What Were They Thinking?!
Lesson 8 of 9
Objective: SWBAT describe the various mental math strategies demonstrated by classmates and begin to understand the properties behind them.
This lesson focuses on the number properties, but doesn't teach them directly. I use a format called a Number Talk, sometimes called a Math Talk. This is a mental math routine that is really good at eliciting multiple methods for solving problems with single operations. I like using this format because it not only gives students "permission" to solve problems using strategies that are comprehensible to them, not just the standard algortithm. This helps build flexibility and fluency as they begin to see options and also connections between the different strategies. I use it here in order to elicit strategies that demonstrate the number properties. One element of Number Talks is that it makes visible the mental processes that usually remain hidden. Asking students to explain their reasoning is an important component of argumentation (MP3).
Students will enter and begin the Warmup which is displayed on the screen. Students are reminded of the strategies they used in recent lessons in which they added numbers mentally (Let's Talk Addition lesson earlier in this unit)
Choose the closest answer choice:
1. 209 + 1.19 (a) 200 (b) 210 (c) 319 (d) 300
Fill in the blanks:
2. 57 + 19 = 57 + 20 - ____
3. 57 + 19 = 60 + 19 - ____
5. 57 + 19 = 50 + 10 + ____ + ____
The first estimation problem tells me a little bit about my students' level of decimal sense. This has been a low point for students in previous years and the lesson for tomorrow involves mental math strategies with decimals (Delightful Decimals in the next lesson). The most common mistake I see students making in regards to problems like this are "lining the numbers up from the back" so that the 9 in the hundredths place in 1.19 is lined up with the 9 in the ones place in 209 and I suspect they will choose (c). I am hoping that many of them will estimate and choose either a or b, preferably b. As we go over these briefly in class I will ask students to share how they estimated in order to arrive at their answer choice. If no one offers an estimate I ask how they think a person may have estimated to get that answer choice and does the estimate make sense (MP3). In this way I hope to reinforce a little more sense of the value of a decimal number as they see that some estimates make sense and some don't.
If students are stuck with the last three there's a few different approaches I may take. I may ask them to work out the first expression and suggest that they try it mentally and see what they need to do to the other expression to make it equal. I may instead ask them to notice how the second equivalent expression has changed from the first and what additional changes we need to make to make it equivalent. When going over these in class I make sure to ask them how they chose to fill in the blanks as they did and what did they notice in the expression that made them choose those numbers. This will help them in their lesson today as they describe what people are doing with the numbers in their heads as they add them mentally.
In order to describe the mental math strategies being used and analyze why they might work we need to model some on the board. We started number talks in a previous lesson (Let's Talk Addition) in this unit. Students have been taught silent signals to show me where they are in their process. (Thumbs up means "I have a solution", forefinger to side means "I'm working on a strategy, but don't have a solution yet", finger and thumb means "I have two strategies"). When I see a lot of thumbs up I ask for solutions which I write down right or wrong and then I ask for strategies. I write down and model the strategies with the student's name and ask who used the same strategy before I ask for another. I do this to give value to their mathematical ideas. I try to model each numerically as well as with an open number line (MP4).
I start with 59 + 33, and I can also move on to 28 + 27 and 58 + 26 if I need to. My goal is to generate enough examples to be able to compare. I am looking for models that show the identity property of addition and also the commutative and associative properties. The identity property is usually demonstrated when students are trying to make one number a "freindlier" number by rounding it up, adding the other number, then backing it up the same amount was rounded up or by rounding up one number and removing that same amount from the other and adding two different numbers. The associative and commutative properties are usually demonstrated together when students break up the numbers into their place value parts and recombine them in another order. Most change the order and regroup them, a few just reorder and add them one at a time (using just the commutative). You may only need one problem, but many of my students are reticent until they have seen what other people think first, so it takes more than one problem. I also found that the closer I got to "midrange" numbers like 25 + 46, the more students tended to revert to counting up strategies, which I did not want modeled, so I avoided using those.
When I have enough models I go through and circle the ones that demonstrate the commutative/associative properties 0n the board and I ask students to take 1 silent minute to look at these methods and notice what these students were doing with the numbers in their heads. Then I ask them to share for two minutes in their group how they would describe what they notice and then we share out to the class. They may only notice that the original numbers are being "broken up" and miss the commutative and associative properties entirely. If this happens I go back and circle just the next part of the strategy that shows the reordering and regrouping of numbers and ask them to repeat the "noticing and sharing" process.
I ask now what that strategy might look like on (20 + 7) + (40 + 5) and ask a student to come up and show on the board or to walk me through the process of reordering and rearranging the numbers. I ask what it might look like on any four different numbers like (a + b) + (c + d). Most may be confused at the letters, but when I put a line underneath it and ask what the next step might look like, especially if I include the parentheses, suddenly some of them see what to do (MP2). Once one person suggests one possibility, others shoot up their hands with other possible regroupings.
Now I move on to the models showing the identity property of addition (the open number lines that show "adding over" then "backing it up" or adding and subtracting to adjust the problem as shown red and purple by Vidisha and Luis in the lesson image) I ask them to repeat the same "notice and share" process. When they share with the whole class I make sure they are specific about the number being added and subtracted. They must explain that it is the same number being subtracted that was added. I ask them why they think this strategy works. I ask now how they might describe using this strategy on any two unfriendly numbers. As they share their explanation I try to model with variables. a + b = a + x + b - x . This again is confusing for many, but I am just exposing them to the idea. To make them more comfortable with the expression I might ask "which number (a or b) are we adding to here to make it friendlier?" Most of them will see that x is being added to a, and then they start relating the letters to numbers. Then I circle the two exes and ask "why is the letter being added and the letter being subtracted the same?" Someone will usually come up with the x is representing the same number being added and subtracted. Then I may ask "if we add x and then subtract x, what are we really adding?"
I think it is so important for students to be able to generalize the relationships so that they conceive of the rule or property that they can then use in any situation.