Yesterday, I blogged about using a 120 chart for adding 2-digit numbers. Today, I’m going to put some names to those mental strategies and show how to record the strategies on an open number line (also called an empty number line).

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Two excellent resources for addition and subtraction strategies are Developing Number Knowledge (Wright, Ellemor-Collins, and Tabor), and Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction (Fosnot and Dolk).

The “red book” is a powerhouse. Incredibly user-friendly, it’s a step-by-step manual that will guide you in understanding mental strategies and provide you with instructional and assessment tasks to support your instruction.

The Young Mathematicians book is a great companion and includes more glimpses into the classroom. One thing to note, while the books describe similar strategies, they use different names, and the “red book” lists out the strategies in a very useful way. The “red book” also suggests using strategy names with your students, so I think what’s important is that you’re consistent. I think the Young Mathematicians strategy names are a little more kid-friendly and the “red book” names are more mathy. In my descriptions below, I am using the “red book” lingo with the Young Mathematicians’ names in parenthesis, where applicable.

For simplicity’s sake, I have copied the strategies from last night’s post and numbered them. The problem was 34 + 28.

- Start on the number 34. Jump down two rows to 54. Count on 8 more to get to 62.
- Start on the number 34. Jump down one ten to 44 and another ten to 54. Add 6 to get to 60. Add another 2 to get to 62.
- Start on the number 34. Jump down three rows to 64. Come back 2 to 62.
- Start on the number 34. Move ahead 6 o get to 40. Jump down two rows to get to 60. Move another 2 to get to 62.
- Start on the number 34. Count on 8 more to get to 42. Jump down two rows to get to 62.

Here are the strategies represented on open number lines.

Numbers 1 and 2 are both versions of the jump (making jumps of ten) strategy. They involve starting on one number, making jumps of 10, and then making jumps of 1. Number 2 is more efficient since Number 1 involves counting on by ones, while strategy Number 2 decomposed the 8 ones into 6 and 2–6 to get to the next decuple (“friendly” number) and then 2 more.

Number 3 is called over-jump. According to Wright, this strategy is good for numbers with 7, 8, or 9 in the ones place. Basically, this strategy involves rounding up to the next decuple, and then coming back to compensate for the rounding. So in the example, 28 was rounded up to 30, with a jump back of 2.

Number 4 is called jump to the decuple. It’s sort of the inverse of the jump strategy. Instead of jumping the tens first, you jump enough ones to get to the next decuple, jump the tens, and then jump the remaining ones.

Number 5 isn’t a strategy described in the book. I think it’s important to note that some kiddos may come up with their own strategies.

The “red book” describes seven mental computation strategies, and I have only listed three here. They are the ones that best describe the strategies students might use when adding on a 120 chart. Remember that you are going to record the kiddos’ strategies, so it’s important that you think about likely strategies and practice drawing the representations.

I don’t know about you, but I SO wish I could have learned math this way!! It’s so much more engaging and makes so much more sense. Enjoy experimenting with these strategies!

It’s funny you mention the Fosnot book! This past summer, we received a training on using Contexts for Learning with Every Day Math and this book was given to us since Twomey-Fosnot helped develop Contexts for Learning with our trainer. What do you think of it? I haven’t read it yet.

I have not read it cover to cover yet. I tend to read books in bits and pieces. It’s very interesting to contrast it with the “red book”, because there’s a lot more verbiage in the Fosnot book, and I like the classroom vignettes. But the “red book”, I think, is more like a user’s manual, if that makes sense. It’s a step-by-step road map.

Donna, I totally appreciate the ideas you’ve presented. I think I have to buy both of these books. I teach second grade now and I’ve finally broken down this week and allowed a few students to use the 100 grid to help them with computation since they just can’t learn the addition facts. However, I taught 6th grade for 4 years and I’m concerned because I had so many 6th graders that just were so dependent on the 100 grid to add and subtract and didn’t know those math facts…I really didn’t want my second graders to use one. I get that the students that are struggling are probably lacking number sense, but what can I do to help them so they can move forward with confidence in math?

I get your concerns, Deb! I think we need to teach facts in the right way. When I learned facts, it was totally memorization. I really struggled with facts like 9 + 5. If taught right, kids now say, “Oh, well 10 + 5 is 15 and 9 + 5 is just one less, so it’s 14.” They also are learning that 6 + 7 is just one more than 6 + 6, a double fact they know. Kids now are learning how to subitize in Kindergarten, so they see 3 dots and 2 dots and know that makes 5 dots–the foundation of basic facts. They are using ten-frames to learn how to make ten from any given number. It really is a different world! When you get a chance to look at the books, you’ll see how all the pieces fit together! If you can only get one, I’d suggest the “red book”.

I just found you through PInterest. I’m excited to follow you. My state (Idaho) has been making a big shift in math instruction in the past three or so years. It’s been so wonderful to see what kids can actually achieve in math. They’re rising to the occasion and we teachers learn how to raise the bar. The Fosnot book is on my list, but I’m unfamiliar with the red one. Thank you for the recommendation.

❀ Tammy

Forever in FirstYou’re welcome, Tammy! The red one is brand new, and it’s good. 🙂

I absolutely LOVE all your ideas. They are so practical! I can use them exactly the way you so clearly explain – what a joy! Thank you!!

My pleasure, Linda! It’s always nice to hear that my ideas are useful. 🙂