Materials: Sold! A Mothematics Adventure by Nathan Zimelman, Illustrated by Bryn Barnard
This book is really a good one to get them thinking about all they have already learned about place value, estimating, adding and subtracting and gets them thinking about multi-step word problems. The estimating, adding, subtracting is sort of inferred and I had to bring it out so that it connected to them and what they have learned. I wanted them to settle into a mindset that connected their math with the real world and this is the perfect book for the goal!
I started the lesson by reading the book aloud and doing a "think aloud" as I read. This helped them catch on to the mathematical concepts in the story. I also stopped and asked questions to see if they could connect themselves to the book; had they ever been to an auction? Have they ever seen one in action? We talked about how auctions work, about different kinds of auctions. I connected myself to the book by sharing that I had just had been organizing items for one. I kept reading and made some predictions, and wondered if the painting he accidentally bought was worth anything more...and sure enough it was!
They begged me to read it again.
They gasped to think it was a VanGogh !
In order to get students really thinking about why we estimate and for me to assess where they are at in their understanding of the purpose for estimating, I started this lesson by referring back to the story and putting the following questions up on my Smart Board to get them to ponder the ideas that math serves us in everyday life. This exercises Math Practice Standard 4 that requires that proficient students apply their mathematical knowledge to solve problems in everyday life. I told them to pick three questions that they thought they would like to talk about in a discussion.
I told them to first jot their thoughts in their journal before discussing them because I wanted them to become aware of their math thinking through their own writing first. I showed them the Blooming Orange site and we discussed how this is model of thinking and learning. I wanted them to understand that it is not just about "doing" but applying their knowledge, like the boy did in the story.
*What can you tell me about estimation? Why do we do it?
*How can the boy use estimation to to help him figure out a total of what he spent?
*In the end, about how much did he receive for his painting?
*How can he figure out about how much money he has left after paying for his items?
*What do you know about estimation that could help you answer these questions?
After about 10 minutes of them jotting down ideas, we took time to share our thoughts. Several students picked the first question. They talked about how we estimate when someone comes to the house to do work on it or when they get a repair done on the car. This gives me an idea that they are paying some attention to math in everyday life, but it was very general.My hope is that they come to understand that estimation will aide them.
The answers I mainly heard were more about the process than about why we would estimate. No one mentioned that we estimate to make adding or subtracting easier. Two talked about "educated guesses." I suspect this response is a carry over from earlier exposure to estimation and rounding. They mentioned that the boy did it to make it easier for him, but couldn't really expound on why he would do that even with a calculator in hand.
These answers helped guide my next steps in instruction.
Using "Carry Over" Teaching: Back to Wisconsin & Learning to Estimate SB file
After briefly reviewing the Smart Board file from our first lesson on rounding, Population of Wisconsin, I asked my students to think about how the the boy figured out how much of his check for $28,000,000 was left. I was trying to connect a real world situation to a prior lesson and direct it into independent practice of real world situations.
One of my high end critical thinkers said that we would add up his purchases and subtract. This helped to lead the class into the line of thinking I wanted. I told them that we would start to practice this skill with smaller numbers because I thought 28 million was a little much to be working with while we are just starting to learn about estimation.
AT this point, I used the grouping strategy "Think Pair Share" to help with support in learning: This strategy works to gain confidence in critical thinking exercises. Students pair up, think over a problem and work together to solve. They then share their thinking with the class or with a larger group. They simply paired with the person next to them in their group, starting with one I chose, and working counter clockwise.
I started by writing this problem on the board.
I wanted to spend $39, $62 and $45 on clothing because it was on sale but, I had two hundred dollars I could spend.
The questions were separated to make it easier to see and pull apart for working. I added a "Think like this" because I know that my students still need to be told how to think. They are developing confidence in their critical thinking, but the majority are really weak. Prior to Common Core, they were used to just being shown and so I am guiding this along as we transition.
*Would I be spending more than $200?
Think like this: *How can I use estimation and subtraction to figure it out?
To model how I wanted them to think about taking the problem apart and solving, I created a KWS chart on the Smart Board Notebook and started a "think aloud". I divided the board in three sections and wrote this:
Know: I know I have $ 200. I know I want to spend $39,$62 & $45 on three clothing items.
Want: What do I want? Would I be spending over $200. ( I know I need a quick answer because I used logic to know that I didn't need an exact answer until I actually bought the items.)
Solve: Use estimation to figure it out quickly. Show rounding, addition and subtraction.
I set the information up for the students and I told students to "Think, Pair & Share" by talking to a partner and be ready to explain how and why you went about solving your problem.
When students were done, I asked them how many operations they used to solve the problem. I discovered through the discussion that they thought estimation is an operation, so I reminded them that + - is an operation and that rounding/estimation is a way to make numbers easier to add and subtract or use an operation to solve a problem.
*This short activity front loads multi-step word problems in the future. It is simple enough, it shows how estimation takes a role in finding a quicker answer, and helps them see that they need to always organize the information. This supports fluency. It supports development and understanding of the purpose of equations.
*I expected the majority of lower students who told me that they rounded to estimate, but then didn't know what to do with the $200.00. This helped me see how I needed to just focus on getting them to round the minuend and subtrahends to estimate and not worry about adding a second step for them. I needed to simplify steps for these students or we would not progress. The KWS chart needs to be continually referred to for these students and I need to teach them to focus on the question first. I plan to have them highlight their "W" part of their KWS charts.
Middle to high students had no trouble and the discussion evolved into them commenting that they needed to practice more story problems. I agreed!
I decided today, we would practice skills. The high end students had some fun in store with using real estate ads!
This independent practice section is complex because three groups are working at once. But, my class is used to working in separate groups on different tasks.
Prepare materials: Real Estate Ads from a Newspaper: Cut them out to stack. They could be laminated on note cards. I didn't laminate mine, but the paper worked out fine for stacking. Here is a little sample of one student's beginning work so you can see what this looks like.
I felt that they have had a good dose of guided critical thinking at this point in the lesson. I found that my students were ready to practice the skill of rounding, estimating and subtracting, but was fully aware that they were at distinctly different levels because of my observations of their pretests and the current classwork. I needed to differentiate now.
I pulled my lower and middle end students aside to work with them. As soon as I got them going, I knew I could start the higher end kids on the real estate game and then bounce back and forth between groups, helping those who needed it. (While I started my lower end students, my high end students worked on eSpark for a few minutes. This ap is a district application that is set up for individual students based on their MAP scores ( benchmark). So,this is a good time for students to work on any catch up work or iPad ap that keeps them thinking until you can instruct their activity.)
Middle to Lower Students: I gave them one on one instruction of how to round three digit numbers using the lead number. We reviewed a Learnzillion lesson that helped us correalate lead numbers with the numberline, reviewed benchmark and midpoints. This helps to reinforce that constant thinking about place value as it relates to estimation. I stopped the video in places to keep the idea that we were estimating and subtracting. So in frames where she talks about rounding the lead number, I would stop, write another number, have a student come up and round that one and then we would subtract. It's great! They really grasp it by combining all they have learned for this connection to subtracting.
RTI 4 Lower end students were assigned a simple set of 10 subtraction problems to estimate on graph paper. I simply wrote them on the board and had them transfer the to the graph paper. They underlined the lead digit and then subtracted. I kept checking back and saw that they were progressing well. I would soon start them on the top section of the worksheet to round larger numbers. At a later time, I will pull them out for help with the word problems instructing them step by step on how to solve them. This always takes extra time and I have to allow for that in instruction if standards are to be mastered. For now, it is only skill work for these students.
Middle students were assigned a worksheet Houghton Mifflin Expressions Estimation Worksheet for Middle Level Students that incorporated some word problems and reviewed addition estimation problems too in areas that they were struggling. It incorporates geography and has them do some written explanations. It supports Math Practice Standard 1 and 4, supporting perseverance and connection to real world problem solving. I quickly reviewed the word " about" and that it means to estimate. I instructed them to read carefully and decide if the word problem wanted the exact answer or the estimate. I wanted them to think " what word lets me know I need to estimate?"
This group of students have mastered rounding, but are not quite fluent enough yet to play the game below and I wanted them to practice more on reading word problems independently and applying rounding to estimation. I let them work in partners at their respective seats. The partner work helps support their thinking. This is all practice and any discussion between them will support thinking and reasoning as well as process. I am hoping they read carefully and I am looking to measure how well my middle level students can do that. From this worksheet, I will be able to tell how much support I will need to give them in multi-step word problems. Word problems tend to be a weakness with my fourth graders, but Common Core's focus will fix that!
* Middle and High Groups are working independently. Groups were working on an iPad math ap while I set up the independent group with instructions and got them going.
4 High Students were told they have become real estate investors because the market prices are down! They were given a budget of $5,000,000 and a real estate ads from newspapers. I cut the house ads apart made a stack to draw from. They took turns drawing off the pile, had to round the number to the nearest lead number and subtract.
They had to spend every dime by estimating the cost of the houses and properties and subtracting. This is a photo of how it looked as the game began: Example of using real estate ads for estimation . The first person to reach zero wins the game. If they go over their budget they had to discard and put that house/property face up so someone could use it. Then, they keep drawing until someones wins. If you have a tie, the person with the least amount of houses or land wins because they have had to work smarter to spend their money. I explained that smart investors try to get the most value for their money, and having too many houses is have several monetary downsides.
*They had a blast! I loved hearing their discussions and their amazement at some of the houses and prices! I had no problem going back and helping my other groups.
This game enriches their skill while connecting real world to math, making them aware of the cost of housing in our area. This activity supports Math Practice Standard 4 because I am having students become aware of large numbers in the real estate market in our area and then giving them a budget to spend as if they were an investor.
In closing with them, I asked them if they thought that $5,000,000 was enough money to challenge them with subtracting rounded numbers.
They responded that they had fun figuring out how to not go over budget because of the three and four million dollar mansions that they were so tempted to buy! I think they really got into this game! All of these activities support Math Practice Standard 1 in teaching them to make sense of problems and teaches them to solve them using strategies.