A Guide to Elementary Story Problems
The ability to tackle complex story problems is the culminating work of each elementary grade. By the end of kindergarten, they’re solving add to/take from
story problems within 10 as well as put together/take apart problems with both addends unknown. By the end of first grade, they’re solving all
addition/subtraction story problem types, include compare problems, within 20. By the end of second grade, they’ve mastered all of the addition/subtraction
story problem types within 100 – and even tackled two-step story problems. In third grade, they begin multiplication and solve equal groups/array story
problems within 100. By the end of fourth grade, they have mastered all addition, subtraction, multiplication, and division story problem types (including
multiplicative compare) with all whole numbers for addition and subtraction and two-digit multipliers and one-digit divisors for multiplication and division.
Furthermore, they master multi-step problems with mixed operations, including measurement contexts.
In order to truly master this content, we know that daily instruction with a story problem focus is key. When we look at our current results and observe students
in class, it is blatantly clear – they are struggling to understand the story problem. This has huge implications for our instruction. We must ensure that operate
(the ability to read a story problem and identify the operation(s) to use to solve) and calculate (the ability to do the math with the numbers to get an answer) are
seen as separate – and that operate is recognized as the most difficult part of a story problem for a child – and therefore prioritized.
To ensure we prioritize operate, we need to elevate the importance of representations as a way for students to show their understanding. We need a series of
representations on a continuum that applies to all story problem situations – that will provide a common language for our students and teachers. By
emphasizing representations, we are prioritizing operate, and this will lead to greater and deeper student understanding of story problems.
On the following page is a table of representations across K-12. The first six (manipulatives, 1:1 drawing, 1:1 tape diagram, tape diagram, bar model, equation)
are the focus in elementary school. The large and bolded Xs denote focal representations for a given grade.
Probability
Model
Proportion Bar
x
x
x
x
x
Verbal
Description
Statistical
Graph
x
x
x
X
x
x
x
x
Area Model
Coordinate
Grid
Number Line
Organized List
Table
Inequality
X
x
x
x
x
X
x
x
x
x
x
x
x
x
Expression
x
x
X
X
Equation
x
X
Bar Model
x
X
Tape
Diagram/
Number Bond
X
x
1:1 Tape
Diagram/
Number Bond
X
x
x
x
x
1:1 Drawing
Manipulatives
K
1
2
3
4
5
6
7
8
9
10
11
12
x
x
X
X
X
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
X
X
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
The rest of this guide works to explain and link story problems throughout elementary school. The guide includes:
- K-4 Story Problem Protocol (page 3):
o The protocol relies on represent as the foundation and builds in complexity over time, while maintaining a link between grades. This will create
a common language for students and teachers throughout their elementary schooling.
- Addition & Subtraction Situations & Representations (pages 4-6):
o We know that representations are the key to understanding story problems. We also know that we need normed representations across story
problem types with clear examples. Those are included.
- Multiplication & Division Situations & Representations (pages 7-8):
o We know that representations are the key to understanding story problems. We also know that we need normed representations across story
problem types with clear examples. Those are included.
- Math Stories Agenda (pages 9-10):
o Daily work with story problems is no longer a nice-to-have. It is essential. This agenda focuses on the group representations early and
transitions to student directed representations in the middle and later stages.
- Elementary Story Problems Scope & Sequence (page 11):
o This further breaks down the story problems by grade into a sequence of introducing each story problem type to maximize understanding,
student’s ability to represent, and ensuring each story problem type has the opportunity to build with/off of other types.
K-4 Story Problem Protocol
Kindergarten
Second Grade
Third Grade
Fourth Grade
“Make a mind movie”
“Picture what’s happening
in the problem”
“Visualize the problem as
you listen/read”
“Visualize the problem as
you listen/read”
“How can we show what’s
happening?” - Represent
“How can we show what’s
happening?” - Represent
“How can we represent
the problem?” - Represent
“What are we trying to
figure out?”
“What are we trying to
figure out?”
-Tell back the story using the
representation.
“Retell the problem.”
-Tell back the story using
the representation.
“Retell the problem.”
-Tell back the story using
the representation.
“Retell the problem.”
“How can we represent
the problem?” - Represent
“How can we represent
the problem?” - Represent
+/- and 2-step +/1) bar model
x/÷ and 2-step mixed
2) other
+/-/x/÷
3) bar model
Multi-Step
4) other
“What do we know?”
“What do we know?”
-Equation
“What equation can we
write?”
-Solve and check
“Figure it out and check
your math with your
representation”
-Equation
“What equation can we
write?”
-Solve and check
“Figure it out and check
your math with your
representation”
-“Write/Tell your answer
in a sentence” – state the
answer in the context of
the problem
-“Write/Tell your answer
in a sentence” – state the
answer in the context of
the problem
Solve
*use K-4 represent continuum
*There were 7 apples on the
table. Some were red and
some were green. How many
of each could there be?”
*use K-4 calculate continuum
Represent & Retell
Visualize
“Make a mind movie”
First Grade
-Include “We need to
know/figure out _____.”
-Include “We need to
know/figure out ____.”
“What are we trying to
figure out?”
-Situation equation
(solution equation if it
happens)
“What equation can we
write?”
-Equation (situation or
solution; move towards
solution)
“What equation can we
write?”
-Solve and check
“Figure it out and check your
math with your
picture/cubes”
-Solve and check
“Figure it out and check
your math with your
picture”
-Solve and check
“Figure it out and check
your math with your
representation”
-“Finish the story” – state the
problem and answer in a
sentence
-“Finish the story” – state
the problem and answer in
a sentence
-“Finish the story” – state
the problem and answer in
a sentence
*There were 7 apples on the
table. 3 were red and 4 were
green.
Addition & Subtraction Situations & Representations
Take From
Add To
*Representations: 1:1 – one to one, 1:1TD – one to one tape diagram, TD – tape diagram, BM – bar model (two-step bar model includes expression in one bar)
Result Unknown (AT-RU)
4 bunnies sat on the grass. 6 more bunnies hopped
there. How many bunnies are on the grass now?
4+6=
Change Unknown (AT-CU)
8 bunnies were sitting on the grass. Some more
bunnies hopped there. Then there were 20 bunnies.
How many bunnies hopped over to the first bunnies?
8 +  = 20
Start Unknown (AT-SU)
Some bunnies were sitting on the grass. 12 more
bunnies hopped there. Then there were 20 bunnies.
How many bunnies were on the grass before?
 + 12 = 20
Potential Misunderstandings: generally not an issue for students;
this is the most simple SP type
Potential Misunderstandings: students often add both numbers,
ignoring that the larger number is the total and not one of the
addends
Potential Misunderstandings: very difficult for students to start
with an unknown; makes it tough for them to represent
1:1
1:1
1:1
1:1TD/NB
1:1TD/NB
1:1TD/NB
TD/NB
TD/NB
TD/NB
BM
BM
BM
Result Unknown (TF-RU)
10 apples on the table. I ate 6 apples. How many
apples are on the table now?
10 – 6 = 
Change Unknown (TF-CU)
20 apples were on the table. I ate some apples.
Then there were 8 apples. How many apples did I
eat?
20 -  = 12
Start Unknown (TF-SU)
Some apples were on the table. I ate 8 apples. Then
there were 12 apples. How many apples were on the
table before?
 - 8 = 12
Potential Misunderstandings: sometimes students add the two
amounts because they’re used to the Add To problems; this is the
most simple subtraction SP type
Potential Misunderstandings: when students read a subtraction
problem they want to “take away”, but this has the “take away”
part unknown; it’s uncomfortable for them; thinking
part/part/whole can help
Potential Misunderstandings: this allows students to “take away”
an amount – but with the start amount unknown, this becomes a
very difficult SP type; part/part/whole helps, because they’ll need
to think “add” to solve the “subtraction” problem
1:1
1:1
1:1TD/NB
1:1TD/NB
1:1
1:1TD/NB
Put Together/Take Apart
TD/NB
TD/NB
TD/NB
BM
Total Unknown (PT/TA-TU)
BM
Both Addends Unknown (PT/TA-BAU)
BM
Addend Unknown (PT/TA-AU)
4 red apples and 6 green apples are on the table.
How many apples on the table?
4+6=
Grandma has 10 flowers. How many can she
put in her read vase and how many in her blue
vase? (Should be able to name multiple
combinations; in some cases, all comb.)
10 =  + 
20 apples are on the table. 8 are red and the
rest are green. How many apples are green?
20 = 8 +  OR 20 – 8 = 
Potential Misunderstandings: students often answer this SP
type correctly, but it can be “right answers for wrong
reasons”; their default is to add; this SP type is challenging
because there is no inherent “action” in the story
Potential Misunderstandings: this SP type relies on the
understanding that there are multiple ways to make the
same number; students struggle because there are
multiple right answers and they may need to find one,
many, or all solutions; manipulatives to show the whole
can help students see how to break it into “parts” –
however, it can also cover up confusion by not making
students ‘choose’ to start with the total
Potential Misunderstandings: this is easy for adults – they
use the subtraction equation to represent and solve;
students struggle because they don’t see a “take away”
action; we must ensure that subtraction indicates
“separating” or “making parts” and not “take away”; this
will help them relate addition to subtraction and use
part/part/whole to understand this SP type
1:1
1:1TD/NB
1:1
1:1
1:1TD/NB
1:1TD/NB
TD/NB
TD/NB
TD/NB
BM
BM
BM
Notes on Representations:
1) Number Bonds and Tape Diagrams can be interchangeably used for Add To/Take From and Put Together/Take Apart stories.
2) Some students understand Number Bonds better because of the line connecting the parts to the whole. For other students, the tight link between the Tape
Diagram and the Bar Model is more meaningful.
3) Number Bonds traditionally have the total on top and the parts below. They are shown here with an opposite orientation to make it clear that both are
acceptable and to show a stronger alignment to the Tape Diagram.
4) The Tape Diagram can also be adjusted to use a brace instead of a bar – as seen in the Engage NY resources; the pro is alignment to that curriculum, the con
is that it’s less directly linked to the bar model
Bigger Unknown – More (C-BU-M)
Bigger Unknown – Fewer (C-BU-F)
Julie has 12 more apples than Lucy. Lucy has 8 apples.
How many apples does Julie have?
8 + 12 = 
Smaller Unknown – Fewer (C-SU-F)
Smaller Unknown – More (C-SU-M)
Lucy has 12 fewer apples than Julie. Julie has 20 apples.
How many apples does Lucy have?
20 – 12 = 
Potential Misunderstandings: this requires students to represent
both quantities and then find the difference – relating this to
subtraction is a challenge but solving with manipulatives helps
students understand
Potential Misunderstandings: because students’ default is to add,
they can get this SP type correct without truly understanding;
representations are important for gauging student understanding
Potential Misunderstandings: the language of “fewer” makes this
particularly challenging; students try to represent an amount that
isn’t there (12 fewer)
1:1
1:1
1:1TD
1:1TD
TD
TD
Compare
Difference Unknown – How Many More/Fewer?
(C-DU-M/F)
Lucy has 8 apples. Julie has 20 apples. How many more
apples does July have than Lucy?
BM
BM
Lucy has 8 apples. Julie has 20 apples. How many fewer
apples does Lucy have than Julie?
8 +  = 20 OR 20 – 8 = 
Lucy has 12 fewer apples than Julie. Lucy has 8 apples.
How many apples does Julie have?
8 + 12 = 
Julie has 12 more apples than Lucy. Julie has 20 apples.
How many apples does Lucy have?
20 – 12 = 
Potential Misunderstandings: similar to the above, students get to
represent both quantities and this helps them; understanding that
x more for y person is the same as x fewer for z person is crucial
Potential Misunderstandings: here, “fewer” actually means the
student needs to add – this is challenging; students need to
interpret and represent this SP type sentence by sentence – and
then work to put it all together to solve
Potential Misunderstandings: similar to the SP type to the left, but
in this one, “more” means to subtract; representing each sentence
and then putting it all together can help students understand
1:1
1:1
1:1
1:1TD
1:1TD
1:1TD
TD
TD
TD
BM
BM
BM
Notes on Compare Situations:
1) Students need to be fluid with the idea that if someone has more, the other has less; that if someone has 1 more, the other has 1 less; that if someone has 3 more, the other
has 3 less; etc.
2) “Julie has 2 more than Lucy”. Students need to hear and say two separate sentences – Julie has more. Julie has 2 more.
Arrays, Area
Equal Groups
Multiplication & Division Situations & Representations
Unknown Product (EG-UP)
Group Size Unknown (EG-GSU)
Number of Groups Unknown (EG-NGU)
3x6=
There are 3 bags with 6 plums in each bag. How many
plums are there in all?
Measurement: You need 3 lengths of string, each 6 inches
3 x  = 18 and 18 ÷ 3 = 
If 18 plums are shared equally into 3 bags, then how many
plums will be in each bag?
Measurement: You have 18 inches of string, which you will cut
 x 6 = 18 and 18 ÷ 6 = 
If 18 plums are to be packed 6 to a bag, then how many bags are
needed?
Measurement: You have 18 inches of string, which you will cut into pieces
long. How much string will you need altogether?
into 3 equal pieces. How long will each piece of string be?
that are 6 inches long. How many pieces of string will you have?
Potential Misunderstandings: this is the easiest multiplication SP type;
occasionally students default to addition – but they also tend to
realize that in G3, they wouldn’t add 3+6 – even if they don’t
understand why they should multiply
Potential Misunderstandings: representing the knowns and then taking the
action required to solve helps students with this SP type; initially they only
divide to solve; as they relate multiplication to division, they begin to
consider using multiplication to solve
Potential Misunderstandings: grouping division problems are the hardest for students;
they like to have an empty group and add to it – they don’t like to pull out an amount
without a visible “group” to put it into
1:1
1:1
1:1
1:1TD
1:1TD
1:1TD
TD
TD
TD
BM
BM
BM
Unknown Product (AA-UP)
Group Size Unknown (AA-GSU)
Number of Groups Unknown (AA-NGU)
3x6=
There are 3 rows of apples with 6 apples in each row.
How many apples are there?
Area: What is the area of a 3 cm by 6 cm rectangle?
3 x  = 18 and 18 ÷ 3 = 
If 18 apples are arranged into 3 equal rows, how many
apples will be in each row?
Area: A rectangle has area 18 square cm. If one side is 3 cm long,
 x 6 = 18 and 18 ÷ 6 = 
If 18 apples are arranged into equal rows of 6 apples, how many rows
will there be?
Area: A rectangle has area 18 square cm. If one side is 6 cm long, how long is
how long is a side next to it?
a side next to it?
Potential Misunderstandings: similar to the above; additionally the
rows/columns vocabulary challenge
Potential Misunderstandings: similar to the above; additionally the rows/columns
vocabulary challenge
Potential Misunderstandings: misunderstanding rows and columns
can make this a challenging SP type
1:1
1:1
1:1
Multiplicative Compare
1:1TD
1:1TD
1:1TD
TD
TD
TD
BM
Larger Unknown (MC-LU)
3x6=
A blue hat costs $6. A red hat costs 3 times
as much as the blue hat. How much does
the red hat cost?
Measurement: A rubber band is 6 cm long.
How long will the rubber band be when it is
stretched to be 3 times as long?
BM
Smaller Unknown (MC-SU)
3 x  = 18 and 18 ÷ 3 = 
A red hat costs $18 and that is 3 times as much
as a blue hat costs. How much does a blue hat
cost?
Measurement: A rubber band is stretched to be
18 cm long and that is 3 times as long as it was
at first. How long was the rubber band at first?
BM
Multiplier Unknown (MC-MU)
 x 6 = 18 and 18 ÷ 6 = 
A red hat costs $18 and a blue hat costs $6. How many
times as much does the red hat cost as the blue hat?
Measurement: A rubber band was 6 cm long at first.
Now it is stretched to be 18 cm long. How many times
as long is the rubber band now as it was at first?
Potential Misunderstandings: all MC problems are
difficult because there aren’t really multiple “parts” to
represent – students who think additively will want to
repeatedly add 6 to solve, and while it will lead to a
correct answer, these problems are asking students to
think multiplicatively which also means thinking past a
representation for “times as much”
Potential Misunderstandings: students must be strong in
the relationship between multiplication and division to
choose to use division to solve – which is the preferred
solution method
Potential Misunderstandings: students must be strong in the
relationship between multiplication and division to choose to use
division to solve – which is the preferred solution method for the
contextual problem; whereas multiplication would be the preferred
solution method for the measurement example
TD
TD
TD
BM
BM
BM
Multi-Step:
A big penguin eats 3 times as many grams of fish each day as a small penguin. A big penguin eats 420 grams off fish each day. Altogether, how much do
both penguins eat?
*The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery
window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.
*Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important
measurement situations.
Math Stories Agenda
Math Stories is a component of the math block in elementary school that is specifically designed to give students daily exposure to story problems at their level as well as
opportunities to apply the story problem protocol to complex problems. When a new story problem is introduced, the focus is on the representation. Over time, the focus
becomes efficiency in solving. This approach intentionally prioritizes understanding the problem and showing that understanding with a representation so that as the
complexity and number of steps in a story problem grows, students will remain grounded in their strong ability to represent their understanding.
While understanding and representing is at the heart of the Math Stories block, manipulatives and organization of work are also indispensable. Students must have access to
appropriate manipulatives (snap cubes and place value blocks) and they must see the importance in organizing their work. Manipulatives help students represent and solve and
well-organized work helps them keep track of what they’ve done and where they’re going.
In K-1, Math Stories is taught to a small group of students (half the class) on the perimeter of the carpet, by the more experienced teacher twice a week for 30 minutes.
In 2nd – 4th grade, Math Stories is taught twice a week for 25 minutes or daily for 15 minutes.
The Math Stories Agenda is on the following page and includes 3 different stages with a different agenda for each stage – early, middle, and later.
Criteria for Stage Advancement:
Math Stories intentionally has 3 stages – early, middle, and later. We know that the level of familiarity with a story problem type impacts the student’s ability to access and
engage with the problem. That said, various stages for the Math Stories protocol are a non-negotiable. For teachers, knowing when to move from the Early stage to the Middle
stage (and Middle to Later) is both necessary and difficult. Below is an early attempt to distinguish the stages and help teachers know when to move to the next stage.
Early: In the Early stage, students use their listening comprehension skills along with scaffolding from the teacher to visualize the problem. Students also work to represent the
problem in this stage. The independent work focus is on representing and retelling. Once they demonstrate the ability to effectively represent with manipulatives and retell the
story problem on a consistent basis, move to the Middle stage.
Middle: In the Middle stage, students are visualizing independently as well as representing on paper/whiteboards. The focus is on efficient ways to represent (moving from
manipulatives to 1:1 drawing, to 1:1 tape diagram, to tape diagram, to bar model). Complex story problems generally require us to represent in more complex/abstract ways.
Once students demonstrate the ability to effectively represent on whiteboards in the ways noted in the S&S for that grade/month/story problem type, move to the Later stage.
Later: In the Later stage, students are responsible for all aspects of the story problem protocol. The teacher still prompts the group, but students visualize, represent and retell,
and solve on their own. By focusing on the representation in the Early and Middle stages, this stage should get to focus on the solution strategy (calculation). If you are working
in this stage and students are struggling to represent, back up to the Middle or Early stage.
EARLY
MIDDLE
LATER
Visualize – 2 minutes
1. T – Read the problem
2. T – Model visualize
Visualize – 2 minutes
1. WG – Read the problem
2. S – Visualize
Visualize – 2 minutes
1. WG – Read the problem
2. S – Visualize
Represent & Retell – 12 minutes
1. WG – Act it out together
2. S – Represent with manipulatives
3. WG – Teacher records drawing of student
manipulative representations (2-3 students)
a. WG/TT – How does that represent the
story?
4. TT – Retell using your representation
a. WG – 1-2 students share retell
Represent & Retell – 12 minutes
1. WG – Act it out together
2. S – Represent on whiteboard
3. TT – Share your representation
4. WG – Teacher records student representations (2-3, least
to most sophisticated)
a. WG/TT – How does that represent the story?
Relating representations to each other
5. TT – Retell using your representation
a. WG – 1-2 students share retell
Represent & Retell – 4 minutes
1. S – Represent on white board
2. TT – Share your representation
a. WG – 1-2 students share representation
3. TT – Retell using your representation
a. WG – 1-2 students share retell
Solve – 6 minutes
1. WG – Solve together (from sharing students’
manipulative/picture representations)
2. TT – Finish the story
a. WG – 1-2 students share their Finish
the Story
Solve – 6 minutes
Solve – 14 minutes
1. S – Solve using your representation
1. S – Solve using your representation
a. Early finishers: tell your turtle how you solved;
a. Early finishers: tell your turtle how you solved; write
write a sentence to tell how you solved; write your
a sentence to tell how you solved; write your
answer in a sentence
answer in a sentence
2. WG – A few students share how they solved (1-3, least to
2. TT – How did you solve the problem?
most sophisticated order)
3. WG – Teacher records how students solved (2-3, least to most
3. TT – Finish the Story
sophisticated) using the calculation strategy name and a
a. WG – 1-2 students share their Finish the Story
picture to show the thinking
a. WG/TT – How do each of these solve the story?
Relating calculation strategies to each other
4.
5.
6.
WG – Everyone tries one calculation strategy together
T – Recap, all of these ways get the same answer
TT – Finish the Story
a. WG – 1-2 students share their Finish the Story
Practice – 10 minutes
1. Final Practice: Students work on 1-2 more problems, independently, aligned to the S&S, to demonstrate their current level of understanding (time permitting – they also write a sentence to tell
how they solved each problem, G1+)
2. On the final day of Math Stories each week, this is a formative assessment to determine next week’s focus for the group; could be done with manipulatives only in the Early stage.
Teacher Prompts: Prompts should focus on helping a student identify an error in their representation or calculation and/or helping them move to a more efficient/sophisticated strategy
- What strategy did you use? Is that the most efficient way to solve? Why did you use that strategy? What if I did this? Where are you confident about what you did? What does this represent from
this problem? Is there another way to represent this? Can you explain what you did here? What in the question makes you think that? Convince me. How do you know? Will that always work? How
did your representation help you decide the operation you chose? Can you represent this portion of the story problem for me? (break-it-down)
Notes:
- T = teacher, WG = whole group, S = students independently, TT = turn and talk
- Timing is built from a 30 minute block; if you have 25, cut 5 minutes from practice; if you have 15 (grades 2-4), you likely won’t have any practice time and you’ll need to speed up other parts
Elementary Story Problems Scope & Sequence
September
Kinder
E-w/in 5
M-w/in 10
L-w/in 15
1st Grade
E-w/in 15
M-w/in 20
L-w/in 30
2nd Grade
M-w/in 40
L-w/in 100
3rd Grade
M-w/in 100
L-w/in 1,000
(mult/div still
within 100)
October
November
December
AT-RU, TF-RU
Early
Manipulatives
AT-RU, TF-RU
Middle
Manip., 1:1
AT-RU, TF-RU
Later
Manip., 1:1
PT/TA-TU
Early/Middle
Manip., 1:1
AT-RU, TF-RU, PT/TATU
Later
Manip., 1:1
January
PT/TA-BAU
Early/Middle
Manip., 1:1
February
AT-RU, TF-RU, PT/TA-TU, PT/TA-BAU
Later; PT/TA-BAU should only be within 10
Manip., 1:1, Number Bond, EQ (equation)
March
AT-RU, TF-RU,
PT/TA-TU
Middle
1:1, 1:1TD/NB,
EQ
AT-RU, CU
TF-RU, CU
PT/TA-TU, AU,
BAU
Middle
1:1TD/NB,
TD/NB, EQ
AT-RU, SU, CU
TF-RU, SU, CU
PT/TA-TU, AU,
BAU
2-Step
Later
AT-CU, TF-CU
Early/Middle
Manip., 1:1,
EQ
AT-SU, TF-SU
Early, Middle
Manip., 1:1,
EQ
C-DU-M/F
Early/Middle
Manip., 1:1
C-BU-M,
C-BU-F
Early/Middle
Manip., 1:1
C-SU-F,
C-SU-M
Early/Middle
Manip., 1:1
C-DU-M/F,
C-BU-M,
C-SU-F
Middle
1:1TD, TD, EQ
AT-SU, TF-SU
Middle
1:1TD/NB,
TD/NB, EQ
PT/TA-BAU,
PT/TA-AU
Middle
1:1, 1:1
TD/NB, EQ
C-BU-F,
C-SU-M
Middle
1:1TD, TD,
EQ
AT-RU, CU, SU
TF-RU, CU, SU
Later
TD/NB, EQ
PT/TA-TU,
AU, BAU
Later
TD/NB, EQ
C-DU-M/F,
C-BU-M,
C-BU-F,
C-SU-F,
C-SU-M
Later
EG-UP, GSU,
NGU
Early/Middle
Manip., 1:1,
EQ
AA-UP, GSU,
NGU
Early/Middle
Manip., 1:1,
EQ
EG-UP, GSU, NGU
AA-UP, GSU, NGU
Middle/Later
1:1TD, TD, EQ
C-DU-M/F,
C-BU-M,
C-BU-F,
C-SU-F,
C-SU-M
Later
TD, EQ
All 2-Step with All 4 Operations
Early/Middle/Later
TD, EQ
April
C-DU-M/F,
C-BU-M,
C-SU-F
Later
1:1, 1:1 TD, EQ
All 2-Step
Early/Middle
1:1TD, TD
May
June
AT-CU, RU
TF-CU, RU
PT/TA-AU, BAU, TU
Later
1:1, 1:1TD/NB, EQ
All 2-Step
Later
TD, EQ
All 2-Step with All 4 Operations
Middle/Later
BM, EQ
4th Grade
TD/NB, BM, EQ
TD, BM, EQ
All 2-Step with All 4 Operations
Middle/Later
BM
MC-LU, SU,
MU
Early/Middle
TD, EQ
MC-LU, SU,
MU
Later
BM, EQ
Multi-Step All 4
Operations
Early/Middle
TD, EQ
Multi-Step
Multi-Step All 4 Operations
All 4
Later
Operations
BM, EQ
Middle
TD, EQ
Note: Be sure to review previous problem types. This can be done as the problem within the agenda and/or in the student IP problems. The IP problems should not all be the same story
problem type as what’s noted in the S&S or what was used for the agenda problem.
Appendix:
1.
2.
3.
4.
5.
6.
First Three – sample story problems for each month of the K-4 Math Stories S&S (pages 12-15)
Math Stories – Teacher Note-Taking Tool (page 16)
Addition & Subtraction Calculation Strategies – by grade and in order of least to most sophisticated (page 17)
Multiplication & Division Calculation Strategies – by grade and in order of least to most sophisticated (page 17)
Addition & Subtraction Calculation Strategies – Illustrated (page 18-19)
Multiplication & Division Calculation Strategies – Illustrated (page 20)
Appendix
1. First Three – Sample story problems for each month of the K-4 Math Stories S&S
Kinder
E-w/in 5
M-w/in 10
L-w/in 15
September
October
November
December
January
February
-Sam had 2 toys.
He got 2 more toys.
How many toys
does Sam have
now?
-Sam had 4 apples.
He ate 3 of them.
How many apples
does Sam have
now?
-Sam had 5 toys.
He gave 3 toys to a
friend. How many
toys does Sam have
now?
-Sam had 8
apples. He gate 2
apples. How
many apples does
Sam have now?
-Sam had 4
pencils. He got 3
more pencils from
the bin. How
many pencils does
he have now?
-Sam had 9 toys.
He gave 4 toys to
a friend. How
many toys does
Sam have now?
-12 bunnies were in
the grass. 4 bunnies
hopped away. How
many bunnies are in
the grass now?
-6 dogs were at the
park. 5 more dogs
came to the park.
How many dogs are
at the park now?
-Sam ate 7 pieces of
candy. Then he ate
6 more pieces of
candy. How many
pieces of candy did
Same eat?
-Sam saw 3
snowmen on the
way to school. Sam
saw 2 snowmen on
the way home from
school. How many
snowmen did Sam
see?
-4 green apples and
3 red apples were
on the table. How
many apples were
on the table?
-4 kids at the park
were on the swings
and 5 kids at the
park were on the
slide. How many
kids were at the
park?
AT-RU, TF-RU, PT/TATU
Later
-Sam has 5
crayons. Some
are red and some
are blue. How
many could be red
and how many
could be blue?
-There are 8 little
dogs and big dogs
at the park. How
many of each
could there be?
-There are 7
vegetables on the
plate. Some are
carrots and some
are peas. How
many could be
carrots and how
many could be
peas?
March
April
May
June
AT-RU, TF-RU, PT/TA-TU, PT/TA-BAU
Later; PT/TA-BAU should only be within 10
Note: Be sure to review previous problem types. This can be
done as the problem within the agenda and/or in the student IP
problems. The IP problems should not all be the same story
problem type as what’s noted in the S&S or what was used for
the agenda problem.
1st
Grade
E-w/in 15
M-w/in 20
L-w/in 30
nd
2
Grade
M-w/in 40
L-w/in 100
-Buddy had 12
bones. Then he
found 6 more
bones. How many
bones does Buddy
have now?
-Buddy had 15 toys
in his bin. He took
out 8 toys to play
with. How many
toys are in Buddy’s
bin now?
-Buddy chased 9
squirrels Monday
and 8 squirrels
Tuesday. How
many squirrels did
Buddy chase
Monday and
Tuesday?
-Buddy had 12
bones. Then he
got some more.
Now he has 19
bones. How many
bones did Buddy
get after the first
12?
-Buddy had 13
toys in his bin. He
took some out to
play with. Now
there are 5 toys in
his bin. How
many toys did
Buddy take out of
his bin?
-Buddy ate 3
treats. Then he
ate some more
treats. In all,
Buddy ate 7
treats. How many
treats did Buddy
eat after the first
3?
-Buddy had some
bones. Then he got
4 more. Now he
has 12 bones. How
many bones did
Buddy have to
start?
-Buddy had some
toys in his bin. He
took out 6 to play
with. Then there
were 5 in his bin.
How many toys
were in his bin
before?
-Some dogs were at
the park. 3 more
came to the park.
Then there were 12
dogs at the park.
How many dogs
were at the park
before?
-There were 12
dogs at the park.
Some were playing
and 7 were lying
down. How many
dogs were playing
at the park?
-Buddy has 13
bones and treats.
How many bones
and how many
treats could he
have?
-14 toys are in
Buddy’s bin. 5 are
balls and the rest
are stuffed animals.
How many are
stuffed animals?
-Buddy has 11 bones.
Fido has 8 bones.
How many more
bones does Buddy
have than Fido?
-Buddy ate 4 treats.
Fido ate 13 treats.
How many fewer
treats did Buddy eat
than Fido?
-8 dogs were at the
park on Tuesday. 14
dogs were at the
park on Wednesday.
How many more
dogs were at the
park Wednesday
than Tuesday?
*make sure to
represent with
addition and
subtraction number
sentence
*toss in a review
type to ensure they
don’t just subtract all
month
-Buddy has 4 more
bones than Fido.
Fido has 7 bones.
How many bones
does Buddy have?
-Buddy at 5 fewer
treats than Fido.
Buddy ate 6
treats. How many
treats did Fido
eat?
-There were 7
fewer dogs at the
park Thursday
than Wednesday.
There were 12
dogs at the park
Thursday. How
many dogs were
at the park
Wednesday?
*toss in a review
type to ensure
they don’t just
add all month
September
October
November
December
January
-Rosa saw 17 ants
on a hill at the park.
Some more ants
came. Then there
were 36 ants on the
hill. How many ants
came to join the
first 17 ants?
-Rosa has 35
flowers. 19 are
roses and the rest
are tulips. How
many are tulips?
-There were 16
forks and 16 spoons
on the table for the
family dinner. How
many pieces of
silverware were on
the table for
dinner?
-Rosa has 34
pieces of candy.
Jackie has 19
pieces of candy.
How many more
pieces of candy
does Rosa have
than Jackie?
-Rosa has 13
more pencils
than Jackie.
Jackie has 18
pencils. How
many pencils
does Rosa have?
-Jackie has 9
fewer library
books than Julie.
Julie has 23
library books.
How many
library books
-Rosa saw some
meerkats at the
zoo popping out of
their holes. Then
13 more meerkats
popped out of
their holes. Then
there were 37
meerkats popping
out. How many
meerkats were
popping out
before?
-Rosa had some
animal crackers in
her box. She gave
16 of them to her
friend for a snack.
Now she has 13
animal crackers
left for herself.
How many animal
-Jackie saw 16
fewer birds than
Rosa at the zoo.
Jackie saw 19
birds at the zoo.
How many birds
did Rosa see at
the zoo?
-Rosa had 8
more fries than
Jackie on her
lunch tray. Rosa
had 33 fries on
her lunch tray.
How many fries
did Jackie have
on her lunch
tray?
-Rosa counted the jelly
beans in her bag. Then
she ate 23 of them.
Now there are 68 jelly
beans in her bag. How
many jelly beans were
in Rosa’s bag before?
-Rosa collects rocks.
She had some in her
bag and then she
picked up 31 more.
Then she had 70 in her
bag. How many rocks
were in her bag before?
-Rosa cleaned up the
fallen leaves in her
yard. She picked up 58
leaves in the morning.
She picked up some
more leaves in the
afternoon. In all, she
picked up 92 leaves.
How many leaves did
-Buddy has 4 more
bones than Fido.
Buddy has 13 bones.
How many bones
does Fido have?
-Fido has 7 fewer
bones than Buddy.
Buddy has 14 bones.
How many bones
does Fido have?
-There were 8 more
dogs at the park
Friday than
Thursday. There
were 19 dogs at the
park Friday. How
many dogs were at
the park Thursday?
*make sure to
represent with
addition and
subtraction number
sentence
*toss in a review
type to ensure they
don’t just subtract all
month
C-DU-M/F,
C-BU-M,
C-SU-F
Later
*review
problems
AT-CU, RU
TF-CU, RU
PT/TA-AU, BAU, TU
Later
*review problems
February
March
April
May/June
PT/TA-TU,
AU,
Later
C-DUM/F,
C-BUM,
C-BU-F,
C-SU-F,
C-SU-M
Later
-When Rosa was at
the zoo, she saw
19 fish, 12
meerkats, and 7
lions. How many
animals did Rosa
see at the zoo?
-When Rosa was at
the zoo, she
counted 13 tulips,
12 roses, and
some daffodils.
She counted 38
flowers. How
many daffodils did
Rosa count at the
zoo?
-Rosa saw 27
monkeys at the
zoo. Her Mom
saw 11 monkeys
and her sister saw
13. How many
-Rosa recycled 32
fewer cans than Ben
and Mark combined.
Ben recycled 61 cans
and Mark recycled 30
cans. How many cans
did Rosa recycle?
-Rosa had 65 jelly
beans. Then she ate
26 of them. The
number of jelly beans
she had left is 5 more
than the number of
jelly beans Jackie had.
How many jelly beans
did Jackie have?
-Rosa gave away the
71 rocks from her
collection. She gave
27 rocks to her sister,
31 rocks to her Mom,
and the rest of her
rocks to her brother.
does Jackie
have?
crackers were in
the box before?
she pick up in the
afternoon?
more monkeys did
Rosa see than her
Mom and her
sister?
How many rocks did
she give to her
brother?
Note: Be sure to review previous problem types. This can be done as the problem within the agenda and/or in the student IP problems. The IP problems should not all be the same story
problem type as what’s noted in the S&S or what was used for the agenda problem.
September
rd
3
Grade
M-w/in
100
L-w/in
1,000
(mult/div
still within
100)
-Mycah counted
paper clips one
afternoon. He
counted 342 in his
3rd grade classroom,
421 in the 3rd grade
room across the
hall, and 781 in the
4th grade room.
How many fewer
paper clips did he
count in the 3rd
grade rooms than
the 4th grade room?
-Mycah had a
baseball card
collection. One day,
he gave 387 of his
cards to his best
friend. Another
day, he donated
268 of them to a
yard sale. After
October
C-DU-M/F,
C-BU-M,
C-BU-F,
C-SU-F,
C-SU-M
Later
*See previous
grades; change
numbers
November
December
January
February/March
April/May/June
-Mycah had 7 bags
with 6 pieces of
candy in each bag.
How many pieces
of candy are there
in all?
-Mycah has 36
inches of string
which he will cut
into 4 equal
pieces. How long
will each piece of
string be?
-Mycah has 28
pencils and he
wants to give 4 to
each of his friends.
How many friends
can he give pencils
to?
-Mycah has a
page of dog
stickers with 8
rows of stickers
with 4 stickers
in each row.
How many
stickers are
there?
-A rectangle has
an area of 21
square inches.
If one side is 3
inches long,
how long is a
side next to it?
-Mycah wants
to put his
baseball cards
on a poster to
display them. If
he puts his 33
baseball cards
-You need 4 bags of
apples, each bag
weighing 5 pounds, to
bring to a party. How
many pounds of apples
will you need
altogether?
-What is the area of an
8 cm by 7 cm
rectangle?
-Mycah bought 42
boxes of juice at the
store. The boxes come
in packages of 6. How
many packages did
Mycah buy?
-There are 7 kids
coming to Mycah’s
birthday party. He
wants to have pizza to
eat. He also wants to
make sure each kid,
including himself, gets
6 pieces of pizza. If
each pizza comes with
8 slices, how many
pizzas does Mycah
need to get for his
party?
-Mycah is 3 years
younger than his
brother Kurt. Kurt is
28 years younger than
his Mom, Nancy.
Nancy is 37 years old.
How old is Mycah?
-Mycah ordered 4
large pizzas from
Figidini’s. A large
-Mycah bought a package of chocolate chips
to bake into cookies. The package he bought
included 220 chocolate chips. He plans on
making 50 chocolate chip cookies, with five
chips each. How many more chocolate chips
will Mycah need?
-Mycah bought a juice for two dollars and 9
hamburgers for he and his friends at the
movie. He spent a total of 47 dollars. How
much did each hamburger cost?
-There were 6 horses on the farm. Each one
eats 8 pounds of oats in one day. How many
pounds of oats are needed to feed the
horses for three days?
that, he had 241
baseball cards left.
How many baseball
cards were in
Mycah’s collection
to start?
-Mycah helped his
teacher clean out
the crayon bins.
There were 427
crayons in all the
bins. They threw
away 249 crayons
that were too
broken or too small
to use. Then they
put 145 new
crayons into the
bins. How many
crayons are in the
bins now?
into rows of 3,
how many rows
will there be?
pizza has 2 more slices
than a medium. A
medium has 8 slices.
How many total slices
did Mycah order?
Note: Be sure to review previous problem types. This can be done as the problem within the agenda and/or in the student IP problems. The IP problems should not all be the same story
problem type as what’s noted in the S&S or what was used for the agenda problem.
th
4
Grade
September/October
November
December
January
February
March/April/May/June
-Kendria had $235 to spend on 7 books.
After buying them, she had $11 left.
How much did each book cost?
-Kendria can ride her bike 12 miles in an
hour. How many miles could she ride if
she rode from 9:00 in the morning to
4:00 in the afternoon?
-Kendria had 873 stickers. She gave 8
stickers each to 10 friends. How many
stickers does Kendria have left?
-A soft-cover book
costs $8. A hardcover book costs 3
times as much as
the soft-cover
book. How much
does the hardcover book cost?
-A grown-up
penguin eats 32
ounces of fish a
day and that is
four times as much
as a baby penguin
eats in a day. How
many ounces of
fish does a baby
penguin eat in a
day?
-A rubber band
was 6 cm long as
first. Now it is
stretched to be 18
-The garden
behind
Kendria’s house
is 9 ft wide.
How wide will
the garden be
when it is
extended to be
4 times as wide?
-The area of a
garden is
extended to be
450 square feet
and that is 3
times as large as
it was at first.
What was the
area of the
garden at first?
-Kendria caught
the ball 42 times
at baseball
practice
-Kendria had 205
stickers. She bought 24
from a store in the mall
and got 48 for her
birthday. Then Kendria
gave 12 of the stickers
to her sister and used
52 to decorate a poster.
How many stickers
does Kendria have left?
-Oceanside Bike Rental
Shop charges $12 plus
$8 an hour for renting a
bike. Sally paid $76
dollars to rent a bike.
How many hours did
she have the bike
rented for?
-Kendria has four times
as many pennies as
Alana and half as many
pennies as Rebecca. If
Alana has six pennies,
-Kendria has 12
crayons. Emma has
twice as many as
Kendria, and Lily has
twice as many as
Emma. How many
crayons do the girls
have in all?
-Kendria bought three
bags of candy with 75
pieces in each one.
She plans to divide all
the candy evenly
among her seven
friends. How many
pieces of candy will
Kendria have left for
herself?
-There are 84 bicycles
and 79 cars in the
garage at Kendria’s
apartment building.
-Jake treaded water for 279 seconds.
Kendria treaded water for 5 minutes and 8
seconds. How many more seconds did
Kendria tread for?
-Kendria planted vegetables in a garden that
was 20 feet long and 20 feet wide. She used
1/4 of the area for corn and 1/2 of the area
for peas. How many square feed are left for
other vegetables?
-A rectangular field is 63 yards long and 21
yards wide. A fence is needed for the
perimeter of the field. Fencing is also
needed to divide the field into three square
sections. How many feet of fencing are
needed?
cm long. How
many times as
long is the rubber
band now as it was
at first?
yesterday and
how many does
How many wheels are
her friend Dante Rebecca have?
there in the garage?
caught the ball
14 times. How
many times as
many did
Kendria catch
the ball at
practice
compared to
Dante?
Note: Be sure to review previous problem types. This can be done as the problem within the agenda and/or in the student IP problems. The IP problems should not all be the same story
problem type as what’s noted in the S&S or what was used for the agenda problem.
2. Math Stories – Teacher Note-Taking Tool
Agenda Focus
Student
Early
Able to ACCESS the
problem?
Early/Middle
Able to REPRESENT
the problem?
Type of representation
Middle/Later
CALCULATION
strategy for solving
Later
ANSWER
Correct?
Number Sentence?
Able to FINISH THE STORY?
3. Addition & Subtraction Calculation Strategies – by grade
*building efficiency from left to right
Kindergarten
First Grade – w/in 20
w/in 100
Count All
Count On
Stick and Dots
Second Grade – w/in 100
w/in 1,000
Sticks and Dots
Third Grade
Fourth Grade
Count On
Count Back
Expanded Notation Addition
Number Line
Count Up
Make Tens
Just Know
Mental
(add tens and tens; add ones and ones)
Expanded Notation Add by Place Compensate
Subtraction
Expanded Notation Add by Place Keep One Number Whole
Subtraction
Flats, Sticks, and
Number Line
Dots
Maintenance of fluency developed in second grade.
Prioritize – Expanded Notation Subtraction and Add by Place/Keep One Number Whole (+)
Expanded Notation (-)/Add by Place (+)
Standard Algorithm
4. Multiplication & Division Calculation Strategies – by grade
*building efficiency from left to right
Unknown Addend
Unknown Addend
Third Grade - w/in
100
Arrays/Groups – Concrete
Arrays/Groups – Pictorial
Count-On/Skip-Count
Distributive Property
Fourth Grade –
4x1 and 2x2 mult.
4x1 division
Place Value Arrays Concrete
Place Value
Sharing/Grouping - Concrete
Place Value Arrays - Pictorial
Area Model
Partial Products
Place Value
Sharing/Grouping - Pictorial
As Group Size
Area as Side Length
7.
Just Know
An Illustration of the Addition & Subtraction Strategies by Grade:
Kindergarten:
1) Within
10/20
Count All: count all
items with
manipulatives,
drawing, or on
fingers
Count On: 5+3 --- put first addend on
fingers, count, number said is total --5…6, 7, 8
First Grade:
1) Within 20
Count On: (see GK)
Count Back: 10-7 --- put total on
fingers, count back the subtract
amount, number said is part/answer -- 10… 9/8/7/6/5/4/3
Count Up: 10-7 --- put up fingers as
you count from the part/lesser
amount to the whole/larger
amount, number of fingers up is
the difference --- 7…8/9/10
Make Tens: 5+7 -- know “ten”
combination,
make a 10, and
add the leftover -- 5 and 5 is 10,
and 2 more is 12
(breaking the 7
into 5 and 2)
Just Know: 5+7 -- I just know
this fact – 5 and
7 is 12
First Grade:
1) Within 100
Count On: (see GK)
-Count on Tens and
then Count on
Ones
- 56 + 30…
56…66, 76, 86
- 56 + 32…
56…66, 76, 86,
87, 88
- 56 + 35…
56…66, 76, 86,
87, 88, 89, 90,
91
Sticks & Dots:
(G1 is 2-digit + 1-digit)
General: Add 10s and 1s separately
1) III… + II.. --10/20/30/40/50/51/52/53/54
/55
Special: Count on by 10s
1) III… + II.. --- 33/43/53/54/55
Expanded Notation Addition:
(G1 is 2-digit + 1-digit)
1) 23+22 --20+ 3
+ 20+ 2
40+ 5 =45
Mental: using general or special
Sticks & Dots method mentally
Second Grade:
1) Within
1,000
Sticks & Dots (1):
(see G1; Flats,
Sticks, & Dots in
G2)
Number Line (2):
1) 327+243
Expanded Notation Subtraction (3):
1) 327-243 (also 327+243)
120
200 +100
300+ 20+ 7
- 200+ 40+ 3
0+
80+ 4 =84
Add by Place (4):
1) 327+243
3 2
+ 2 4
5 0
6
1
3 7
Keep One Number Whole (6):
1) 327+243
Unknown Addend (7):
1) 327-243
2) 327-243
Compensate (5):
1) 264-198
2) Add 2 to both 264 and 198 --- 266-200 --- 66
+
3
2
5
5
2
0
2
4
6
5
7
+
+
7
0
7
0
7
3
0
7
3
0
0
0
0
3) 264+198
4) Give 2 from 264 to 198 --- 262+200 --- 462
2) 327 – 243 (w/out regroup)
-
3
2
1
2
0
2
4
8
8
Third Grade:
1) Within
1,000
Fourth Grade:
1) Within
1,000,000
8.
7
0
7
0
7
3
4
Maintenance of fluency developed in second grade
Expanded Notation (-) and Add by Place/Keep One Number
Whole (see G3)
Standard Algorithm
An Illustration of the Multiplication & Division Strategies by Grade:
Third Grade –
w/in 100
Arrays/Groups –
Concrete: use counters to
build arrays and/or
groups to model, count
all/on
Arrays/Groups – Pictorial: draw arrays
and/or group to model, count all/on
Count-On/Skip-Count:
Count by Zs or Ys to solve Z x Y; I know 5x2
is 10, count on 5 more – 15 is 5 x 3
Fourth Grade –
4x1 and 2x2
multiplication
Place Value Arrays –
Concrete:
1) Build the frame
with PV blocks
and multiply the
parts
2) 13 x 12
Place Value Arrays – Pictorial:
1) Draw the frame with Flats,
Sticks, & Dots and multiply the
parts
2) 13 x 12
Area Model:
1) Use numbers to make the place
value frame; then multiply the
parts
2) 13.12
4x1 division
Distributive
Just Know:
Property:
I know 5x7 is 35 and I
5x7 is the
can explain it if asked
same as 5x5
and 5x2 – so
that’s 25 and
10 more – 5x7
is 35
Partial Products:
1) See the value of the digits
2) Multiply all of the parts; ones
x ones, ones x tens, tens x
ones, tens x tens
Place Value
Sharing/Grouping –
Concrete:
1) Gather the total
in place value
blocks
2) Share into the
number of
groups, or pull
out groups by
size
3) Regroup
hundreds for
tens, tens for
ones as needed
Place Value Sharing/Grouping –
Pictorial:
1) Draw the total using Flats,
Sticks, & Dots
2) Share into the number of
groups, or pull out groups by
size
3) Regroup hundreds for tens,
tens for ones as needed
4) 156 ÷ 12
As Group Size:
1) Draw the number of groups
2) Share tens/hundreds (with
known fact), record with version
of standard algorithm
3) Regroup as necessary
4) Share tens/ones (with known
fact)
Division as Side Length:
1) Think about division as
multiplication
2) 12 times what equals 156?
3) Find a factor for the
hundreds/tens
4) Multiply, subtract
5) Find a factor for the tens/ones
6) Multiply, subtract
7) Add up the place value side
lengths