MACC.912.A-CED.1.1
Create equations and inequalities in
one variable and use them to solve
problems. Include equations arising
from linear and quadratic functions,
and simple rational and exponential
functions.
CCSS Mathematics Practice
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of
a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make conjectures about the form and
meaning of the solution and plan a solution pathway rather than simply jumping into
a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They
monitor and evaluate their progress and change course if necessary. Older students
might, depending on the context of the problem, transform algebraic expressions or
change the viewing window on their graphing calculator to get the information they
need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important
features and relationships, graph data, and search for regularity or trends. Younger
students might rely on using concrete objects or pictures to help conceptualize and
solve a problem. Mathematically proficient students check their answers to problems
using a different method, and they continually ask themselves, “Does this make
sense?” They can understand the approaches of others to solving complex problems
and identify correspondences between different approaches.
At 8:00 AM, the art teacher, Mr. Burns
put some scissors out for his students in
the scissor rack. By 10:00 AM, Mr.
Burns found that one-half of the
scissors were gone. By noon, Mr. Burns
found that one-third of the scissors that
were left from when he checked before
were gone. In the afternoon, Mr. Burns
found that one-fifth of the remaining
scissors were gone. If at the end of the
day there were at least 15 scissors left,
what is the smallest possible number of
scissors he could have started with?
Entry Points
Mathematically proficient students start by
explaining to themselves the meaning of a
problem and looking for entry points to its
solution.
– Guess, Check, and Revise
– Make a Model or Diagram
– Work Backwards
– Use a Formula or Equation/Inequality
Guess, Check, and Revise
Let’s try 30.
1
´ 30 = 15
2
2
´15 = 10
3
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
The problem states that there are at least 15
left in the afternoon…this is already too small.
Let’s try a larger number to start with.
Guess, Check, and Revise
Let’s try 40.
1
´ 40 = 20
2
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
2
´ 20 = ?
3
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
I cannot find two-thirds of 20.
Let’s try a different number. I can find two-thirds
of 21, so start with 42.
Guess, Check, and Revise
Let’s try 42.
1
´ 42 = 21
2
2
´ 21 = 14
3
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
The problem states that there are at least 15
left in the afternoon…this is already too small.
Let’s try a different number. I know I have to divide
by 2 and 3, so let’s try multiples of 6.
Guess, Check, and Revise
Let’s try 54.
1
´ 54 = 27
2
2
´ 27 = 18
3
4
´18 = ?
5
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
In the afternoon, Mr. Burns found that one-fifth
of the remaining scissors were gone.
I cannot find four-fifths of 18.
Let’s try a different number. I know I have to divide
by 2, 3, and 5, so let’s try multiples of 30.
Guess, Check, and Revise
Let’s try 60.
1
´ 60 = 30
2
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
2
´ 30 = 20
3
4
´ 20 = 16
5
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
In the afternoon, Mr. Burns found that one-fifth
of the remaining scissors were gone.
The answer is 60.
Make a Model or Diagram
At the end of the day there were at least 15 scissors left.
In the afternoon, Mr. Burns found that one-fifth
of the remaining scissors were gone.
4
4
4
15
16
20 pairs of scissors
4
4
Draw a Diagram
By noon, Mr. Burns found that one-third of the scissors that were left from when he checked
before were gone.
10
10
20
30 pairs of scissors
10
Draw a Diagram
By 10:00 AM, Mr. Burns found that one-half of the scissors were gone.
30
30
60 pairs of scissors
30
Work Backwards
Afternoon: at least 15 scissors
One-fifth of the remaining scissors were gone…
this means 4/5 is left.
4
x ³ 15
5
75
x³
4
Work Backwards
Noon: one-third of the scissors that were left
from when he checked before were gone…
this means 2/3 is left.
2
75
x³
3
4
225
x³
8
Work Backwards
10:00 AM: one-half of the scissors were gone
1
225
x³
2
8
225
x³
4
1
x ³ 56
4
Use a Formula or Equation/Inequality
1
2
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
2 1 1
´ =
3 2 3
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
4 1 4
´ =
5 3 15
In the afternoon, Mr. Burns found that one-fifth
of the remaining scissors were gone.
4
x ³ 15
15
225
x³
4
1
x ³ 56
4
Mathematically Proficient Students…
… monitor and evaluate their progress and
change course if necessary.
Solutions
• Using our multiple methods, we came up with
2 different answers:
– 60
– 56 ¼
• Can we have a fraction of a scissor?
• Do we round down or round up?
Solutions
By 10:00 AM, Mr. Burns found that one-half of
the scissors were gone.
57
58
29
60
Find ½ of 57. 58
By noon, Mr. Burns found that one-third of the
scissors that were left from when he checked
before were gone.
30
Since 1/3 is gone, 2/3 must be left. Find 2/3 of
29. 30
20
In the afternoon, Mr. Burns found that one-fifth
of the remaining scissors were gone.
16
Since 1/5 is gone, 4/5 must be left. Find 4/5 of
20.
Guiding Questions
How might you represent the number of pairs of scissors that are in the rack
at:
–
–
–
–
–
8 a.m.?
10 a.m.?
12 noon?
In the afternoon?
End of the day?
Would an equation or an inequality be the better choice for representing this
scenario?
Do you need one or two variables to represent this scenario?
What kinds of numbers make sense for the unknown(s) in this scenario?
What does the 15 at the end of the day represent?