We worked with exponent power rules again today, but you learned two new things. First, we talked about the difference between (–11)^{2} and –11^{2}. Most people think that these are the same, but if you do them on your calculator you will find out that they are not! Here is how you should think of them…

(–11)^{2} = (–11) × (–11) = 121

–11^{2} = –(11^{2}) = –(11 × 11) = –121

These have subtle differences, but important differences! You had a chance to expand these expressions out in a table to really look at how and why these work the way that they do. There are also a couple of tricks that you might pick up as you do them! Here are the notes that we wrote down for them:

The other new thing we did today were two-step power rule problems. On #3 of the classwork, you needed to apply the power rules twice in order to simplify the expressions. Use the order of operations to determine what your first step is. I want you to show your work on these, and then cross out your answer from the list of possible answers on the side margin.

Your homework is to finish #1-7.

]]>We continued to work with power rules today. We divided powers by expanding them out, crossing out the numbers that undid each other, and then simplifying the numbers that were left as a single power. We quickly found a rule that worked for all of these!

The harder rule was the second rule. We know that when you do not have an exponent next to a number, it means that it is raised to the first power. But what happens when you have a number that is raised to the power of zero? You got a chance to explore that and you found out that any number to the zero power always comes out to the same value… one!

We wrote these two new rules in our notes:

For classwork, you should have finished #1-6. Also, we will have our first quiz on exponents on Friday of next week. Have a great weekend!

]]>Today we were computing with powers. In the first case, you were multiplying two powers with the same base. In the second case, you were multiplying two powers with the same exponents. In the third case, you were raising a power to another power.

In all three cases, you had to expand them out and then simplify them down to a single power. A table and an example were included for each to help you. You then had to come up with a rule that would allow you to figure out the simplified power without having to expand it out!

What we found was that each had a simple rule to get the simplified single power. We wrote them in our notes:

Your homework is to finish #1-6.

]]>Today, we continued our work with scientific notation. However, instead of working with very big numbers, we worked with really small numbers! I’m talking about numbers that are so close to zero that they have a lot of zeroes after the decimal point! When we write these numbers in scientific notation, the power of ten will be negative to show that we have to move the decimal point in the other direction. It basically means that we are dividing the number by ten several times!

Here are the examples that we wrote in our notes. Use the bouncing arrow to help you keep track of the number of times that you are moving the decimal!

Remember two key facts: A negative power of ten means that the number will be small and a positive power of ten means that the number will be big. Remember, the power of ten tells you how many places to move the decimal, not how many zeroes there are!

Your homework was to finish #1-10.

]]>Today, we looked at scientific notation. Numbers in science are often very large. For example, the mass of the earth is 5,970,000,000,000,000,000,000,000 kg. Big numbers like this are overwhelming and too time consuming to write out, so there is a shorthand way to write them: 5.97 x 10^{24} kg. This is called scientific notation. Here are the notes that we wrote:

To write a number in scientific notation, take the digits on the left side until all that are left are zeros. Next, put the decimal after the first digit. Finally, multiply the number by the power of ten required to move the decimal from its original spot to where it needs to be now.

When given a number in scientific notation, take the first number and move the decimal over as many times as the power of ten. The common misconception with kids is that you just “attach zeros” on the end based on the power of ten. This is not the case! We are moving the decimal based on the power of ten!

Your homework is to finish #1-12.

]]>We started a new unit today! This unit is all about exponents. Today, we started with some basic review of exponents:

In class, you should have finished #1-8. Have a great three-day weekend!

]]>Today, we took the last quiz of this unit! Your homework is the Unit Reflection, where you have to rate your understanding, answer some reflection questions, and finish the pre-assessment.

We will start a new unit tomorrow!

]]>We have a quiz tomorrow! For the quiz, you should be able to:

• Solve a system of equations using elimination method

• Use systems of equations to solve story problems

• Use systems of equations to solve linear relationship problems

Today was our review day! Your homework is to finish #1-7 from the Quiz Review.

]]>Today, we had to apply our skills with systems of equations in a variety of different situations. We had story problems, like we did yesterday. We also had problems involving the linear relationship work that we did earlier this year. For these problems, you had to write linear equations for pairs of situations and then use algebra to find the point of intersection. You need to find the y-intercept and the slope – it was really good review!

Your homework is to finish #1-6. Don’t forget that we have a quiz coming up on Thursday!

]]>Today our focus was on solving story problems by setting up and solving a system of equations. For the past week or so, you have had story problems where I gave you possible equation options to choose from to match. Today, you had to write them on your own!

There is no magical formula that works for every single story problem. You have to read the problem and figure out what the two equations are. Sometimes each of the equations equal a total amount of items. Sometimes the equations will equal a variable.

Your homework is to finish #1-6. Don’t forget that we have a quiz on Thursday, so make sure that you are understanding!

]]>Today, we used the elimination method again, but we had to do a step first. This time, we couldn’t just add or subtract the two equations because it wouldn’t eliminate any of the variables! Instead, our first step was to multiply one of the equations by a number so that one of the variables matched up with the variables in the other equation. Here is the example that we did in our notes:

Notice that we multiplied the first equation by 3. This was strategic because it allowed us to have the same number of y’s! Now, we can just add the two equations together to solve for x like we did yesterday!

If you would like to see me explain it, check out the video below:

Your homework is to finish all of #1 (although I assigned #1-2 for 3rd period). Have a great weekend!

]]>We continued using the elimination method today, but it got a little trickier. Yesterday we learned that we can add two equations together to eliminate one of the variables. Today, we found that we can also subtract two equations to eliminate one of the variables.

Notice that our original equations both have 4y. If we were to add the two equations, it would not eliminate any of the variables. However, if we subtract them, we would eliminate the y’s, because 4y minus 4y leaves you with nothing! We can then get the other variable just like we did yesterday.

If you’d like to see me explain it, check out this video:

Today’s assignment had elimination by adding and subtracting all mixed up. You need to look at the equations first and determine if you need to add them or subtract them to eliminate one of the variables.

Your homework is to finish #1-3. Also, we will have a quiz on all of this work on Thursday of next week!

]]>Up until now, we have been using substitution to solve systems of equations. Today we learned a new method! We have had situations where both equations were in standard form, but one of the variables was always easy to solve for and then substitute into the other equation. Today, that was not the case! However, all of today’s problems could be solved by adding the two equations together. See the example we wrote in our notes:

Because the first equation has 6x and the second equation has -6x, adding the two equations together leaves us without any x’s! This makes it easy for us to solve for y! Then to get x, we substitute the y-value into one of the two original equations. Notice, that this method only works because adding the two equations together eliminates all of the x’s! If we could not eliminate a variable by adding the two equations together, then we would not do it!

If you want to watch me solve it, check out the video below:

In class, you should have finished at least #1-4 (just #1-3 for fifth period), which includes some story problems!

]]>Welcome back to school! Today was a review day to refresh our memories on the systems of equations work that we were doing before the break. We will be learning new stuff tomorrow though!

Your homework is to finish #1-6 (although I assigned #1-7 for third period).

]]>We had a quiz today! There is no homework! Have a great winter break!

]]>We have a quiz tomorrow! Today was the quiz review. For the quiz, you should be able to solve systems of equations using algebra and by graphing. We have learned four different situations that involve substitution, so you should be familiar with them all!

Your homework is to finish #1-7 of the Quiz Review. Remember, you have to have the review complete in order to take the quiz!

]]>Today we learned how to solve a new type of system of equations. This time, instead of having an equation with y=, we had an equation with x=. We can solve this with substitution, just like before, but this time we are substituting things in for x instead of y. Here is the example that we did in class:

Notice that when we substitute in for x, we get an equation with just y’s. This means that we solve for the value of y first. To find x, we take our y value and put it into the equation with x=.

If you want to see it in a video, you can check it out…

Your homework is to finish #1-4. Don’t forget that we have a quiz on Friday!

]]>We continued our work using algebra to find the point of intersection of two lines. So far, we have always had at least one of the equations with y=, making it easy to substitute into the other equation. Today, that was not the case… but we can make it so!

Notice, the first thing we do is take one of the equations and get y alone on one side of the equals sign. That gives us our y= equation! Now we can solve the system of equations like we have been doing all week! Once we find x, put it back into one of the original equations to get y – but notice that we have to solve the equation to get the value of y!

If you want to see it in a video, you can check this out…

Your homework is to finish #1-4.

]]>We solved systems of equations again today, but this time we had some situations that we have seen earlier this year…

If the two lines run parallel to each other, then they never intersect, so there is no point of intersection and we say that there is “no solution”. In algebra, we have no solution when we end up with a number that equals a different number. We did an example in our notes of what it looks like on a graph and what it looks like with algebra:

If the two lines are on top of one another (they are the same line), then every point on the lines are points of intersection! In that case, we say that there are “infinite solutions”. In algebra, we have infinite solutions when we end up with a number that equals itself. We did an example in our notes of what it looks like on a graph and what it looks like with algebra:

Your homework is to finish #1-5. Also, we will have a quiz on systems of equations on Friday!

]]>Today’s algebra wasn’t that much different than yesterday’s algebra, except now you will be using the distributive property with negatives!

Notice that when you use the distributive property, you have to multiply everything in the parentheses by -4. This is where a lot of kids mess up, so work carefully!

Your homework is to finish #1-3. Have a great weekend!

]]>Yesterday we learned that we can use algebra to find the point of intersection of two lines if they are both in slope-intercept form. Today, we learned that we can also do it when one of them is in slope-intercept form and one is in standard form.

To do this, we use substitution again, but it looks a little different…

In the example above, we have an equation in slope-intercept form (y=3x+2) and an equation in standard form (-5x+3y=14). Since we know that y is equal to 3x+2, we can replace y in the second equation with 3x+2. This gives us an equation that is very long… but solvable! Use the distributive property, simplify, and then solve for x. To find y, put the value of x (in this case, it is 2) into the equation with y=.

If you’d like to watch me explain it in a video, check it out here:

In class, you should have finished #1-3. The algebra is getting a little harder each day, so make sure that you are working hard to stay with us!

]]>The focus of the next few weeks will be on systems of equations. A system of equations is when you have two or more equations at the same time. When we solve a system of equations, we are looking for the coordinates of the point of intersection.

We spent time this week graphing pairs of equations and finding where they intersect, but today, we found the point of intersections by using algebra! To find the point of intersection algebraically, you have to use substitution…

The first equation tells us that y equals 8x-11. We can therefore substitute 8x-11 in for y into the second equation, which gives us an equation with only x’s that we can solve! Once we find the value of x, we can put the value into either of the original two equations to find the value of y!

If you’d like to see this done in a video, check out the one I made below…

Your homework is to finish up #1-3.

]]>We had a quiz today! Your homework is the Unit Reflection, where you have to answer some reflection questions and correctly answer each of the problems on the attached pre-assessment. We will start a new unit tomorrow!

]]>We have a quiz tomorrow! For tomorrow’s quiz, you should be able to:

• Determine if a relationship is a linear

• Determine if a relationship is a function

• Write a function rule for a set of inputs and outputs

• Take equations in standard form and write them in slope-intercept form

• Find the solution to a system of equations by graphing

Today was our review day. You need to finish #1-9 of the review in order to take tomorrow’s quiz!

]]>Today we learned about systems of equations. A system of equations is when you have more than one equation. When we work with a system of equations, we are usually looking for the solution to the system… which is a fancy way of saying that we want the point of intersection!

When two lines intersect at one point, the coordinates of the point of intersection is the solution to the system of equations. When two lines are parallel, they never intersect! Then we say that there is “no solution”. When one line is on top of the other line, they intersect at every place on the line! Then we say that there are “infinite solutions”.

Don’t forget that when graphing equations in standard form, you have to use algebra to rearrange the equation into slope-intercept form before you can graph it. One equation in each part of #1 is in standard form. Show your algebra work on these! Here is the example that I showed in my Powerpoint in class:

In class, you should have finished #1-3. We will have a quiz on Tuesday. Have a great weekend!

]]>We have already learned how to distinguish between relations that are functions and not functions using the vertical line test. Today, we had to distinguish between linear and non-linear graphs. A graph is linear if it is one straight line. So if the graph is curved or has corners to it, it is not linear.

In the classwork, you had to determine if the graphs were linear or non-linear. Then you had to determine if they were functions or not functions. You found that a graph can be both linear and a function, be neither linear nor a function, or be one or the other!

Your homework is to finish #1-5.

]]>Today we worked with linear relationships again, but the equations looked different than what we are used to. We are used to seeing equations that have y alone on one side of the equal sign. This form of an equation is called slope-intercept form, because the equation shows you the slope and y-intercept.

Today’s equations had both x and y on the same side of the equal sign. This form is called standard form. This form does not tell you the slope or the y-intercept, so you have to do some algebra work before you can graph it. You will need to use algebra to get y alone on one side of the equal sign, then you can see what the slope and y-intercept are. Here is what we wrote in our notes:

If you’d like to watch me explain it, check out the video…

You are required to show your algebra work! Work carefully, because there are a lot of little steps that can trip you up (negatives, reducing slope, etc)

Your homework is to finish up #1-6.

]]>We continued our work with functions today, but our focus was on their graphs. We know that a graph is not a function if there is an input that gives us different outputs… which means that the points have the same x value and different y values. If you look on a graph, this shows up as points lined up on the same vertical line! So if we have more than one point on the same vertical line, then the relation is not a function!

Now we can easily test to see if a graph is a function by just looking at it! If there is a vertical line that touches the graph in more than one place, then it is not a function! We call this the Vertical Line Test. Here are the examples we wrote in our notes:

In class, you should have finished #1-8. Your homework is the Homework Worksheet. You also got your quiz back today, so Quiz Corrections are due a week from today!

]]>Welcome back from break! Today, you took a math assessment in the library! Your homework is the After Assessment worksheet.

]]>Today we learned a new concept, functions. A function is a relation where every input gives you exactly one output. Within each function is a rule, so when you put a value into the rule, you get an output.

The example I showed you in class was a website with an interactive function machine. You input a value and it gives you the output. You have to figure out what the function rule is!

Here are the notes that we took in class:

Notice that we think of tables for functions a little differently than we did for tables of linear relationships. Instead of finding the rise and run for the rows in the table, we look at each x value and try to determine the rule that gets its corresponding y value. Also, remember that the definition of a function is a relation where each input has exactly one output. That means if you put the same input into the function, you should get the same result both times. If not, then it is not a function!

In class, you should have finished #1-5. You will turn this in on Tuesday 11/28.

On Monday, you will go directly to the library for math class because we are taking an assessment on the computers! It will not affect your grade, but it will give you a chance to show what you have learned so far!

Have a great Thanksgiving break!

]]>We had a quiz today! Your only homework was the After the Quiz worksheet.

]]>We have a quiz tomorrow! We spent today reviewing the work that we have been doing. For tomorrow’s quiz, you should be able to:

• Write an equation for a table

• Determine if a table represents a linear relationship, and prove it

• Write an equation for a line given a point and the slope

• Write an equation for a line given two points

• Make a graph to match a story

• Match the parts of a story to parts of a graph

Your homework is to finish the rest of the Quiz Review #1-7. Remember, in order to qualify to take the quiz, you have to have the Quiz Review done!

]]>Today was a chance to review the variety of work that we have been doing. We have a quiz on Tuesday, so it was almost like having an extra quiz review day!

Your homework is to finish #1-8. Have a great weekend!

]]>Today’s assignment was very similar to yesterday’s except it was a little harder. Pay attention to the scale of the graphs, because they were not always going up by 1. There were also some graphs that didn’t have a scale at all! You can interpret the graph based on the shape and the relative steepness of each line segment.

Your homework is to finish #1-8. Don’t forget that we have a quiz on Tuesday of next week!

]]>Today we had graphs that were made up of different line segments. The graph told a story, and it was your job to figure out how each part of the graph matched the story. This was a little trickier than you might think. Every line segment and point between the segments represented something important in the story. You had to label those parts of the story onto the graph, then answer questions based on the graph.

Here is the example I showed in class:

In class, you should have finished #1-4. Your homework is the Homework worksheet.

]]>Today, we again had to write an equation for a line given just two points, but this time we used a new technique to do it… algebra! The process isn’t difficult, but there are quite a few steps, so I broke it down into four major steps. Here is the example that we wrote in our notes…

If you’d like to see and hear me explain it in a video, check it out below…

For classwork, you should have finished #1-4. Your homework is the Homework worksheet.

]]>Today, we had to write equations for lines, but this time it was a little trickier. Instead of being given the slope or y-intercept, all we were given were two points that are on the line. We learned two methods for doing this.

The first method used a graph. Simply plot the two points, carefully draw the line through them, and then write an equation for the line. Make sure that you use a straight edge so that you can get an accurate line, otherwise the y-intercept might be off. Also make sure to reduce your slope!

The second method used a table. Put the two points in as rows on the table, complete the table based on the pattern, and then write an equation for table!

Here is an explanation in a video…

For classwork, you should have finished #1-6. Your homework is the Homework Worksheet.

]]>We worked with tables again today. We started by taking tables and turning them into graphs. All you have to do is plot each row of the table as a coordinate, because they are points on the line!

We also filled in a couple more table examples in our notes. One of the tables did not show the y-intercept (when x is zero). To find the y-intercept, we had to continue the graph until x got to zero! The other table in our notes was not linear! We know this because the slope changed from row to row. If this was really a line, then the slope would be the same throughout!

In today’s assignment, you had to determine whether a table was linear. I had you do this in two ways… One way is to plot the points and see if they form a straight line, because if they do, then the table is linear! The other way involved slope – find the rise and run between every pair of rows in the table and then calculate each slope (reduced). If the slope stays the same throughout the entire table, then the table is linear!

Finish up #1-6 by Monday. Have a great weekend!

]]>Today, we started with tables of linear relationships. You had to make the graphs, equations, and stories to match them. You can find the y-intercept and slope of the linear relationship pretty easily from a table. Here are the notes we took on tables:

The y-intercept in a table is the y value when x is zero. Be careful, because this does not always have to be the first row in the table! Also, there were times when you had to extend the table to find when x was zero. Show your work on these!

For the slope, you need to find the rise and run from the table. Pick two rows. The rise is the change in the y column of the table. The run is the change in the x column of the table. You can pick any two rows in the table to do this, because you should get the same slope (after reducing).

In class, you should have finished #1-4. Your homework is the Homework Worksheet!

]]>We had a quiz today! Your only homework is the After the Quiz worksheet!

]]>We have a quiz tomorrow! For the quiz, you should be able to:

• Solve problems using an equation

• Solve problems using a graph

• Make a table, equation, and story to match a given graph

• Find the y-intercept when it is a fraction

• Find the y-intercept when it goes off of the shown graph

• Work with graphs that have scales other than one

• Understand the slopes of parallel and perpendicular lines

Your homework is to finish #1-7 from the Quiz Review (you do not have to do the optional bonus problems!). Remember, you have to complete the Quiz Review in order to take the quiz!

]]>Today, you got more practice with graphs. You had to write equations for graphs, complete tables for graphs, and solve problems using graphs.

You should have finished #1-4 in class. Also, we will have a quiz on Tuesday. Have a great weekend!

]]>We have been solving problems using equations everyday for the POD for weeks. Today, we learned that we can also solve these same problems using a graph. It is actually faster and easier to use a graph!

Here is a video explaining how to solve problems using both methods:

The advantage of using a graph to answer these types of questions is that it is so much faster and easier than using the equation! However, the advantage of using the equation is that your answer is more accurate!

The hardest part of today’s assignment was finding the slope of each line, because the scale on each axis was not necessarily just 1. When you find the rise and the run, you cannot just count the squares anymore… you have to look at each axis to see how much you are actually increasing/decreasing!

In class, you should have finished #1-3. Your homework is the Homework worksheet. Also, we will have a quiz on Tuesday.

]]>Today’s assignment gave you more practice writing equations and making tables for graphs. Some of them were tricky though, because the y-intercepts were not whole numbers! Use the slope of the line to figure out what fraction of a square the line is going up/down by!

Your homework is to finish #1-5.

]]>Today we worked with pairs of parallel lines and pairs of perpendicular lines. Parallel lines are lines that never intersect and remain the same distance away from each other. Perpendicular lines are lines that intersect at exactly a 90 degree angle (right angle). You had to write an equation for each line and then look for relationships… the relationship between the slopes of parallel lines and the relationship between the slopes of perpendicular lines.

We found that parallel lines have the exact same slope! This makes sense, since the slope controls the steepness of the line and parallel lines must have the same steepness! The relationship between the slopes of perpendicular lines was a bit more difficult – their slopes are the opposite sign and the reciprocal of one another (the fraction flipped upside down)!

You should have finished #1-5 in class, there is no additional homework! Have a happy and safe Halloween!

]]>We continued our work with graphs again, but today we learned how to make a table to match a graph. The key to making the table is understanding that each row in the table is a point on the line… so find the coordinates and you have the table! Here is the example that we wrote in our notes:

Don’t forget that naming coordinates and finding slope are in the reverse order of one another… coordinates list the horizontal first and then the vertical, while slope has the rise over the run! It is easy to get them confused if you aren’t paying close attention!

For classwork, finish #1-6. Your homework is the Homework Worksheet. You also got your quiz back today! The Quiz Revision is due in one week!

]]>Today we combined the linear relationship work that we have been doing with the algebra work that we did last unit! We had to write linear equations for situations, and then use them to solve problems. All you have to do is take the number the problem gives you and replace it for the appropriate variable. The key is knowing which variable to replace… and that requires you to actually READ THE PROBLEM!

Once you substitute your value into the equation, you need to find the value of the other variable. Sometimes, it means just computing one side. Other times, it means solving the equation like an algebra equation! If you would like to see two examples, watch the video below:

For classwork, you should have finished #1-8. Have a great weekend!

]]>Today, you were given the graphs of several linear relationships and you had to write equations for them. This meant you had to find their y-intercepts and calculate their slopes! Remember, you can pick any two points on the line to find the slope, just make sure that you reduce it!

Here are the examples that we wrote in our notes:

Notice how I showed my work on the graph. I recommend doing this so that you can see the y-intercept and see the pattern of the slope!

There are several things to keep in mind when writing equations for lines:

- Slope is the rise over the run. This can be confusing because when you plot coordinates, you always move horizontally first and then vertically.
- Always move to the right while finding the slope. Sometimes you will have to go up and to the right (positive slope) and sometimes you will have to go down and to the right (negative slope).
- Write your fractions vertically and not horizontally. In other words, don’t write them like this: 1/2
- Do not forget the y= at the start of your equation and the x next to your slope!

For homework, finish #1-7.

]]>So far in this unit, I have given you an equation and you had to draw the graph. Today, it was the reverse situation – I gave you graph and you had to figure out the slope!

To do this, you had to pick two points, then draw in the rise and run from one point to the other. But does it matter which two points you pick? That was the exploration today!

What we found was that it does not matter which two points you pick! Once you write the slope as a fraction and reduce it, you will get the same slope! Be sure to draw in the rise and run on your graph, so that you can see the triangles on the line. The triangles will be similar (same shape, but different size).

Your homework is to finish #1-9, there is no additional homework!

]]>We took a quiz today! Your homework was the After the Quiz Worksheet.

]]>