Arizona
The Arizona Adult Education Mathematics Standards are guided by NCTM’s Process Standards, including Communication, Connections, Problem-solving, and Reasoning and proof.
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
I like to have students share thoughts, ideas and previous knowledge and listen to each other when I bring up a new concept. Algebra is something students think of like space travel, maybe: Interesting, difficult to comprehend, unimportant or irrelevant to their lives. I failed to find good ways of making algebra relevant but students loved trying to understand once we established it as a possibility. They liked helping each other when a “neighbor” wasn’t getting it. There was a lot of talk among them following the presentation of signed numbers about ways of looking at it ( we used the idea of a balance, of weights (balls) of pos and neg, thermometers, etc.) They liked it I think because none of them had understood it so all of them shared the learning. They wanted to help each other “get it.”
Before completing the algebra lesson, I asked each student to complete the algebra mind map.
As expected, the students had many negative thoughts and ideas. However, after the mind maps and algebra lesson were completed, the majority of the students were excited to be able to say they could do algebra! The students were overjoyed and surprised by their skills. The lesson acted as a huge confidence builder!
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
When I have taught Algebra in the past, I have tried to help students see the connection between graphing and real-life situations. For example, we have graphed the cost of renting a U-Haul truck depending on how many miles you drive it. The last TIAN institute really helped me understand more completely, not only how graphs are related to real life, but also how those two are related to equations and tables. As my knowledge increased, and as they worked with the TIAN materials, these connections really come through for my students.
Yesterday I was helping 2 students with the test practice on page 75 of the student book:
Which graph shape below could most likely represent change over time in the height of water in the rain barrel over weeks of drought?
I had the students explain to me what each of the graphs would mean in real-life terms: evaporation, rain, neither of these, or some combination over time. I would never have asked them to do that before TIAN.
When we did the “How Many Calories Am I Burning” activities (tables, equations, graphs) my students made connections between math and their own lives. The math was suddenly interesting to them. They realized they use math with their physical fitness and weight loss activities. Many found it easy to calculate the values for the tables because it was connected to something they already knew about and understood. It seemed to de-mystify “algebra”.
While I had my doubts about how well some of my students understood in & out tables and developing equations from them, I thought I would assess them to determine where their understanding was.
I chose problems from 2 pages of the Top 50 GED questions book. On one page they had to figure out the pattern and predict several steps ahead. On the next page they were given the description of the pattern and had to choose the correct equation. One student who was represented in my work sample as the “low achiever” had the best results! 7 out of 8 correct! The problems were based on possible real life situations—fines for late return of library books, for example. They all had at least 5 out of 8 correct and actually did the best on the equation part. I was happy to see them making the connection between the book (EMPower) and other questions.
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
While doing the “toothpick row houses” activity, students became especially involved as we had set up teams to do the activity and the teams came up with quite different methods of determining the number of toothpicks needed for any number of row houses.
For some taking the leap from stating in words what to do to using symbols to show an equation of what to do created something of a “eureka!” experience. For them it was the first time that an algebraic expression was “real.”
I asked the students if they wanted me to bring toothpicks to class so that we could actually “build” the houses. They said “Oh no!—“Let’s just do more problems like this this way!”
In figuring out the tables in Chapter 1 at first the students were baffled and no one had an inkling on how to go about finding the patterns.
Fortunately we had one student in the group who is very bright. It was thanks to her that we were able to solve one table. Solving this table gave us skills to solve the next table, and then the next. As we solved each table more and more students got the “Eurekss”. I wish I could say that all the students achieved the “Eureka” moment, but even though not all were able to solve the tables on their own, I can say that everyone could see how the concept worked and many were able to solve the tables on their own by the end of the chapter.
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
Most of my students still have problems with their multiplication tables. We tried using gumball manipulatives to reason out the process of multiplying:
3) 3 groups of four gumballs is the same as 3 x 4
4) 3 is related to 12 using the number 4
5) This provides a reasoning and proof to make a visual connection to the numbers
in the multiplication tables.
When we were working on the diameter and circumference Lesson 6, we constructed the table showing the measured diameter, measured circumference, calculated circumference using 3.14 and the calculated circumference using the calculator. At the end of the lesson one student raised their hand and commented in amazement that after we calculated our circumference we could divide by the diameter in each row and get pi. She saw this herself.