Consider the initial value of coins in the piggy bank, and the remaining value after coins have been removed. Based on the remaining value of money in the piggy bank, predict the value of each coin that was removed through this missing subtrahend scenario. Represent the context using a model and as a single equation.

In this task, students explore the relationship between fractions and decimals. In particular, we will explore the equivalence between coins, expressed as fractions of a whole dollar, and their corresponding decimal representation.

The purpose of this lesson is to encourage students to think flexibly about these two representations. When operating with fractions and decimals, at times, it will be more convenient to think of the fraction as a decimal, and vice versa. In today’s task, students will tackle a missing subtrahend that may elicit a linear model to determine the difference between the initial value and the remaining value. The context expressed as an equation will create an opportunity to explore the use of variables.

All tasks within this unit are intended to be completed without the use of a calculator to support reasoning and thinking.

Some of the big ideas that may emerge through this task include:

• Fractions can be represented in a variety of ways;
• Fractions (and their decimal representation) represent values relative to a whole (for example, one whole dollar);
• Quantities represented as a decimal are fractions limited to base ten denominators (i.e.: tenths, hundredths, thousandths, etc.);
• Standard representation of coin values are expressed as decimal hundredths;
• Missing subtrahend is one structure of subtraction; it can be modelled on a number line;
• Missing subtrahends can be revealed by counting on or through a part-part whole model;
• The missing subtrahend subtraction structure is active subtraction (as opposed to passive);
• A variable can represent an unknown or changing quantity.

Show students the following video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• I notice a piggy bank.
• I notice a bar on the side.
• I notice there was $2.25 to start.
• I notice that they shook out 5 coins.
• I notice that there is still money left in the bank.
• I wonder what coins they removed.
• I wonder if all of the coins have the same value.
• I wonder how many coins are left.

At this point, you can answer any wonders that you can cross off the list right away. For this particular lesson, you will want to keep most of the information hidden at this time. By sharing too much information, you may rob students of their thinking. One thing you can confirm is that the piggy bank was empty before the first coin was deposited.

After we have heard students and demonstrated that we value their voice, we can land on the first questions we will challenge them with:

How much money was in the piggy bank to begin with?

How much money is left in the piggy bank?

Make an estimate.

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to use spatial reasoning alone to try and come up with an initial estimate and to share it with their neighbours by trying to articulate why they believe their prediction is reasonable.

Consider asking students to think about a value that would be “too low” and a value that would be “too high” before asking for their best estimate in order to help them come up with a more reasonable estimate.

Let them chat with their neighbours and challenge them to an estimation duel or a math fight.

Monitor student thinking by circulating around the room and listening to the mathematical discourse.

Encourage students to use precise mathematical language. You will likely hear students discussing coins using the names of the coins (quarter, dime, nickel), as well as the number of cents. Based on the quantities represented by the metre.

For example, “I think it is less than half a dollar, so it is less than 50 cents”. “I think it is more than one quarter”. “I think they had two dollars and one quarter to start, because it looks like there are four equal parts of that quantity between 2 and 3.”

After students had an opportunity to share their best guess, tell them that you will share some information with them that might help to refine their estimate.

Show students the following image:

At this point you can celebrate the student estimates that were closest to the actual value, 2.25 and 0.40.

Prompt students with:

What five coins were removed from the piggy bank?

Represent this context using a model of your choice and as a single equation.

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Show the denomination of the five coins shaken out of the piggy bank with the following video:

US Coins:

Canadian Coins:

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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