Great Ideas of Primary Mathematics

Classroom Activities

Mathematical content and background knowledge has been provided previously in this unit. Teachers should feel free to make their own activities based on the ability of their class. The activities are broad in scope to allow for differentiation. Information provided relating to the decimal system and the concept of zero might be good introductory material.

Exploding Dots

In this activity students will be introduced to the concept of exploding dots. Credit should always be given to James Stanton the creator of the concept exploding dots.

Students should be shown a box and then dots added until they explode! The first machine that should be demonstrated is a 10 -> 1 machine. The teacher should not explain what is happening but should instead just place dots until they explode.

Frequently the teacher should ask students how many dots are actually represented in the box. Once the students seem capable of determining accurately the numbers represented the teacher can move on to writing a list of numbers and having students draw boxes to represent the numbers.

Discussion is important at this point as it should be obvious the numbers represented are the numbers we use every day in our number system. The concept of digits and what they are as well as the importance of zero. I would then explore other number machines such as 2 -> 1. Binary as this is called is critical to computer programmers as is base eight and base sixteen. Exploring numbers written in these different bases could be beneficial to students in understanding base ten.

The final activity I would do with exploding dots would be to do operations with the dots. Stick with just addition and subtraction using exploding dots as it will model operations with numbers written in expanded form later on. Use the problems in Appendix A labeled Practice A as support for instruction.

Expanded Form

Write the number 1,234,649 on the board and ask students to write out this number in expanded form. Students may inquire as to what expanded form is. I would tell them to write out what they say when saying the number, for example one million, two hundred and thirty four thousand, six hundred and forty nine. Now tell them to write the number out with only one digit then all zeros added to the other numbers. This would be 1,000,000 + 200,000 + 30,000 + 4,000 + 600+ 40 +9. We could then move on to break down the numbers further by writing them as powers of ten times the one digit. This would give us 1(1,000,000) + 2(100,000) + 3(10,000) + 4(1,000) + 6(100) + 4 (10) + 9. Finally we could show the powers of ten in their exponential form which would be 1(10 ⁶) +2(10 ⁵) + 3(10 ⁴) + 4 (10 ³) + 6(10 ²) + 4 (10 ¹) + 9(10 ⁰).

It is important that students see the order to numbers in base ten and the importance of zero as a place holder. Students should practice writing out numbers in expanded form.

Operations with Expanded Forms

The teacher should start with addition and move on through subtraction and then multiplication. Teachers can make problems progressively more difficult or differentiate the problems based on the academic level of the students. I would start off with some simple problems and addition. The teacher must be careful to add numbers that will require carrying or the purpose would be defeated. Having students discover what to do in order to be successful might be made easier if the teacher used exploding dots and modeled actually what was going on. Students would then try to do model this when they performed operations with the numbers written in expanded form. Please note some sample problems in practice B.

Integers and Place Value

The final concept of the unit is performing operations without carrying and borrowing or using place markers. It will reflect the students' ability to understand place value by performing operations using their own expanded form version of the numbers. Students should use problems from Practice C as a tool to help them understand.