1. Bellwork Find each difference. -8 – (-2) 12 – (-5) -7 – 9 -3 – (-4)
2 - 10

2. Modeling Subtraction With Number Lines and Counters
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Course 2
7.NS.2a

3. Today’s Lesson
When working math problems, it is important to understand the meaning behind the math. In other words… why does it work?
Today, we will solve subtraction problems and will learn how to justify our answers using a number line and integer tiles.

4. Why Learn?
Well, let’s take a look at how a hike and a little “butterfly watching” ties into today’s lesson!

5. Butterfly watching …
(also called butterflying) is a hobby concerned with the observation and study of butterflies!
If you are in to butterflying, then you must add The Monarch Butterfly Biosphere Reserve to your bucket list of must-do’s.
Every year hundreds of millions of butterflies undertake a great journey of up to 3,000 miles in their annual migration from Canada and the United States to a protected reserve located just outside of Mexico City, Mexico.
The reserve is a 200 square mile protected area which serves as a winter home for up to 60 million to a billion butterflies!

6. Serious butterfly watchers should be prepared to hike anywhere from 20 minutes to over an hour (or to ride a donkey).
You can only reach the butterflies on paths laid by the reserve, and they congregate at extremely high altitudes — between 9,000 and 11,000 feet — so visitors should be in good enough physical condition to handle steep inclines.

7. Henry loves backpacking and has always wanted to hike the mountainous
Meet Henry the hiker!
Henry loves backpacking and has
always wanted to hike the mountainous
terrain of Mexico to see the annual migration of the monarch butterflies.
¡hola
“¡hola” translates to “Hi” in spanish.

8. Real World Application Henry and his best friend start their
MAP
Real World Application
Henry and his best friend start their
hike at an elevation of 8,500 feet.
They ascend 375 feet, then stop and take pictures of a cluster of butterflies on a tree. They hike another 1,000 feet on the same path where they next stumble upon a field of thousands of butterflies in flight. After enjoying the view for awhile, Henry and his friend decide they would like to tour the butterfly museum and gift shop. They turn the opposite direction, and descend 1,700 feet to the museum.
Write an expression that would determine the elevation of the butterfly museum.
8, – 1700
Butterflies in flight
Start Hike
8,500 ft
Butterflies on tree

9. What mathematical symbol is this?
Most will probably say it’s a minus or subtract sign. Prompt students for other possible mathematical meanings of the symbol.

10. Well… the symbol can have several meanings.
Subtract
Take Away
Minus
Opposite
Negative
Which meaning you use depends upon the given situation.

11. When modeling subtraction on a number line we will use the term:
Opposite

12. 7 ─ 3 1) Solve: 7 ─ 3 = ____ 4 Turn the “opposite” direction
Think of Henry the hiker.
Always start at zero facing the positives.
The first number is positive 7, so walk forward 7 steps.
The minus sign means to turn the “opposite” direction.
The 3 is assumed positive, so walk forward 3 steps.
Answer: 4
Turn the “opposite”
direction
3 steps forward
7 steps forward

13. Modeling Subtraction
When modeling subtraction with
counters we will use the term:
Take Away

14. 7 ─ 3 4 1) Solve: 7 ─ 3 = ____ 7 positives “take away” 3 positives
The first number is positive 7, so start with 7 “+” tiles.
When you see the minus sign, think of it as “take away”.
The 3 is assumed positive, so take away 3 “+” tiles.
Answer: 4
7 positives
“take away”
3 positives

15. -7 ─ 3 2) Solve: -7 ─ 3 = ____ -10 Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is -7, so walk backward 7 steps.
The minus sign means turn the “opposite” direction.
The 3 is assumed positive, so walk forward 3 steps.
Answer: -10
Turn the “opposite”
direction
3 steps forward
7 steps backward

16. ─7 ─ 3 -10 2) Solve: ─7 ─ 3 = ____ 7 negatives “take away” 3 positives
The first number is negative 7, so start with 7 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 3 is assumed positive, so take away 3 “+” tiles.
Since there are no positives to take away, we must add zero pairs.
Add 3 zero pairs. Now we can “take away” 3 “+” tiles.
Answer: -10
7 negatives
“take away”
3 positives
There are not 3 positives to take away, so we must add enough zero pairs so that 3 positives can be taken away.

17. 2 ─ 5 -3 3) Solve: 2 ─ 5 = ____ Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is positive 2, so walk forward 2 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 5 is assumed positive, so walk forward 5 steps.
Answer: -3
Turn the “opposite”
direction
5 steps forward
2 steps forward

18. 2 ─ 5 3) Solve: 2 ─ 5 = ____ -3 2 positives “take away” 5 positives
The first number is positive 2, so start with 2 “+” tiles.
When you see the minus sign, think of it as “take away”.
The 5 is assumed positive, so take away 5 “+” tiles.
Since there are not enough positives to take away, we must add zero pairs.
Add 3 zero pairs. Now we can “take away” 5 “+”.
Answer: -3
2 positives
“take away”
5 positives
There are not 5 positives to take away, so we must add enough zero pairs so that 5 positives can be taken away.

19. Your turn to practice…
Take sixty seconds to try problem #4 on your own.
1 Minute

20. 8 ─ 1 4) Solve: 8 ─ 1 = ____ 7 Still having trouble?
Imagine your pencil is
Henry the Hiker! 8 ─ 1
- Always start at zero facing the positives.
The first number is positive 8, so walk forward 8 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 1 is assumed positive, so walk forward 1 step.
Answer: 7
Turn the “opposite”
direction
1 step
forward
8 steps forward

21. 8 ─ 1 7 4) Solve: 8 ─ 1 = ____ 8 positives “take away” 1 positive
The first number is positive 8, so start with 8 “+” tiles.
When you see the minus sign, think of it as “take away”.
The 1 is assumed positive, so take away 1 positive tile.
Answer: 7
8 positives
“take away”
1 positive

22. You got this… Take sixty seconds to try problem #5 on your own.
Don’t forget… Your pencil is Henry the hiker. Everyone hold up Henry!
Put Henry on zero and face him towards the positives…. Now GO!
Your Henry!
1 Minute

23. -3 ─ 6 5) Solve: -3 ─ 6 = ____ -9 Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is negative 3, so walk backward 3 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 6 is assumed positive, so walk forward 6 steps.
Answer: -9
Turn the “opposite”
direction
6 steps forward
3 steps backward

24. -3 ─ 6 -9 5) Solve: -3 ─ 6 = ____ 3 negatives “take away” 6 positives
The first number is negative 3, so start with 3 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 6 is assumed positive, so take away 6 positive tiles.
Since there are not enough positives to take away, we must add zero pairs.
Add 6 zero pairs. Now we can “take away” 6 positives.
Answer: -9
3 negatives
“take away”
6 positives
There are not 6 positives to take away, so we must add enough zero pairs so that 6 positives can be taken away.

25. Now let’s look at some tricky ones!
Try the problems on the next page. There are five so I will give you 5 minutes. We will go over them together.
5 Minutes

26. -10 6) Solve: -8 ─ (+2) = ____ -8 ─ (+2) 2 steps forward
─ (+2)
- Always start at zero facing the positives.
The first number is negative 8, so walk backward 8 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 2 is positive, so walk forward 2 steps.
Answer: -10
2 steps forward
8 steps backward
Turn the “opposite”
direction

27. -8 ─ (+2) 6) Solve: -8 ─ (+2) = ____ -10 8 negatives “take away”
The first number is negative 8, so start with 8 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 2 is positive, so take away 2 positive tiles.
Since there are not enough positives to take away, we must add zero pairs.
- Add 2 zero pairs. Now we can “take away” 2 positives.
Answer: -10
8 negatives
“take away”
2 positives
There are not 2 positives to take away, so we must add enough zero pairs so that 2 positives can be taken away.

28. 5 ─ (-2) 7 7) Solve: 5 ─ (-2) = ____ Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is positive 5, so walk forward 5 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 2 is negative, so walk backward 2 steps.
Answer: 7
Turn the “opposite”
direction
2 steps backward
5 steps forward

29. ─ (-2) 5 7) Solve: 5 ─ (-2) = ____ 7 5 positives “take away”
The first number is positive 5, so start with 5 “+” tiles.
When you see the minus sign, think of it as “take away”.
The 2 is negative, so take away 2 negative tiles.
Since there are not enough negatives to take away, we must add zero pairs.
- Add 2 zero pairs. Now we can “take away” 2 negatives.
Answer: 7
5 positives
“take away”
2 negatives
There are not 2 negatives to take away, so we must add enough zero pairs so that 2 negatives can be taken away.

30. -3 ─ (+6) 8) Solve: -3 ─ (+6) = ____ -9 Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is negative 3, so walk backward 3 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 6 is positive, so walk forward 6 steps.
Answer: -9
Turn the “opposite”
direction
6 steps forward
3 steps backward

31. -3 ─ (+6) -9 8) Solve: -3 ─ (+6) = ____ 3 negatives “take away”
The first number is negative 3, so start with 3 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 6 is positive, so take away 6 positive tiles.
Since there are not enough positives to take away, we must add zero pairs.
Add 6 zero pairs. Now we can “take away” 6 positives.
Answer: -9
3 negatives
“take away”
6 positives
There are not 6 positives to take away, so we must add enough zero pairs so that 6 positives can be taken away.

32. -5 ─ (-3) 9) Solve: -5 ─ (-3) = ____ -2 Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is negative 5, so walk backward 5 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 3 is negative, so walk backward 3 steps.
Answer: -2
Turn the “opposite”
direction
3 steps backward
5 steps backward

33. -5 ─ (-3) 9) Solve: -5 ─ (-3) = ____ -2 5 negatives “take away”
The first number is negative 5, so start with 5 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 3 is negative, so take away 3 negative tiles.
Answer: -2
5 negatives
“take away”
3 negatives

34. -7 ─ (-8) 10) Solve: -7 ─ (-8) = ____ 1 Turn the “opposite” direction
- Always start at zero facing the positives.
The first number is negative 7, so walk backward 7 steps.
When you see the minus sign, it means turn the “opposite” direction.
The 8 is negative, so walk backward 8 steps.
Answer: 1
Turn the “opposite”
direction
8 steps backward
7 steps backward

35. -7 ─ (-8) 10) Solve: -7 ─ (-8) = ____ 1 7 negatives “take away”
The first number is negative 7, so start with 7 “-” tiles.
When you see the minus sign, think of it as “take away”.
The 8 is negative, so take away 8 negative tiles.
Since there are not enough negatives to take away, we must add zero pairs.
Add 1 zero pair. Now we can “take away” 8 negatives.
Answer: 1
7 negatives
“take away”
8 negatives
There are not 8 positives to take away, so we must add enough zero pairs so that 8 negatives can be taken away.

36. Practice