Adding Multi-digit Whole Numbers

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SWBAT use tape diagrams and place value understanding to fluently add multi-digit whole numbers.

Big Idea

In this lesson, students are introduced to a new model - a tape diagram - and use this model to add numbers.

Number Hook - Magic Trick

5 minutes

There are many interactive number magic tricks available on the internet.  If you click here, and then click on the Refrigerator Trick, this is the interactive magic trick I have ready to go as students enter the classroom today.

I particularly like this trick because it works similar to a card trick I will do in tomorrow's lesson. I am hopeful that my students will notice patterns between the two tricks. 

I know some will figure this trick out with time. 

Warm Up

10 minutes

I start this warm up with a fluency practice. Students practice the skill of rounding using their personal whiteboards.   This fluency reviews rounding skills that students are building as they continue towards mastery.  

For this fluency practice, I  shout out a number for students to round, ie. 68,254 to the nearest thousand, 169,458 to the nearest thousand, etc. 

I do this very quickly and energetically. One management strategy I use in order to keep students engaged and energetic is that I have students say a random word when they are ready.  For example, I have them shout "Twinkie," "rutabaga," "carnival," or anything that pops into my head.

This strategy really gets their attention and lets me know who's ready. For some teachers, this random word is distracting.  Since this is a fast paced, energetic warm up, it helps get the classroom climate revved up.  

Concept development

40 minutes

One potentially useful tool to help students visualize the relationship among quantities in a problem is a diagram. In many Japanese elementary mathematics textbooks, a linear model called a “tape diagram” is used.  Math Practice 4 states that Mathematically proficient students are able to identify important quantities in a practical situation and map their relationships using such tools and diagrams, two way tables, graphs, flowcharts, and formulas.

In this lesson, I introduce a tape diagram to students  as a tool that can help them visualize addition with large numbers and identify the unknown using a variable. Drawing models that represent concrete objects provides a bridge to help students connect their concrete representations to the abstract world of mathematical symbols.

I start this lesson by giving students a piece of cash register tape to illustrate  a tape diagram.  Students label the parts and the whole and then TAPE their tape diagram in their notebooks. 


I emphasize the word tape in hopes that my students remember this tool.

Then I model a tape diagram on the board.  I display 3,257 + 4,568 and use the tape diagram to show that each part equals the whole.  I tell students we can use a letter, or a variable to represent our unknown which is the whole in this problem. I ask students to add these two numbers. 

Students work on mastering standard 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm, when I ask them to then add the two addends (3,257 + 4,568).  

As students add the two addends, I circulate around the room and observe the various methods students are using.  At this point, most of the students will use new groups above, new groups below, or a subtotal strategy when adding. Each of these strategies were practiced in third grade for student using Houghton Mifflin's Math Expressions textbook.  For more information about these methods, click  here or check out the resource section. 

I ask 5 students to add the two numbers on the board for all to see.  This allows students to practice MP.8 as they look for similarities and differences among strategies and begin looking for generalized methods and or shortcuts.  Students who are still using a subtotal of sums strategy will likely benefit from seeing the new groups above or below strategy.  While the subtotal strategy is effective, it is not very efficient for adding large numbers. 

 Note:  I did not have any student using a subtotal of sums strategy. If you do have a student using this strategy it will benefit them seeing others' shortened strategies for adding large numbers.

Next, students work on the independent  practice page completing the first 6 problems and then ONLY page 1 of the word problems  page.  I remind students that the tape diagram is a way to visually represent or model the problem. 

Note: Some students did not want to use the tape diagram because they knew they were going to add the two numbers listed in the word problem.  I had students re-read the directions with me and note how the directions state that a tape diagram is part of solving this problem. I also referred students back to our math journal rubric scoring guide and reminded students about the importance of modeling. I also told students that using a tape diagram with "less challenging" problems is a great way to experience a new math tool.  Using a tape diagram will be more useful when students begin wrestling with multi-step word problems and additive compare problems in the near future.

Once most students are finished, I go over the answers to the problems.  I ask student volunteers to share their strategy on several of the problems.  I am very specific with naming the digit's correct place.  For example, in the first set 3,257 + 4,568, I correct students when they say, "I added the 6 and the 5 and I put a 1 down here and a 1 over(up) here."  

I am specific and explicit and remind students that the 5 represent 5 tens and the 6 are tens as well, which explains 11 tens, or 1 ten and 1 hundred.  At first this is somewhat confusing for my students.  This language and emphasis is new for them, but the more I use it, the easier it gets and they in turn start using this terminology too. 


Wrap Up

5 minutes

Students complete an exit ticket to wrap up this lesson today.  On a post-it note, students solve 23,548 + 9,473 and 45,029 + 16,254


I will use this exit ticket to design lessons for small groups of students needing more support, guidance or scaffolds.  For example, students not yet comfortable with the new groups above or below or subtotal strategy will be taught in a small group setting using proof drawings so they may visualize what is happening to the digits at a more concrete level before continuing with the abstract level.