Wanted Posters Revisited February 4, 2014

Doing wanted posters for numbers is a great way to have kids think about specific properties of particular numbers. The posters make a great display, too —  and I am always looking for ideas for math bulletin boards!

The idea is useful at lots of grade levels. You could have older students choose a proper fraction (e.g., 5/8), a mixed number (e.g., 4 1/3), or a square or cube root (e.g. the square root of 50).I wrote about these posters before — click here for the link to the previous write-up where you can download the template.

Give them a try and send me a picture of your display!

Mathematically yours,

Carollee

Math Club for Elementary or Middle School Students November 5, 2013

You may have heard of math clubs in high schools, but math clubs are a wonderful idea for elementary or middle school students as well. For a number of years I ran a successful weekly math club at the inner city elementary school where I was then teaching. The club was mainly targeted at students in grades 4 to 7 (my school was a K-7 school) although if a grade 3 student were interested in coming, I never turned the child away.

“Euclid Club”, named of course for the Alexandrian Greek mathematician/geometer, met for 30 minutes one day each week after school. It came about simply because I felt there were so many interesting math ideas that just did not fit into my classroom time (or curriculum!) and I wanted the opportunity to share those ideas with kids. Thus Euclid Club was born! I can get pretty excited about mathematics (as anyone who knows me can testify to!) and did not have too much difficulty getting kids to come give the club a try. Our numbers certainly varied over the months depending on what other after-school activities were happening or what other out-of-school activities students were involved in, but we consistently had a pretty good group out at our meetings. I always provided some kind of small snack as well! Certainly not enough to be the main draw, but it was always welcomed by the kids.

There are many benefits to engaging children in math club. For me, first and foremost was that it gave students a chance to build a different perspective about mathematics. Many of them thought of themselves as not being “good” in math and tended to disengage in math in the classroom. The atmosphere in Euclid Club was welcoming, engaging, and lively, and many not only became comfortable with exploring math ideas, but additionally they built a sense of belonging within the club.

So, what kinds of ideas and activities can be explored in that context? Here are some of the things we explored in Euclid Club:
• We learned and played math/thinking games such as Nim, chess, cribbage, etc.
• We examined other number bases, such as base 4, base 2, and base 12. We wrote the value of base 10 numbers in the different bases. We figured out the base 10 value of numbers written in other bases. We added and subtracted in other bases.
• We worked with pentomines, trying to fits sets of pieces into given frames: 6 x 10, 5 x 12, or 8 x 8 (with either the four corners “removed” or the four center squares “removed”).
• We created designs with tangrams.
• We made pattern placemats using cut-out pattern block pieces to make interesting borders on construction paper.
• We used pattern blocks to create designs with one or more lines of symmetry.
• We measured our bodies and compared ratios (e.g., height to arm span; circumference of thumb to circumference of wrist; circumference of wrist to circumference of neck; circumference of neck to circumference of waist, etc.)
• We solved logic puzzles (using ones commercially produced).
• We created tessellations: we found shapes that would tessellate as well as creating our own unusual shapes that would tessellate.
• We made paper quilt squares in a variety of patterns and calculated the fractional part of each colour we used.
• We examined the Fibonacci sequence and looked at real-life examples of where it appears in nature (such as on a pinecone, on flowers, leaves, pineapples, seeds in fruit, etc.
• We created Moebius strips, and marked and cut them to discover interesting properties about them.
• We assigned each letter of the alphabet an amount (a = 1 cent, b = 2 cents, c = 3 cents, etc.) and looked for words whose letters would total $1.00.
• We solved magic squares and then created our own.
• We examined Pascal’s Triangle and looked for patterns on it.
• We created designs with exactly one metre of string glued on to paper (easier to do the basic designing first with dry string, then dip the string in white glue to create the final project).
• We created our own codes using numbers and wrote secret messages to each other.
• We made designs on 100-grid paper using a specific amount of coloured squares (e.g., what designs can be made colouring exactly 50% of the grid? 60%? etc.)
• We created “Guess My Number” puzzles for each other to solve. Each puzzles was to have 3 to 5 clues, first starting with a broad clue and getting more specific each time. (E.g., 1 My number is a prime number less than 30. 2 My number is not part of a pair of twin primes. 3 My number is even.)

I am sure there are other things we did, but those are the ones that I remember at the moment! I am sure you can find other ideas and topics to explore as well.

I hope you will consider giving Math Club a try (but give it a cool name! Kids love that!)

Mathematically yours,
Carollee

Math Bowling October 1, 2013

This “Math Bowling” activity is one that students tend to love! It is great for practicing math facts as well as for stretching students’ thinking.

The activity is done as follows (students alone or in pairs):
Roll three dice (your choice whether to use regular six-sided dice or include one or more different dice, such as a ten-sided die). Write the numbers in the boxes marked “strike”. Using all three numbers each time exactly once, students work to write equations to equal each of the numbers 1 to 10 of the “pins” marked on the sheet and thus “knock them down”. Students may use whatever operations they understand: addition, subtraction, multiplication, and division are standard, but students may also use exponents, roots, and factorials if those are in their realm of mathematical knowledge.

For instance, if the numbers rolled are 2, 3, and 6, students might “knock down”
1 = 6 – 2 – 3 OR 1= 6/(2 x 3)
3 = [( 3!)/6] + 2
4 = (2 x 6) ÷ 3
5 = 6 + 2 – 3
6 = (6 ÷ 2) + 3
7 = 3 + 6 – 2
9 = (6 ÷ 2) x 3

If the students did equations for those 7 numbers/pins, that would constitute the first throw of the ball. Since all the pins are not knocked down, the player may roll the dice a second time, record the numbers in the boxes marked “spare” and try to knock down the three remaining pins using that second set of numbers to score a spare. If that is not accomplished, the student scores the number of pins knocked down in the two “throws”.

If you wish, as players take multiple turns, you can calculate scores in the manner that 10-pin bowling is actually scored. As someone who was on a youth bowling league in my younger days, I know the scoring system well. There is some good math in the score keeping, too! If you are not familiar with that scoring system, here is a website which will walk you through the scoring process.

http://www.bowling2u.com/trivia/game/scoring.asp

Download the “Math Bowling” sheet here.
Download the score sheet here.

I hope you will give math bowling a try with your class.
Mathematically yours,
Carollee

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.

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

Math Bulletin Board: Square Number Towers May 23, 2013

Recently I had two of my classes represent visually the idea of “squaring” a number: namely, that a number times itself is literally the area of a square with side length of that beginning number. The students cut squares from centimetre grid paper representing 10 x 10, 9 x 9, … 1 x 1 and them glued them onto construction paper. To each square they added the multiplication fact represented, as well as showing the exponential form of the number. Square numbers show up quite a bit in secondary mathematics, and helping students understand these numbers (as well as memorizing the sequence of them!) is beneficial for them as they move on.

I am always looking for math ideas to display on a bulletin board, and I think this is a good one!
Mathematically yours,
Carollee

Factoring: A Visual Representation of Numbers January 8, 2013

Are you interested in factoring, prime numbers, and composite numbers? If so, this is the link for you!

Not long ago someone posted this link on the BCAMT list serve. When I first accessed the link I was fascinated as I watched the progression of dots on my screen, each representing the next natural number. The configuration of each number of dots revealed information about the make-up of that particular number.

It made me wish that I had had access to such a visual when I was teaching about factoring, prime numbers, and composite numbers. I thought I would pass the link on to you folks as I know some of you are, indeed, teaching these concepts associated with number theory.

Many of you will be familiar with exploring these particular concepts through the process of creating rectangles from square tiles. In this method, for example, seven can be shown to be prime because seven square tiles can be made into only one rectangle: 7 x 1. Eight, however, can be shown to be composite because eight square tiles can be made into more than one rectangle: 8 x 1 and 4 x 2.

The visual presented here offers another way for students to literally see whether or not a number is prime, and, for those which are not prime, to be able to deduce some or all of the factors from the grouping of the dots.

I hope you will use the link and the accompanying picture here to explore primes, composites, and factors.

Here is the link to the animated factorization diagrams.

Mathematically yours,
Carollee

PS: Thanks Kelli Holden for commenting on the picture and sending along a link to Malke Rosenfeld’s blog where the picture has been turned into a game! Check out this link.

 

 

Approximating the Square Root of Numbers — Play the Game! October 13, 2012

For those teachers who are looking for a way to have their students practice approximating square roots in an interesting manner, Richard De Merchant (a friend and colleague) has created a game for just such a purpose.

He explains how to play the game and adds some other helpful tips on his blog Math in the Middle (click here for the link to this particular post).

If this is something your students need practice with, then I heartily recommend that you check out this game 🙂

Mathematically yours,
Carollee

Math Camp (gr 6-8) 2011 August 31, 2011

What a great group of participants we had at yesterday’s gr 6-8 math camp! I was delighted with all of the group sharing that we were able to facilitate — there is always so much that teachers can learn from each other.

The workshop yesterday focused on several things: mental math, integers, fractions, and algebra. These areas, I believe, are important for students to master if they are to be successful in subsequent levels of mathematics.

Mental math is a skill which is only developed if practiced, and we discussed some particularly useful strategies that might be incorporated into regular practice sessions. First, mental math begins with basic facts! From there we can have students practice thins such as these:

• adding multiples of tens (MOT’s) (20 + 50)
• subtracting single digits from MOT’s (50 – 8 )
• adding single digits to non-MOT’s (39 + 6)
• adding to get 100 or – from 100
• adding any two-digit #’s (47 + 39)
• doubling numbers
• halving numbers
• multiplying by 10, 100
• dividing by 10, 100
• multiplying by 20 (by doubling and then multiplying by 10)
• multiplying by breaking up numbers (using “nice” numbers)

Response boards are a great way to do immediate full-class assessment during mental math practice.

Our focus on integers was in using “chips” to have students learn about positive and negative numbers in a very visual way. It is helpful for students if they understand the power of zero in adding and subtracting integers, and zeros can be visualized by an equal quantity of positive and negative chips. Using the chips to solve addition, subtraction, multiplication, and division problems involving integers allows students to build a conceptual understanding of integers which goes far beyond memorized rules.

For fractions, we discussed the need for students to develop number sense regarding them. Using benchmarks to estimate fractions is one way to help facilitate this. We talked about making pocket charts for both the teacher and the students to use in practicing this. (Pocket chart directions.)

As for algebra, once again, visualization was the key to helping students make sense of this generally abstract area. By having students display and manipulate equations in a concrete way on a “balance scale”, they have the opportunity to learn what are acceptable or “legal” moves in solving algebraic equations. (Balance scale.) A hands-on, visual approach to algebra allows every student to be successful in this area!

One of the teachers at the workshop is going to email me a rubric that she used in her math classes last year, and I will up-date this post with that rubric once I have received it. (later) Here is the link for the rubric. When we discussed this, the “traffic light” part was really important. Remember that students can self-assess their understanding and record it as red (“I am totally stuck.), yellow (“I am able to work some on the problem but not I am not really sure about it.), or green (“I understand this well enough that I could teach someone else.)

In the meantime, I hope you will think about how you might better teach these areas of mathematics that are critical for students in these grades.

Mathematically yours,
Carollee

Get “Illuminated”! May 13, 2011

I was talking to some math teachers yesterday, and one of the things I mentioned to them was a great site run by the NCTM (National Council of Teachers of Mathematics) called Illuminations. The site is a treasure trove of lesson plans and activities for grades K-12. Many of the lessons have interactive applets embedded in them and, as such, are suitable for using with a white board.

Searching for particular lessons is easy. You can create a specific search by choosing a grade band, math strand, and then adding specific terms (for example, gr 3-4, geometry, with the words “right angles” typed in) or choose to cast a wider net and use broader criteria in your search. There are some really wonderful ideas there, and I encourage you to bookmark the page and try some of the ideas presented.

Remember as you look at lessons and/or activities that they are grouped in four grade bands: K-2, 3-5, 6-8, and 9-12. You will need to be aware of the curricular requirements of your province or state and check the lessons and/or activities against your curriculum. It is, of course, never a problem to “stretch” students beyond the given curriculum, but for assessment purposes, it is important to assess against the standard set by the grade level curriculum.

So, here is the link:
http://illuminations.nctm.org/

Go have a look! I know you will find some great ideas there!
Mathematically yours,
Carollee