Mandi Edwards
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the Leadership for the Common Core in Mathematics project at the
University of Wisconsin-Milwaukee.
Grade: 3
Domain: Number and Operations in Base Ten
Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic
Standard: 3.NBT.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place
value, properties of operations, and/or the relationship between addition and subtraction.
Explanation and Examples
Unpacking this standard for students encompasses a lot. Students must first have fluency with and
understanding of single-digit addition computations and the relationship those have with subtraction
(number bonds/fact families). In addition to that, they also need to understand base ten place value
concepts. For example there are 10 ones in a ten, 100 ones or 10 tens in a hundred, and 1,000 ones or
100 tens in a thousand. With the understanding of the above, students should become fluent more
quickly in problem solving through computations within 1,000. Students will be successful and proficient in
this standard when they can apply known strategies to efficiently solve the problem. There are many
ways to arrive at the desired answer, but students should find the most efficient one to demonstrate their
ability to fluently add or subtract within 1,000.
Besides breaking apart these numbers based on place value, students should also be familiar with
properties of operations to justify the answer they arrived at. Operations may include Commutative
property, Associative property, and Distributive property to help students efficiently compute their answer
as well as show their understanding of the value of the numbers they are working with. For example, if a
student were solving a problem and add ones to ones, tens to tens, and hundreds to hundreds, they are
using a place value method as well as the Commutative and Associative properties (745+126=871,
700+100=800, 40+20=60, 5+6=11, and 800+60+11=871). This same type of algorithmic strategy can be
used a little differently as we look at the ‘traditional’ way we learned to add and subtract with regrouping
(745+126=871, 5+6=11 (1 ten and 1 one) regroup 1 ten, 4 tens +2 tens+1 ten = 7 tens, 7 hundreds + 1
hundred =800 hundreds, 1 one+7 tens+8 hundreds =871
745
+ 126
800
60
11
871
1
745
+ 126
871
745 + 126 = 745 + 100 + 20 + 6 = 845 + 20 + 6 =
865 + 6 = 865 + 5 + 1= 870 + 1 = 871
Students can also use the relationship between addition and subtraction to solve equations in a fluent
manner. If I use the above problem and switch numbers around to read 871-745, I can turn it into an
addition problem to state 745+____=871. I can then use my base ten strategies and understanding to get
me to my answer: 745+5=750, 750+50=800, 800+71=871 therefore 5+50+71=126 giving me the
difference for 871-745=126.
Student ability to manipulate and understand that numbers can be broken apart and that they can take
one equation and make it several smaller equations to accomplish the same task, seems less daunting to
do. If students knew that and had permission to do that on a regular basis, they would become more
fluent mathematicians much quicker. And, as we see, this NBT standard works jointly with the OA
standards that students have been working to master as well. As teachers, we must understand that
these standards and domains don’t work in tandem, but rather hand-in-hand to make our students
successful, life-long learners of math. Once we see that and can teach through standards and strategies
cohesively, students will also obtain cohesive understandings of mathematics, not disjointed segments
that we see currently. I believe that through this, they will one day astound us with their ability to relate
strategies to unknown problems.
School Mathematics Textbook Program: Math Investigations
Textbook Development:
I think Investigations does a great job of introducing this standard in that the whole first unit is devoted to
developing ideas around operations with whole numbers, fluency with computation, as well as place value
and base-ten system. It also puts these concepts into different scenarios and contexts such as coins,
travel distances, and temperature. I appreciate that this is hit hard at the beginning of the year, because it
sets the foundation for what students will do going forward with larger addition and subtraction problems
as well as into multiplication and division.
It starts out with a review from grade 2 in which the students practice with strips of 10 stickers. They use
‘number of the day’ exercises to show numbers based on tens strips and singles. For example, if the
number was 37, the students would put out 3 ten strips and 7 singles. They then write an equation to
solidify this concept. Therefore, the above example’s equation would then read 30+7=37. This is also
done as a quick image activity to help students subitize those larger numbers without having to count
them all out prior to working with them. Different tools such as number lines, 100 charts, cube towers, and
sticker strips are introduced to students as ways that they can physically manipulate these larger numbers
to arrive at their answer. Students who don’t need the tangible manipulatives are encouraged to draw out
these examples as they write equations.
I especially like that Investigations arranges story problems so that students have to figure out the value of
a number written in place value form before going on to add or subtract. I think it makes students read
more carefully to figure out what they need to do to be successful. Example: Alex had 72 stickers. He
went to the store and bought 3 strips of 10 and 5 singles. How many stickers does he have now?
This beginning unit also incorporates ten-minute math practices on given numbers where students need to
represent that number in as many different ways as they can using hundreds, tens, and ones. If they were
given the number 123, a student could say that they need 1 sheet (sheet represents 100), 2 strips (a strip
represents 10), and 3 singles. They could also say that they need 12 strips and 3 singles. Any
combinations of sheets, strips, and singles that make the given number are acceptable. This shows
students that it is ok to break numbers apart (decompose), and that by doing so they start to understand
what numbers really are made of and how they can use them to arrive at an answer when numbers
become larger than they are used to working with.
Not only are students given addition and subtraction problems, but they have an opportunity to practice
finding unknown addends when they are given a number and asked ‘How many more to get to 100?’.
Students begin to count by tens when they see full rows of the hundreds grid empty and then go back and
count up with the singles left over. Most of the time students arrive at this point; some start out counting
each individual square, but as they get more and more practice they see the 10 in the grids.
To help students understand the concept of groups of ten, they also use a chart that relates single digit
addition problems that equal 10 to groups of tens equaling 100. For example, 1+9=10 and 10+90=100,
2+8=10 and 20+80=100 etc.
As we progress through the curriculum, numbers extend to 1,000 as well as adding two and three-digit
numbers within 1,000. It helps students understand the importance of landmark numbers such as
multiples of 10 or 100 to help them compute efficiently and fluently. Construction of 1,000 charts, using
hundred grids, help students to locate and visualize these larger numbers as well as landmark numbers to
help in computational fluency. The activity of the ‘number of the day’ is repeated throughout the
curriculum as numbers become larger and increase with place value knowledge. Students also gain
fluency with subtraction facts related to addition combinations up to 10+10. Tougher problems that
encourage number line usage to solve them are splattered all over this curriculum and increase in value
as well as language complexity. Students go from answering simple sticker problems to answering travel
problems where they needed to pull out what is necessary to solve their problems. Their number line
jumps go from increments of 10, to larger increments that got them beyond moving to the next 10 and
going to maybe the next hundred or a ten closest to the total asked to reach. Example: If I traveled to mile
36 the first day and to mile 70 the second day, how many miles did I travel on day two? Early on it is
acceptable for students to ‘jump’ the following jumps: start at 36, jump to 40, jump to 50, jump to 60, jump
to 70 to arrive at the answer of 34 (4+10+10+10). By the end of the year, students should be able to
make larger jumps and go from 36 to 66 (knowing that is 30) and then go the 4 more to 70 (30+4).
One last thing that I really liked was the introduction of negative numbers and using a number line strategy
to solve subtraction problems. Every Wednesday throughout the year we keep track of the temperature
and record it on a graph. Students are asked throughout the curriculum to note changes in temperature
and describe the shape of the graph that we create alongside the temperature chart. Towards the end of
the year, students have to do some computation of the highs and lows on the chart. One day last year
we had a negative temperature and students were stumped by that. Because the thermometer is like a
number line, and they already knew that strategy, it made it easy for me to use the thermometer to explain
how I find the difference of a positive number and a negative number.
Conclusions and Suggestions:
Looking through the curriculum for evidence of alignment, I see that Investigations aligns itself really well
in the way of base-ten strategies and algorithms. The curriculum does a lot with breaking numbers into
place value pieces, writing expanded notations, and using number lines and 100 grids to show values of
numbers and relationships to the values based on position in a given number. They use several different
properties of operations as well, but neglect to call them that. Because of that, teachers need to be
familiar with the properties and make notes to be intentional about calling the strategy processes by the
property names so that students are familiar with them for succeeding grade levels. While the CCSSM
standards note that third graders don’t need to call these by the operation names, it is important for
students to have exposure to them.
In regards to relating addition and subtraction, the curriculum doesn’t do much in the way of subtraction.
They pose subtraction problems and show students how to use an addition strategy to solve the equation.
The relationship is there, but I think teachers really need to be more intentional about ‘teaching’ this
relationship and not assume that the students fully understand that they are related. It goes back to the
idea that the beginning of the OA progressions document points out; important ideas need to be taught in
ways that are engaging and meaningful for our students. Though some things may seem obvious to us,
our students may not draw the same conclusions because of their limited exposure to mathematic
concepts and ideas.