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bers as it excludes them from being able to use their intuition about relationships between num bers. Figure 2 shows a problem that can be used to help students develop their own understanding of com paring and ordering integers in sixth grade. In this problem, students use thermometers to first model different integer temperatures and then order the temperatures from coldest to warmest, or least to greatest. After providing time for students to work through the problem either individually or in small groups, the teacher can then facilitate a class dis cussion on student strategies for determining the coldest and warmest cities. Consider asking stu dents why Anchorage, Alaska is not the coldest city. This will address a common misconception about comparing and ordering integers as students sometimes overgeneralize a rule from elementary school that one - digit numbers are less than two - and three - digit numbers. It is important to think about order from least to greatest as well as great est to least. Figure 3 uses a different context to or der integers. In a follow - up question, students arrange integers from greatest to least using the water level model. This could easily be adapted for hands - on learning by utilizing seven graduated cylinders. Tape a new scale ranging from 8 to negative 8 on the outside of the graduated cylinder and allow students to fill them with water to the given amounts. Then stu dents can physically rearrange the cylinders so they are in line from the highest sea level to the lowest sea level, or greatest to least.

Graphing Inequalities

The traditional orientation for graphing solutions to one variable inequalities is on a horizontal number line. However, context should again be taken into account when determining if a horizontal or verti cal orientation should be used. For example, when graphing the scenario, “ You must be at least 54 inches to ride the roller coaster, ” students are most likely to make a connection to a vertical number line as height is a vertical measurement. A good strategy for introducing students to finding solu tions to an inequality is by having them place points that could be true on a number line (see Fig ure 4a). Students would place a point for the height of any person that would be able to ride the ride. In the classroom, the teacher could post 6 - 8 scenar ios around the classroom that fit a vertical orienta tion of a number line like the one provided in Fig ure 4. Students would walk from scenario to sce nario and place a point where they think a solution to the scenario lies. Assign a few students to each scenario and they can discuss the location of the points on the number line and identify the points that do not make sense to them and discuss why. They will cross off these questionable points only after the discussion and all group members are con vinced the points incorrect (see Figure 4b). Next, groups will shade in all of the possible solutions and add an arrow to indicate that their shaded line continues to infinity (see Figure 4c). Finally, the teacher leads a class discussion determining the difference between an open and a closed point (i.e. circle) and how they determine whether the point is included or not included in the solution (see Figure 4d). When continuing to develop student understanding about graphing inequalities in other activities, show students the relationship between graphing solu tions on a horizontal number line and a vertical number line by rotating the vertical number line 90 ͦ to the horizontal orientation, or vice versa. Then, ask students if the graph still makes sense. Another strategy that may be helpful, is to have the students place the variable on the number line before draw ing the arrow (see Figure 5). To continue to devel op students flexible ways of thinking about the re lationships between numbers, have students write an inequality statement from a graph. Be sure to in clude both a vertical and a horizontal orientation so students can practice with both. When providing choice in the classroom for graphing inequalities, I am always surprised how many sixth - grade stu-

Figure 3: Another context for ordering integers

Virginia Mathematics Teacher vol. 46, no. 2

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