algebraic thinking Archives - Page 4 of 5

Algebra, High school mathematics, Math investigations

Teaching combining algebraic expressions with conceptual understanding

Posted on by

In Math investigation about polygons and algebraic expressions I presented possible problems that students can explore. In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding. Like the rest of the lessons in this blog, this lesson is not so just about learning the math but also making sense of the math and engaging students in problem solving.

The lesson consists of four problem solving tasks to scaffold learning of adding, subtracting, multiplying and dividing algebraic expression with conceptual understanding.

Problem 1 – What are the different ways can you find the area of each polygons? Write an algebraic expression that would represent each of your method.

The diagram below are just some of the ways students can find the area of the polygons.

1. by counting each square

2. by dissecting the polygons into parts of a rectangle

3. by completing the polygon into a square or rectangle and take away parts included in the counting

4. by use of formula

The solutions can be represented by the algebraic expressions written below each polygon. Draw the students’ attention to the fact that each of these polygons have the same area of $5x^2$ and that all the seven expressions are equal to $5x^2$ also.

polygons and algebraic expressions — Multiple representations of the same algebraic expressions

Problem 2 – (Ask students to draw polygons with a given area using algebraic expressions with two terms like in the above figure. For example a polygon with area $6x^2-x^2$ .

Problem 3 – (Ask students to do operations. For example $4.5x^2-x^2$ .)

Note: Whatever happens, do not give the rule.

Problem 4 – Extension: Draw polygons with area 6xy on an x by y unit grid.

These problem solving tasks not only links geometry and algebra but also concepts and procedures. The lesson also engages students in problem solving and in visualizing solutions and shapes. Visualization is basic to abstraction.

There’s nothing that should prevent you from extending the problem to 3-D. You may want to ask students to show the algebraic expression for calculating the surface area of solids made of five cubes each with volume $x^3$ . I used Google SketchUp to draw the 3-D models.

3-D models using Google SketchUP — some possible shapes made of 5 cubes

Point for reflection

In what way does the lesson show that mathematics is a language?

Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

Posted on by

Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers — Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers — Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions. Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

integers and variables — Task 3 – Integers and Variables

Algebraic thinking and subtracting integers – Part 2

Posted on by

Algebraic thinking is about recognizing, analyzing, and developing generalizations about patterns in numbers, number operations, and relationships among quantities and their representations. It doesn’t have to involve working with the x‘s and other stuff of algebra. In this post I propose a way of scaffolding learning of operations with integers and some properties of the set of integers by engaging students in algebraic thinking. I will focus on subtracting of integers because it difficult for students to learn and for teachers to teach conceptually. I hope you find this useful in your teaching.

The following subtraction table of operation can be generated by the students using the activity from my algebraic thinking and subtracting integers -part 1.

Now, what can you do with this? You can use the following questions and tasks to scaffold learning using the table as tool.

Q1. List down at least five observation you can make from this table.

Q2. Which of the generalizations you made with addition of table of operation of integers still hold true here?

Q3. Which of the statement that is true with whole numbers, still hold true in the set of integers under subtraction?

Examples:

1. You make a number smaller if you take away a number from it.

2. You cannot take away a bigger number from a smaller number.

3. The smaller the number you take away, the bigger the result.

Make sure you ask students similar questions when you facilitate the lessons about the addition of integers. See also: Assessment tasks for addition and subtraction of integers.

Number Sense

Subtracting integers using tables

Posted on by

In my earlier post on this topic, I discussed why teaching subtraction using the numberline is not helping most students to learn the concept. In this post I describe an alternative way to teaching operations with integers that would help students develop a conceptual understanding of the operation and engage their mind in algebraic thinking at the same time.

The table of operation is one of the most powerful tool for showing number patterns and relationships among numbers, two important components of algebraic thinking. It is a pity that most of the time it is only used for giving students drill on operation of numbers. Some teachers use it to teach operation of integers but more for mastery of skills and to show some beautiful patterns created by the numbers. Below are some ideas you can use to teach operation of integers conceptually as well as engage students in algebraic thinking. I promote teaching mathematics via problem solving in this blog so this post is no different from the rest. Use the task below to teach subtraction and not after they already know how to do it. Of course it is assumed that students can already do addition.

The question “Which part of the table will you fill-in first?” draws the student attention to consider the relationships among the numbers and to be conscious of the way they work with them. It tells the students that the task is not just about getting the correct answer. It is about being systematic and logical. Engage the students in discussion why they will fill-in particular parts of the table first.

Students will either subtract first the same number and this will fill the spaces of zeroes or they can subtract the positive integers. They will of course have to define beforehand which will be the first number (minuend) and which will be the second number (subtrahend).

Surely most students will get stuck when they get to the negatives except with the equal ones which results to zero. You may then ask them to investigate the correctly filled up parts of the table that could be of use to them to fill-in the rest of the table. Students will discover that the numbers are increasing/decreasing regularly and can continue filling-in the rest of the spaces. This is not a difficult task especially if the process for teaching addition was done in the same way. Encourage the class to justify why they think the patterns they discovered makes sense.

The discussion of this topic in continued in Algebraic thinking and subtracting integers – Part 2

Point for reflection

Share!

Share!

Share!

Share!