Representing Decimals September 25, 2013

For those of you teaching decimals, here is a sheet you might make use of to have your students represent the decimals in a variety of ways. The separate place value part at the bottom allows students to visually see the relative size of each of the place values in tenths, hundredths, and thousandths. In using it in a classroom here in my school district I was surprised how challenging it was for students to actually fill in — which of course means we are surfacing misunderstandings or lack of understanding in the students.

I would be interested to hear how your students do in the task of representing decimals.
Mathematically yours,
Carollee

Download the “Representing Decimals” sheet here.

Take Graphing to Great Heights—One for the Ceiling!! September 18, 2013

I have been thinking about writing about this graphing idea, and with the fall equinox upon us, I wanted to make sure and get this posted! One of the best graphs I ever did in my classroom was one that used data collected over time: namely, we measured the length of the shadow of a metre stick each month, at noon, around the 20th/21st of the month (there would be some slight variations of the date as we needed full sunlight to see the shadow, so if it were cloudy for some days, we measured on the first available sunny day!)

I would ask two students to do the outside measuring. One student would hold the metre stick perpendicular to the ground. Under the tip of the metre stick would be the end of the roll of string. The second student would roll out the string to the length of the shadow, and then cut the string.

They would bring the string inside and measure the length of the string. Additionally they would cut a piece of masking tape (blue or green work best) the length of the string which we used to construct a graph on the ceiling. Across the front of the room, close together, were tags for each month of the school year (September through June for us). For each month we put up the length of masking tape showing the actual length of the metre stick’s shadow for that month and recorded the measured length on the tape.

It was amazing the difference in length of the shadows, especially comparing December and June, the winter and summer solstice months respectively. I live and work in northern British Columbia where we have relatively short winter days (though certainly not as short as in Dawson City, Yukon where I lived for over three years!) Still, even here in Fort St John it was interesting to note the change in the shadow over the months. The first time I did this with a class, I went home that evening excited to tell my husband how long the December shadow was, but he did not believe me! He thought it seemed too long, and he was sure the students had made an error in measuring. He and I had to measure the shadow of a metre stick at home that weekend so he could see its length for himself before he would believe how long it was!

I wish I had taken picture of one of the “shadow graphs” my classes had made on the ceiling of the classroom. It was a powerful representation of the combined effect of the earth’s orbit around the sun along with the tilt of the earth.

I would love to hear about this if you do it with your class. I have a couple of classes in my district that are about to embark on the measuring task in the next few days. I am looking forward to seeing the graphs on the ceilings of those classrooms!

Mathematically yours,
Carollee

PS: Alternately, one can graph the shadow of a metre stick over the course of a single day, measuring the length of the shadow every hour or every two hours from sun rise to sunset. That is also fascinating data to collect. It might even be interesting to gather both sets of data!

Developing Math Vocabulary September 12, 2013

Mathematics, like many subject areas, has some terms specific to discipline. Additionally, there are words that have uses in everyday language but a specific meaning in math (like “product”, “root”, and “obtuse” just to name a few). Within mathematics itself are some strands that are particularly vocabulary rich, such as geometry and measurement.

There is also the issue in most schools where some portion (in some cases a large portion) of the student population are English Language Learners , ELL, (or termed English as a Second Language students, ESL).

Clearly there is a need for teachers to be proactive regarding helping students learn the various terms that we use regularly in the mathematics classroom.

One easy way to support math vocabulary is a make a Math Words chart that hangs in the classroom, always visible to students. Now, some teachers, particularly in the elementary grades create word walls of general vocabulary terms for young learners, and this is a great idea. Many that I have seen have individual words written on cards and placed alphabetically on the wall. That is a great idea, but I must confess one that for me was not very easy to keep up with on a regular basis.

I am suggesting, instead, that you give math its own sheet so you can add words easily at any time. You need only start with a few words at the beginning of the year and ask your students for suggestions of words to be included. As new words come up in the course of the year, add them. I have often had students in my class prompt me to do just that – they would stop me during our math work and inform me that a certain word needed to be added to the chart. Students used the chart regularly when writing about their thinking. In fact, many times I would see a student sitting, not knowing what to write, scanning the Math Words chart. Finally one term would spark something for him, and the writing could begin.

Having the words posted also reminded us all to use the words in our math discussions. Instead of calling a blue or tan Pattern Block a “diamond”, we would use the correct mathematical term “rhombus”.

The picture of the chart posted here is clearly one used in an primary grade, but the chart is easily adapted to any level. If a phrase is used (such as “ten frames”) one colour is used to show it is a phrase. Otherwise words are written in any colour, multiple words to a line. If possible a small symbol or “cue” is added beside a term to prompt students to remember the meaning of the term.

In particular units of study (such as angles) where there are many new terms, it may be helpful for students to do deeper vocabulary work with the various terms. Using a Frayer Model is helpful for that. (Click here for information about that.)

I hope you will put up a Math Words chart today if you do not already have one up!
Mathematically yours,
Carollee

Ten Uses for Sticky Notes in Math Class September 9, 2013

 Sticky notes (that wonderful — though accidental — invention that 3M first marketed as Post-It Notes©) are a wonderful tool to use in mathematics. I have always found that students enjoy using those little sticky pieces of paper! So here are ten ideas for incorporating them into math lessons:

1) Ordering numbers: Write 4 or 5 different numbers on sticky notes and have students work in pairs to put them in order from least to greatest. This is great practice for multi-digit numbers, for decimals, for fractions, etc.
2) Creating numbers: Write 4 or 5 different digits on sticky notes and have students work to create specific numbers: greatest possible; number closest to a particular given number (whole number or fraction); a number between two given numbers, etc.
3) Using operations: write several digits on sticky notes. Then use them and operations that the students know how to use to make number equations with as many different answers as possible or to get as close as possible to a particular answer.
4) Writing word problems: Give students numbers, operation signs, and possibly other math symbols (such at a percent sign) on sticky notes and have them create a word problem that uses all of the notes.
5) Graphing: Create the axis of a graph on chart paper or the chalkboard. Add your categories (4-6 work well). Give each student a sticky note and have them create a bar graph with then .(I usually have kids make their choice at their seats and write it on their notes so they don’t just add to a single category to make it “win”.)
6) Scavenger hunt: each student has a sticky note with math geometry word on it. Students must find an example in the room to represent the term and place the sticky note there (e.g., perpendicular lines, acute angle, sphere, etc.).
7) Measuring area: Cover a book or other object with sticky notes and calculate the area using the notes as the unit of measure. A particular book or surface may be covered by notes of one size, then by notes of another size and the area calculations compared.
8) Estimating on a number line: draw a number line on the board with only the endpoints marked (endpoints may vary according to grade: 0-10, 1-100, 0-1, 20-80, 1-1000, etc.). Give each student a number that appears on between the endpoints and have them come and place their number where they think it would go and explain their reasoning for placing it there.
9) Commenting on each other’s work: teach students to peer-evaluate problem solving work. Students can exchange their papers after working on an open-ended problem. The evaluator can make comments and ask questions regarding the strategies, visual representations, etc. by writing their comments on sticky notes.
10) Transformational geometry: Use sticky notes to show transformations, often called “flips”, “slides”, and “turns”. Light coloured sticky notes tend to be translucent. Using a sticky note, student can trace a shape from its original location on a grid and then use the sticky note to show the desired transformation (e.g., down two, left three; 90 degree clockwise rotation; reflection over a particular line).

I hope you try one or more of these in the next weeks. Let me know how it goes!
Mathematically yours,
Carollee

Helping Your Young Children Learn Math September 4, 2013

 Most parents are aware of the need to begin reading to their children early in order to lay a good foundation for formal education in reading (literacy). However, fewer parents are aware of the importance of laying a good foundation with numbers (numeracy). Learning to count, to say the numbers in order correctly is a great start, but only a beginning! Here are some other things to do to build that important math foundation:

• Count everything! Children need to know that the numbers relate to things no matter their size – that the number tells how many. Count blocks, forks, steps, socks, noodles, etc. Count forward and backward.
• Sing counting songs, recite counting rhymes, and read books with numbers in them. Some song and rhymes are well known, like “One, Two, Buckle my Shoe”, “The Ants Go Marching”, and “Ten in the Bed”. But if you need more inspiration, there are lots of counting songs on YouTube that you can access. Look for books that have numbers in them such as Canada 1, 2, 3; How Many Feet in the Bed?; The Doorbell Rang, and Bean Thirteen, just to name a few.
• Use dice (with spots, not numbers), dominoes, and playing cards to look at visual arrangements of numbers (have you noticed the arrangements of hearts, diamonds, spades, and clubs on playing cards?). Recognizing five as it is shown on a dice or nine as it is shown on a domino is a helpful skill as children begin to manipulate numbers.
• Look for numbers in the environment and talk about them: look for them on signs, on buildings, on phones, on remotes, etc. Don’t just throw away junk mail and flyers – have your children use them as number search pages (for example, put a circle around each 7 that you find).
• Practice skip counting with children. Count by 2’s, by 5’s and by 10’s. Be sure to practice counting past 100. Although 100 seems a natural place to stop, children need to hear how the counting sequence goes beyond that, so keep going!
• Compare things. Describe things as longer, shorter, heavier or lighter. Compare containers to discover which hold more, which hold less. Line up cereal boxes or cans of vegetables from tallest to shortest.
• Talk about time. What time do you eat lunch each day? What time is bedtime? For how many seconds can each person hold his breath?
• Estimate things. How many steps do you think it will take to walk from the car to the store’s door? How many glasses of water will fit into the pitcher? How many minutes will it take to read this story?
• Identify shapes, both 2-dimensional and 3-dimensional. Don’t be afraid to use words like rhombus, hexagon, pyramid, and sphere with young children. They love learning names of dinosaurs that are 27 syllables long (ok, maybe that is an exaggeration!) so using appropriate geometric terms is not a problem.
• Sort things. Sort buttons, clothes, blocks, etc. Find a rule that can be used for sorting and then sort by it. For buttons, you might sort by colour, by size, or by the number of holes. Sort crayons by color families. Sort blocks by geometric shapes. You can have just 2 sorting piles (for instance, socks are white or not white) or multiple sorting piles.
• Make patterns. Use any items around the house to arrange in patters. You might make patterns with silverware: fork, spoon, fork, spoon, fork, spoon. You might make patterns with sounds: clap, clap, snap, clap, clap, snap, clap, clap, snap. You can draw patterns, too. Consider starting any kind of pattern and having your child continue it. Go on a pattern hunt in the house. Clothes, floors, and walls are great places to find patterns.

There are many wonderful ways to build a math foundation with your child. I hope you will try some of these ideas soon!

Mathematically yours,
Carollee