Content Objectives
This unit will take approximately three weeks to teach. I will present it during the first quarter, and it will share time with other classwork. The first week will focus on the concept of place value and how the base ten system is an efficient way to represent whole numbers. As a pre-assessment activity, I will give students a target number, 467, and having them think of all the ways they can represent that number. They may create math equations using various operations or properties (e.g. 467 x 1), they may decompose the number into smaller addends (400 + 60 + 7), or possibly even create a graph to show their understanding of that target number. Additional work will center on the Oklahoma fourth grade standard (4.N.2.7), “Comparing and ordering whole numbers using place value, a number line, and models such as grids or base 10 blocks.” I do not anticipate much misconception during this initial week as it should be review of third grade topics. Confusion may arise if I use a larger number moving into the thousands and ask them to repeat the task. This will not be a review of previous work. I will stretch their thinking by helping them grasp the concept that the largest place where the numbers differ, determines which is greater. I will ask them to compare numbers with 4, 5, and 6 digits.
The second week will be focused on the texts, Great Estimations and Greater Estimations, as students move towards using very large whole numbers and plotting these big numbers on a number line. An anchor activity will be to have jelly beans in cups of various sizes (1/4 cup, ½ cup, 1 cup, and 2 ½ cups), have students estimate the number of jelly beans in each cup starting with the smallest cup first. Then use their knowledge of the base ten system to estimate how many jelly beans will be in a cup that is twice as big, four times as big, or ten times as big. Students will count the actual number of jelly beans in each cup. I will use class discussion to compare and contrast estimations and actual counts. Similar scenarios will be used in classroom demonstrations and the estimates will assuredly become more precise and accurate. Activities like this will address Oklahoma math standards in both fourth (4.N.1.5.) and fifth grade (5.N.1.4.), “Solving multi-step real-world problems requiring the use of different operations with whole numbers.” Students will be writing and illustrating in their math notebook daily to cement their thinking and reasoning about the relationships between our number system and effective estimations. The writing in their notebooks may be numerical, pictorial or narrative forms of their learning. The use of writing or drawing is encouraging students to see the mathematical ideas. As Jo Boaler states in What’s Math Got to Do With It, “Mathematicians draw all the time.”(5) My use of the notebooks will address the fourth and fifth grades standard for language arts of listening to others’ ideas and asking and answering questions to clarify meaning. By writing it down, sketching it out, or creating a similar math problem, they are participating in the productive struggle of mathematics.
The third week will focus on the connection between estimating and determining placement of large multi-digit numbers accurately on a linear number line. The primary focus text will be Millions, Billions and Trillions, Understanding Big Numbers by David Adler. A number line around the school that will depict the orders of magnitude by showing the relative space between numbers will be their culminating project. This number line will begin at my classroom door and the initial tick marks will be determined by using a paper image of base ten blocks cut apart and placed end to end, thus establishing the mark for ten and the mark for one hundred. Students will use these markings to then estimate where numbers such as one thousand, ten thousand, and possibly one hundred thousand will be located on the number line. As we continue through the unit, students will be adding to the length of the number line by establishing the possible position of one million and one billion. Students will also be placing individually researched quantities, such as the lifetime earnings of differing vocations or how many seconds old they are, correctly on this number line to show their use of integers in real-world situations. Additional number lines will be added to our classroom where one represents one million and the other will represent one billion. Start by subdividing into ten equal subintervals, and discussing what the tick marks would represent. Then subdivide one of the subintervals into 10 equal smaller intervals, and discuss what the tick marks there would represent. I will propose some meaningful (populations, expenditures) multi-digit numbers that will fit in the interval you have just discussed, and ask my students to locate it. Hopefully, they will see that after the 3rd digit, or maybe the 4th, the intervals have gotten too short for them to distinguish further digits. These additional number line representations will allow students to see that the first few digits determine almost all of the location no matter the length of the number line. These activities will reflect the Oklahoma standards from fourth (4.N.2.7) and sixth grade (6.N.1.1) that require them to use a number line to represent integers and recognize the concept of magnitude. The order of magnitude will definitely be a new way to discuss numbers. I anticipate some misunderstandings around this topic and so the classroom activities will address the possible misconceptions. Students have not had previous exposure to order of magnitude and thus will have a gap in their grasp of the scale of numbers.
Even though fourth graders should have already mastered place value and how to estimate for small numbers, this unit will focus on creating a deeper conceptual knowledge of both skills. Also a significant emphasis will be placed upon the concept of the order of magnitude. To that end the video The Powers of 10 will expose them to much larger numbers than they have been used to, and will be the initial introduction to using exponents for the base ten units, which will appear again when we are writing larger numbers in expanded form. Class discussion will center on then using information from the video to write large numbers using scientific notation. (E.g. 1,000 can be written as 1 x 10³.) A strong robust foundation in number sense will be required for them to continue to learn mathematics at a more conceptual level. This unit is designed to have students take a deep “dive” into using the above mentioned skills and fully develop their number sense and ability to work with numbers in a strategic and flexible manner as well as to make a direct connection to their daily lives.
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