Before I even get started I want to say that I’m so excited because I received Tracy Zager’s book: Becoming The Math Teacher You Wish You’d Had in the mail yesterday! I can’t wait to start reading it. Go buy it if you haven’t already!!
Algebra 1 – We are continuing with sequences this week. I love this first problem the kids had to do (this was 2 day lesson):
“Samantha and her teacher have been busy creating new sequence generators and the sequences they produce. Below are the sequences Samantha and her teacher created.Your Task: Working together, organize the sequences into families of similar sequences. Your team will need to decide how many families to make, what common features make the sequences a family, and what characteristics make each family different from the others. Follow the directions below. As you work, use the following questions to help guide your team’s discussion.
|a. −4, −1, 2, 5, …
||b. 1.5, 3, 6, 12, …
|c. 0, 1, 4, 9, …
||d. 2, 3.5, 5, 6.5, …
|e. 1, 1, 2, 3, 5, 8, …
||f. 9, 7, 5, 3, …
|g. 48, 24, 12, …
||h. 27, 9, 3, 1, …
|i. 8, 2, 0, 2, 8, 18, …
||j. , , 5, 10, …
How can we describe the pattern? How does it grow?What do they have in common?
(1) As a team, initially sort the sequence strips into groups based on your first glance at the sequences. Remember that you can sort the sequences into more than two families. You will have a chance to revise your groups throughout this activity, so just sort them in a way that makes sense to start out with. Which seem to behave similarly? Record your groupings and what they have in common before proceeding.
(2) If one exists, find a sequence generator (growth pattern) for each sequence and write it on the strip. You can express the sequence generator either in symbols or in words. Also record the next three terms in each sequence on the strips. Do your sequence families still make sense? If so, what new information do you have about your sequence families? If not, reorganize the strips and e
xplain how you decided to group them.(3) Get a set of resource pages, saving a copy of each of the tables and graphs for your team. Then record each sequence in a table. Your table should compare the term number
, to the value of each term
). This means that your sequence itself is a list of outputs
of the relationship and the inputs
are a list of integers! The first term in a sequence is always n
= 1. Attach the table to the sequence strip it represents. Do your sequence families still make sense? Record any new information or reorganize your sequence families if nece
(4) Now graph each sequence on a the graphs your teacher gave you. Include as many terms as will fit on the existing set of axes. Be sure to decide whether your graphs should be discrete or continuous. Use color to show the growth between the points on each graph. Attach the graph to the sequence strip it represents. Does your sequence families still make sense? Record any new information and reorganize your sequence families if necessary.”
There was a lot of discussion on grouping, and some kids persevered on the rule and others had some troubles. We discussed the rules and I had students share how they figured out the harder ones. The next day they “learned the language” of sequences – arithmetic, geometric, sequence generator, common difference and were reminded of discrete vs continuous and domain. It was also new to write the sequence as t(n) = 3n+4, and to understand that n is the term number, 1st, 2nd, 3rd. The next day was explicit equations vs. recursive equations. It’s the first time I’ve taught recursive equations as this didn’t used to be in algebra 1! The kids didn’t see the point of it at first, why should they write an equation depending on the term before the current term if they could just write the rule? But…. a few days before we’d had the Fibonacci numbers in one of the sequences, so they remembered they couldn’t write a rule for it. However, they could write a recursive rule for it! Another great CPM Algebra lesson. Today we had a quiz, that was all on review material (which is given daily in the homework). I told them if they don’t do well on the quiz it’s because they aren’t doing their homework very well (CPM has help for every problem, with hints, for free. Some check their work, some don’t…) I’ll grade the quizzes this weekend and see how they do!
Math 7 – We started our Probability Unit this week. We’re using the lessons from the 7th grade CPM book (CC2). If you buy a Black Line Master from CPM you can make as many copies of it as you like – so since I had one we could copy the lessons for the kids (it’s in ch 1 and ch 5). And, since the homework help is FREE, they can get help if we assign those problems…which we aren’t since we do the weekly hw page which is AWESOME.
We have worked on experimental vs. theoretical probability, we investigated probability by making a spinner and then having each student spin the spinner (bobby pin) 10 times. We totaled the spins by color and compared the class results to the theoretical results. Today we wrote about the difference between experimental vs. theoretical probability, and then calculated the theoretical probability for different things if it was possible.
“Look at the situations below and decide with your team if you can find a theoretical probability for each one. If you decide that you can find the theoretical probability, then do so.
- Picking an Ace from a standard 52-card deck.
- Not rolling a 3 on a standard number cube.
- The chances of a thumbtack landing with its point up or on its side.
- Getting the one red crayon from a set of eight different-color crayons.
- The likelihood that you will run out of gas on a long car trip.”
Initially the kids said #2 was 1/6, then some said NO, it’s Not rolling a 3! so 5/6.
After that we worked on what would happen if we “modified the sample space”. They had a bag with different colored blocks in it and calculated the theoretical probability for drawing the different colors (naturally 1 bag was missing a green, and that was the group I called on for their first probability…. so it was 4/11 rather than 4/12… so i had to run across to where I’d left the box ‘o blocks to get another green!). Then I gave them another (identical) bag and they calculated the probabilities again. They were the same!
The initial questions in the lesson were “If you want to have the best chances of getting a red gumball from a gumball machine, is it better if the machine is full of gumballs or half empty? How do the chances of getting an ace in a deck of playing cards change if you have three or four decks of cards to choose from instead of only one deck? In this lesson, you will think about the size of the sample space (the collection of all possible outcomes of an event).” We will talk about this again on Tuesday (Martin Luther King, Jr. day is Monday) because it’s natural to think that if there are more you have a better chance. It was a fun start to the unit.
Math 7 Accelerated – We are finishing up chapter 4 in the CPM CC3 book so we were putting together the pattern, table, rule and graph. They graphed a line without an xy table. There was a lot of discussion on the “growth” (we haven’t said slope yet), and why if the pattern increases by 4 each time you graph that as up 4 and over 1. I wrote “change in y / change in x” then introduced it as delta y/delta x and we talked about and drew the “growth triangles”. It’s hard for me to remember to do the actual drawing “up and over”, I’m so used to just “counting” up and over. But I think it pushed the kids and that was good. Yesterday we went to the computer lab and the kids did Desmos’ Marble Slides-Lines. Last year we had done this at the beginning of the unit and the kids figured out “how” to make the lines move, but didn’t know “what” they were doing. This year it was better, but some were still not sure how to move the lines up and down. We’ll be doing a lot more with the lines, and I think we’ll go back and do Marble Slides again because most of the kids didn’t get to the “hard” ones. It was a great way to review slope and y-intercepts though! Today they had their group quiz. One of the problems gave them an equation then a tile pattern from a different equation. They needed to write the rule, draw 3 or 4 patterns, make a table and graph the line for each of the patterns, then tell me if they thought the 2 patterns would ever have the same number of tiles. Yep – we looked at it and said, “these are great!” So last night I did the quiz – and looked at the answer key. The second pattern was triangular (okay) and decreased (okay), removing one layer, or column each time. So pattern 0 had 7 columns, with 7, then 6, then 5, then 4 etc. tiles per column. Pattern 1 had 6 columns. I told them they needed to draw the pattern first – at least 3 more – then do the other representations. If they didn’t know how to write the rule they could describe what was happening. This is because the rule is [(x-8)(x-7)]/2. Yep. I think I’m going to change the instructions to tell them to describe the rule because I like having them do this problem….but so many were starting in slope intercept form, with -28 as “b”, since pattern 0 had 28 tiles! I’ll grade those this weekend too.
That was the week!