In the previous exercise, you saw how to code random-effect intercepts. You will now see how to code random-effect slopes. 
With lme4 syntax, lmer() uses (countinuous_predictor | random_effect_group) for a random-effect slope. When lme4 estimates 
a random-effect slope, it also estimates a random-effect intercept. scale() rescaled the predictor variable mathkind to make 
the model more numerically stable. Without this change, lmer() cannot fit the model.

In the previous exercise, you estimated a random-effect intercept for each classroom and one slope for all data. Here, 
you will estimate a random-effect intercept for each class and a random-effect slope for each classroom. Like a 
random-effect intercept, a random-effect slope comes from a shared distribution of all random-effect slopes.

# Rescale mathkind to make the model more stable
student_data <-
	student_data %>%
    mutate(mathkind_scaled = scale(mathkind))

# Build lmer models
lmer_intercept <- lmer(mathgain ~ mathkind_scaled + (1 | classid),
                       data = student_data)
lmer_slope     <- lmer(mathgain ~ (mathkind_scaled | classid),
                       data = student_data)
                       
# Add the predictions to the original data
student_data_subset <-
    student_data %>%
    mutate(lmer_intercept = predict(lmer_intercept),
           lmer_slope = predict(lmer_slope)) %>%
    filter(schoolid == "1")

# Plot the predicted values
ggplot(student_data_subset,
       aes(x = mathkind_scaled, y = mathgain, color = classid)) +
    geom_point() +
    geom_line(aes(x = mathkind_scaled, y = lmer_intercept)) +
    geom_line(aes(x = mathkind_scaled, y = lmer_slope), linetype = 'dashed') +
    theme_bw() +
    scale_color_manual("Class ID", values = c("red", "blue"))

The model with fixed-effect slopes has parallel lines (solid lines) because the slope estimates are the same. 
The model with random-effect slopes (dashed lines) does not have parallel lines because the slope estimates are different. 
The model with random-effect slopes (dashed lines) has lines that are shallower than the other model. This occurred because 
slopes are being estimated for each classroom, but include a shared distribution. This shared distribution pools information
from all classrooms (including those not shown on the plot).