We are working on fractions now!
Today’s problem (June 12th):
To activate our brains we reviewed the vocabulary of fractions (half, third, quarter, fifth, sixth, etc…)
How many different ways can you represent the fraction 2/6:
Most of the students work could be divided into three categories
Consolidation: We talked about why the fractions were organized into these groups.
1. In the first group students noticed that all the fractions were represented as lines (like the 10’s stick from base ten blocks). They were told that this representation is called a “linear model” .
2. In the second group of work the students noticed that all the fractions were drawn as pieces of a whole (e.g.: parts of a pizza). When the students were asked what is it when we talk about the surface of a shape (like a pizza, or a garden), they answered area. They were told that this representation of a fraction is known as an “area model”.
3. In the third group of answers the students noticed that all the fractions were drawn as sets of things, like balloons, or marbles, or people. They were told that this kind of a representation of a fraction is known as a “set model”.
Today’s problem (June 13th):
From this problem the students were given the 7 broken pieces of the square tile and asked if they could put it back together:
Many of the groups succeeded:
Consolidation: When they were done they were asked, “If there are seven pieces in this puzzle, what fraction does the big triangle represent (of the whole square)?”
The students then had a debate as to whether the big triangle represents 1/4 of the square or 1/7.
They realized that the big triangle represents 1/4 of the square because it would take 4 equal pieces to make the square. It does not represent 1/7 because all 7 pieces are not the same size.
Today’s question (June 14th):
To activate our brains we reviewed the 3 ways to represent fractions:
Some students displayed their work using drawings and fractions:
Some students organized their work more efficiently using tables:
Consolidation: When all the work was on a class table, the students were asked if they noticed any patterns. They noticed that the numbers on the top (numerator) and on the bottom (denominator) doubled each time. Another student pointed out that all the fractions in the red column meant (were the same as) 3/4. One student called them equivalent fractions.
So we came up with a rule, that in order to create an equivalent fraction, whatever to do to the top number (multiplication or division) of a fraction, you also have to do to the bottom number (i.e.: 1/4 = 2/8 = 4/16).
Today’s problem (June 15th):
To activate our brains we reviewed the 3 models used to represent models and we reviewed the rule we discovered yesterday about equivalent fractions (if you double/multiply, or cut in half/divide the numerator and denominator by the same number, you will get an equivalent fraction).
Using the “tile” problem from the other day, we talked about how the large triangle represents 1/4 of the square (The porcelain tile broken by Mr. Tan), because if you divide the entire square into large triangles, it will take 4 equal large triangles to cover the whole shape:
So the question is, what fraction of the tile do all the other pieces represent?
Some students included fractions, but no explanation as to where their answers came from:
Some students used one piece to fill in the entire tile, and then counted how many pieces it took:
Some students included a good explanation about how they compared the sizes of different shapes to come up with the answer (2 medium triangles fit into a large triangle. It takes 4 large triangles to make the tile. Therefore it will take 8 medium triangles (4×2) to make the tile… answer – medium triangle = 1/8):
Consolidation: We had an interesting discussion about how students solved the puzzle (focusing on some equivalent fractions). For example 4 small triangle fit into one large triangle. Therefore one small triangle equals 1/16 of the tile (4 small triangles x 4 large triangles). Two small triangles make up the small parallelogram, therefore the parallelogram = 2/16 or 1/8 of the tile.
Today’s problem (June 19th):
To activate our brains we asked:
The students figured out it was 1/4. It began a discussion about how you break up parts of a whole into smaller parts.
Some students answered the question using visuals (drawings of all the pieces of pizza):
Some students tried using calculations to answer the problem (e.g.: 8+8+8+4=28 pieces of pizza; 1/4 of 28 is the same as 28 divided by 4 = 7)
Consolidation: Our discussion did not go in the direction I expected. The conversation looked at how do we compare fractions (what fraction is bigger, 2/5 or 2/3) if they have a different denominator. Similar to today’s question, one way is to look at fractions visually:
(As long as the 2 wholes are the same size)
Another way is to try to use some sort of formula. Our discussion looked at finding equivalent fractions that had the same bottom number (denominator). In order to do that we had some students come up with the idea of doubling the fractions until you get a common bottom number (denominator). Other students had the idea that they could just move up by using multiples of the denominator (in other words multiply the top and bottom by the same number – x1, x2, x3, x4 – until both fractions have a common denominator.
It was an interesting discussion, and it moved in a very interesting (and unexpected) direction – that of common denominators.
Today’s problem (June 20th):
To activate our brains we looked at this question:
The students came up with many answers. It opened into a discussion about equivalent fractions and common denominators. Then we looked at fraction lines as a visual way of comparing fractions with different denominators:
Some students broke the chocolate bar into pieces and tried to divide them into 3 even groups and 5 even groups (this worked for groups that used 15 or 30 pieces, it did not work as well for groups that used 32 or 24 pieces – numbers not divisible by 3 and 5):
Some groups used exact measurements and calculations:
Some groups used only calculations to find a common denominator:
Consolidation: We reviewed the idea that when trying to compare fractions (or add or subtract them), the most efficient way to do that is to find a common denominator. Some students reiterated that you can find a common denominator if you multiply the two denominators of the two fractions you are comparing. (1/3 and 2/5, you would multiply 3 x 5 to get 15, and both fractions can be changed into equivalent fractions that have a denominator of 15 🙂 )
Today’s question (June 21st):
To activate our brains we continued to focus on comparing fractions (although today’s question does not ask the students to compare fractions).
Last class we looked at comparing fractions by using visual representations of fractions and by finding a common denominator.
To continue with where our brains left off, the students were asked to compare fractions by seeing how they relate to 3 benchmarks: 0, 1/2, and 1.
By sorting fractions in our heads according to how close they are to 0, 1/2, or 1 whole, it becomes easy to compare fractions that have different denominators.
The discussion also included the fact that when we are comparing fractions, we do so knowing that the whole is always the same size (in other words when we compare 2/5 of a pizza to 5/8 of a pizza, both pizza’s are exactly the same size). But the denominator determines how many pieces that pizza is broken into.
Today’s problem (oddly, a question that does not ask the students to compare fractions):
Some students answered the quesion by drawing 120 squares and then breaking them up into 4 equal pieces:
Some students broke the number 120 into 100 + 20. The 100 breaking up into 4 equal pieces of 25, and the 20 breaking up into 4 equal pieces of 5 (25+5=30 or 1/4 of 120):
Some students used addition. They knew that 12 could be broken into 4 equal groups of 3, so 120 could be broken up into 4 equal groups of 30:
Some groups used division (very efficient) and simply divided 120 by 4:
Consolidation: Our discussion centred around the idea that whenever you need to find a fraction of a set, you simply divide the set (in this case 120 square metres) by the denominator (in this case 4) and then multiply that answer (120 divided by 4 = 30) by the numerator (3) giving us the answer (30 x 3 = 90)
To activate our brains we looked at the three models used to represent fractions, and connected it with a decimal (7/10 = 0.7)
Some students broke up the 50 students into 10 equal groups of 5 (apparently some students lost their paper?) 🙂
Some students connected the decimals to common fractions (0.5 to 1/2 and 0.3 to 1/3)
Some students looked at 0.3 as 3 out of every 10 students (and added up 5 fractions to represent 50 students):
Some students found equivalent fractions until they found an equivalent fraction with a denominator of 50:
Consolidation: We looked at and discussed how the different ideas were connected to each other: