Another Internet Resource

 I was out looking for additional information on using project based learning in Math because that is apparently one of the goals of the school.  In the process, I stumbled across the K-12 Internet resource center site. 

According to the website, it has over 3500 web and video resources along with over 240 lesson plans which sounds quite impressive.  So I checked the lesson plans.  They have the lesson plans subdivided into English, math, science, history, and other subjects. So of course, the first one I looked at was math.

The math lesson plans offer me the opportunity to drill down even further by grade levels. If you don't filter before going further, you are taken to a page with the filters across the top and a list of places with grade levels and if there are lesson plans or videos associated with the place.

After checking out several sites are actually sites that have some free materials but may ask one to subscribe to get all the benefits. However, many sites are totally free with worksheets, videos, and activities.  There are quite a few nice resources to get ideas, find lessons, etc.

I also did a general search on math to see what that brought up. The first couple or three listings were sites to show how certain math words are used in real life.  Many of the words are associated with physics but there was also a link for words used in geometry. Then I came across a nice place for math worksheets but they offer both lessons and practice so that makes it nice if you have a student who is a bit more advanced and needs work beyond the others in your class.  They can read the lesson that has a couple of problems and the problems are explained with quite a bit of detail. 

One interesting link was to the Math Learning Center which has lots of free materials including books, worksheets, and so much more.  The one thing that caught my eye was two books on algebra through visual patterns.  I checkout the free downloadable pdf and liked the way every lesson is detailed with every step you need.  I've often wondered about how to present certain concepts and this offers some different ways.  I like it.  This site has materials for grades K to 12 and it is well worth checking out.

There is even a link for a math dictionary for the younger children.  I will say that the definitions listed are not always easy to read for the really young children, it would be nice for older kids.  In addition, there is a link to a place with all sorts of math posters that could be downloaded and used in the classroom.  

This is a site well worth checking out because it has math resources that I've never seen before.  There are just too many to talk about here in my blog so check them out yourself and see if you find activities and such that you can use yourself.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If you laid all the iPhones and iPads end to end that are sold in one year, they will cover half the circumference of the earth.  So if the circumference is about 25,000 miles, how long would the line of iPhones and iPads be in kilometers?

Warm-up

 If an Olympic swimming pool is 164 feet long and a beaver can swim the length 16 times without taking a breath, how far can he swim in miles before taking another breath.

Ways To Improve Student Engagement.

 Sorry about yesterday.  I live in a small place in Alaska that has at best 2G phone service and internet that can be problematic.  It went down after I'd gotten my first entry for the week and went down right after so I couldn't do anything, even at work. So I am back and on time with my Friday entry. Today, I'm revisiting student engagement since that seems to be an ongoing struggle.

Fortunately, there are some things teachers can do to engage students so they are more involved with our lessons. One of the easiest things is to use manipulatives either virtual or physical.  Don't be afraid to use manipulatives in middle or high school since they can be used for multiple situations. 

For instance, grab some of those manipulatives that are small cubes, strips of 10, and squares of 100.  Instead, use the small cubes to represent numbers, the strips represent the x's and squares to be x^2.  You can use them to teach binomial multiplication and trinomial factoring.  You can also use them to teach combining like terms in algebra.  If you have two different colored sets, one set represents positive and the other negative.  I've have also used these for x^3 by combining x^2 with a single x to make x^3.   This type of manipulative can also be used to help students visualize properties of exponents, the distributive property, and even basic fractions.

In addition, it is important to provide differentiated instruction since most of us have students at a variety of levels even though they are in the same class. Sometimes, it can be instruction so one starts with the basics for those who need it, moving up to the more complex application of the same concept for those who are more advanced.

Don't forget to include math games.  When students are having fun, they really enjoy themselves.  One can use Kahoot, Quizlet, or other type of online game.  One can also arrange scavenger hunts, gallery walks, and so much more. 

Furthermore, take time to show how math is related to the real world.  Let students do activities where they are involved and consider assigning project based learning to have them more involved.  Due to a Covid outbreak, I have less than half of my normal number of students, so I am having them take recipes, enlarge them, calculate ingredients in cups, pounds, gallons, and quarts as if they are a real bakery.  Once they figure out what they need each day to bake 1025 loaves of banana bread, they will calculate the cost of purchasing the supplies needed to make that.

Finally, provide opportunities for students to explain their thinking to others. When you use word problems, substitute names of students so it is more relatable and use places they know.  When I have automobile problems, I usually have to rewrite them to use things like snow machines or boats since that is the preferred method of transport in places I work.

Give it some thought as to how you can get students more involved.  Let me know what you think, I'd love to hear.  Have a great weekend.

Ways To Teach Fractions In Middle School.

 

One quick way to find out if you middle school and high school students know fractions is to ask them to make a square that is 3.5 by 3.5.  I did that for art and a few of my students didn't know they should start at zero.  They ran their lines from one to three and something.  Others weren't sure where the 1/2 mark was, if there was no 1/2 written.  This showed me they didn't have a firm grasp of relating fractions to a ruler which acts as a number line.

Although, many of the methods mentioned are used in elementary school, they can be used in middle or high school with a bit of modification.  It is still highly recommended that students connect from concrete, to visual, to abstract for the best way to fully understand the topic.  We know that when students use physical and visual representations, it helps them build fluency.

One thing we often forget when teaching fractions is that in addition to being part of a whole, they are also units in and of themselves.  So the denominator is the unit and the numerator is the number of units. So when we see 3/2, it means three units that are say 1/2 inch wide.  It is also 1 1/2 or one whole and half of a whole so many fractions have multiple identities. The reason to show fractions on a number line is that it shows they can be counted.  It is important for students to understand that 1/denominator is the basic unit for this fraction.

Furthermore, it is important students understand equivalent fractions for instance, if you fold a paper in half and color in the half, it is the same area as if you folded a paper in four and colored in two of the squares, or folded it in eight and colored in four.  Although they look different, they are really the same amount.  In addition, this shows that there are multiple representations of the same number which is important for changing fractions so they have the same denominator.

Many of these activities can be done via paper folding, tape diagrams and circles, area models, and number lines so students see fractions as units and numbers, as equivalent fractions so they see that one fraction can be represented in more than one way, and then add, subtract, multiply, or divide fractions.  When I researched this, I ended up with some great ideas to help me provide scaffolded instruction for a young man who seems to have to idea on how to do fractions without a calculator.  So I will be applying them. Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

In 2010, a study was done to see who texted more, girls or boys?  They found that girls texted about 80 messages per day while the boys only texted 30 per day.  What percent of the girls texting rate did the boys text.  

Warm-up

 If there are 27 bones in each hand, including those in the wrist and there are 206 bones in the body, what percent of the bones are found in the hands?

Fact Family Triangles

 

Since I teach high school, I am not always aware of the latest developments in the younger grades.  A friend turned me on to fact family triangles.  I actually like them better for showing the relationships between numbers since the standard addition or multiplication charts do not.

If you have not seen them yet, they are triangles with the sum or product at the top leg of the isosceles triangle with the two numbers that combine to make them on the other two legs.  I am more familiar with the multiplication ones since those are the ones my students use.  

The triangles make it easy for students to multiply or divide and they can easily see the relationship between the product and it's factors.  In addition, students who have difficulty distinguishing rows and columns in a regular chart, have only the three numbers to deal with and there is less confusion.  Furthermore, fact family triangles make it easier for students to write mathematical sentences with only three given numbers.  Students can also develop their critical thinking skills and ability to analyze while learning about and expanding their knowledge about inverse operations.  The triangles allow them to see that subtraction is the inverse of addition and vice versa or division is the inverse operation of multiplication and vice versa.  

Thus using fact families help students learn their basic facts, builds their confidence and knowledge of basic number facts while developing a solid foundation of algebraic thinking.  Consequently, students have an easier time when they begin solving one and multiple step equations because they are already familiar with the concept of inverse operations. 

In addition, the use of fact families helps students build more flexibility into their number sense because they are exposed to seeing more than one operation at a time.  Now, there is some argument that learning each operation individually is better because they are more fluent with the facts but due to COVID, many students did not accomplish the fluidity so they need scaffolding.  

I think using fact family triangles with middle and high school students is a really awesome way of scaffolding.  It helps them see the relationships they might have missed and it allows them to build number sense.  Let me know what you think, I'd love to hear.  Have a great weekend.

How Important Is Factoring?

 

As I wrote the blog on the importance of fractions, it made me wonder how important it was for students to be able to factor in both elementary and in the upper grades.  We know that factoring helps students simplify fractions,  find lowest common multiples or lowest common denominators, ratios, and so much more. Unfortunately, we tend to teach factoring as something that is mostly done on worksheets with much less connection to how it could be used in a much more interesting manner.

In addition, factoring which is the breaking down of a number or mathematical expression or equation into smaller parts.  It allows people to reverse engineer equations and this skill is used in many careers including data analysis and application. So factoring is extremely important to teach but not everyone has learned the skill.

Often factoring is taught as an extension of multiplication by asking a student who is trying to figure out the factors of 6 - "What two numbers do you multiply together to get 6?".  If a student is not good at multiplication, they will struggle to figure out the answer and this struggle continue into high school where they are expected to know how to factor more complex terms.

I've found, it is worth taking time to teach students different ways of factoring when they get to high school.  I am a firm believer in using the rectangle as a visual to show how to factor numbers.  I am not sure what else it is called except the area model where you draw a rectangle using the factors that make the number.  If you have 6, then you arrange it as either one by six or two by three.  If you have 24, you might arrange it as six by four, eight by three, two by twelve or twenty-four by one.  So if you want to find the greatest common factor it would be six because there is a six by one in both examples.

Then when moving to binomials and trinomials, I have a method of binomial multiplication has students use a visualization, the same visualization I've used to teach elementary children two digit by two digit multiplication. (I will see if I can cover that in another entry).  The terms go on the outside and the product on the inside. When it comes to factoring I start with the visualization by drawing the products inside and working outward.  From here, I can move the students to the grouping method since that is one of the best ways to teach factorization of quadratics and some polynomials.  The visualization actually shows the grouping so it is easier to see. The visual step is important because it helps tie standard multiplication to binomial multiplication

If students can find connections between multiplication and factoring visually, they often find it easier to transfer knowledge from one concept to the next. Let me know what you think, I'd love to hear.  Have a great day.

Why Knowing How To Do Fractions Is Important.

 

I have a student who wanted to do his fraction worksheet with a calculator.  He didn't see any reason to know how to do the problems if he could use his calculator and get the right answer.  He even showed me a website that shows the steps but he was unwilling to copy the steps down since he'd just looked at them.  He felt the answer was the only important part of the problem.  I explained that showing his work was the way to communicate to me, how he solved the problem from start to finish.  So today, I am looking at why it is important for students to learn to do fractions.

It turns out that fractions are the most frequently found numbers used in mathematics.  In addition, fractions provide an important foundation for when students study more advanced mathematics. If you teach high school mathematics, you know that students have difficulty when they have to solve an algebraic problem containing fractions.  Many students never get a solid foundation in working with fractions.

In addition, learning fractions is really a student's first introduction into the abstraction that exists in mathematics. In other words, Fractions provide the best foundation for algebra in later years.  Furthermore, developing a number sense about fractions, helps us better understand division, and really small numbers or really large numbers broken into manageable sizes. Fractions also help students understand numbers and their interactions.

Students need to develop an understanding of fractions so they see why you have to change fractions with unlike denominators into fractions with the same denominator, how to compare fractions so we know which one is bigger or smaller, and so many other things.  This is important because we use fractions in cooking, in construction, with tools, and so much more.  If students cannot operate in fractions, how do they know that 7/16 is smaller than 5/8 when looking for a screw?

This is why many of the current instructional materials include lessons with number lines, or models.  It helps them "see" or picture fractions better which leads to a better understanding of the concepts.  Furthermore, number lines allow students to compare fractions in a way that is not possible using the traditional pie charts.  

There have been studies done to show how important it is for students to gain a good fundamental knowledge of fractions.  One study shows that how well a student understands fractions in fifth grade is a good predictor on how well they will do later in mathematic courses.  Furthermore, it is important for students to intuitively understand the concepts rather than just memorizing so they have a connection between things.

Due to COVID, many students missed out on learning fractions when they should have.  For this student, I've gone back to the basics, finding material that he can use that has number lines, models, and walks him step by step through things.  I'm also going to assign him some BrainPop materials to help him with his learning.  

I'm sure we all have students who are in this same spot.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If 1,000,000,000,000,000,000,000,000 snow crystals fall from the sky every year, how many would that average per day?

Warm-up

Did you know the worlds richest man is worth $60 billion dollars. If you can buy one million sports cars, how much does each car cost?

 

Squares And Rectangles.

 I came across this article published in the New York Times on a topic that is always quite interesting.  There is a person who is working on a Ph.d. The topic is on "computing motives of moduli stacks of vector bundles on stacky curves." Quite a mouthful but when it is translated into plain English, it is about trying to figure out the different ways squares can be divided into similar rectangles.

The topic came from a puzzle that was originally posted on Mathstodon stating that a square can be divided into three rectangles with the same proportions.  They were able to divide a square into three equal rectangles, a square into rectangles that are 1.5 times as long as they are wide, and the last where they are 1.75847 times as long as wide.  The author posed the question of what would the proportions of the rectangles if a square is cut into four rectangles. 

Another mathematician sketched out how to find the solution but didn't have the time to actually do it so others did.  In fact, people came up with 11 ways that a square could be divided into 4 rectangles with similar proportions. Some of the people created pictures to illustrate the 11 solutions so people could see them rather than trying to visualize them from a description.

The student, Ms Taams,  who is using the idea for her Phd at one point posted an 11 part thread explaining the math behind each solution. She also took time to show how this problem is connected to certain formal math.  In plain English, her solution showed how the ratios are connected to algebraic numbers which is a major topic in Number Theory. 

One person discovered you can use electrical circuit theory to help solve this if you think of the width and height as voltage and current. This way you can "square the square". Furthermore, it lead to people wondering about cutting a square into 5 or 6 rectangles of similar proportion. Upon further inspection, people came up with 51 solutions for 5 similar rectangles in a square and 245 possible solutions for 6 similar rectangles in a square. The number of possibilities jumped to 1,371 solutions for seven rectangles but they are still working on eight rectangles.

So now you know more about the topic of finding the number similar rectangles cut from a square.  Let me know what you think, I'd love to hear it. Have a great weekend.

Valentine's Day Math Opportunities

 

Valentine's day is sneaking up on us. Instead of doing this column the day before or the day of, I thought I'd share some resources early enough to plan. It is nice to have a choice of activities to choose from so students have options. 

Many of these range from fairly easy things like graphing a heart to something more that requires a higher level of math but there should be something for every level.

Let's start with a heart shaped pizza activity at Emergent Math. Back in 2011, Papa Johns pizza chain offered two different heart shaped pizzas, one with one topping, and a second one offering two heart shaped pizzas for about $6.00 more.  The author could not find any information on size but he does suggest asking open ended questions such as how would you cut this pizza, or how would you go about finding the area of the pizza, and so many more ideas.  It is one of those activities with no "correct" answer but so many possibilities.

As of last year, Yummy Math had activities for both Valentines and Presidents day since they are both fairly close to each other. One note here, Yummy Math will be retired beginning January of 2024 so you might want to check everything out and download what you want.  For 2022, they have a web page offering 11 different math activities for Valentine's day.  One deals with the return of sweet heart candies.  The original company went bankrupt in 2018 but another company took over so students will look at this, the number of candies sold, etc.  There is an exercise on where flowers come from for this special day, along with so many other possibilities.  Check it out.

If you don't want to spend all class doing a full math lesson, what about incorporating math into art?  This site has a clever twist on the Sierpinski Triangle activity that uses hearts instead of triangles.  It also looks at different ways of making cartoids, and a mobius strip for Valentines day.

If you didn't know, Science Friday did a wonderful lesson on hearts.  The lesson covers from how to draw one using a square and one circle to translation of the hearts, dilation, reflection, and rotation.  In addition, each section has a challenge and at the bottom, it explores combining some of these such as rotation and dilation.  In addition, someone created a lesson at Geogebra based on this so it is already to go if you want to do it.

So have fun choosing which one of these activities you want to do in your class.  Let me know what you think, I'd love to hear.  Have a great day.

How To Facilitate Mathematical Discussion

 

You just started working at a new school.  The students you have are used to listening to the teacher, taking notes, and doing the work. They look at you strangely when you ask them to talk to each other because their last teacher never had them do it.  How do you get them to start talking to each other in a meaningful mathematical way.  I've faced this exact scenario and it can be difficult.

So today, we are going to look at ways to encourage student discussion in your classroom because we know how important it is.  We know that mathematical discussion helps teachers assess student understanding, improve student progress and proficiency, and helps students see themselves as able to do math.

One of the most important things is to foster community within the classroom so students feel safe in expressing themselves.  Students also need to become active listeners so they can listen to what others say along with being celebrated with then communicate. In addition, we have to encourage discourse and help students learn to share how they completed a problem, or how they figure out the approach.  Many students arrive in your classroom without knowing how to share their thoughts. It is important to let them know that engaging in mathematical discourse helps them better understand mathematical concepts. 

As far as posing a question to encourage mathematical discussion?  The question does not always result in a conversation because talk begins with student interaction. When students interact in pairs, small groups, or whole class, they have to be taught to construct viable arguments and to critique each other's ideas. So we have to train students to get the high level of conversation they need. So we have to provide low-risk opportunities for them to 

One way to do it is to encourage something called "rough draft" math thinking.  This is where students are encouraged to share their efforts at solving the problem with their peers.  Students share this information with each other, comment on the attempts, and then revise their work based on feedback and then explain how their thinking changed during this process. In addition, when students talk about how their thinking changed, it gives teachers the opportunity to highlight the value of their work. This in turn shows students how their contributions helped someone else grow.

Another option is to use small groups because many students are not ready to share their ideas with the whole class and this provides a safer situation for them. Small groups is the perfect setting for students to tackle word problems.  When the teacher checks in on the group, they need to ask open ended questions such as "Why do you believe that?" because it helps guide the discussion while deepening their thinking.

Of course, as a teacher we need to model math talk but we don't have to do it out loud all the time.  We can also think about using those cartoon thought bubbles to help model our thinking process.  The teacher models how to solve the problem, then circles a certain part of it before drawing in a thought bubble.  Teachers ask students "What am I thinking here?" pointing to the thought bubble and students can write down their ideas on paper in a text box.  

I will revisit this topic at another time to share more information.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

So if an Olympic sized swimming pool contains about 660,000 gallons of water, how many gallons spill over Victoria Falls in one hour if the amount can fill 1635 Olympic sized swimming polls in that time?

 

Warm-up

 

Victoria Falls in Africa spills enough water during the rainy season to fill 1635 Olympic sized swimming pools.  So how many swimming pools could be filled in one day? One week? One month?

Why Teachers Let Students Use Multiple Methods To Solve Problems.

 

I tend to check a few websites to see the latest in math news and I found an article explaining why teachers allow students to use a variety of methods to solve problems on Physics.org website.  It begins by discussing the change from telling students how to solve problems to letting them explore and use multiple ways to solve problems. 

There has been a change from using the exact same resources and assessments to providing a variety of resources and assessments to see what they've learned.  Furthermore, most of the methods used to teach in the past were based on the group in charge so in the past, teachers showed their way of solving problems as the only way but not everyone see's math in the same way. This sometimes put students at a disadvantage. 

There may also be cultural differences in what method is chosen to teach students. For instance, in Japan, students learn to multiply using times tables while in the Philippines, students use finger multiplication.  I remember in upper elementary/middle school, they had a method for multiplication that I just couldn't get so my father taught me the way he learned and to me, that made so much more sense. 

When teachers draw on the student's prior knowledge, it helps them determine the best way to present the material to them. In addition, the use of collaboration is considered a good tool because it allows students to have their individual voices heard when they work with others.  

Overall, the focus is on a classroom that allows equitable experiences for all students.  It includes what one should look for in an equitable classroom, what it sounds like, and what it feels like.  Some of the ideas include focusing on how students get the answer rather than focusing on the "right" answer, instructional time happens only after students are engaged, teachers ask students what they are thinking about, and they wait till the students ask for help rather than offering it.  

I enjoyed the article as it made me think.  Let me know what you think. Have a great day.

New Prefixes For Numbers

 

In today's world, we are producing more and more data so the current descriptive prefixes we have are not enough. This past November, members of the General Conference On Weights and Measures met just outside of Paris to discuss adapting several new prefixes for extremely large and extremely small numbers.  These prefixes will become part of the International System of Units (SI) and should now be a part of the system.   

These numbers are so big and so small that we as humans cannot picture them.  I have no grasp of them, even with examples so they fall under the category of mindboggling. 

Let's start with the biggest numbers.  The two new ones are ronna and quetta.  Ronna is the prefix representing 10^27 and quetta is the new prefix designating 10^30.  To put it in perspective, the earth weighs one ronnagram.  The other two are the ronno which is the new prefix for 10^-27 and quecto for 10^-30 and the mass of a single electron is one quercto.

These are the first prefixes added since 1991 when they added zetta at 10^21, Zepto at 10^-21, yotta for 10^24 and yocto for 10^-24 as a way of helping meteorologists adapt to the chemists Avogadro number.  The lastest additions come from the world of data science.  The amount of data produced world wide is already in the zetta range of 10^21 or roughly 1000000000000000000000 pieces of data.

As it becomes necessary to add these prefixes, there are certain naming rules to follow such that all large numbers end in a and the small numbers will end in o. In addition, the words chosen need to sound similar to the way Greek or Latin numbers sound.  The zepto and yotta are similar to the Greek words for nine and ten.  The original word for 10^-27 was quecca which was thrown out since it was too close to a Portuguese swear word. This is one reason why it takes so long to come up with the names is that it involves a lot of discussion before anything is settled on.

Right now, they only need the prefixes for the really large numbers but they still make sure there is an equal one for the smaller numbers so that everything is balanced.  As for the future, it is proposed that scientists start combining already established terms to create the new terms such as kiloquetta rather than coming up with new alphabets.

So we have four new prefixes and it may be a while before the next ones are proposed and accepted.  Let me know what you think, I'd love to hear.  Have a great day.