Activities
Assessment
In order to gauge how well my students grasp the concepts addressed in the unit, I will administer a pre test. This will be a written test and will include a sampling of prompts that I will use throughout the unit. They will require students to offer written responses. The pretest data will allow me to determine the most common misconceptions and the range of ability levels. This information will be important when planning discussion groups and the amount of time that I should plan to spend on each concept. This same assessment will be administered at the end of the unit in order to gauge which concepts need to be revisited to reach mastery. It is important to note again that a sufficient mastery of concepts addressed in the preceding unit must be reached before moving on to this unit.
In order to raise the stakes, hold students accountable and gauge student understanding throughout the unit I will give weekly concept checks as formative assessment. These will be short checks that require students to demonstrate their understanding of the week’s concepts. In general they will ask students to represent an addition or subtraction sentence on the number line or generate an expression for a given number line model. This task will be accompanied by a prompt that requires students to justify their answer. In my experience, these concept checks are needed to reinforce to students the importance of daily full participation. When students know they will be asked to demonstrate their learning at the end of the week, I find that they are more motivated to offer their ideas and attempt to make sense of what other students say. It is also an important opportunity for students to track their progress through the concepts.
Problem solving contexts
Adding and subtracting signed numbers is a major work of the middle school math curriculum. In order to supplement their work with the number line, I will also present students with problem solving tasks where they can use the number line to solve them. These problems will give them a chance to deal with adding and subtracting integers in contexts that support their understanding. In order for problem contexts to be supportive they should (1) have a meaningful origin, (2) include two distinct objects or movements to act as positive and negative integers and (3) result in a net value. Some good contexts include elevation compared to sea level, movement back and forth on a city street, the rise and fall of the balance of a bank account and movement up and down a football field. To detail how a problem solving context could support concept development of adding and subtracting integers, I will describe an activity that I will return to throughout the unit: the hot air balloon game.
Hot Air Balloon Game
One useful problem solving activity that I will incorporate throughout the unit is the Hot Air Balloon Game (NCTM 2016). The premise of the game requires students to imagine having a basket with a bunch of attached hot air balloons hovering in the air at a starting altitude. Students will understand this starting altitude as the origin. To visualize this, students will have paper cut outs of hot air balloons set next to a vertical number line. The game begins once students start drawing cards that direct them to add or take away helium balloons and sandbags. Although not scientifically accurate, the assumption will be made that for every helium balloon added the basket will rise one unit and for one taken away it will fall one unit. Sandbags, if added, will be assumed to cause the basket to fall in altitude by one unit and if taken away will cause the basket to raise one unit. Partners take turn drawing their cards that either say plus or minus a given number of balloons or sandbags. Students are then forced to reason with what the action will ultimately do to their basket in terms of the vertical number line. The value in this context is that it provides a concrete object to associate with positive and negative numbers. Students can see positive numbers as balloons that cause an increase in value and negative numbers as sandbags that cause a decrease in value when added. Students will also benefit from being able to easily conclude that if sandbags are taken away, the basket will rise. This supports the abstract concept that if negative numbers are taken away it results in positive movement on the number line.
It should also be noted that the number line represented in a vertical fashion supports students understanding of value going up and down. A common point of confusion for my students is the fact that the numbers on a horizontal number line seemingly increase going to the right of the origin and to the left. In my experience, this muddles students’ understanding of what up and down mean in certain word problems. The vertical number line better supports their understanding of up and down and will afford them an easier visualization of adding and subtracting integers that they can then translate to the horizontal number line.
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