IMP Overland Trail - Water Conservation and stopping to find balance...

My 5th period class is Algebra IA year-round. They should be covering Overland Trail and Cookies before the end of the year. I just realized that they are at a point of Overland Trail where I had to skip some of the activities with my Algebra IA Fall Block classes. I was determined to finish Overland Trail by Christmas in order to start Cookies in January.

The Water Conservation activity is another good one to put on chart paper. Most students will generate a chart in order to answer questions 1 and 2. However, I had some students who graphed the lines first and then tried to go back and answer how many gallons each family had left after a given number of days. This did give us the opportunity to discuss how it is easier to be exact when you use a table instead of the graph due to the estimation that is required when looking at the graphs.

I loved that the group that generated the work shown above included the table and the graph along with their answers to the questions. One of the questions that this group did not address is how to find how much water each family has after x amount of days. This forces them to think about the starting point and rate of change. They know to muliply the amount of water used per day times the number of days but then they have to consider that they are trying to give an expression for the amount of water LEFT not the amount of water consumed.

After covering this lesson I pulled some practice problems where graphs were given and we were asked to find equations. I also talked about the slope formula and had them find the slope given 2 points. I know that I need to take time to stop and explicitly teach topics that we have covered but they may not know the "math vocabulary" so I am going to list them here:

• graphing from slope-intercept form
• writing the equation of a line when given a graph
• finding the slope of a line given 2 points
• finding the slope of a line given a graph

Our students have done so much more with this curriculum and creating graphs given real-world circumstances. However, when they take the end-of-course exam and the ACT they are going to need to be familiar with how to do these problems when there is NOT a context. After having taught through this unit (Overland Trail) the first time I realized that there are times I needed to stop and teach the students what types of "traditional algebra" problems they might see from the content we have been covering. The problem was that having never been through the curriculum before I did not want to "steal the thunder" of a lesson that we were going to cover in the future. I feel that to teach using the "exploration/inquiry/problem-based" method you have to also find a balance with showing students what the "traditional/worksheet/drill and practice" version of the algebra looks like. I KNOW that I can do a better job of finding a balance between the two. 

IMP Cookies - Rock 'n' Rap, formative assessment and collaboration

Rock 'n' Rap is an activity that talks about producing rock and rap albums. The students will once again be led to find profit lines which should help them on solving the Cookies unit problem. There are 3 constraints this time. Also some of them shade below but one shades above so there is a little variety.

When graphing the inequalities Sonya New, my IMP teaching buddy, has stressed with the students that one way to graph the lines is to find values for x and y that make the equation true (they really do this by trial and error using number sense). I love that she has done that (even though I have talked mostly about using intercepts or slope-intercept form). She has taught SI form and graphing using intercepts also, but she says alot of her students still graph by finding ordered pairs that work.

When I did the problems with my class we said that there had to be more rock albums than rap albums. However, smarty pants Sonya corrected me (I hate when I'm wrong) and told me that it should have been rock is greater than or EQUAL TO rap. When you read the paragraph about this constraint it says that the company promises that it would not release more rap than rock. Then it goes on to say that the company is more closely related to rock music in the public mind. Even though I know she is right I could really make an argument about why I chose to publish more rock than rap! (But the mathematician in me knows she is right...dang it!)

Lastly, I had my students put this activity on chart paper. I have been in such a rush lately I haven't done this as much. It made me realize that having them put it on chart paper "makes it real" for the students and they invest more in the problem. Also, I get the opportunity to formatively assess at a glance by reviewing their work. The picture below is of a group of students explaining their findings from the Rock 'n' Rap activity.

IMP Day 44 - Quiz on writing equations given 2 points

Today I gave my algebra students 30 minutes to work on POW 4 - On Your Own. This is a great POW that makes the students create do some research on the types of jobs they could get right out of high school. It also has them to find out how much living expenses would be and create a budget for living on their own. They also have to decide whether or not they will need a roommate in order to pay their bills. This is a research POW instead of a math or logic problem.

I gave the students a quiz on writing the equation given 2 points. One of the points was negative and one was positive. For this first quiz both slopes were integers. Out of 50 students who took the quiz 16 students aced it. another 3 students made between a 35 and a 39. 14 students made between a 24 and a 24 (a 24 is passing with a 60%). 17 made less than a 24. There were only 5 students who were absolutely clueless. So...67 % of my students passed the quiz. Now I have always said that having making a 60 in algebra does NOT mean that you are proficient. However, 38% of my students did extremely well on the quiz after working on this for 2 days. I wish I had this type of data for the first time I covered this topic in previous years. All I can do is tell you that my algebra grades in general were declining over the last 3 years (with the exception of last year when I started trying to find new teaching methods...even before the new IMP books). It was not unusual for my average quiz or test scores to be in the 40s or 50s. (My average score for this one was around 63.) I also feel like my students have a much better conceptual understanding of what we are doing. Several of the students who didn't perform well filled in their tables correctly and found the correct slope and had the coordinates for the y-intercept they just haven't gotten everything to connect just yet. However, I really think I can help them to get there! I intend to give them the opportunities to correct their quizzes so that they can see how close they are!

IMP Day 43 - Making "formal" connections

I have thoroughly enjoyed the Moving Along activity and decided to "sit down" here and make some formal connections to the slope formula and slope-intercept form. However, we first did the activity without using either formulas and it was AMAZING! We actually have been using slope-intercept form but they just didn't know it as y=mx+b. The IMP book uses y=ax+b and calls b the "starting point" and a the "rate of change."

Today I gave the class a "warm-up" where they were asked to find the slope of a line and write the equation of the line given 2 points. There was no context given and they did great! There were no formulas on the board but most of the students proceeded to put the 2 points in a table and then fill in the x-values from 0 to the highest x-value given. They go back to 0 for x because we have drilled the fact that x-coordinate of the starting point is always 0. We have also tried to drill that the starting point is always on the y-axis but they seem to forget that sometimes...

After the warm-up, which most of them got with a table, I had them add the slope formula and slope-intercept formula to their notes. I told them that they would receive a reference page on their end-of-course (EOC) algebra exam and that I wanted them to be familiar with the formulas. We went over how we could have used the formulas for the warm-up and then I gave them another problem.

I used this weird effect on my picture to make it a little more readable. 

He writes so light I know it is a little difficult to read. However, Raul (whose paper is above) was the first student finished finding the slope and equation of the line! Then I gave the class a problem where the slope was a fraction thinking that they would resort to using the formula (and most of them did). However, Raul and one other student STILL used the table to get the equation. Their "number sense" is very good and thinking in fractions (or decimals) did not bother those 2 a bit! We have only been writing equations of lines given 2 points for 2 days and there are many more students who do it correctly than I have seen in the past.

It is exciting to realize that there are students who really benefit from their exposure to the different methods that can be used to write equations. I guess that since I am so accustomed to using the formulas I thought they would automatically start using them. However, the majority of my students are still using tables! I don't think I have ever used a table to find the slope or y-intercept so I am getting an education too!

Math teachers are "Formula Babies" - we need to be more natural!!

My teaching buddy, Sonya New, and I are writing this post together! We have learned this week that we are FORMULA BABIES!! We ran across some problems where students need to write equations given 2 points and thought the students would just HAVE to have slope and point-slope formulas because that is how we learned to do it ourselves. BAHAHAHAHAHAHA!!

What an education we have received! I was so concerned about how they would find the "starting point" or y-intercept without formulas! Earlier this week Sonya's honors algebra realized that once they found the rate of change (slope) they could multiply the x coordinate by the slope and find the b (or starting point) by figuring out what to add or subtract to get y. I know that makes no sense when you read it! However, it took me and Sonya 2 WHOLE DAYS to realize that what they are doing is using the slope-intercept formula to solve for b. We felt STUPID!

ALSO, I had students to use tables to find rate of change and then extend the table "back" to zero to find the starting point or y-intercept. I had one student who hated the formulas yet got EVERY SINGLE PROBLEM correct using tables. Even problems with fractional slopes!! IT WAS AMAZING AND EYE-OPENING!!

Then we laughed about the fact that we are "formula babies" and have come to the conclusion that we need to understand that the logical (or NATURAL instead of FORMULA) way to write equations makes more sense to our students.

P.S. - Sonya is a new mother and I have had 3 breast-fed babies myself...so we couldn't resist the analogy.

IMP Day 42 - Moving Along - making connections!

The teacher's guide for Moving Along says that all algebra teachers love it because it has types of  problems that we are used to seeing. The students are given two points and asked to write the equation of the line. The interesting part of this assignment is seeing the different approaches that the students will take to solving it since we have not discussed the slope formula or the point-slope form of an equation.We have explored graphing through starting points, rate of change, and in-out tables. We have used slope-intercept form (without calling it that) by discussing that when the equation is y=ax + b that b is the starting point and a is the rate of change. The majority of my students write their equations in the form y=b + ax which seems more logical when we talk about b as the starting point.

I started class by assigning each group 2 of the 4 problems (everyone did #4). I gave them 10 minutes to brainstorm on how to approach the problem. The first group who got their equation explored the problem by using a table and finding the rate of change. The two points given in this problem were (0, 36) and (6, 24) so they remembered that the starting point had to be 36 because we had discussed for several days that the x-coordinate of the starting point is always 0. They realized the values for y went down by 2 each time and wrote the equation. One of the girls actually said, "I am getting good at this!" Here are their In-Out tables:

After the first 10 minutes I had the girls share with the class how they came to find their equation for #2. I took a graph from one of the students in a group that was working on #3 where the 2 points given were (2, 300) and (10, 196). He had graphed the 2 points and drawn a line through it which touched both the x and y axes. By looking at the graph the starting point was estimated to be 325. I gave the groups another 10 minutes to work. There was another group working on #3 and they were trying to build a table that worked with a starting point of 325. One of the students at that group blurted out, "I got the rate of change. It's 13." I directed the students to try that rate of change and adjust the starting point as needed and they found an equation that worked! I was surprised that the student's rate of change was correct. I went to the boy's side and asked how he found it. He had subtracted the y values and then divided the answer by the difference of the x values!!! Shazam! He used the slope formula and didn't even know it! The slope was actually -13 instead of 13 but he figured that out when he wrote his equation because the y values were getting smaller. I had him share with the class how he found the slope and then I went to the board and wrote the slope formula there. I asked them if they knew what it was and none of them remembered it from last year. I showed them how my students from previous years found the slope using the formula. I also showed them that if they "plug into" the formula correctly they would get the correct sign for the slope of the line. I was so excited at the connections that we were making to the "traditional" algebra. The majority of the groups came up with the correct equations...on the first day that we wrote equations given 2 points!!! I was amazed!

This is Hunter sharing how he found the rate of change using 2 points. He did an incredible job of figuring it out AND explaining it!

Lastly, #4 in this activity was the first situation/problem that we have encountered where the starting point is negative. The 2 points given were (3, 12) and (7,32). I had 2 different students to use a table to find that the rate of change is positive 5. I had to give them a prompt to help them to find the starting point. The students who figured out the rate of change had created tables that started with an input value of 3 and went up to 7. All I had to do was put a 2, 1, and 0 in the input column (I put them above the 3 to help them see the pattern) and asked them if they could continue the pattern to find the starting point. It ended up being -3. 

Now, my teaching buddy, Mrs. New, and I have been talking about whether or not teaching the formulas is important. AND...we still think that students need to be exposed to the "old-school" formulas and even practice some problems using the formulas. However...I think I had more students to get their equations correct given 2 points (on the 1st day!!!) than I ever have. The exploration and work that we have been doing over the last several weeks gives them such a firm conceptual understanding of writing equations by finding the starting point and rate of change that I believe the "old-school" work will have more meaning!!!  Mrs. New keeps asking me to consider whether or not the majority of our students remember how to use the formulas when they see these problems on standardized tests and my answer is NO... the majority do not remember! Now our students have a "hook" (as our instructional partner, Dr. Montgomery calls it) on which to hang that concept so that they will hopefully retain the information. They have also been shown how to use various approaches to examine and solve these problems.