Misconceptions and Considerations 8.SP.1 8.SP.2 8.SP.3 8.SP.4
3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.
Students may mistakenly believe there is only one right line of best fit for each scatterplot. While this may be true, it is dependent on the method used to find the line of best fit. In 8th grade, students use the “eyeballing� method, which means that every student may draw a line of best fit a little differently. Although there are more sophisticated methods of finding a line, they are not discussed in 8th grade. The reason a line of best fit is used is to determine the relationship between the variables. The line helps us stay focused on the middle of the data or the average relationship of the two variables.
WHAT TO DO:
Give students a scatterplot, or have them make one, and a dry spaghetti noodle. Use the spaghetti noodle to form the line of best fit. Discuss the similarities and differences between each line of best fit. Although the slopes and y-intercepts are different they should be relatively close in value. The relationship and correlation between the variables should be the same.
Interpreting the slope and the y-intercept can be confusing for many students. What exactly does a slope of ____ mean in context of the data? What does the y-intercept mean? Are the slope and y-intercept even useful when discussing relationships between two variables? The slope always has meaning! For example, using the scatter plot below the slope is approximately 1.67 cm. The slope describes the change in the stride (y-axes) as the length of legs changes stride (x-axes). So the slope is approximately 1.67 . That means that for every length 1.67 cm our stride increases our leg length increases 1 cm. The y-intercept is the value of y when x = 0. In this case, the y-intercept is 19.8. Could that mean when a person’s leg is 0 cm long their stride is 19.8 cm? That’s not possible! Students need to consider whether or not the y-intercept makes sense. Although we may need a y-intercept to write a line’s equation, it may not always be meaningful.
Two-way tables can be designed showing frequency or relative frequency. Typically, students understand how to read a frequency table because it shows counts. However, occasionally they misread the totals. In the table below, 54 students were polled about their favorite activity. Dance
Sports
TV
Total
Boys
3
12
9
24
Girls
16
10
4
30
Total
19
22
13
54
Relative frequency tables are a little more difficult for students to calculate and interpret. Using the data above we calculate the relative frequency of the entire table by dividing the individual numbers by the total. Each piece of data tells us something different. For example, 19% of all the students polled are girls that prefer to play sports in their free time. Dance
Sports
TV
Total
Boys
.05
.22
.17
.44
Girls
.30
.19
.07
.56
Total
.35
.41
.24
1.00
Data can also be calculated to show the relative frequency in each column and each row. The following relative frequency table has been calculated by row; this shows the relationship between just the boys or just the girls. The probability of a girl saying she prefers to play sports in her spare time is 33%. Dance
Sports
TV
Total
Boys
.13
.50
.38
1.00
Girls
.53
.33
.13
1.00
Total
.35
.41
.24
1.00
The following relative frequency table has been calculated by column. This calculation shows the relationship between gender and activity. The probability of picking a girl that prefers to play sports is 45%. Dance
Sports
TV
Total
Boys
.16
.55
.69
.44
Girls
.84
.45
.31
.56
Total
1.00
1.00
1.00
1.00
Students often get confused about what numbers to divide to find the relative frequency and then get even more confused about how to read the data once they have calculated it. It is important to give students multiple opportunities to manipulate data and interpret the data.