Prep: Have a personal story prepared to tell the students about an experience, funny or otherwise, about estimation.
Personal Connection: Story Telling in Math
I opened this lesson with a model of using a personal story with why estimation saved me $65.00. I have learned over the years that if I can make some connection in my life, students love it. This supports MP 4 practice standard of modeling with mathematics.
The $65 Corned Beef Story
A few years ago, I went to the grocery store to do my weekly shopping. I always have a grocery budget and I know how much I should spend, so when I go, I round each grocery item and add it in my head as it goes in the basket. I had stopped at the deli and bought 1lb of corned beef for lunch meat @ $6.50 per lb. While the clerk scanned each item, I heard the beep of the register and added my estimated numbers in my head. My total came up and I paid her. I thought to myself " Gee, that's about $50.00 off. Hmm...I must have been wrong." So, I checked the cash register tape when I pulled my cart away from the counter. I had been charged $65.00 for the corned beef! Someone had programmed the scale incorrectly. I learned that technology is only as good as the person is running it and that my estimation skills were essential in catching that mistake. I wasn't wrong!
I used this story as a base to build number sense by asking a series of questions.
*I asked my students what $6.50 and $65.00 had in common? One student responded saying that $65 was a lot larger than $ 6.50. Another said that it was ten times as much. From their response about place value and value, I could tell that there was some number sense in my class about money.
*What did the person who programmed the scale do? A student said that she pushed the wrong buttons.
*How does that impact the math? ( How did it make my grocery bill go up?) Another student said that because she pushed the wrong buttons, I was the one who suffered from the mistake. My beef "costed" to much.
*How was it important to use estimation in my example to avoid paying too much for groceries when you only have so much money to spend? We discussed this together and talked about how their parents use estimation. They talked about getting roof estimates because their parents only had so much to spend. This indicates that some are paying attention to money, budgets and estimation in the real world I am hoping they connect it to their math today. I like to develop questioning that leads the student to realize that estimation has an impact on their lives helps them see that logic is part of the function of math in our lives.
*Why is technology sometimes not dependable? I told them that we have to be smarter than our technology! That we have to be more math savvy than the technology we use.
I could see they were ready to talk about more estimation and so we returned to our Wisconsin Rounding Smart Board lesson.
I closed this section of the lesson by asking: Do you think you are ready to learn this important life skill? You need to master estimation by using rounding skills you have already learned.
Prep: Prepare a Smart Board Notebook File Regarding Your State's Population prior to 2014, preferably from up to 10 years ago.
To help my students understand better how estimating sums and differences helps us understand data, I turned on my Back to Wisconsin & Learning to Estimate. This topic is overlapping with some map studies we are doing in social studies. I used their student Atlases to build this SB file.
Transferring the skill of rounding to actual real world application is the goal of this section of the lesson. It is an example of practicing MP2 as we make sense of quantities and use that to achieve a mathematical goal. We need to have mastered 4.NBT.A.3 fairly well first to understand this. I sent the above grade level achieving students off with atlases and an iPad and asked them to find the populations using the atlas and internet of 3 or more different states from 2003 and 2013 . I wanted them to tell me which state had grown the most in ten years. *These students showed me mastery on a pretest in rounding prior to this lesson and so it was important to get them busy applying their skills and knowledge right away, rather than making them sit through more of the lesson.
I worked with my at and below grade level students in front of the Smart Board. The next steps for these students involved teaching them to round to the highest place value or lead number. I began with the Rounding Smart Lesson in the Hundreds Place. We reviewed slide 3 of Rounding Review and reviewed the Rounding song together. I started with hundreds place to teach rounding the lead number because it is so easy to do with smaller numbers first. Rounding and Estimating shows how the circle reveals the correct answer.
After the 100's place exercise, I had them practice rounding 2 hundreds place numbers in a standard alogrithm in their notebooks (rounding to the lead number) and then add. I heard voices singing the rhyme. They under lined the lead number and rounded. I could see four of them struggling with rounding. One student rounded the answer. One student rounded the estimation! Next: We practiced rounding a thousands place algorithm together and then kept moving up through hundred thousands. This gradual practice seemed to get smoother with each step.
Leading them forward: To get them caught up with the other students, I suggested we look up another state and find its population both in the atlas and for 2013. We looked up California and estimated the difference (together) ten years had made on the population. They were amazed at the population differences between Wisconsin and California. I was glad to see they were thinking about the vast size difference on their own and making sense of numbers. I asked them to add the total estimated population of Wisconsin and California together using the lead number to round. I checked notebooks to evaluate. I had to still support them in their thinking and rounding process.
I checked on the above grade level achieving group who were working well independently in partners. Two students asked to look up more. They were engaged. (They can work on their own quite well. I attribute this independence to the community we have established through TRIBES. I think this independence is essential to effectively reach all levels and master standards)
*This research activity also practices the short research and informational text reading standards in LA. I like CCSS because it allows me to teach standards across the curriculum.
I predicted that many of my students after the lesson will revert to the common mistake of adding and rounding the sums. I referred back to the Smart Board Lesson to ward off the temptation. I took about 5 minutes to discuss and talk about good math sense and why we need to think about why we are rounding first and then adding or subtracting. I referred back "corned beef" story again.
I questioned: What did I do to quickly add up my grocery tape? How did I discover my mistake? Would it have made sense to add the tape up exactly and then round the answer? Why?
Two students took turns in a discussion of why I would not want to add up the exact amount. It would take me forever! And they pointed out that the idea was to make it easier. They concluded that estimation makes math easier!
I continued by asking: Can we think of other examples of real life situations where it doesn't make sense to add the numbers and round the answers? This supports MP1 by working to make sense of problems. One student shared about going to McDonald's and estimating the bill after his order, knowing he had $20.00 to spend.
I asked students to return to their desk to work independently in their notebook. I gave them each three dice with instructions that were differentiated according to my understanding of ability from their pretest and from my class observation during the lesson.
Below Grade Level Achieving Students: I gave the 1 cm. graph paper because some cannot keep numbers lined up. Instructions: *Roll three dice and create the largest number they can. Roll again and do it again. Create a standard algorithm. Round the three digit addends to estimate, rounding the lead number. Add. Write a sentence at the end of the exercise explaining why you round the addends first and not the answer. This helps them think about the common mistake.
At Grade Level Achieving Students: They got 5 dice. Instructions: Roll 5 dice and create the largest number. Roll again, and round the addends to add in a standard algorithm . Ask them to add the exact number to practice addition skills. I asked them to compare the two numbers. Did the estimate make sense? These students really honed their estimation skills and had to think about whether their answer made sense. I asked each of them if they could see how estimation can be used to see if their exact answer was correct. I think they will need more practice at understanding that.
Above Grade Level Achieving Students: I had them take the populations of the states they had been researching and round the total amount of people in those states using the lead number. They had to know then to subtract to find the total estimated growth. I like how this sets the up for thinking about multi step word problems later.
No iPad? I would suggest this to challenge them because it gives them a real life scenario: Give them a budget of a $300,000 and car ads from a newspaper and tell them they can buy as many cars as they can and not to exceed the amount given. Real Estate ads are great for this too but you will have to increase the budget! Grocery Store ads work well, but now you are working with decimals to round. If they are ready, this works well too! But, at first, I stick with the whole numbers. (I keep newspaper ads on hand for all kinds of math exercises.)
I assigned them to research University of Wisconsin Madison (43,275) and University of Wisconsin LaCrosse's ( 10,427) population of students for 2014. They were to estimate about how many students are in both universities. I assigned this to all levels of students to challenge the mid and lower groups. I am certain, after the high group worked with large numbers today, that this will be very easy for them.
The next part of the assignment was simply 5 problems in the addition section B.1 of 4th grade IXL.com with varying place values for them to find just the estimated amounts. They had to add the exact number to answer and then show me the work on their paper for estimation. I could have assigned a worksheet, but they like using IXL and they need to practice the exact answers too to be fluent. I felt that the skill practice was important. Students who have trouble lining problems up had graph paper.