Subtracting by Adjusting
4NBT4
A student had to solve these problems.
β’ Why might they have thought the middle problem was the easiest to solve?
β’ What makes the other two problems challenging?
- 3,000
- 1,463
- 931
- 198
- 5,329
- 1,217
Iβm going to start with that middle one because it will be the easiest to solve.
Today we will talk about a strategy you have used in earlier grades and how it can make problems like these easier to solve!
- 3,000
- 1,463
- 931
- 198
What is the difference between the two cube trains?
In other words βΒ how much longer or shorter is one than the other?
What is an equation that shows the difference between the lengths?
Build two cube βtrainsβ: one with 10 cubes and one with 7 cubes.
Original Trains
What if you added one more cube to each train?
- 10 long
- 7 short
- 13 difference
Keep adding pairs of cubes. Record the difference of each new pair.
Original Train | +1 +1 | +2 +2 | +3 +3 | +4 +4 | +5 +5 |
| | | | | |
β 10 long
β 7 short
-β 3 difference
β 11
β 8
-β 3
What do you notice about the subtraction problems?
Debrief
When you adjust two numbers by adding the same amount to each, they stay the same distance apart.
The numbersβ difference stays the same.
Debrief
β 10
β 7
-β 3
β 12
β 9
-β 3
β 15
β 12
-β 3
What happens if we try adjusting numbers by taking away cubes?
Keep removing pairs of cubes. Record the difference of each new pair.
Original Train | -1 -1 | -2 -2 | -3 -3 | -4 -4 | -5 -5 |
| | | | | |
β 14 long
β 8 short
-β 6 difference
β 13
β 7
-β 6
What do you notice about these subtraction problems?
Debrief
When you adjust two numbers by adding or subtracting the same amount to each, they stay the same distance apart.
The numbersβ difference stays the same.
take away 4
take away 4
add 2
add 2
Debrief
β 10
β 7
-β 3
β 12
β 9
-β 3
β 6
β 3
-β3
β 10
β 7
-β 3
β 10
β 7
-β 3
When you adjust a problem by adding or subtracting the same amount from both numbers, you can make a problem easier to solve.
The numbersβ difference stays the same.
12 βΒ 9 might not be easy
to solve, but 13 βΒ 10 is!
14 βΒ 8 might not be easy to solve, 12 βΒ 6 and 10 βΒ 4 are!
Debrief
Adjust 83 βΒ 56 by adding and subtracting to create eight problems equivalent to 83 βΒ 56.
Practice
Which problem(s) are easiest to solve in your head?
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 83 -56 | | |
|
|
Adjust 121 βΒ 97 by adding and subtracting to create eight problems equivalent to 121 βΒ 97.
Practice
Which problem(s) are easiest to solve in your head?
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 121 - 97 | | |
|
|
Adjust 172 βΒ 38 by adding and subtracting to create eight problems equivalent to 172 βΒ 38.
Practice
Which problem(s) are easiest to solve in your head?
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 172 - 38 | | |
|
|
Try adjusting these problems to make them easier to solve. Then solve them!
HINT: Think about how to make two numbers that donβt require regrouping to subtract!
Practice
Original Problem
Adjusted Problem
- 3,000
- 1,463
- __ , __ __ __
- __ , __ __ __
- 931
- 198
- __ __ __
- __ __ __
Keep adding pairs of cubes. Record the difference of each new pair.
Keep removing pairs of cubes. Record the difference of each new pair.
Name: _____________________________________________
Date: ________________
Original Train | +1 +1 | +2 +2 | +3 +3 | +4 +4 | +5 +5 |
| | | | | |
β 10 long
β 7 short
-β 3 difference
β 11
β 8
-β 3
Original Train | +1 +1 | +2 +2 | +3 +3 | +4 +4 | +5 +5 |
| | | | | |
β 14 long
β 8 short
-β 6 difference
β 13
β 7
-β 6
Adjust 83 βΒ 56 by adding and subtracting to create eight problems equivalent to 83 βΒ 56.
Which problem(s) are easiest to solve in your head?
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 83 -56 | | |
|
|
Name: _____________________________________________
Date: ________________
Adjust 121 βΒ 97 by adding and subtracting to create eight problems equivalent to 121 βΒ 97.
Which problem(s) are easiest to solve in your head?
Adjust 172 βΒ 38 by adding and subtracting to create eight problems equivalent to 172 βΒ 38.
Which problem(s) are easiest to solve in your head?
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 121 - 97 | | |
|
|
-4 -4 | -3 -3 | -2 -2 | -1 -1 | Original Problem | +1 +1 | +2 +2 | +3 +3 | +4 +4 |
|
| | | 172 - 38 | | |
|
|
Try adjusting these problems to make them easier to solve. Then solve them!
HINT: Think about how to make two numbers that donβt require regrouping to subtract!
Original Problem
Adjusted Problem
Original Problem
Adjusted Problem
- 3,000
- 1,463
- __ , __ __ __
- __ , __ __ __
- 931
- 198
- __ __ __
- __ __ __