How coin counters work

how coin counters work

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Draw 15 circles in a row on the board. Ask the students to count them. It is ideal if some of the students get a total other than 15, because this will show how easy it is to lose track when counting by ones. Try to elicit ideas for better ways of handling this task.

If possible, give 15-40 counters to each student. If this is not possible for logistical or budgetary reasons, just have them use drawn circles. Ask the students to arrange 15 counters or circles in an orderly way. Guide them to the idea of making a 3x5 or 5x3 grid, but first let them try to figure out on their own that if they try to make 2 or 4 rows, they won't be even. When they are done, emphasize the fact that some students made a 3x5 grid, and some made a 5x3.

Ask the students to add up their rows. Depending on how they arranged their counters, some will add 3+3+3+3+3, and some will add 5+5+5. Emphasize that in each case, it adds up to 15. Discuss why this is the case, and whether it will always be the case for a similar problem. Ask a 3x5 student if the 5x3 student next to him/her was "wrong," and vice-versa.

Discuss the fact that adding 3 five times was a bit time consuming. Ask what would happen if we had 99 rows of 3. Hopefully the students will recognize that it would take all day if we needed to add 3 to itself 99 times, and will be happy to learn that there is a faster way. Most students love the prospect of a shortcut.

Explain that we have an operation called multiplication, which is nothing more than repeated addition. Instead of adding 3 five times, we can multiply 3 times 5. Elicit the fact that this represents three rows of five counters each. For the students who arranged their grids the other way, elicit that they added five rows of three counters each. The main thing is to stress that either way is the same thing. You could try explaining that this is called the commutative property. It doesn't matter we group the counters in a 3x5 or a 5x3 grid. It's still 15 counters.

At this point, introduce the students

to a 6x6 multiplication table. Don't just "fill it in." Work with the counters or circles on the board to compute the value that belongs in each cell of the table. Continue to emphasize the concept of commutativity, meaning that once we have 6x4, we automatically have 4x6. Emphasize that once we compute a cell of the table, we don't have to "reinvent the wheel" every time we need to do that computation.

Convince the students that they simply must memorize the multiplication table, and understand where all the values come from. Teach them that the answer to a multiplication problem is called the product, and that they will hear this word constantly throughout high school. Emphasize that if they cannot do simple multiplication problems in their head quickly and easily, they will have tons of trouble in later math. Students need to understand that multiplication is not just something given to them to keep them busy.

Explore the concepts of multiplying by 0 and 1. Why do the answers work out the way they do? Explain to students that a typical multiplication table allows them to find products up through 12x12, but that they will later learn a procedure which will allow them to multiply any two numbers, including very large ones. Discuss the fact that a table larger than 12x12 is not very practical.

Begin to introduce students to word problems that are solved using multiplication. A typical problem is, "There are 5 children at a party, and each child will be given 6 candies. How many candies are needed?" Discuss the fact that the keyword "each" implies multiplication in this case. Ask students for some other real-world scenarios where multiplication would be helpful.

Note that in an ideal world, multiplication should not be taught until students are fully comfortable with addition, but this is often not possible. Multiplication should not be taught "by rote," meaning that students need to understand what multiplication really is. It is meaningless to give students a page of practice exercises which they will solve without any thought at all, but will simply solve by way of a multiplication table or a calculator. With that said, once students fully understand the concept of multiplication, and when to use it, you can reinforce basic multiplication facts by having them use flashcards, perhaps in pairs or small groups. Good luck!


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