Dear Families,
Your child is starting a new unit in math, Ages and Timelines, which is all about subtraction. This
project-based unit is from Contexts for Learning Mathematics, the same publishers who brought us
Measuring for the Art Show, an earlier second grade unit about addition. You will find that your second
grader will have homework to find the ages of at least four people at home or in the neighborhood, at
least two of whom must be 20 years old or more.
In Ages and Timelines students focus on subtraction by examining the differences in ages of people
within both a family that they have read about and their own family. Students solve problems using a
variety of strategies. The open number line, a number line with no markers, is used to model these
strategies. The strategies students may explore include Counting On and Counting Backward; Taking
Leaps of Ten; Decomposing to get to a Landmark Number; Adding on vs. Removing; and Using
Constant Difference. Please see the addendum for an explanation of these strategies.
Research has shown that the exploration of different methods and strategies is the best route to
students’ deeper understanding of subtraction. The Massachusetts Math Frameworks do not include the
traditional algorithms for double digit addition or subtraction in second grade. The first time those
algorithms will be assessed is in grade 4. Instead of memorizing steps to a procedure, second graders
are expected to develop strategies to subtract and add numbers. These strategies should be based on:

the relationship between addition and subtraction

understanding subtraction as removal, difference

constant difference as equivalence; i.e. 28 – 9 = 29 – 10

understanding of patterns that occur when adding and subtracting groups of tens
Please give your second grader time this year to explore problems and develop ideas without
memorizing steps. Don’t work on regrouping or “borrowing” methods with your child. When children have
developed a solid understanding of subtraction concepts, the algorithms will have meaning as well as
being efficient procedures. Thank you for partnering with us in our efforts to help children become
mathematical thinkers!
Thank you,
Mary Eich, K-8 Mathematics Coordinator
Elementary Math Coaches
Subtraction Strategies
Counting Backward and Counting On:
To solve 43 - 7, students might count backward 7 steps to find their answer:
 Start at 43.
 Count their 7 steps backward: 42, 41, 40, 39, 38, 37, 36.
 Say the answer is 36.
To solve 43 - 38, students might count on instead of counting back since 38 is such a large
number:
 Start at 38.
 Count their steps up to 43: 39, 40, 41, 42, 43, so 5 steps
 Say the answer is 5.
Taking Leaps of Ten:
To solve 72 - 23, students might go backwards by tens ands ones as follows:
 Start at 72 and take 2 leaps of ten backward to 52
 Count 3 more backwards by ones: 51, 50, 49
 Say their answer is 49.
This process is shown on the open number line below.
Decomposing to Get to a Landmark Number
To solve 72 - 36, students might decompose the 36 into 30 and 6, and decompose the 6 into 2
and 4. Then they would use those numbers to go backwards from 72 as follows:
 Start at 72 and take a leap of 2 backwards to 70
 Take a leap backward of 30 and get to 40
 Take a leap backward of 4 and get to 36
 Say their answer is 36
Adding On versus Removing
When numbers are close together as in 72 - 68, it is helpful to think of Adding On to find the
difference between the two numbers. Students add on chunks of numbers as illustrated below:
 Add on 2 to 68 in order to get to a landmark number
 Add on 2 more to get to 72
 See that the answer is 4 by adding the twos together
Using Constant Difference
As students work in this unit, they see that the difference between their age and the age of
another family member will always be the same, even as they get older. This understanding of
difference being constant can help students see that if you want to find the difference between
two numbers, you can add the same amount to each number to change a hard problem to an
easier problem. They may see that as sliding a pair of numbers up or down the number line to
keep the difference the same.
For example, to solve 84 - 29, students might do the following:
 Add 1 to each of those numbers to make the problem 85 – 30
 Solve 85 – 30 which is a much easier problem