When I first read through the sample problems illustrating the different variation of addition and subtraction problems (Table 1 in the Appendix of the Mathematics CCSS), it struck me that the majority of these problems do not contain the "key" words that we teach students to look for when solving a word problem. I also noticed that several of the problems expressed the quantities in words instead of digits. Coincidence?... I don't think so.

Consider the following two examples from Table 1:

1) Julie has

2) Julie has

If a student attacked these problems using key words, they would think that they should perform the same operation to solve both problems. However, this is not the case. In the first problem, Lucy's two apples plus the three more that Julie has gives you Julie's number of apples. In the second problem subtracting the three more that Julie has from Julie's five apples gives you Lucy's number of apples. To solve the problem the students must actually be able to make sense of how these numbers are related.

So how do we teach students to make sense of problems so they can persevere in solving them? Tape diagrams can be a great tool for representing these relationships (part/whole or comparison). In the attached document, you will see each of these problem types modeled with a tape diagram. The tape diagram is not an end in itself. It is a tool to help students see that while there are many different variations in wording, there are only two basic relationships in their addition and subtraction word problems (part/whole and comparison). If a student can picture these relationships, they will also have a great foundation from which we can build their understanding of multiplication and division word problems. The models will highlight the fact that multiplication and division involve equal parts. The tape diagrams also sets the stage for understanding the difference between additive comparison (3 more) and multiplicative comparison (3 times as many).

The file below shows each problem within Table 1 modeled with a tape diagram.

table_1_tape_diagrams.docx |

After trimming each section at a point that made it easy to line up, I taped the pieces together. If you have an easier process or an inexpensive source of meter strips, please comment and share your suggestions. When you encounter a "high prep" lesson in the modules, consider what you have that you can just repurpose (possibly just demonstrate whole class with a meter stick). If you taught it whole class, could your students practice with this task in a small group at a center? A room-full of kids wielding meter sticks might get crazy, but I would think a couple kids could sit next to a meter stick on the floor and practice the subtracting a ten task. This way the kids can have the experience without you having to make or purchase a class set of the materials.

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place_value_cards_1-999_extra_label.docx |

A number path is a tool that young children (up to 2nd grade) can use to model addition, subtraction and comparison problems. You can find an explanation of a number path and a template you can print at this site

http://www.mathematicallyminded.com/downloads/1-20%20Number%20Path.pdf

This model provides a bridge between concrete manipulatives and abstract numeric symbols. If you laminated the number paths, the students could mark out problems on their number path and then erase it. For comparison problems it may be useful to have two number paths so students can easily see how the two quantities compare. Use this model to lay the foundation for the use of the bar model and the number line to solve problems with a larger range of numbers in the later grades.

The link above takes you directly to the information on number paths, but the Mathematically Minded website has a number of printable resources that are worth checking out. http://www.mathematicallyminded.com/book1.html

]]>http://www.mathematicallyminded.com/downloads/1-20%20Number%20Path.pdf

This model provides a bridge between concrete manipulatives and abstract numeric symbols. If you laminated the number paths, the students could mark out problems on their number path and then erase it. For comparison problems it may be useful to have two number paths so students can easily see how the two quantities compare. Use this model to lay the foundation for the use of the bar model and the number line to solve problems with a larger range of numbers in the later grades.

The link above takes you directly to the information on number paths, but the Mathematically Minded website has a number of printable resources that are worth checking out. http://www.mathematicallyminded.com/book1.html

useful when you need to make sense of the given information and figure out how to attack a complex problem (MP1). Once students know how to create and apply these models, all they need is a surface to write on and a writing utensil.

This model shows how the part and whole are related through multiplication and division. I could show that if I had 15 stickers and I divided them evenly among 3 kids, each kid would get 5 stickers. Later we can use this diagram to model fraction relationships. This picture shows that the part is 1/3 of the whole.

So what can you do with these basic models?

I haven't decided what I think about using computer or iPads to create bar models (Erin). I love the simplicity of just being able to draw a quick sketch anytime anywhere. The best argument for using a program to create a bar model is a situation that requires you to split a bar into equal pieces. You'll see how easily that was done in the video.

The Youtube video below shows a solution modeled using the "Modeling Tool" on the Thinking Blocks website.

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The rekenrek (rek- en- rek) is a tool that was developed in the Netherlands to support the development of number sense in young children. The picture shows a homemade rekenrek that I created with string, beads, and the bottom of a shoebox. This rekenrek has 20 beads. Much of your work with students in grades PK-2 could be done with a rekenrek of this size. You will also see larger versions that have 10 rows of beads to work on numbers within 100. The larger version is great for counting patterns within 100. The larger version would also be useful for 3rd grade teachers because you can easily create arrays on the rekenrek that model multiplication.

If you are looking for a rekenrek app for the iPad, I like the one called

"Number Rack" You can adjust how many strings of beads you have showing up to 10 strings. The movement of the beads is pretty similar to the actual rekenrek. It's nice that I can pull beads to the right at the same time I'm sliding beads to the left. It also lets me slide a whole group of beads over at once. It has a screen, but I can't seem to be able to move the beads under the screen like I could with a real rekenrek. However, you could use the screen to decompose a number. If 3 beads are showing, how many beads in the string of 10 are hidden under the screen, etc.

__http://itunes.apple.com/us/app/number-rack-by-math-learning/id496057949?mt=8__

Want to see the rekenrek in action? Check out this video on Youtube created by Bill Davidson. He was one of the presenters that came to Albany to share information about the upcoming math modules. In less than 5 minutes you can see many ways to use a 20 bead rekenrek... enjoy :)

"Number Rack" You can adjust how many strings of beads you have showing up to 10 strings. The movement of the beads is pretty similar to the actual rekenrek. It's nice that I can pull beads to the right at the same time I'm sliding beads to the left. It also lets me slide a whole group of beads over at once. It has a screen, but I can't seem to be able to move the beads under the screen like I could with a real rekenrek. However, you could use the screen to decompose a number. If 3 beads are showing, how many beads in the string of 10 are hidden under the screen, etc.

Want to see the rekenrek in action? Check out this video on Youtube created by Bill Davidson. He was one of the presenters that came to Albany to share information about the upcoming math modules. In less than 5 minutes you can see many ways to use a 20 bead rekenrek... enjoy :)

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