6 minute read
Why
Addition and Subtraction Within 200
Why are two topics devoted to simplifying strategies for addition and subtraction?
By the end of grade 2, students are expected to add and subtract fluently within 100 by using strategies based on place value, properties of operations, and the relationship between addition and subtraction. Fluency means being able to operate with numbers flexibly, efficiently, and accurately.
Because students are not expected to work fluently with the standard addition and subtraction algorithms until grade 4, topics A and C are intentionally devoted to Level 3 addition and subtraction methods, in which students use simplifying strategies to make simpler problems. This gives students time to work through and to make connections between various strategies. As students apply place value understanding from module 1 and leverage familiar tools, they develop confidence and flexibility. While students are not expected to master all of the Level 3 strategies, they are expected to reason about the numbers in a problem and to consider efficient solution paths by using tools and written recordings. This builds their capacity toward mental math.
Addition and subtraction problems are presented horizontally throughout grade 2. A vertical presentation implies use of the standard vertical form notation. In contrast, a horizontal presentation is more conducive to students thinking flexibly about number relationships to choose the most efficient strategy.
Why is the standard vertical form for addition and subtraction not introduced in this module?
After much consideration of our students’ learning, teachers’ input, and a review of the research around how students learn and how mathematical concepts progress, we decided it makes the most sense to hold off introducing the standard vertical form until module 4. Why?
1. This module focuses on the conceptual understanding of addition and subtraction through the use of concrete models, drawings, and strategies. By intentionally delaying
Methods for Addition and Subtraction
Level 1: Count all
Level 2: Count on by ones
Level 3: Make a simpler addition or subtraction problem. These methods often use the associative property:
• Decompose addends to add or subtract like units
• Use benchmark numbers to count on or count back
• Use compensation to adjust numbers • Decompose addends to make or take from a ten or a hundred
• Think of subtraction as an unknown addend problem.
the introduction of the vertical form, we create more time and space for students to explore a variety of strategies, which encourages them to reason about number relationships and efficiency, rather than jump to one specific strategy.
2. Grade 2 math standards call for students to relate their strategies to a written method.
When students compose or decompose units by using models or drawings, they connect actions and language to corresponding steps in a written recording (e.g., expanded form, totals below, unit form). In contrast to the vertical form, these written methods have one thing in common—they explicitly highlight place value units.
Why do some word problems in lesson 27 involve single-digit addends?
Lesson 27 is the students’ first formal experience with two-step word problems. Due to the added cognitive lift of solving a multi-step problem, many two-step problems involve single-digit addends. By using single-digit quantities and easier problem types, students can focus on representing number relationships with a drawing and with an equation. This builds their confidence as problem solvers.
Ms. Bell wants to give each student a pencil in their favorite color. How many pencils does Ms. Bell need for the class?
She already has 5 yellow pencils. How many more pencils does Ms. Bell need?
Favorite Colors in Ms. Bell’s class
14 19 9 3 5 2
Number of Students 11
10
9
8
7
6
5
4
3
2
1
0
Red Blue
YellowPurple Color
Which word problem types, or addition and subtraction situations, are used in this module?
The table shows examples of addition and subtraction situations.1 Darker shading in the table indicates the four kindergarten problem types. Students in grades 1 and 2 work with all problem types. Grade 2 students reach proficiency with the unshaded problem types.
Grade 2 students are expected to master all addition and subtraction problem types by the end of the year. This module focuses on the following problem types.
• Add to with result unknown: Both parts are given. An action joins the parts to form the total.
27 cars are in the parking lot. 39 more cars pull into the lot. How many cars are in the lot now? (Lesson 7)
• Take from with result unknown: The total and one part are given. An action takes away one part from the total.
63 people are on a bus. 48 people get off the bus at the park. How many people are still on the bus? (Lesson 19)
• Put together with total unknown: Both parts are given. No action joins or separates the parts. Instead, the parts may be distinguished by an attribute like type, color, size, or location.
125 students are sitting in the cafeteria. 69 students are standing in the lunch line. How many students are there in all? (Lesson 7)
• Compare with difference unknown: Two quantities are given and compared to find how many more or how many fewer.
The vet checks the pets. She checks 74 dogs and 28 cats. How many more dogs than cats does she check? (Lesson 16)
1 Common Core Standards Writing Team, Progressions for the Common Core (draft), Grades K–5, Counting and Cardinality &
Operations and Algebraic Thinking, 9.
Take from with change unknown: The total and the resulting part are given. An action takes away an unknown part from the total. The situation equation (e.g., 57 – = 28) can be rewritten as a related solution equation (e.g., 28 + = 57 or 57 – 28 = ).
There are 57 tacos in the lunchroom. Then some tacos are eaten. Now there are 28 tacos left for the next class. How many tacos were eaten? (Lesson 13)
• Put together/take apart with addend unknown: The total and one part are given. No action joins or separates the parts.
Hope picks 63 apples. 47 of the apples are green. How many apples are not green? (Lesson 21)
• Compare with smaller (quantity) unknown: The bigger quantity and the difference between the quantities are given.
There are 28 fewer plums than lemons in a bin. There are 73 lemons. How many plums are in the bin? (Lesson 19)
The following problem types tend to be among the most challenging subtypes for grade 2 students.
• Add to with start unknown: The total and one part are given. An action joins one part with the unknown start to form the total. Since one part is unknown, the problem can be thought of as a subtraction problem. The situation equation (e.g., + 35 = 90) can be rewritten as a related solution equation (e.g., 35 + = 90 or 90 – 35 = ).
Alex has some money in his bank. He finds 35 cents. Now he has 90 cents in his bank. How much money did have Alex have in his bank to start? (Lesson 26)
• Take from with start unknown: Both parts are given. The starting quantity, or total, is unknown. There is an action that takes away one part from the unknown total. Since the total is unknown, the problem can be thought of as an addition problem. Note that the situation equation (e.g., – 25 = 20) can be rewritten as a related solution equation (e.g., 25 + 20 = ). Add to and take from with start unknown problems are two of the most challenging subtypes for students.
Ming has some money. She spends a quarter on an eraser and has 20 cents left. How much money did she have before she bought the eraser? (Lesson 26)
