Seeing the connection between math and the world:
Thank you, Currency System!
I was really sad that I was not there to witness this lesson in action. My Associate Teacher and I were having a bit of a hard time with our Grade 7 kids with fractions, decimal, and percentage. After some brainstorming, my AT had a brilliant idea to use tipping as a way to bring the idea of decimals and percentage closer to home, as it were.
To give a bit of background information, the students are mostly okay with proper fractions but as soon the fraction becomes an improper fraction or a mixed fraction, somehow, things get all jumbled up in the heads for some students.
One thing in common with these students who are struggling with this is this:
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They can’t quite figure out the “howmuchness” of a given fraction;
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They feel that something is off but they don’t know quite know how to describe what exactly feels off.
This is a tough one because neither my AT nor I can answer to a feeling! What I am discovering is that it is a bit easier to explain things to an “argumentative” type, because they have a way of logically constructing ideas (this is why they can twist your words back at you!) and applying one pattern to another that they see in the world. Some of the less sceptical students seem to have a harder time because they cannot argue their way out when they get stuck. One such student got a bit turned around when we were working on figuring out the percentage of 4 out of 100 cubes in a bag. This student initially thought 4/100 was 40% and would not be convinced that it was 4%, until we showed her that 4 cubes take up much smaller portion in a bag than 40%, which is close to 50%.
Continuing with the cubes in a bag problem, many students struggled when we changed the question from “let’s figure out the percentage of x-number of cubes out of 100 cubes” to "let's now consider how the percentage changes if we change the whole from 100 to 80 cubes." I sat with many pairs of students going over this and guided them to keep on multiplying or dividing the numerator and the denominator with the same number until they got the number 100 for the denominator. Some students really wanted to add the same numbers to both the numerator and the denominator instead of multiplying both by the same numbers. When I showed them that adding would change one fraction (½) to a complete different fraction (¾), they were rather stunned.
I was particularly apprehensive as to how one of the students would address this problem because he kept mixing up the numerator and denominator and it was a hit and miss whether he could represent a proper fraction. The concept of a “whole” was not quite grasped and the student couldn’t pinpoint exactly what it was he was not understanding. Usually, it is quite difficult to have this student stay focused even in a pair because he shuts down when he feels overwhelmed. For some reason, he was able to pay attention on this particular day when he was paired up with a quiet and earnest student, who needed a little guidance before she was able to understand what she needed to do to tackle the 12 out of 80 cubes question. She and I manipulated the numerator and the denominator together to get 100 for the denominator - X20, ÷8, ÷2. It appears that the student was watching our endeavour, and at one point, he made an "a-ha" noise and when his partner asked him to work on figuring out 9 out of 80, he was able to get the equivalent fractions for 9/80 to get the denominator to be 100 on his own! It was a big moment for him.
What I observe from the math classes is this: Jo Boaler is right! Number Sense seems to be a key difference between students who can wrap their heads around the fact that they can manipulate the numbers at will to get to the solution.
How could we go about bringing all of the “kiddos” (as my AT calls them) to have this number sense - a sort of an elasticity of mind?
Well, while I don’t have an answer to this question, the fact that the Canadian currency system uses the dollar-cent combination seems to have made it easier for many students to grasp the concept of decimals. Relating 0.25 to a quarter (25 cents) and 4 quarters making up a loonie (1 dollar) seems to come naturally. In other words, they got what my AT calls the howmuchness of this.
In short, the world had already done the job of teaching the kids the concept of quarters (fractions) and decimals (one cent = 0.01 dollar).
In light of this, the next lesson that my AT planned on decimals and percentage was built around something adults do all the time - calculating tips after dining out. There is a fine balance between being generous and not being taken and there are a lot of thoughts that go into determining the amount: Should we tip on an amount before tax or after tax? Should we calculate the tip amount differently if we only had liquor? Should the tip amount be lower or higher than the standard 15% when you have your food delivered to your house? My rule of thumb is that if the service rendered is decent, I stick with 15%. If the service was phenomenal, I go 20 to 25% and leave a big “Thank you, you were great” note. If the service was utterly horrendous, I do NOT tip and leave a comment as to why. Enough about my own tipping custom. Back to the math class.
This is how the lesson went. Unfortunately, I do not have pictures of students’ work so I tried to recreate their group work below.
The receipt the students were given:
Observation 2.0:
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Restaurants lie and screw you over.
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If you wish not to be taken, do not trust the “suggested tip amount” unless you can confirm that they are accurate.
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You get different amount depending on whether you tip before or after tax. The difference is likely to get larger when the original price goes up. (Imagine tip amount on a $200 meal!)
Additional Prompting Question:
Where do numbers on the receipt come from? With this, some of the students got to use more fractions and decimals to calculate where the suggested tip amount came from!
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6.63/39.21 = 0.1690 = 16.90% of before tax amount [The restaurant claims that it is a 15% tip amount & they are charging extra 1.90%]
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6.63/41.95 = 0.1580 = 15.80% of after tax amount [The restaurant claims that it is a 15% tip amount & they are charging extra 0.80%]
What was incredible was the amount of writing that each group had on their 25 X 30-inch grid paper. It was an inquiry-based math problem with multiple entry points and multiple investigative points, with true relevance to a real world problem.
And it was a big success!