Fractions, Decimals, Percents
Grade 9s are going to be working with fractions and decimals and percents coming up. Today we worked on putting them in order, as a diagnostic collaborative activity. I’m uncertain which gaps exist in this class due to what work would have been missed during COVID lockdowns years ago. There have been different gaps this year that I have noticed, with perimeter, and I wondered if fractions may have also fallen through the cracks more than in the past.
Students worked in their randomly selected table groups. Groups were given a handful of assorted cards. White cards are fractions, green are percents and yellow are decimals. They needed to work together and use their strategies to sort the cards from smallest to biggest.
Groups chatted about how to tell if fractions were big or small (if they were in mixed numbers they looked for the whole numbers, they also looked for improper fractions where the numerator is bigger than the denominator). They were very confident ordering decimals, and ordering percents, but putting decimals and percents together took some thinking.
Each group then sent a volunteer with their smallest value to chat and decide what the smallest smallest value is, and we started a communal number line. They sent a different volunteer with their largest values to chat about which one was the largest largest value and put the other end point of our number line on the board. Next everyone placed their cards in order, and made sure the number line made sense.
We talked a bit about what fractions mean, and how to represent them, and we chatted a little about what numerators and denominators are. We learned that if we divide numerator by denominator we can make a decimal, so we can convert fractions to decimals and compare the numbers nearby. We also noticed that decimals and percents differ by a factor of 100, so to convert decimals to percent we multiply by 100% and to change percent to a decimal we divide by 100%.
We talked about comparing fractions as well. If the fractions have the same denominators we know that the unit rectangle is split into an equal number of pieces (the denominator) but we are discussing different quantities of pieces. E.g, 3/5 compared with 2/5 we know that the unit rectangle is split into 5 pieces. One fraction talks about 3 of the 5 pieces, and the other fraction talks about 2 of the 5 pieces. I know 3/5 is bigger than 2/5.
If the numerators are the same and the denominators are different we can compare them easily as well. E.g. 3/8 compared to 3/9. I can imagine a rectangle split into 8 pieces and a rectangle of the same size split into 9 pieces. If it’s split into more pieces the pieces are smaller. If I have 3 larger pieces compared to 3 smaller pieces, I can conclude that 3/8 is larger than 3/9.
We then worked on correcting the order in the number line.
The final challenge was to space our numberline properly. We set benchmark values of “friendly numbers” like 0, 0.5, 1, 1.5, 2, 2.5 and then each table group sorted and spaced one of those sections of the number line.
We have some work to do, but during this activity students were able to explore and try and reactivate ideas from years ago. Some expressed that they were very unsure of fractions in general. We will work on these topics for a while, then try a similar task later and see if they feel better about the strategies we have learned.
I captured photos of their finished numberline. We may look back at these photos next week and see if we can find things to correct, or discuss.
Math Outside!
It has been such lovely weather that we decided to do math outside today! My grade 10s joined with the grade 9s from across the hall, in intermixed groups and we represented algebra with toothpicks.
We provided each group with a page that had written expressions, then groups built physical representations and showed them on a concept circle.
It was great to see the grade 9s and 10s work together to build their models. Students were problem solving and challenging each other’s misconceptions. We had a lot of interesting discussions about what brackets mean, and what (2x)(1/2x^2) would mean. Would it be just the same as building an x cube, since if you multiply it that’s what it gives you, or if you need to build a rectangular prism with the base area as (1/2x^2) and the height of (2x) (as seen above).
Since we were working outside we had loads of space, so we made our concept circles bigger, but also contain more slices. Here is the total circle.
We had a spicier one ready to go, but didn’t need it.
We also had ideas of asking as an extension to build an (x+y)^3 which I think would be fun to see.
Thanks for the great fun and collaborative work today students!
Exponent Laws Continued
Today grade 9s finished working on exponent laws. We looked at power of a power today. We noticed that you could write out (5x^3y)^2 as (5x^3y)(5x^3y) and then use the multiplication law to get to the answer.
After a few questions it got a bit boring to write things out long form, and we noticed some patterns. We need to apply the outside exponent to the coefficient in the brackets. We also can multiply the outside power with the inside power to get to the answer.
We got really good at this and then upped the spice level to try some questions with power of a power AND multiplication as well. This picture was taken partway through the problem. The final answer is 432x^10y^6.
We’re practicing how to simplify using a mixture of the laws now.
Exponent Laws Continued
Grade 9s were hard at work today continuing on their exponent laws journey.
We explored some examples with numbers.
We learned that if we have a fraction with the same numerator and denominator it simplifies to 1. Each of the circled pairs will simplify to 1. We can now express the answer as an exponent. When we divide and the bases are the same, we can subtract the exponents.
This works if the base is a variable as well.
The next example includes coefficients and 2 variables. We know that we divide the coefficients as normal. 12/6=2. We look at the xs next. We have x^(4-2) which is x^2, and something neat happens with the ys. We see that when we subtract the exponents we get 0. We also see that we can circle the y^3 in the numerator and denominator and that this fraction will equal 1. Anything to the power of 0 will be equal to 1.
The next example leads us to explore negative exponents. We can easily manage the fraction and the as, but when we have b^(4-6) that leaves us with b^(-2).
We explored how to write negative exponents another way. When we have a negative exponent, it has come from a case where the exponent in the denominator is higher than the exponent in the numerator. If we write it in developed form and circle all the fractions that equal 1, then we will be left with variables in the denominator. a^(-2)=1/(a^2)
We consolidated our work, and made a note in quadrants. Students made their own examples, and we co-created a list of things to remember and write down.
I sent a mass email to students and families with this homework task: Students should explain how to do this question tonight or over the weekend.
Exponent Laws Introduction
Today grade 9s were working hard to represent multiplication in many ways. Here’s a sequence of questions that led us to the exponent law for multiplying.
Represent 4 times 2 times 3:
Students showed in many ways that they understood what (4)(2)(3) is.
Represent 3 squared.
Following what we did yesterday, we know that the exponent 2 means to make a square. The side lengths are 3. We know we can write (3)(3) in many ways as well.
Represent 4 cubed:
It will take 64 little cubes to build a 4x4x4 cube.
Represent x squared.
Represent 2x^3
2(x)(x)(x) is another way to show it.
Represent 2x^3y^4
Represent 1/2x^4z^2p^3
We noticed that there is a hidden multiplication between all the letters.
Represent (x^4)(x^2)(x^3)
We can see so many (x) multiplied together, and we can represent this as x^9. When we multiply and the bases are the same we can add the exponents.
This next question was more challenging because we now have coefficients involved.
(-4)(x)(x)(x)(y)(y)(-2)(x)(y)(y)(y)(y)
this can be written as (-4)(-2)(x)(x)(x)(x)(y)(y)(y)(y)(y)(y) since the order doesn’t matter when we multiply.
It was a busy fun class today! Well done grade 9s.
Representing Exponents
Today my grade 9 class built exponent models using toothpicks and plasticine. The mild version included x^2, x^3, 2x^2, 2x^3, (2x)^2, (2x)^3, 1/2x^2 and (1/2)^2.
The x is a toothpick, so x^2 is a square (x by x). We made connections between the exponent being a 2 meaning we build a square, and if the exponent is 3 we build a cube. The base of the exponent represents the side length of the cube or square, and the coefficient will tell you how many to build.
We had a very carefully prepared (2x)^3. We can clearly see that it is made up of 8 little x cubes. This helps us see that if we double the side length of a cube, the volume will be multiplied by 8 (2^3=8). In the same photo you can see a (2x)^2 which helps us see that there are 4 x squares in it. When the side length of a square is doubled, the area is multiplied by a factor of 4 (2^2=4).
We moved on to a spicy concept circle, including skewers for y and toothpicks for x.
We had some interesting conversations about where to start. Some groups gravitated to the (x+y)^2 since they knew they were making a square with side lengths of (x+y).
We were able to split up the (x+y)^2 into x^2+y^2+2xy. We also showed that (x+y)^2 is NOT equal to x^2+y^2.
Other groups got into building the rectangular prisms with dimensions (x)(y)(y) or (x)(x)(y)
We noticed that an x^2y and an xy^2 were not the same. We could identify which was which based on the square that we saw. This one in the picture is an xy^2 since we see the skewers making a square.
I think we had many lightbulb moments today as we explored what exponents mean. We will continue exploring exponents for the next few days as we look at exponent laws.
Distributive Property
Today several grade 9 classes were working on the distributive property and how that applies to multiplying monomials by binomials.
We started with representing multiplication with tiles. Here is an example of (2)(3), a 2×3 rectangle involving 6 unit tiles. The area is the product and the dimensions are the factors.
Next we tried 3x, which can be seen as 3 groups of x (below)
or we can make a rectangle with one dimension as 3 and the other as x.
The next interesting thing to try is to make as many rectangles as possible for 12x.
we can write these as products. (1)(12x) is the long thin rectangle (top left), then (2)(6x) which is bottom left, then (4)(3x) bottom right and (6)(2x) top right. There could also be (3)(4x) and (12)(x) which are not shown.
Here is another challenge. There are different ways of writing 2(3x+1)
many students see and understand this as 2 groups of (3x+1) which is great. We know there will be 2 groups of 3x, and 2 groups of 1. We can also show this as a rectangle with dimensions of (2) and (3x+1).
The rectangle idea is important when we get to a situation of x(x+2). We struggle to understand x groups of x+2, but we can create a rectangle with dimensions of (x) and (x+2).
We next cut up a puzzle and tried to match the sides that have equivalent expressions
We got them sorted out! It took a while, but our teams worked hard and mastered the challenge.
For more puzzles like this check out monclasseurdemaths
Introduction to Spreadsheets
We were working with spreadsheets today in grade 9 math. Students measured their foot size and their height and we put the information into a shared spreadsheet.
We looked at how google sheets can make a scatter plot, and add a trend line with an equation and R-squared value. We also used our graph to make some predictions.
We also explored how we can use coding to calculate the maximum and minimum values in a column of data. We also used coding to calculate the median, mode, and mean, and used the max and min to calculate the range.
Introducing Algebra
Today we started algebra in several grade 9 classes. We started by using algebra tiles to make some pictures.
Next we eliminated all the zero pairs and simplified the expressions into polynomials.
We talked about the vocabulary of algebra, introducing the words “like terms”, “coefficient”, “constant”, “variable”, “exponent”, “trinomial”, “binomial”, “monomial”, “polynomial” and then started creating some polynomials, then adding polynomials together and combining like terms, and simplifying.
Later we had challenge questions to create the following:
(Binomial)+(binomial)=(trinomial)
(trinomial)+(trinomial)=(monomial)
With tiles the algebra is more tangible and less abstract.
Distance and Midpoint
We were up at the walls working on determining length of a segment, and midpoint of a segment today.
The sequence I used is as follows:
Plot the points (0,2) and (8,2). Determine the length of the line segment. (This is pretty simple to do by counting).
Plot the points (-1,-2) and (-1,4) and determine the length of the line segment. I asked students to think of how they could calculate the distance with operations. They decided that subtracting works 4-(-2)=6.
The next challenge was to find the length of the segment between (0,0) and (3,4). There was a bit of a struggle, until some students started to draw a right triangle to help.
I asked some groups to try to find the distance between any two points (x1,y1) and (x2,y2). It was neat to see students use their idea of rise and run to create an equation.
We consolidated as a group and derived the distance formula that they will be given.
The sequence for determining the midpoint was similar. We looked at the midpoint of a horizontal segment, a vertical segment that had end points in different quadrants, and then for a segment that is on a slant.
It was interesting to see the connections that students made to their prior learning. Some were discussing the fact that a horizontal segment has a y value that is always the same, and a vertical segment has an x value that is always the same. Other students used the idea of finding the middle by taking the average. Pretty soon we were all taking the average of the x values, the average of the y values, and putting that in (x,y) form as the midpoint.
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