Tuesday, May 19, 2015

Discovering Slope

I have never taught slope before. This year is a first for several topics, among which was graphing functions - we did this last week, by using function tables to generate points, and I mentioned positive and negative slopes in passing. In getting more specific about slope, I knew that I didn't want to just tell the students about slope (and about y-intercept) - I wanted them to figure out how the equation of a line can help them understand aspects of the graph of the line. But, I didn't know quite what to do. So, here's what I decided to do: I created a simple worksheet with four equations and their graphs and I simply asked students to find relationships between the numbers (and symbols) in the equations and the graphs of the equations. I didn't give much more direction than that. I had them each think about this, study the equations and graphs, and write their observation on their papers, without discussing with anyone, for about 5 minutes.

Then, I had them choose a partner to discuss their observations with, and to search for more ideas, for about five minutes. As they discussed, I circulated, listened, and asked questions. For the most part, they had written down how the negative/positive sign in front of the x relates to the slope, and many had identified the "added or subtracted number" as the y-intercept. Some had noticed that when the coefficient is higher, the slope of the line is steeper.

Next, I re-paired the students using popsicle sticks, to allow them to share more ideas. At this point, I wrote on the board: "# that is added or subtracted" and "# in front of the x," and asked them to try to figure out what these numbers could tell them about the line (if they hadn't figured it out already).  There weren't many students who made the connection that the slope tells how far to move horizontally and vertically between points, but there were several student whose observation was that the "m" is "how far apart" the points on the line were (they identified the points as where the line crossed the intersection of grid lines - I didn't put points on the lines for them).  After the second pairing, I asked student to write their observations on the board and then we went through and discussed whether they were correct or not.  Then we looked at the same lines graphed on the Smartboard, and we went through what the "m" tells us - we started with the fractional slopes and moved to the whole number slopes.  In all, the entire lesson took about 35 minutes. I was really happy with the students' perseverance (for the most part) in trying to find what I wanted them to find:) I enjoyed their "a-ha" moments!
Click to download

One "mistake" I made in the equations was that both equations with negative slopes also had negative y-intercepts...this led some students to incorrect conclusions, so I changed that for next year. The fixed worksheet is here, if you'd like to use it:)

Today's thinking day is my favorite kind of day:)

Sunday, May 17, 2015

Sailing Into Summer Blog Hop!

I am so excited to be part of the Sailing into Summer Blog Hop! I hope you will "hop" through all the blogs and read the wonderful ideas that everyone has to share! My "fast four" are:

1) One classroom thing I want to do again next year: I did a much better job of using my "Ticket out the Door" poster this year, and I want to be sure to continue that practice next year (and improve it even more).

Click to see on Pinterest

2) One classroom thing I want to change next year: I need to super-organize my materials into binders, as I found on Pinterest (Ms. DeCarbo at Sugar and Spice). I do have my materials in binders already, but I have larger binders that separate materials more by the type of activity than by topic. So I want to reorganize by topic....it could take me a while!

Click to see on Pinterest
3) One gift idea for instructional assistants: I have been thinking about this one a lot lately, because my instructional assistant's last day is Thursday. I usually have trouble thinking of cute, unique ideas, so I searched Pinterest, and found this one from "Life is What You Make It." This is a great idea - quick and easy (and useful), which is perfect for me!

4) One classroom organization tip: This may seem basic, but I still have to remind myself to do it every day - put things back where they belong! In the course of my day, I use so many different materials and "accumulate" such a variety of new papers that things can get disorganized very quickly. Immediately putting materials back where they belong and putting new materials in their appropriate place (even if it's the trash can!) helps keep that potential chaos of papers at bay.

Have a fantastic end of the year, and before you sail into summer, check out the rest of the blogs in the blog hop!

Thursday, May 7, 2015

Graphing Functions, with a Freebie

I can't believe it's May already! Time is moving so quickly (as usual), and we are down to about one month left of school!

Our state testing wrapped up and we have started working with function tables and graphing function equations, on a pretty basic level.  We used some practice from our textbook, and the students created functions for each other to graph, but I was feeling that it just wasn't enough practice. I couldn't find anything to suit my needs "exactly," so I decided to make a shorter Footloose activity to give the students some extra practice (along with the movement that Footloose provides).

Click to download for free:)
I created 15 cards that give the directions and the functions. The cards all have the same directions, but they have different functions to graph.

The answer grid for this activity is actually 2 pages - one for students to choose x-values and find ordered pairs using a table and a second one for them to graph the functions. The grids on the graphing page are definitely small, and I was a little worried that they might be too small, but overall, the students had no trouble with the tiny grids. I had one student (out of 125 students) who asked if he could use bigger graph paper, which was fine. The rest of the students did well with the grids.

Before they began choosing x-values for their tables, we talked about the fact that the grids only went up to 10, in both the positive and negative directions. Knowing this, they needed to be careful to choose x-values that would result in the y-values being less than 10. As the students worked, it was interesting to see which students purposely chose negative x-values, to challenge themselves to work with negative numbers (we haven't officially studied operations with negative integers), while others stayed with the comfortable positives.

The students really enjoyed this one!  Feel free to download and use it with your students:)

Sunday, April 26, 2015

Guest Post on Minds in Bloom

Click to go to post
I'm so excited  - I was given the opportunity to guest post on Rachel Lynette's blog, Minds in Bloom. The post went up on her blog today, if you'd like to check it out:)

Many thanks to Rachel!

Tuesday, April 21, 2015

How Does Coloring Improve Math Skills?

Today, as my students were working on a color by number in math class (which I thought was a fun, different way to practice math), one of them asked "How does coloring help with math?" The question was asked with a "there's no reason I should have to do this" attitude. I explained that it helped with motor skills and helped one to use the brain in a different way, and that exercising the brain in different ways could help in all things that require thinking (not just math). I don't think he really appreciated my answer:)

Integer Operations Color by Number - freebie
So, I decided to do a little research, to see what I could find. Most of what I found (not a super-long time of searching, because I didn't have that much time!) was mostly related to the benefits of coloring for young children (and did relate to math skills) and for adults. Here are a few things that I found, as coloring relates to adults:

According to the Huffington Post (10/13/14), coloring benefits adults (and I would assume children as well) because it "generates wellness, quietness and also stimulates brain areas related to motor skills, the senses and creativity." In addition, psychologist Gloria Martinez Ayala states that when we color, we activate different areas of our two cerebral hemispheres. "The action involves both logic, by which we color forms, and creativity, when mixing and matching colors. This incorporates the areas of the cerebral cortex involved in vision and fine motor skills [coordination necessary to make small, precise movements]. The relaxation that it provides lowers the activity of the amygdala, a basic part of our brain involved in controlling emotion that is affected by stress."

According to PenCentral, coloring benefits adults in helping them to maintain fine motor skills -this requires extra work by your brain to coordinate your actions and muscle control in your hands and arms. Coloring can help delay loss of fine motor skills as people age. Coloring may also help fight cognitive loss, especially if challenging pieces are completed every so often.

I didn't necessarily find research to answer my student's exact question, but what I found was quite interesting! If anyone knows of other articles or published research to support the role of coloring in improving math skills, please let me know!

Monday, April 20, 2015

Fantastic Birthday!

Yesterday was my birthday, and it was a great day! I got up early and went out for a walk/run, worked in the garden and had a great dinner with my husband and kids. Among my wonderful gifts were these fantastic "coloring books for adults" from my daughter. They are AMAZING! They were created by Johanna Basford (maybe you've already seen them) and are just incredibly detailed. The picture above is the one I started to color....it's a little hard to see the detail, but I think it's just incredible.

Now all I want to do is color.....:)

Monday, April 13, 2015

Learning to Love Math - Chapter 3

Chapter 3 - Examples of Differentiated Planning for Achievable Challenge

This is a continuation of Chapter 3, from a couple of weeks ago (I had my notes written, but it has taken me a while to type them!!). In the previous Chapter 3 post, I reviewed a couple of the examples of differentiated planning and activities that the author offered. In each example, students are learning the same basic concepts, but at different levels of challenge, which should lead to maximum success and should minimize their frustration.

This example is called Exploring Number Lines, and the author states that it is a helpful activity for both "explorers" and "map readers." As a preliminary activity, students explore number lines without any specific assignment; the author suggests using large number lines that can be rolled out on the floor. Students meet in groups and create KWL charts. In working with the number line, students will predict where they will end up with certain movements (2 places to the left, 5 places to the right, etc....on the positive side of the number line). As students move to a higher complexity, they will move toward exploring the negatives on the number line.

Another example she gives is related to understanding division, with the goal being the understanding of the concept of division as a way to break larger amounts into specific numbers of parts. The low-complexity group plays games/"sharing activities," in which students are given 10 manipulatives and are asked how they can be shared among their group of 5 members. Next students are given 15 items and asked if they can be shared evenly - if so, how? This activity continues, using different numbers of items and different numbers of group members.

The medium-complexity division
group (of 5 students) is given 100 pennies (or plastic pennies) and is asked to determine how many ten-cent pencils each group member could "buy" (equal number for each member). Students are then asked to determine how many 20-cent or 15-cent items could be purchased for each member.

The higher-complexity group would also work with ten and 20-cent items, would evaluate the worth of the items, and would use newspaper ads to find the unit rates of products.

A whole-class activity related to the division concept is to place students into groups of 3, give each group 7 large blocks, and ask them to determine how the blocks would be divided so that each person gets an equal share - the author states that this leads to the concept of fractions without necessarily calling them fractions.

All examples are very interesting! On to Chapter 4:)



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