Cubes in a Bag

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Students will be able to analyze a sample of data to predict how may cubes of each color are in a bag.

Big Idea

Can you predict how many cubes of each color are in a bag using replacement sampling?

Gathering the Data

20 minutes

Probability is a number between 0 and 1 that tells the chance an event will happen.  The closer the probability is to 1, the more likely the event will happen. 

Students need to practice and experience a concept in many different ways.  This is the third activity we have done with probability, but the difference this time is it is replacement probability – once a cube has been pulled from the bag it is returned into the bag.

For this lesson, you will need a paper bag (make sure you cannot see the cubes through the bag) and 20 cubes of different colors.  I used my centimeter cubes in yellow, green, blue and purple but you could use anything.  I didn’t use a specific number of cubes of each color but just made sure I had a total of 20. 

To do this, I walk around the room and each student has a chance to pull a cube.  We record colors in a chart, in individual math journals, and on the board (MP4).  After having about 1/3 of the class draw out cubes I ask them to predict which cube (by color) they think has the most number of cubes and which color has the least number of cubes. 

I record my own answer on the board, modeling my thinking.  I speak out loud about how I think yellow is the highest number, because at the time more yellow cubes are drawn (MP7).  I do the same for red, stating it has the lower number of tallies and that leads me to predict fewer red cubes in the bag.  We continue on until the last student had drawn a cube out of the bag.  We predict again, recording in a chart, and I continue to model my thinking. (MP4)

Analyzing the Data

5 minutes

Students need firsthand experience analyzing data.  In today's lesson every student participates, but this time everyone arrives at the same outcome. 

Red      1

Yellow   11

Purple   6

Blue      7

When I ask the students to analyze today’s data collection, they did not hesitate is using the correct vocabulary because we have been using it in context, not as a separate list of words presented outside the hands-on lesson (MP3).  One student's analysis is, “Because of the outcome of 11 yellow cubes drawn, it is the highest number of cubes in the bag.  Red has the lowest number of outcomes, so its probability of having the fewest number of the cubes is closer to 0.”  I have a multiage classroom of 4th and 5th grade students; this more complex answer is from a 5th grade student. This video shows a 4th grade student who shares a different way to say the same thing. 

Extending into Fractions with Reducing

10 minutes

I take every opportunity to integrate math concepts and not teach them in isolation from each other.  In this lesson, an opportunity to talk about fractions, parts of a whole and reducing fractions arose. 

After the totals are tallied, I ask my students if this reminds them of fractions.  I provide students time to talk at their tables so more will make connections to fractions.  As soon as I notice a lull in conversation, I know they are finishing the topic and will move on to other things if focus isn't brought back into the conversation.  I ask for someone to share what the discussion at their tables. The reply is "This is like when we did fractions and talked about parts of a whole.  Each color is a part of the whole number of cubes." (MP3)

If you don’t have a student who makes this connection, you could write the first fraction on the board and look to see if any hands go up, and the next, and the next.  If no hands go up after you have written all the fractions on the board, I would draw a picture of a bag with 20 cubes written on the front and then start writing, “There is one red cube out of 20…..11 yellow cubes out of 20…”.

Once the fractions are written, I ask, “Is this your final answer?”  (Borrowing the phrase from a game show.)  I want my students to always look at a fraction to see if it can be reduced or changed from an improper fraction to a mixed number. It is also another opportunity to review using benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (5NF.A.2)

The following video is a student explaining how they reduce the fractions.