diff --git a/04-One-Dimensional-Kalman-Filters.ipynb b/04-One-Dimensional-Kalman-Filters.ipynb
index aaff20d..022ae32 100644
--- a/04-One-Dimensional-Kalman-Filters.ipynb
+++ b/04-One-Dimensional-Kalman-Filters.ipynb
@@ -335,7 +335,7 @@
     "\n",
     "$$ x = v \\Delta t + x_0$$.\n",
     "\n",
-    "This is a model of the dog's behavior, and we call it the **system model**. However, clearly it is not perfect. The dog might speed up or slow down due to hills, winds, or whimsy. We could add those things to the model if we knew about them, but there will always be more things that we don't know. this is not due to the tracked object being a living creature. The same holds if we are tracking a missile, aircraft, or paint nozzle on a factory robotic painter. There will always be unpredictable errors in physical systems. We can model this mathematically by saying the dog's actual position is the prediction that comes from our **system model** plus what we call the **process noise**. In the literature this is usually written something like this:\n",
+    "This is a model of the dog's behavior, and we call it the **system model**. However, clearly it is not perfect. The dog might speed up or slow down due to hills, winds, or whimsy. We could add those things to the model if we knew about them, but there will always be more things that we don't know. This is not due to the tracked object being a living creature. The same holds if we are tracking a missile, aircraft, or paint nozzle on a factory robotic painter. There will always be unpredictable errors in physical systems. We can model this mathematically by saying the dog's actual position is the prediction that comes from our **system model** plus what we call the **process noise**. In the literature this is usually written something like this:\n",
     "\n",
     "$$ x = f(x) +\\epsilon$$\n",
     "\n",
@@ -416,7 +416,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The constructor `__init()__` initializes the DogSimulation class with an initial position `x0`, velocity `vel`, and the variance in the measurement and noise. The `move()` method has the dog move based on its velocity and the process noise. The `sense_position()` method computes and returns a measurement of the dog's position based on the current position and the sensor noise.\n",
+    "The constructor `__init()__` initializes the DogSimulation class with an initial position `x0`, velocity `vel`, and the variance in the measurement and noise. The `move()` method moves the dog based on its velocity and the process noise. The `sense_position()` method computes and returns a measurement of the dog's position based on the current position and the sensor noise.\n",
     "\n",
     "We need to convert the variances that are passed into `__init__()` into standard deviations because `randn()` is scaled by the standard deviation. Variance is the standard deviation square, so we take the square root of the variance in `__init__()`."
    ]
@@ -959,7 +959,7 @@
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
-    "\\mu&=\\frac{9 \\sigma_\\mathtt{z}^2 \\mu_2 + \\sigma_\\mathtt{z}^2 \\mu_\\mathtt{prior}} {9 \\sigma_\\mathtt{z}^2 + \\sigma_\\mathtt{z}^2} \\\\\n",
+    "\\mu&=\\frac{9 \\sigma_\\mathtt{z}^2 \\mu_\\mathtt{z} + \\sigma_\\mathtt{z}^2 \\mu_\\mathtt{prior}} {9 \\sigma_\\mathtt{z}^2 + \\sigma_\\mathtt{z}^2} \\\\\n",
     "\\text{or}\\\\\n",
     "\\mu&= \\frac{1}{10} \\mu_\\mathtt{prior} + \\frac{9}{10} \\mu_\\mathtt{z}\n",
     "\\end{aligned}\n",
@@ -2190,7 +2190,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It is easy to see that the filter is not correctly responding to the measurements which are clearly telling the filter that the dog is changing speed. I encourage you to adjust the amount of movement in the dog vs process variance. we will also be studying this topic much more in the following chapter."
+    "It is easy to see that the filter is not correctly responding to the measurements which are clearly telling the filter that the dog is changing speed. I encourage you to adjust the amount of movement in the dog vs process variance. We will also be studying this topic much more in the following chapter."
    ]
   },
   {