs.)
= -7
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Build the following addition problems:
1) -7 + 2 =
2) 8 + -4 =
3) 4 + 5 =
4) -6 + (-3) =
-5
9
4
-9
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Write a “rule”, in your own words, for adding
integers.
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What is subtraction?
There are different models for subtraction, but when using the
two-colored counters you will be using the “take-away” model.
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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
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Here is another example:
–8 – (–2) = –6
take away
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Subtract : –11 – (–5) = –6
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Build the following:
1) –7 – (–3)
2) 6 – 1
3) –5 – (–4)
4) 8 – 3
= –4
= 5
= –1
= 5
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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
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Build –6.
Now try to subtract +5.
Can’t do it? Think back to building collections in
different ways.
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Remember?
+5 =or
or
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Now build –6, then add 5 zero pairs.
It should look like this:
This collection still has a value of –6. Now subtract 5.
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–6 – 5 = –11
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Another example:
5 – (–2)
Build 5:
5 – (–2) = 7
Add zero pairs:
Subtract –2:
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Subtract: 8 – 9 = –1
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Try building the following:
1) 8 – (–3)
2) –4 – 3
3) –7 – 1
4) 9 – (–3)
= 11
= –7
= –8
= 12
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Look at the solutions. What addition problems
are modeled?
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1) 8 – (–3) = 11 = 8 + 3
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2) –4 – 3 = –7 = –4 + (–3)
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3) –7 – 1 = –8 = –7 + (–1)
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= 9 + 3 4) 9 – (–3) = 12
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These examples model an alternative way to solve a subtraction
problem.
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Subtract: –3 – 5 = –8
–3 –5+
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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
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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)
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What is multiplication?
Repeated addition!
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3 × 4 means 3 groups of 4:
3 × 4 = 12
++
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3 × (–2) means 3 groups of –2:
3 × (–2) = –6
+ +
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If multiplying by a positive means to add groups, what
doe it mean to multiply by a negative?
Subtract groups!
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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.
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What is the value of this collection?
Take away 2 groups of 3. What is the value of the remaining collection?
–2 × 3 = –6
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Try this:
(–4) × (–2)
(–4) × (–2) = 8
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Solve the following:
1) 5 × 6
2) –8 × 3
3) –7 × (–4)
4) 6 × (–2)
= 30
= –24
= 28
= –12
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Write a “rule” for multiplying integers.
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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.
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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
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If 3 × (–5) = –15, then –15 ÷ –5 = ?
and –15 ÷ 3 = ?
If –2 × –4 = 8, then 8 ÷ (–4) = ? and 8 ÷ (–2) = ?
3
–5
–2–4
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Write a “rule” for dividing integers.