Mathematics for Teaching Portfolio 2015
Julia Chamberlain BA Honours, B. Ed, OCT
Table of Contents
Overview (Ontario Curriculum/Introduction).......................................................... 3-5 Big ideas: 5 Strands of Mathematics........................................................................................... 3 The Mathematical Processes ........................................................................................................ 4 Teaching Mathematics Philosophy............................................................................................. 5
1. Number Sense and Numeration .............................................................................. 6-11 1.1 Fractions................................................................................................................................... 6-7 1.2 Integers ................................................................................................................................... 8-10 1.3 Ratio, Rate, proportional Thinking ................................................................................... 11
2. Patterning & Algebra ................................................................................................12-13 3. Geometry & Spatial Sense........................................................................................14-15 4. Measurement...............................................................................................................16-17 5. Data Management & Probability ................................................................................18-19
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The Big Ideas The Ontario Curriculum 5 Strands of Mathematics
Number Sense and Numeration General understanding of number and operations as well as the ability to apply this understanding in 6lexible ways to make mathematical judgments and to develop useful strategies for solving problems.
Measurement Learn about the measurable attributes of objects and about the units and processes involved in measurement.
Geometry and Spatial Sense Spatial sense is the intuitive awareness of one’s surroundings and the objects in them. Geometry helps us represent and describe objects and their interrelationships in space.
Data Management and Probability Patterning and Algebra Recognize, describe, and generalize patterns and build mathematical models to simulate the behaviour of real-‐world phenomena that exhibit observable patterns.
Learn about different ways to gather, organize, and display data. Learn about different types of data and develop techniques for analysing the data that include determining measures of central tendency and examining the distribution of the data. Students also actively explore probability by conducting probability experiments and using probability models to simulate situations.
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The Ontario Curriculum Mathematical Processes: Problem Solving
Reasoning and Proving
Reflecting
Selecting Tools and Computational Strategies
Connecting
Representing
Communicating
Learning to solve problems and by learning through problem solving, students are given numerous opportunities to connect mathematical ideas and to develop conceptual understanding. The process involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results. They are able to recognize when the technique they are using is not fruitful, and to make a conscious decision to switch to a different strategy, rethink the problem, search for related content knowledge that may be helpful, and so forth. Students need to develop the ability to select the appropriate electronic tools, manipulatives, and computational strategies to perform particular mathematical tasks, to investigate mathematical ideas, and to solve problems. Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. In elementary school mathematics, students represent mathematical ideas and relationships and model situations using concrete materials, pictures, diagrams, graphs, tables, numbers, words, and symbols. Students are able to reflect upon and clarify their ideas, their understanding of mathematical relationships, and their mathematical arguments.
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My teaching for mathematics portfolio is a culmination of mathematics activities conducted by teacher candidates in the Brock University Teacher Education program. These activities were derived from the textbook, Making Math Meaningful by Marian Small. The activities relate back to grade and expectations based on the Ontario curriculum and anecdotal notes of personal reflections on the activities have been made. When it comes to teaching mathematics what I have come to explore is that the most successful activities demonstrated by the teacher candidates were those that supported a social constructivist model, including collaboration between students to engage learning. Many of the activities also asked students to use information they were familiar with or to gather from their own lives (i.e. shoe sizes, heights), or uses visual models that students can relate to (i.e. deck of cards). Using resources or information that students can relate to helps them to better contextualize problems and deepens learning by having them internalize and respond. Through creating this profile and curating these activities I have come to see the importance of differentiated instruction and how that would help particular students in their understanding of mathematics. Teaching math should enable students to have an experience, to be tactile, learn visually, and orally communicate their processes. Digital games encompass these learning styles, and also friendly competition adds another level to the learning process that allows students to commonly make mistakes without academic penalty. This portfolio encompasses the type of mathematics teacher I plan to be, where students are allowed to explore different algorithms and share those ideas with the class to reason and prove their thought process. It is also important to have students learn through experiences, connect to prior experiences to new contexts and reflect back on their learning process and redefine it.
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1. Number Sense and Numeration: 1.1 Fractions •
Fractions describe relationships between a part (the numerator) and a whole (the denominator).
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Although there are two numbers we have to think of them as one idea, and the relationship between the two numbers.
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Fractions can be used to represent parts of a region, parts of measurement, parts of a set or group, and division and ratios.
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Fractions have different meanings and we should be able to put these meanings together to compare their equivalency.
Activities: Use pattern blocks and grid to introduce equivalent fractions and have students visually compare. Ask students to represent a whole in halves, thirds, eighths, etc. Grade 4 & Grade 5 Specific Expectations: Demonstrate and explain the relationship between equivalent fractions, using concrete materials. (p 67 & 78) Process Expectations: Reasoning and Proving, Selecting Tools, Representing * This activity allows students to visually compare shakes to uncover fractions. Comparing visually allows for a more in depth understanding.
Give students a grid of 100 squares and ask them to use 3 different colours to make a picture using only a specific amount of squares (ex. 32). Each square equals 1/100, use this to relate hundredths to decimals and back into fractions. Grade 5 Specific Expectations: Demonstrate and explain equivalent representations of a decimal number, using concrete materials and drawings
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(CONT.) Grade 6: Determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions decimal numbers, and percents. * This activity helps to visualize representations of decimals out of one hundred. This activity allows for learners interests to be accommodated (by what they would like to draw). Relating activities to what the learners prefer will help instill the knowledge.
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Number Sense and Numeration: 1.2 Integers Adding a negative is the same as subtracting a positive Subtracting a negative is the same as adding a positive •
The negative integers are the “opposites: of the whole numbers. Each integer is the reflection of its opposite across a line perpendicular to and cutting the number line at 0.
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Integer operations are based upon the zero principle, the fact that (-1) +(+1)= 0.
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The meanings for the operations that apply to whole numbers, fractions, and decimals also apply to negative integers. Each meaning can be represented by a model, although some models suit some meanings better than others.
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If I love to love you, I love you (++) If I hate hating you, then I love you (- -)
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If I hate to love you, I hate you (- +) If I love to hate you, then I hate you (+ -)
Activities:
Create an Integer line using a large-scale deck of cards *Homemade cards After introducing integers, go through some rules for addition and subtraction. Model these equations on a real lifeline! Have students come up and take a card. Here, clubs are negative and hearts are positive, aces are 1 and joker is 0. Go through some equations and have students walk up and down the line Use the deck of cards number line to introduce Integro.
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Integro: Rules: 1. In groups of 2 or 4, a student shuffles and deals cards equally to their group (Using only numbers 2-10 and Aces -- Reds cards are positives, Black cards are negative, Aces are 1, Remove face cards and jokers) 2. In a round, each player places one card face up on the table. 3. The first person to call out the sum of the cards wins all the cards in the turn. These cards go into the players bank pile. 4. Tied players play additional rounds until someone wins. 5. When a player runs out of cards, the player shuffles his or her bank pile and continues playing. If the player’s bank is empty the player is out. 6. The game ends when one player has won all the cards. Grade 7 Overall Expectations: - Represent, compare, and order numbers, including integers - Demonstrate an understanding of addition and subtraction of fractions and integers, and apply a variety of computational strategies to solve problems involving whole numbers and decimal numbers Specific Expectations: - Represent and order integers, using a variety of tools (e.g., two-colour counters, virtual manipulatives, number lines) - Add and subtract integers, using a variety of tools (e.g., two-colour counters, virtual manipulatives, number lines) *A real-life integer line is impressionable to students who need extra engagement. Involving them in the lesson will allow for learning through experiences. Integro should be used after the class has a clear sense of how to add and subtract integers and should be grouped appropriately by level because this game can get unfair or frustrating quickly if others are faster at mental math. _______________________________________________________________________
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Have students toss a coin; if the coin lands heads they gain a point (+1). If the coin lands tails, they lose a point (-1). After the students have tossed the coins 20 times, they indicate their final score. *Ask how many tails someone whose final score is (-2) or (+5) could have tossed and why. Grade 7 Overall Expectations: - Represent, compare, and order numbers, including integers - Demonstrate an understanding of addition and subtraction of fractions and integers, and apply a variety of computational strategies to solve problems involving whole numbers and decimal numbers * This seems like a good Minds On activity to get students thinking about ordering numbers in a simple, yet memorable way. Asking questions after the activity such as “how many tails would this person have flipped for a score of (-2) allows students to apply their knowledge and logic of numbers in a more in depth context.
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Number Sense and Numeration: 1.1 Ratio, Rate, and Proportional Thinking •
Ratio, rates, and percents, just like fractions and decimals, are comparisons of quantities. A rate compares quantities with different units, for example, distance to time, or price to number of items. A percent always compares a quantity to 100.
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Proportional reasoning in the deliberate use of multiplicative relationships to compare quantities and to predict the value of one quantity based of the values of others.
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Solving rate, ratio, or percent problems generally involves representing the rate or ratio in a different form.
Activities: Ask students to double the size of the object. In this case double the size of the pumpkin using proportional thinking. Think of each square as one unit. So since the eyes span over 4 units in the given image, in the new image they will span over 8 units. Grade 4 Overall Expectations: - Demonstrate an understanding of proportional reasoning by investigating whole-number unit rates. (p 66) Specific Expectations: - Demonstrate an understanding of simple multiplicative relationships involving unit rates, through investigation using concrete materials and drawings. (p 68) *This activity could be modeled first by simple shapes (squares, rectangles, triangles) and could be modeled first by the teacher. Some students may find a drawing of a pumpkin difficult. They would have to be taught to look for points where the line of the drawing intersects with the corners of the grids.
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2. Patterning & Algebra: •
Patterns represent identified regularities and can be represented in a variety of ways.
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Some ways of displaying data highlight patterns.
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Much of other strands in mathematics are built on pattern foundation.
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Algebra is a way to represent and explain mathematical relationships and to describe and analyze change.
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Relationships between quantities can be described efficiently using variables.
Activities: Each group will be given 3 cue cards with different questions and 1 baggie of M&Ms per person at the table. Individually the students will answer the questions by separating the corresponding answer colour from the rest of the M&Ms. Once all 3 questions are answered and the M&Ms are separated, students will design a pattern using the M&Ms and fill in the chart of the steps their pattern takes and also defining what the pattern is.
Grade 5 (all page 83) Overall Expectations: - Determine, through investigation using a table of values, relationships in growing and shrinking patterns, and investigate repeating patterns involving translations. Specific Expectations: - Create, identify, and extend numeric and geometric patterns, using a variety of tools. - Build a model to represent a number pattern presented in a table of values that shows the term number and the term.
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-Make predictions related to growing and shrinking geometric and numeric patterns -Make a table of values for a pattern that is generated by adding or subtracting a number to get the next term. * Direction will have to be very clear with what to do and activity will take longer than expected. It is a good activity to connect to real life because it relates to students interests and the pattern represents those interests. Perhaps telling them this will make them more eager to do the task.
Give students a visual where they must figure out what the pattern is. Give students chart paper and markers and in groups draw and explain how they see the pattern forming within the image. They will be able to see all the different ways we can visualize patterns. (Ex. In image: X formation and Box formation) Grade 6: Overall Expectations: - Describe and represent relationships in growing and shrinking patterns (where the terms are whole numbers),and investigate repeating patterns involving rotations (p 93) Specific Expectations: Grade 5: Make predictions related to growing and shrinking geometric and numerical patterns (p 83) Grade 6: Determine the term number of a given term in a growing pattern that is represented by a pattern rule in words,a table of values, or a graph. (p. 93) *This allows for students to see how others in their group may see patterns differently then they see patterns. They will present to whole class and reflect and communicate with each other to justify the patterns they have visualized.
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3. Geometry & Spatial Sense: •
Properties and attributes of geometric shapes become the object of study in geometry. Many of the same properties and attributes that apply to 2-d shapes apply to 3-d shapes.
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Every shape can be cut up and rearranged into other shapes is a fundamental part of this study.
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Many geometric properties and attributes are related to measurements. For example: when two sides are the same length, two faces are the same area, and lines are equal distance apart.
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Although properties of shapes are often the focus of attention in geometry, development of skills in describing and predicting location is also an important aspect of spatial sense.
Activities: Present vertical, horizontal and slanted flips. Ask what is a flip or reflection? Relate reflections to what we know in everyday life. A flip is always made over a line called the flip line or line of reflection. Use transparent mirrors, or MIRA math geometry tool, as a tool for performing reflections. Give students graph paper with three types of lines of reflection and different shapes and ask them to perform each type of flip.
Grade 6: Create and analyze designs made by reflecting, translating, and/or rotating a shape, or shapes, by 90 degrees or 180 degrees (p.93) Grade 7: Create and analyze designs involving translations, reflections, dilatations, and/or simple rotations of two-dimensional shapes, using a variety of tools (concrete materials, Mira, drawings, dynamic geometry software) and strategies (paper folding) (p.104) * This activity, especially with shapes of unequal sides, such as hearts, triangles, or even the smiley faces allow students to see how images are flipped or reflected, more so than the square which has an equal reflection.
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Have students create skeletons of threedimensional shapes using marshmallows and toothpicks. Grade 4: Construct skeletons of threedimensional figures, using a variety of tools (eg., straws, and modeling clay, toothpicks and marshmallows, Polydrons), and sketch skeletons (p. 71) * This would allow students to visually represent and hold three-dimensional objects before being asked to draw them, so they have a deeper understanding of the shape. You could also ask how many faces does the cube or prism have? How many toothpicks (lines) did you use? How many marshmallows (vertices) did you use? Have a chart prepared for students to fill in this information for each cube or prism they make to refer back to.
Give students centimeter graph paper and have them, with different coloured pencil crayons; create a figure using all different types of 2-d shapes (polygons, quadrilaterals). Grade 4 Overall Expectations: Identify quadrilaterals and three-dimensional figures and classify them by their geometric properties, and compare various angles to benchmarks. - Construct three-dimensional figures, using twodimensional shapes. * This allows students to be creative and include the arts in mathematics.
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4. Measurement: •
Different measurement attributes of the same object are not always related, so it is possible for an object to be large in one way and to be small in another
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Sometimes you measure directly and sometimes you measure indirectly
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Most measurements can be determined in more than one way
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Familiarity with certain measurement referents helps you estimate
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There is more than one possible unit that could be used to measure an item, but the unit chosen should make sense for the object.
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In order to measure, a series of uniform units must be used, or a single unit must be used repeatedly
Activities:
Have a group of students line up shortest to tallest in front of the board and ask the class to estimate how tall they believe the students to be. Another student comes up to mark the height of each student lined up. Have those students look back at their marked heights 16
and record the estimates in M/CM. When they are done recording, ask those volunteers their actual height. Grade 4 Overall Expectation: Estimate, measure, and record length, perimeter, area, mass, capacity, volume, and elapsed time, using a variety of strategies. Specific Expectation: Estimate, measure, and record length, height, and distance, using standard units (i.e. millimetre, centimetre, metre, kilometre. * This activity puts emphasis on identifying measurement of height through the Metric system as opposed to the Imperial system. This is important to ask students because usually they would state their height in feet and inches and ask them why they would do that. After working on examples of calculating area and perimeter, divide students into groups and give them some short questions and initiate competition. Have students find the area and perimeter within their group and race to the board to write out their answers when they are done. First group to finish wins! Grade 7 Overall Expectations: Report on research into real-life applications of area measurements. Determine the relationships among units and measurable attributes. * This activity would be a good way for students to begin to apply their knowledge and understanding of area and perimeter and is fun competition. The correct answers are on the board afterwards so even if the group of students is wrong, they will be able to have feedback and improve. This is a better setting for learning rather than sending students to their textbook questions right away.
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5. Data Management & Probability: •
Most data collection activities are based on the prior sorting of information into categories.
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To collect data, you must create appropriate questions and think about how best to gather the data.
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It is often useful to “summarize” data by using one or two statistics that, in some way, reflect the whole data set.
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Probability is a measure of likelihood. It can be expressed qualitatively or quantitatively as a fraction or decimal between 0 and 1 or an equivalent percent.
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Unless an event is either impossible or certain you can never be sure how often it will occur.
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To determine an experimental probability, a large representative sample should be used.
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To determine a theoretical probability, an analysis of possible equally likely outcomes is required.
Activities: Have students choose a partner and have each person roll a set of dice 10 times. Students will record data, the data being the number they have rolled. Hand out graph paper. Have students use their data to create a double bar graph between the data collected from each student’s outcomes. Grades 4 & 5 Overall Expectations: - Collect and organize discrete primary data and display the data using charts
and graphs, including double bar graphs. - Read, describe, and interpret primary data and secondary data presented in charts and graphs. * This activity helps to shows ways in which students can record and display data. It deepens understand of what data is by having the students record the own data they are creating. This is also a fun activity to do with a partner and enhances learning.
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Using an understanding of data management, we can understand the relationship between mean, median and mode with regards to a set of data. Have students in their table groups explore mean, median and mode of data regarding shoe sizes of the students in that group (regardless of gender) and discover the range.
* This is an effective way to have students explore mean, median and mode through experiences with others and with something that relates to them. To make this activity a part of a real world problem, you would ask questions after such as, “What would the most important piece of data be that a shoe factory would need to know?� It helps to contextualize the problem.
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