# Constant Difference meets Broken Ruler

One of my biggest struggles in teaching math has to be in teaching the big picture. There are so many math connections to be made, but I don’t even see them all yet, so I struggle to help students see those connections too.

Recently, I was observing a subtraction number talk, and it brought about so many questions that I brought back to my amazing team of Math peeps. We were having all sorts of conversations about making connections across the grade levels for subtraction. That’s when the wheels started turning.

I read portions of Making Number Talks Matter and Powerful Numeracy. One example asked the reader to use a constant difference strategy. I was really struggling. The more I added to one number, the less it helped. I was trying so hard to make the digits in each place match up that I realized I didn’t really understand the strategy at all. It was only through listening to my colleague think aloud about her way of solving it that the idea of a method finally clicked for me (shameless plug for more math discourse!) Then, I started to think about what I’d read/heard about the idea of the broken ruler, and how I’d seen it used in the primary grades! It seemed like the perfect math connection for me to make: using a ruler partitioned into fractional parts to make sense of subtracting fractions with a constant difference.

For the lesson:

I printed out rulers divided into eights for each student. I handed them out to the students and I asked the best math question ever, “What do you Notice?” Here’s the list I got! I was so thrilled! They were already starting to see where I was headed with fractions!

After I recorded all the noticings, I asked if students had any questions. They had some questions and comments about other’s thinking, but it started to get away from math, so I asked a question. I underlined the statement in red and asked what the student meant by this. That was a fun discussion and proved to me that the students had really noticed a pattern with the idea of lines and spaces creating fractional parts. Now that the majority of students were seeing that the ruler was broken into eighths, it was time to make connections!

I challenged them to figure out the length of a “pencil” (it was printed on paper so all students had the same “pencil”), without beginning at zero. I wrote the number 6  3/8 on a poster. I asked the students to put the pencil’s eraser at 6  3/8 inches and told them to point the tip toward the zero. I had two questions. (1)What number did the tip of the pencil land on? (2)How long is the pencil?

It was a great place to start. Most students had access and could use the numbers to find the length. They told me the tip landed at 3 inches and the pencil was 3  3/8 inches long. I recorded some strategies to explain how they figured out how long it was, then I moved to the next one.

I wrote 5 1/4 on the board and asked them to do the same. Put the eraser at 5 1/4 and the tip pointing toward zero and answer both questions again. Surprisingly, there were only a couple of students who realized before measuring that the pencil would be the same length. So, we answered both questions again and students proved the length with strategies they used.

Finally, I asked students to do the same starting at 5. The majority of students this time were catching on that it would be the same length, but I still asked them to convince me as I walked around.

When we came back together, one student said he found the length of the ruler by subtracting 5 – 1  5/8, and a student said, “What? How did you subtract?” (Hello disequilibrium!)  This is exactly what I was hoping for! We started to have great discussion around whether or not this was subtraction, and once students agreed that we could subtract, I added the subtraction and equals symbols to out poster.

I was visiting a classroom that wasn’t my own, so I had to wrap up my lesson here. I introduced the teacher to this amazing animation by Steve Wyborney to get them talking more about difference and the broken ruler. It was a fun lesson and got our K-5 team talking about building greater understanding of progressions and connections K-5 instead of hyper-focusing teachers on their standards alone.

What do you think? How would you use it? Where would you go next?

# Quick Images Number String

I walked into a 5th grade classroom a couple weeks ago, ready to show a brand new teacher how to do a number talk. Little did I know, her kid’s would be helping me to learn that day!

I attempted to facilitate a multiplication number string (2 x 7, 4 x 7, 4 x 8) and noticed that a few students were successful, but most lacked either the vocabulary or the conceptual understanding to explain what was happening in the multiplication number string and why the facts were related. As I struggled through the number talk, I showed a visual relating the last two, I tried to ask the best math question I know, “What do you notice?” and I asked students to compare and contrast the problems and the pictures I had made, but still I did not see a lot of “Aha”s. I was crushed and wished that I hadn’t demonstrated such an unsuccessful number talk for this brand new teacher. As I left, I contemplated what I could do to make it better. I knew I couldn’t leave those students like that, and I knew it was important for the teacher to see how successful a number talk can be. I had spent the morning watching a colleague of mine, and listening to TKers and Kinders explain dot cards. Then I remembered this blog post about multiplication subitizing cards from Graham Fletcher, and it just seemed to click. I decided to do a number string with dot cards, and the lesson was inspiring!

I created the string found on this PowerPoint, utilizing the subitizing cards from Graham Fletcher and a PowerPoint I found 3 years ago, which I can no longer find to cite. (If you know where it comes from, please let me know so I can give proper credit).

This time, the number talk was exactly what we want. Kids were engaged, all students had access and were participating. Even those who struggled with language could use the visual of the dot cards to help them explain. I had students using the words “groups of” and truly explaining a conceptual understanding of multiplication as a number repeated multiple times. Some were using repeated addition, others were using the 1st fact (2×4) two times to solve 4×4. The best part was on the final card (7×4), they were using the distributive property and partial products to explain the number of dots on the card. They were using a previous dot card (5 x 4) to help them explain and it all gave a great opening to take it back to the number sentences. I saw an instant connection between what these 5th graders were doing to explain the visual patterns in the dot cards and what the Kinders did earlier in the week with dot cards. The quiet, unsure class transformed into a talkative class, asking for more. Thanks to those who inspired this lesson and continue to share what they know with the rest of us!