Warmup

 If you need 3.5 pounds of apples to make a quart of applesauce, how many quarts of applesauce will you make from a bushel (48 pounds) of apples?

Warmup

 If there are 3 apples in a pound and 48 pounds of apples are in a bushel, how many apples are there in a bushel

Cool Math Stuff At Getty Villa

 As you read this column, I am in Los Angeles area going to a weekend full of educational things.  As part of the offerings, we could pay a bit extra and enjoy a tour of the Getty Villa.  I'd never been there but I found so much math stuff there that it was great. I've chosen some wonderful pictures showing certain mathematical concepts or principles.

As you enter, there is a great garden area that is symmetrical.  The symmetry runs down the middle of the water area so the left and right sides are the same.  If there is a grape vine covered walkway on the left, there is another one on the right.  So very nice. The photo is taken from the end and side so you can see both.

Every garden that was built around the  villa is totally symmetrical.  They usually have one line of symmetry running the long way as if you could fold it in half.

Then along the walkways running along the edge of the garden at the beginning are several wonderful patterned squares with so many nice geometric  patterns. There are four different squares that repeat at least twice in the walk way on each side of the garden.

If students every wonder when they will have to read Roman numerals?  I found some at the entrance and exit of every room.  The rooms are numbered with roman numerals so it was really cool.

I found some really cool floor patterns that were made of repeated patterns such as this cool one which uses a bunch of Isosceles triangles repeated to form a circular pattern.  Following them are other interesting patterns I found throughout the villa.

I spent so much time oohing and awing over the symmetry, patterns, and Roman numerals I saw throughout the whole building.  Have a great weekend.  

What Are Some Of The Benefits Of Using Technology To Teach Math

 

I love using technology in my classroom and I am always able to provide reasons for using it to the administration should they ask.  Yes, I've had one or two administrators ask why I used it because they didn't see the advantages of technology.

First of all, there are so many videos, animations, movies, and other multimedia presentations available to use in the classroom.  These types or presentations allow students to review sections as many times as needed to develop understanding and skills.  They are also great for students who are absent or traveling and need to keep up with class.

Furthermore, students are often more engaged and excited when viewing multimedia presentations because they might involve raps, music, puppets, cartoons, etc which tend to be the type of videos students like watching.  When they find these presentations capture their attention, they are more likely to go back and review the material as needed and they are willing to watch new ones as they come up.

Many presentations are interactive, allowing students to explore topics to help learn.  Research indicates it is important to student learning to make mathematics visual because visual pathways in the brain are involved when working with symbolic number calculations. If possible, it is important to provide visual representation for all mathematical concepts either through multimedia games, apps, simulations or other digital tools.  For middle and high school students, Desmos or Geogebra provide some wonderful visualizations and activities. Desmos has so many activities designed to let students explore concepts, share and collaborate.  

If you want to provide more individualize instruction, technology allows this to be done.  Students can be assigned tutorials, videos, screencasts, and other instructional media to their digital device so they can work through the material at their own pace.  It is easy to assign addition support material to a student who is having trouble with the concept while extensions can be given to the more advanced student.  There are several sites that provide instruction, quizzes, work, and reinforcements so each student can have their needs met.

It is much easier to find real life examples to connect math to the real world.  It can be done by recording things on your phone and having students watch the video and have a question to answer. For instance, if you are looking at rates per unit, you might record something in the cheese department of the grocery store showing several different brands and asking students which one is the best price.

Technology also allows teachers to set up zoom or google meets with architects, bank managers, construction workers and others so the guest can discuss how they use math and the type of math they actually use in their jobs.  Students are hearing from real people who use math in real life situations while being able to interact by asking questions.  Talking to people is much better than reading general articles on the internet because articles don't interact with the student.

So these are some reasons one should use technology in the math classroom.  I realize that some places have limited internet bandwidth and you might not be able to do everything you want but you do as much as you can.  Let me know what you think, I'd love to hear.  Have a great day.

Want Deeper Learning, Think About Interleaving.

 

Unfortunately, too many times, we have students learn one skill or concept at a time.  Due to the demands of the administration and others, we don't always have time to provide the time to practice the skills in such a way as to help them use them all.  What is better is to have students practice related ideas and concepts in various combinations.  This is referred to as interleaving.

When students practice the information via interleaving, they are more likely to store the topics and concepts more deeply, while asking them to figure out patterns, and form connections.

When students learn the material through interleaving, research indicates better retention, longer term skill development, and it promotes the ability to transfer learning to other contexts.  The transference is extremely important for students to move past the mindset that every problem is different and unique.  

In addition, students are forced to remember prior learning as they work through a mix of related concepts while selecting the proper strategy needed to solve each problem.  This method helps secure the learning in long term memory, improves problem solving skills, and improve mastery.  It does not replace the blocked practice where students practice lots of practice when first learning a skill.

When integrating interleaving into classroom instruction there are a few things that can be used to make it more effective.  First, when using interleaving, it is best to assign fewer problems because lots of the same type of problems are actually used when students learn the original skill such as when they learn addition, they will practice a ton of addition problems. When students have learned how to do addition, subtraction, multiplication, and division, then interleaving occurs when all four types of problems are assigned, the teacher might assign 20 problems instead of 30.  When students work mixed problems, they have to slow down, think about how to solve each problem, and make connections between the problems.  It is important to have fewer mixed problems to avoid mindless repetition.

Another way to integrate interleaving is by using rotating stations. This allows students to work through different related concepts in small groups with teacher lead instruction to scaffold as needed, online or collaborative activities. It is suggested that the first stations introduce students to related concepts at the introductory levels and later stations have students working more complex problems. For instance, if students are studying volume of three dimensional shapes, the first stations will have students explore the formulas for the simpler shapes. They would then move to the more complex formulas before applying the formulas to a mix of shapes.  They might have to calculate the volume of a cube, a cylinder, a different sized cube, a triangular prism, etc.  This helps students develop a longterm proficiency in this topic.

When you plan the next unit, think about incorporating activities which have students work with both the new and previous materials. This helps students understand connections of related ideas, refresh knowledge, and it extends their learning. Finally when wrapping up a unit, do not just move on.  Make sure to include a few problems from previous learning every day so as to regularly trigger previous learning.   This can be done in the warm-up or as an exit ticket.  

It is important to include mixed problems so students are encouraged to practice a type of productive struggle resulting in better learning.  Although students will make mistakes and will struggle, it will result in better pathways being created in the brain.  Let me know what you think, I'd love to hear.  Have a great week.

Warmup

 If this is the net for a six sided dice or cube, what would the net look like for a four sided dice or a triangular prism?

Warmup

Estimate the number of cubes in the picture and explain how you came up with your estimation.