**Intermediate Level Math**

Fraction, Decimals, and Percents

Algebra

Geometry

Measurement and Proportionality

Harry Potter and the Sorcerer's Stone by J.K. Rowling

**Problem Centered**

Learning Classroom

Learning Classroom

In Problem Centered Learning Classrooms...

Students are learning through real contexts, problems, situations, and models.

Students are learning to grapple through a problem instead of being taught how to solve the problem.

Students learn mathematics as a result of solving problem.

Mathematical ideas are a result of solving problems.

Van DeWalle pg. 33

Shapes and Properties

- a study of the properties of 2D and 3D shapes, as well as the relationships built on properties.

Transformation

- a study of translations, reflections, rotations, and symmetries, and the concept of similarities.

Location

- coordinate geometry or other ways to specify how objects are located on a plane or in space.

Visualization

- recognition of shapes in the environment developing relationships between 2D and 3D objects, and the ability to draw and recognize objects from different viewpoints (VdW p. 403).

Geometric Content

Van Hiele Levels of Geometric Thought

Level 0 - Students begin to identify shapes by what they "look like". A square is a square because it “looks like” a square, similarly a diamond is a diamond because it “looks like” one. Even though diamonds are squares rotated 45 degrees it is still a diamond to them. Students will say that the shape looks like a house instead of a pentagon. Teachers can informally suggest it is a pentagon but we should focus on what properties shapes have in common with other shapes. Students should be able to feel, observe and build shapes at this stage to better grasp the concept of classes of shapes.

Level 1 - Students begin to start thinking about classes of shapes rather than individual shapes. They begin to refer to all rectangles, rather than one rectangle and see that a collection of shapes go together because of their properties. Students that operate at level one might be able to list the properties of squares, rectangles, and parallelograms. They will continue to use models and drawing of shapes.

Level 2 - Students are able to develop relationships between properties of geometric shapes. They reason using informal deduction. For example, a shape has all four right angles then it must be a rectangle. All four angles on a square are right angles, therefore a square is a rectangle. As teachers we can ask "Why?" and "What if?" questions.

Level 3 - Students reason and prove abstract properties of shapes instead of thinking about properties. They build on definitions to create theorems. They work with abstract statements to come to conclusions. This is high school level thinking.

Level 4 - Students appreciate the distinction and relationships between different axiomatic systems (Van DeWalle pg 406). This is beyond intermediate level and is taught in college.

Activities

Quadrilateral Concept Map - Gets students thinking about the relationships between squares, rectangles, rhombi, parallelograms, and trapezoids.

PASS Standard: Reasoning and Connections

CC Standard: Model with mathematics.

Triangle Sorts - Students sort different types of 2D triangles into equilateral, scalene, isosceles, and acute, right, and obtuse groups or use a Venn diagram to compare and contrast them.

PASS Standard: Reasoning and Connections

CC Standard: Use appropriate tools strategically.

True or false statements - Students determine if statements on shapes are true or false and make an argument to support the decision. Ex: If it has exactly two lines of symmetry, it must be a quadrilateral (Van DeWalle pg. 416).

PASS Standard: Reasoning and Communication

CC Standard: Construct viable arguments and critique the reasoning of others.

Concept map of Quadrilaterals

Have students measure out Hagrid, who is said to be twice as tall and five times as wide as the average man. Students in grade 4-5 can make a table of the students height and width and find a pattern. And students in grade 6-7 can make a scatter plot of widths and heights and see where Hagrid would be placed.

Shape Sort

Spatial Sense

Spatial Sense is "an intuition about shapes and the relationships between shapes and is considered a core area of mathematical study" (Van DeWalle pg 403).

Students can be taught to have good spatial sense, it is not something we are particularly born with. Teachers need to provide meaningful experiences with shapes and spatial relationships.

Why Teach Mathematics with Literature?

Provides a meaningful context for the problem centered classroom.

Celebrates mathematics as a language.

Demonstrates that math develops out of human experiences.

Addresses humanistic, affective element of math.

Fosters the development of number sense.

Integrates mat into other curriculum.

Restores an aesthetic dimension to math literature.

Supports the art of problem solving (Whitin & Wilde, 1992, pg. 4-16).

Mystery Math: A first book of algebra by David Adler

This book introduces algebra and finding the mystery number. It also suggests that you use a scale to balance the equations, which is a great technique for solving algebra problems.

Full House: An introduction to fractions by Dayle Dodds

I used this text with my tutees and it helped them understand parts of a fraction. We did an activity and cut our cookies into fractions to share with 4 guests, and then 8 guests.

Length

Area

Volume

Weight and Mass

Start teaching measurement by making comparisons, then move to physical models of measuring units, and then go to using measuring instruments.

Early on, you can introduce nonstandard units.

Benefits: it provides a good rationale for standard units, avoids conflicting objectives, and makes it easier to focus on the attribute being measured.

Doing estimation activities can better help your students focus on the attribute being measured and provides an intrinsic motivation for measurement activities.

Van DeWalle pg 376-379.

Length Scavenger Hunt: Give pairs of students a strip of card stock, a stick, a piece of rope or some other object to find five things that are shorter than, longer than, or same length as the item. They can draw or write what they find.

PASS Standard: Reasoning and connections.

CC Standard: Reason abstractly and quantitatively, attend to precision, and use appropriate tools strategically.

http://www.mathplayground.com/area_perimeter.html

Have students go through online games like this one on Mathplayground.com to help them learn area.

PASS Standards: Representation and problem solving.

CC Standard: Look for and express regularity in repeated reasoning.

Have students make predictions of which container has the most volume. Students can write the containers in order from least to greatest. Then, have the students fill the containers one at a time with colored water and then into a 500ml beaker to find the volume. Once they find the accurate volume students can rewrite in their journals the containers from least to greatest.

PASS Standards: Representation

CC Standards: Attend to precision and construct viable arguments and critique the reasoning of others.

http://www.ashleigh-educationjourney.com/2014/03/monday-math-stations.html

Balances and scales are great for measuring weight and mass. But for using models of weight and mass teachers can also have students use a collection of uniform objects as weight units. For example, using wooden blocks or for heavier objects metal washers.

Proportion

Fractions, decimals, and percents all intertwine. To have students that are good at fractions, they need to work with and be comfortable with division. One great way to link all three concepts, is working with money. All students are familiar with the context of money.

Fractions

Learning fractions begins in 3rd grade for common core. In 4th grade, students should be able to add and subtract fractions with like denominators and multiply whole numbers. In 5th grade, students are developing fluency of adding and subtracting fractions and learning to multiply and divide a whole number with a fraction. In 6th grade, they are continuing division of fractions.

Students need to understand fractions represented in area, length, and set models. So, a teacher needs to provide activities that use all these models.

Types of Fractions

Part-whole: part of a whole, like a fraction of a group, a part that is shaded, or part of a length.

Measurement: identifies a length and using it as a measurement piece.

Division: dividing whole into parts for groups.

Operator: indicates an operation, like how much is 2/3 of the audience?

Ratio: part-part or part-whole comparisons.

Five Talk Moves that Support Mathematical Thinking in Classroom Discussions

Effective fraction computation suggests:

Use contexts

Use a variety of models

Include estimation and information methods

Address misconceptions (Van DeWalle pg. 316)

Activity

Slice Fractions App

• Part-whole partitioning

• Numerator / Denominator notation

• Equivalent fractions

• Fraction ordering

• Subtracting fractions from 1

Based on Common Core Standards

Decimals

Three concepts that should be taught: 1) A two-way relationship (10-makes-1 rule) 2) Regrouping 3) The role of the decimal.

Work with manipulatives to learn the procedural thinking and then students can use 10x10 grids and number lines to move to the conceptual learning of decimals.

Link decimals to fractions by using base ten fraction models.

Percents

Percents are the same as hundredths and fractions out of one hundred.

Using based ten models help students see the link between all three.

Estimating percents can be useful practice for students.

Three real-world contexts are: tips, taxes, and discounts.

Kaput's Five Areas of Algebraic Reasoning

Number System

Patterns

Mathematical Modeling

Functions

Meaningful Use of Symbols

Early on, students learn to create generalizations of numbers and computation. Basic facts and meaning of operations are important.

Make structure of the number system by teaching addition/subtraction and multiplication/division.

This is important, so students know how to make sense of numbers and can easily manipulate them the figure out math problems.

Discussions must be a part of the daily experience for students to learn algebra fully.

Activity

Find four or five different ways to count the tiles around the border.

It is important to teach the meaningful use of symbols because many students don't have a full understanding of them, for example the equal sign and variables.

The meaningful use of symbols means that students understand that symbols represent real life events and used for problem solving.

Activity

Doing balance problems helps students learn that an equal sign is showing equivalence, rather than an answer after it.

Functions are important for students to learn so that they can see the relationship between numbers and a graph, and how variables affect growing patterns.

Functions are the equations of a rate of change. It can show the pattern and how the variable changes it.

Activity

Make a table of the number of square tiles and the perimeter of the string of square tiles and then graph it.

Real-world Context

Learning to look for patterns and how it extends is important because it is in all areas of math.

Patterns are a repeated design of objects or numbers.

Activity

Draw the next step of the growing pattern.

Mathematical modeling is linking math and statistics to everyday life, work, and decision-making (CCSSO 2010, p. 72).

It is important to teach this so that students use appropriate mathematics and understand statistics better and improve decisions.

Activity

"A car gets 23 miles per gallon of gas. It has a gas tank that holds 20 gallons. Suppose that you were on a trip and had fille the tank at the outset. Determine a mathematical model that describes the gallons left for given miles traveled" (Van DeWalle 2012, p. 281).

Algebra is the study of patterns and functions.

Algebraic thinking or reasoning is:

Forming generalizations of number and computation.

Formalizing ideas with use of meaningful symbol system.

Exploring concepts of patterns and functions .

5 representations of a function are the pattern, the table, the verbal description, the symbolic equation and the graph.

What's my Rule is an activity where the students find the rule of a pattern. This is important because it shows students that patterns follow a rule. The students need to be able to identify it so they can see the relationship between a pattern's equation, graph, and table.

Recursive Pattern

A recursive pattern is a pattern that can only be found by finding the prior step to find the next step.

There is an iterative rule to find the pattern. It is how you continue on the pattern.

The pattern cannot be found by changing the variable in an equation like you can in a growing pattern.

Algebra and Technology

Students can play with the algebra interactives on the Illumintations Website.

http://illuminations.nctm.org/Default.aspx

Students can play on apps on a tablet.

Students can play games on the computer.

http://www.math-play.com/Algebra-Math-Games.html

These are important to use so that students are clearly communicating and understanding what is being said in discussions.

Example of Problem Centered Learning

Problem Centered Learning

Class begins with a problem posed by the teacher.

Class is organized into small groups.

Students work collaboratively in groups on the task.

After 25 minutes, the class comes together for a class discussion where the students present their solutions.

Classmates validate if the answer is correct. (Wheatley p.34)

Teachers Role

To select appropriate tasks based on her knowledge of the students.

Organize the groups and listen carefully as they work.

And facilitate the class discussion (Wheatley p.34).

PASS Process Standards

CC Standards for Mathematical Practice

1. Problem Solving

2. Communication

3. Reasoning

4.Connections

5.Representation

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.