The purpose of comparing multiple strategies is to use the student's invented method that is comprehensible to him/her in order to make sense of more sophisticated strategies. It helps students see connections between a strategy that makes sense to them and one that is new to them. It also exposes the mathematical foundation. Students then have the option or are directed to try out the new strategy.
On the white board is modeled the strategies 005.JPG that students are comparing in their warmup. There are four strategies modeled by four pretend students (Jenna, Adam, Cesar, and Yolanda). I tell students these are strategies that they have been using and seeing in class during our number talks ("Let's Talk Addition" and "What Were They Thinking?!" in previous lessons in this unit). I ask them to use the sentence frames for Comparing mental math strategies projected on the screen to help them look for multiple similarities and differences in the strategies. While they are working I periodically ask for a show of hands who has come up with more than one similarity so that they don't stop at just one.
I ask them to share their comparisons with their math family group and write down any additional comparisons they came up with as a result. Then I ask them to share with the class. I follow up with several questions including: "who else noticed that?", or "where do they see that?", or "does anyone have another way of saying that?", "why do you think he/she chose to do that?", "how does that help us solve it in our heads?".
Some sample responses may include that all of them used the open number line, they all got 73 as an answer, Jenna and Cesar both start with 30, Jenna & Adam both add 44, Jenna and Adam both took one away, Adam, Jenna, and Cesar all added one to 29, Cesar added in one jump on the number line and all the others used more than one jump. Most will notice that the strategies shown by Jenna, Adam, and Cesar are very similar because they add and subtract the same amount and that Yolanda's strategy does not do this. Most also notice that Yolanda's strategy involves more steps and may be harder to manage in your head. If they don't notice that all of them changed the numbers in some way to make it easier ask if any of them actually added 29 + 44. Once they do notice it, if they didn't say that they all changed at least one of the numbers to make it easier to add, ask them why they think the numbers were changed.
Ask them where the friendly numbers are in each of the strategies. In Adam's, Jenna's, and Cesar's it is 30, in Yolanda's they may see 20, 40, 60, 10. Then ask if "both" of the original numbers have to be made "friendly". You really want them to see that they only need to make one of the numbers friendly to make it easier to add mentally, so they can apply this to the decimal addition.
Number Talks were introduced to the class in a lesson earlier in this unit (Let's Talk Addition). The problem string for this number talk is:
4.9 + 4.1
6.8 + 3.2
5.7 + 2.3
6.8 + 3.3
3.7 + 5.5
I write 4.9 + 4.1 on the board and ask them to estimate first to help us predict a reasonable answer. I ask what a good estimate for 4.9 and 4.1 would be then we add the estimates to predict an answer near 9. Now I ask students to try either Jenna's, Adam's, or Cesar's strategy on this first problem by trying to make one of the numbers a friendlier number. When I see mostly thumbs up I ask for solutions and write them on the board right or wrong. Then I ask for their strategy. Since they were asked to use one of the first three strategies they may start by using 5 as a friendlier number than 4.9 and then adjust the result by removing the .1 (that was added onto the 4.9 to make it 5). Or they may use compensation and add .1 to the 4.9 and remove it from the 4.1, thus adding 5+4. I model the strategies on the board with their name.
Now I ask what it might have looked like using Yolanda's strategy and I take suggestions from the class and we work it out as a whole class. I don't want them trying the strategy on their own yet, because my students' decimal sense is usually so low that this is the one that confuses them. They are likely to get 8.10 instead of 9.
Next we try 6.8 + 3.2 first estimating to predict and then students are told they can use any mental strategy they like and I continue to model each strategy either with an open number line or with the unit bars in tenths (MP4). We continue with each of the problems in the string. During this number talk I try to look for students who are not engaging, because these are the ones who may shut down automatically when they see decimals. If I see anyone in particular I may go over and draw an open number line and start it for them and suggest they try to complete the strategy on paper. I try to walk away to find others, because I don't want to spend all my time with one student. This small intervention may be enough for some students. After one or two I will get a better sense of who will need more intensive help when we are working on homework.
Before distributing homework I ask them to begin with the problems that involve decimals first, which are problems 1, 7, and 10. I tell them I also have some blank diagrams of unit bars in tenths if they want to use them. If I have noticed any students particulary struggling with the decimal addition or any who were not participating I check in with them to see if they need some help. I ask which problem they are starting with and ask how they want to start. If they don't start with one of the decimal problems that's fine I wait until they work it out and then get them started on number 1, which involves decimal addition and ask first to estimate to predict a reasonable answer and then move on to a strategy. Even if they don't have much decimal sense they can usually say that 6.7 is closer to 7 than it is to 67, but if not then we skip this part and just model with the bars. Most of my students who need intensive remediation don't understand the difference between a numeral in front of the decimal and one behind, so we diagram with the bars. They can only understand the open number line strategy after they understand that 7 + 2.3 increases the whole number part by 2 instead of 3.If they are able to estimate accurately then I use this to help support the idea that 7 increases by 2.