This session is the first of five in which you will focus, in turn, on each of the operations: Addition, Subtraction, Multiplication, and Division. During these sessions you will look at the meaning of the operation, explore contexts and representations, and consider several computational strategies for solving problems. You will watch short segments of videotapes that provide examples of students working on whole number computation and demonstrate how students build their understanding of the operations. This session will focus on Addition.
In this session you will:
Please have the following available for this session: 48 Unifix or linking cubes in one color and 25 in another color.
Note: The Common Core Standards states that the U.S. Standard Algorithm for addition and subtraction are to be studied in grade four. Think about the foundational concepts that are practiced in the grades before so fourth grade students can, not only perform the steps of the U.S. Standard Algorithm, but also understand the meaning behind the steps in these algorithms.
Ten-Frame Quick Images
Ten-Frames help students organize sets of objects in order to make them easier to count and combine. Students describe the spatial representations, and develop visual images for various quantities. Using Ten-Frames helps students establish 10 as a unit, and pushes them to use it as a benchmark - seeing numbers in terms of ten. Ten-Frames are a great representation to help students to develop fluency with number combinations to 10 + 10.
Second Grade Ten-Frame Classroom Routine
Respond to the following questions in your notebook.
Mentally solve the following problem.
Kim has 6 cars.
Her friend gave her 7 more cars.
How many cars does Kim have now?
You may have found the sum (13) using a math fact. The following are examples of ways.
These examples remind us that even with a problem as simple as 6 + 7 there is more than one way to solve the problem and arrive at a correct solution. Young students commonly use one of three approaches to solving an addition story problem: counting all, counting on or using numerical reasoning. It’s important to observe and listen to students at work to understand their approach.
The ability to work with chunks greater than one and numerical reasoning develops gradually over the early elementary years. As students build their understanding of number combinations, place value and number relationships, as well as their ability to visualize the structure of the problem as a whole, they will begin to develop more efficient and flexible strategies.
Kim has 6 cars.
Her friend gave her 7 more cars.
How many cars does Kim have now?
Watch the class discussion as they share their strategies and their teacher notates three approaches to solving the same problem: Counting All, Counting On and Numerical Reasoning. Pay attention to how she guides the discussion about strategies.
(Insert Video V1.O1) Counting All
(Insert Video V1.O2) Counting On
(Insert Video V1.O3) Numerical Reasoning
Discussing Addition Strategies from Number Games and Crayon Puzzles (Grade 1)
Discussing Addition Strategies, from Landmarks and Large Numbers (Grade 4)
In what ways does discussing and sharing strategies help students?
Discussing Addition Strategies from Number Games and Crayon Puzzles (Grade 1)
Students’ Addition Strategies, from Partners, Teams, and Paper (Grade 2)
Adding 2-Digit Numbers, from How Many Tens? How Many Ones? (Grade 2)
Addition Strategies, from How Many Hundreds? How Many Miles? (Grade 3)
The Case of Ezra Who “Just Knows” the Answer from Implementation Guide (Grade 3)
Discussing Addition Strategies, from Landmarks and Large Numbers (Grade 4)
The readings above are all published in Russell, S.J.; Economopoulos, K.; Wittenberg, L.; et al. Investigations in Number, Data, and Space®, Second Edition. Glenview: Pearson, 2012.
In this activity, you will:
You will begin by mentally solving an addition problem. Often our mental strategies—like the ones we might use while at the grocery store - are the ones that best show our flexibility and use of number relationships.
48 + 25
Below are three different strategies. Look for similarities among the strategies, particularly those that start with the same first step, but have subtle differences in the final steps. There are three or four basic strategies, not dozens.
Note: Students sometimes use “run-on-equations.” When you record student strategies, make sure not to use “run-on-equations.”
Students often record “run-on-equations” such as:
40 + 20 = 60 + 8 = 68 + 5 =73
This is incorrect.
While 40 + 20 = 60, 40 + 20 ≠ 60 + 8 and 60 + 8 ≠ 68 + 5.
The correct way to record this thinking is using three separate equations:
40 + 20 = 60
60 + 8 = 68
68 + 5 = 73
Read the following Teachers Notes – they further describe addition strategies across the grade levels.
Choose a strategy (yours or one of the three that are illustrated above). Using Multilink cubes or Unifix cubes, take 48 cubes in one color and 25 cubes in another color. Act out the steps of the strategy using the cubes.
Focus on just the first step of the strategy. Use a blank paper as a mat. Model the first step of your strategy on the mat. (Any cubes that weren’t part of the first step can be left off to the side.)
Review the different strategies in the following slideshow.
In this activity, you will view video clips of students demonstrating different strategies to solve 48 + 25.
INSERT video clip V1.2
Use each of the students' strategies to solve 54 + 37.
INSERT video clip V1.3
Use Chris and Isaiah's strategies to solve 250 + 687.
As students share their strategies, they:
In this activity, you will work on Addition Starter Problems. You will solve the same problem with different “starts,” or first steps.
One goal for students as they develop computational fluency with the operation of addition is to be flexible. We want students to have several strategies to choose from depending on the number relationships presented in a particular problem. Starter Problems encourage students to expand their repertoire of strategies. As you solve these problems, think about how starter problems help to develop flexibility.
288 + 456
Students might say things like:
Record how you solved 288 + 456 using each of the following starter problems:
Think about…
Record your response in your notebook.
Optional: If you would like to try more starter problems, here are a few from grade 4.
We want students to be accurate, efficient, and flexible. As you saw in the video and know from your own experiences, efficiency varies from one individual to another.
Think about how the work you did in this session links to the work your students do with the operation of addition.
How do contexts and representations help students develop understanding of the addition operation? Cite examples from the session activities, readings or video.
How do Starter Problems help students identify, describe and compare addition strategies?
Remember to post early and return to the discussion forum throughout the week to read and respond to others’ posts.
The content in this session is aligned to the following Common Core State Standards and Math Practices:
MP1: Make sense of problems and persevere in solving them.
MP2: Reason abstractly and quantitatively
MP4: Model with mathematics
MP5: Use appropriate tools strategically
MP6: Attend to precision
You may also wish to review the full set of Common Core State Standards in Math and/or videos of the Math Practices enacted in Investigations.
This chart was created by Bill McCallum in an attempt to provide some higher order structure to the practice standards, just as the clusters and domains provide higher order structure to the content standards.
K.OA.1-5
1.OA.1-8
2.OA.1, 2
3.OA.8
4.OA.3
K.NBT.1
1.NBT.2abc
1.NBT.4-6
2.NBT.1ab, 2.NBT.5-9
2.MD.5
3.NBT.1
3.NBT.2
4.NBT.1-4
5.NBT.1