YPS Math Start UP! Workshop 2010
Presenter: Ann-Marie Hunter, Dufferin Elementary - Kamloops
To make your Math program more successful, please consider adding these activities to your regular Math units as integral parts of your teaching.
1) Mastering the Basics - a program devised by Trevor Calkins
Ø Handouts in the workshop outline the program.
Ø The CD you receive has all the files and explanation you need.
Ø Discussions in the workshop cover reasons for using the program and specific methods that will enable you to successfully implement the Mastering the Basics program.
2) Basic Facts/New Concepts Review: “Number Magic”
Ø This is a daily activity that is one way of focusing on the learning and representing of Basic Facts as well as New Concepts Review. It also offers an opportunity for students to talk about their understanding of math and helps to create a risk-free environment.
Ø Each day, a question for Number Magic is written on the board or overhead. I will usually make up one that is either a basic fact or is related to the unit we are currently studying. Often, I will ask students to devise questions and hand them in beforehand (beginning of the day or the start of math class, perhaps). All students write their solutions in their booklets. One student per day presents his/her solution, explaining his/her thinking. This activity is one of my “start up math class” activities, organized by students. It becomes a very smooth, teacher-friendly, opportunity to review basic facts and current topics as well as an opportunity for students to be presenters.
Ø Students use half page Basic Facts Journals (cut key-tabs in half) to record their process for finding the answer. The teacher needs to outline how he/she would like the pages set up.
Ø I request that students show the mental math used or a visual representation of their thinking. As well, they are encouraged to include a real life application for the question.
Ø The student leader, “Math Expert of the Day”, chooses the student who presents his/her work on the board or overhead. Others are encouraged to comment on how the given solution is like or different from theirs.
Ø Throughout the term, I collect these notebooks and use for assessment and evaluation.
3) A Vertical Number Line is a tool used to help students visualize numbers and their relative values. The vertical number line is not used as a counting board, just a place where they can see how the numbers are related. Their skills of comparing and ordering are more easily developed when the vertical number line is revisited often. I also find it to be an amazing tool for estimating. For example: You can ask students, “566 is closest to which hundred?” Students can actually see it (as they can for smaller numbers on a regular horizontal number line. The vertical number line is extended to 1000, so I find it more useful as it can be applied to larger numbers.) Please see Wiki for copies of this resource.
4) Teach Multiplication Facts: “After learning to read, learning the multiplication facts is probably one of the most important things we learn in elementary school.” - Trevor Calkins. This statement rings so true with me, as I have experienced many years of watching students suffer from math anxiety because they have not mastered the multiplication tables. Here are some ideas to help your students learn the multiplication tables:
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Ø Help students to see the patterns in the times table and to focus on which facts they can easily know, using those to generate their knowledge of the other facts. Following the pattern of learning each times table question as the smaller digit times the larger digit reduces the number of facts by 50%. Students are simply shown that, for ex., 8 X 3 is equal to 3 X 8. Therefore, they do not need to learn 8 X 3, since they already mastered 3 X 8. The 12 times table becomes only 12 X 12 = 144!
Ø Their goal is to learn the facts, but it is important that within the learning, they focus on the visual. Use patterns on the times tables to focus on what the students already know and then expand this knowledge by looking at related questions. For example. The 5 and the 10 times tables are easily learned and can help students to find answers to other times tables questions. Model and discuss strategies like doubling or halving to find answers. Some Times Tables Patterns as well as some cool tricks are explained on this website: http://home-ed.info/maths/multiplication_tricks.html
Ø Classroom discussion is so important in helping students realize their understanding of the value of numbers and relationships between numbers. Talking with others (either whole class or in pairs) will enable students to experience the use of mental math to develop their confidence with the facts. Making your classroom a place where students can openly, in a risk-free environment, describe how they reached an answer to a question will encourage students to talk about their understandings and will give them the outward practise for their internal thinking.
Ø Trevor Calkins’ book Power of Ten - Learning to Multiply and Divide Strategies outlines the steps for teaching the multiplication facts using episodic memory and the visual brain. I highly recommend this book! www.poweroften.ca
Ø Other ways to represent multiplication facts are:
1) Cross-lines - count the intersections 2) Arrays Colour in graph paper to represent
to calculate e.g. 3 X 6. specific multiplication questions. (E.g. 2 X 7) Then find the
These do not neet to be drawn exactly at answer either by counting squares or identifying groups of
all - students just draw the lines with the ten (as in power of ten cards) and adding on other squares.
vertical and horizontal likes intersecting 2 X 7 = 10 + 4 = 14.
3) Draw groups of items and then count them to calculate the total xxxx xxxx
xxx xxx
4) Build models of groups of things with manipulatives
Ø Be sure to point out, with diagrams and models, the related division facts at the same time as examining multiplication facts.
Additional Resource to make learning Timestables Fun:
Timestables the Fun Way! - Judy Rodriguez - a contextual representation of the multiplication facts.
Kids love them!
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5) Mental Math Instruction
Ø Make a point of mentioning, aloud, specific calculations that you do without pencils to give students more experience and more confidence with mental math skills.
Ø Teach multiplication of powers of ten early in the year. Ex: multiplying any number by 10, 100 or 1000 as well as multiplying two (or more) multiples of ten together.
Example: 30 X 200 = (3X2, the significant digits) with the total number of
zeros in the question (3) ‘placed’ (do not say ‘added’) on the end. Ans. = 60 000.
Ø Use mental math to ‘debug’ the multiplication algorithm. See the file: Multiplication Algorithm Station on the Wiki.
Ø Offer students examples of mental math tricks. The first two examples are actually like tricks. Including the last two examples as ‘tricks’ motivates students to learn to use them.
a) Multiplying by 11 -
§ the last digit of the number is the last digit of the answer
§ add the two digits to find the middle digit of the answer
§ the first digit of the number (plus any carry from the middle sum) is the first digit of the answer Example: 24 X 11: the last digit is 4, the middle digit is 6 and the first digit is 2, so 24 X 11 equals 264.
b) Squaring 2-digit numbers that end in 5 -
§ multiply the first digit times one higher and complete the answer with 25. Example: 35 X 35: 3X4 = 12, so the answer is 1225.
c) Adding and subtracting from left to right - uses place value principles. For example:
24 + 63. Help students to see those numbers as (20 + 4) and (60 + 3). Then the mental math calculation becomes (20 + 60) + (4 + 3) = 80 + 7 = 87. Another example: is
17 + 54 = (10 + 7) + (50 + 4) = (10 + 50) + (7 + 4) = 60 + 11 = (60 + 10) + 1 = 70 + 1 = 71.
d) Multiplying from left to right - uses place value principles and multiples of 10, 100, etc. skills - This is a big concept for mental mathematics! This example shows the meaning of the question rather than the multiplication algorithm at work. Include an estimation of the answer before beginning.
Example: Calculating 34 X 26, encourage an estimate of the answer by rounding one number up and the other number down. A very general estimation might be 40 X 20, which students would easily calculate as (4 X 2) with two zeros placed at the end, or 800.
To use mental math to actually calculate, we can multiply 30 X 25 = (3 X 25), with a zero placed on the end, = 750
Plus . . . 4 X 25, which is easily seen as 100 (using money examples)
Plus . . . 34 X 1, and the answer becomes, 750 + 100 + 34, which is 884. Then check this with your initial estimate to see that your estimate was in the same hundreds! Good.
Offering students examples like this opens up their understanding of the value of the numbers and makes them feel confident about using their heads rather than a calculator.
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Ø Involve students in making up mental math questions; this way, the mental math quizzing is less of a mystery and more fun. Encourage students to use the “Quick Think” methods discussed: adding numbers from left to right, doubling, or using the proximity to 10 to find answers to calculations.
Ø Developing students’ mental math abilities is a huge step to reducing math anxiety!!!!
Games and Tricks Resources:
Ø You will find many games in your textbooks. Take the time to learn them and prepare to give students time for playing during class time or when they are finished other assignments. If you don’t have time to learn a game, assign the learning and teaching of the game to one of your students!
Ø Family Math is an excellent resource for games that develop skills. The games are presented for a parent to be playing with his/her child, but can easily be adapted to the classroom setting.
Example: Palindromes page 214 to 216. (see the files on the Wiki) To order the book, here is the website address: http://lawrencehallofscience.stores.yahoo.net/familymath.html
All other resources mentioned in the workshop are listed on the Wiki
in the file: ‘2010 Math Workshop Resources’
6) Teach Place Value to help students read and write numbers. Stress that they need only know how to read any 3-digit number (e.g. 475) and they can then read any big number by saying the 3-digit number and reading the name of the ‘space’, which identifies the value of the number (thousand, million, billion). Stress the importance of saying the word ‘and’ only when they come to a decimal. The decimal indicates the beginning of the fractional part of the number, or the part that is less than one.
The ability to read and write numbers easily helps students to feel confident with numbers. After they have developed the skills of reading and writing big numbers, they can be guided to identify the value of any specific digit by reading its place in the period (the 3-digit section) and then reading the name of the first space after the number to give its value. See the files listed on the Wiki for handouts.
7) Problem Solving methods - to help students feel empowered to think aloud and to share their ideas for Problem Solving:
Ø I use the term ‘word question’ rather than ‘problem solving’ because I think it is less anxiety-producing; problem and solving are terms that sometimes create stress in people! As well, the word questions are just the same as what the students do in the daily “Number Magic” activity.
Ø Begin with students creating the word questions and later on introduce the idea of students answering word questions. They develop a more comfortable sense of how they can do it! Encourage students to use ideas from other subject areas in devising their word questions. The activity can be set up to exchange questions for responding. Building on the regular Basic Facts drawing/modelling, encourage students to answer word questions the same way. Allow time for students to show/talk about how they worked it out; the modelling/sharing provides students with many different examples of processes used.
Ø Within each math unit being covered, try to give opportunities for students to use the new math skills to answer word questions. Look for word questions in the text, or ask students to make up questions that would use the skills they are working on. Again, this is reflected in “Number Magic”.
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Ø You will need to be clear about how to write word questions.
1. Teach students to recognize mathematical terms: factor, product, sum, difference, quotient, multiple, and total (let the students add their own). These can be taught while you are discussing the basic facts. You might want to use the Word Question Vocabulary Poster (included on the Wiki) for student reference. Make a rule that the word question must use at least one of these terms.
2. The word question must be clear and understandable, containing all the information needed to find the answer.
Ø One activity that is quite powerful and that encourages discussion and sharing of ideas is the ‘PLACE MAT’ activity. One question is written in the middle of a large piece of paper and the paper is then divided into the number of sections needed to be used by the number of students. Students take a section of the paper to write, draw and describe their interpretation of the questions and how they approach finding the answer. Encourage students to ask others to explain their processes. Place Mats can be posted to further enrich learning.