Text Selection
Even though we will be working on math concepts, I have selected three main texts to use during this unit. These texts are similar to each other in structure, and the mathematical theories in each are pedagogically sound. The bold graphics and illustrations will appeal to the visual learners and each text has opportunities for students to engage with it interactively.
The first text is Great Estimations by Bruce Goldstone. The text introduces estimating in a fun way with colorful found objects illustrations. Students usually follow the “Goldilocks pattern” when estimating: either too small, too large or just right. However, they struggle when trying to determine if their estimation is reasonable. This picture could be used to introduce how to cluster items to estimate the total number of bears in this picture. If you cluster all the orange bears together they represent ten bears. Students can then use this number to estimate the total number of bears, regardless of color, in the figure 1.
Figure 1
The text starts out using groupings of items arranged in a myriad of ways to work on the technique known as eye training. This technique trains your eye to look for groupings of smaller numbers and then adding those small groupings together to arrive at a reasonable estimate. Another technique used to facilitate estimating reasonably is termed clump counting. It requires students to count a specific number of items in the picture, like 10, and then notice the amount of space required for that clump. Then use that clump to estimate the total number by multiplying by ten or by one hundred to arrive at an estimate. This strategy reinforces the powers of ten to support a student in estimating appropriately. Clump counting allows students to choose numbers that are easy to work with to find a close estimate. The final strategy introduced is box and count. If the picture has lots of things spread out then students can draw, or imagine, boxes overlaid on the image to divide it into manageable pieces. Once the boxes are in place, each individual box is given an estimated number and then the total number is estimated by multiplying the small estimate by the number of boxes overlaid on the picture. My example is if I draw 100 boxes on a graphic, then estimate the number of items in one box, I then multiply the single box estimate by one hundred to arrive at an estimate for the total number of things pictured. (Or written mathematically: total = 100 ´ X, where X is the estimate from one box).
The second book I will be using is Greater Estimations by Bruce Goldstone. Greater Estimations uses the same three techniques previously described, but it also moves into estimating volumes and measurements. The most significant differences in the two texts are the variance in numbers. The first text uses numbers reaching into the hundreds of thousands. This text focuses on really big numbers, millions and billions. An example of a picture representing possible blades of grass which may be estimated in the billions is Figure 2.
Figure 2
Students would need to isolate a small portion of the picture using the box and count method, estimate the number of blades in that portion and then use information about the size of the picture to estimate a total number of blades of grass. This requires working with very large numbers and the use of area to calculate the approximate number of blades of grass. Both texts reinforce the five stages of place value by encouraging students to also use exponents when working with really big numbers. The author’s letter at the end of this text highlights an important point for students--that estimating is an everyday activity. Estimating has implications vocationally, politically and socially. It affects how parents prepare meals, how planners set up public events, and how governments allocate funds. This direct connection will enable students to see the correlation between math and real world application.
The last text will be Millions, Billions, and Trillions, Understanding Big Numbers by David Adler. The illustrations make these numbers accessible for students. The problems presented show how these large numbers are used in our everyday life. An example of how a picture could be used to assist students in developing their understanding of these very large numbers is provided beneath figure 3.
Figure 3
This picture is a herd of sheep. Students can use their estimating skills to find out how many are in this picture; however, if the question then becomes how many sheep are located in the United States, then students must do some further calculations. This would require some research and then a knowledge of using multiplication using the orders of magnitude to provide an approximation of the total number of sheep in the United States. An extension of this problem would be to calculate how many sheep would be in a square mile or how many per acre? This example would provide a chance to discuss how many significant digits should we keep in a large number? Do we need to keep all of them or would rounding to the two or three significant digits be a reasonable estimation? This discussion will also be informed by the previous number line activities.
The text is presented in a logical sequence to assist students in arriving at the understanding that each of these numbers is based on the powers of ten concept. The examples of where these large numbers are useful also provide chances for students to delve deeper into realization of how immense these numbers are. A misconception I believe students will have is how these very large numbers make a direct connection to their daily life. So we would proceed to work in small groups to answer the following questions: How many minutes old are you? How many million seconds old are you? How many billion? How many billion seconds old are your parents? I believe their first estimates would be significantly too small. This would then lead to that “productive struggle” when students would need to reason through the steps to arrive at a reasonable approximation of their ages in minutes and seconds. Students will also talk about how many people are in the school, and in Tulsa, and in Oklahoma, and in the US, and compare these numbers, and perhaps also compare US or Oklahoma population to the whole world.
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