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[students clapping]
What could be bringing these students such joy in a math class?
This video will discuss MGSE6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems
utilizing strategies such as tables of equivalent ratios, tape diagrams or bar models,
double number line diagrams, or equations.
In fifth grade, students are expected to interpret fractions as division of a numerator by a denominator,
as well as solve real-world problems
involving division of whole numbers resulting in fractions or mixed numbers.
This connects to sixth grade ratio concepts such as:
Students can view ratios as fractions and apply division to help determine the multiplicative relationship within ratios.
Students develop the meaning of ratios and proportions using equivalent fractions.
Students develop the concept of a ratio where the comparison is a part to whole.
These concepts will be used to solve problems involving ratios.
As students transfer their sixth grade knowledge to seventh grade,
they should be able to compute unit rates of ratios with fractions,
represent proportional relationships,
and determine if relationships are proportional.
[students talking among themselves]
What makes his table incorrect?
Ohh!  I like that!
So, if we were to correct his table, then what would that look like?
These students are discussing Question 1 of this task from Illustrative Mathematics Open Resources.
The consensus is the table is inaccurate
because the 3000 is associated with the wrong quantities.
The progression of strategies for solving proportions is to begin with ratio tables.
It is essential for students to understand the structure of tables
as they will eventually graph the pair of values displayed in ratio tables on coordinate grids.
As students progress in their strategies, they will discuss relationships among quantities within the table.
Listen as this student shares how he believes the table should be set up based on the relationship he identifies.
What do you all think back here?
Um...So, we think it is wrong because it starts off by dividing by 2, so when it gets to 10, that should be a 5,
and where it says 50, then that should be a 25 because it's dividing it by 2.
Ah!  So, you're thinking it should be divided by 2.
Let's consider this task.
[clapping]
One, two, three, four...
[whistle blows]
[fast clapping noises]
Using this helpful information,
let's consider what strategies students might use to determine how many times he would clap in a minute.
Here are a few examples of strategies: using a ratio table as seen in the video for MGSE6.RP.3a,
using concrete materials,
using a bar model or tape diagram,
or using a double number line.
In this clip, the student explains how he
used the provided information to determine how many claps were completed in one minute.
Okay, so tell me what you all were doing.
I divided this and this, and then, I got this.  And, then I multiplied by 60, and then I got this.
Why did you divide?
I divided to find the unit time.
Okay!  So, you're trying to figure out how many claps he can do in a second?
So...
Where did the 107 come from?
Oh!  We're trying to figure out how many claps he can do in a second.
That's how much he did in 12 seconds.
How much he could do in 12 seconds....
So, in order to change that 12 seconds to one, what would we need to do?
Divide by 12.
Divide by the twelve.
And so, in this case, what should you divide, then?
Wait, so you would divide...
12 and 107?
So, which one would be the divisor?
12
The 12 would be the divisor.
The student said he divided to find the unit time.
It is important for students to focus on the meaning of the terms like "for each one" and "per"
because these
equivalent ways of stating ratios and rates are the heart of understanding the structure in ratio tables,
and provide a foundation for learning about proportional relationships in seventh grade.
You may find that students will use proportional reasoning without setting up an actual equation.
This student explains how she and her partner determined 107 claps were made in 7 point 9, 9, 4 seconds,
which helped them identify the unit rate.
And...um...This is how many seconds he's clapped so far.
And then, we divided 107, and then that equaled 13 point 3.
And then, we multiplied 13 point 3.  13 point 3 was the amount he clapped for each second.
And then, we multiplied 13 point 3 by 60 and got 7 hundred 98.
Did you notice cross-multiplication was not mentioned in this video?
Teachers tend to believe the only way to teach solving proportions is to cross multiply.
Students must first conceptually understand the relationship between the ratios in order to see why that algorithm works.
Research shows that students who have conceptual understanding of proportional relationships are better problem solvers than students who lack
conceptual understanding and instead go to cross multiplication as a procedural strategy.
Therefore, cross multiplication is not an expectation of this standard.  Instead, students should be able to find
solutions by using equivalent ratios.
Which strategies do you think you need a better understanding of before discussing with students?
These equipment animations from nzmaths.co.nz will offer some insight.
The aim of this animation is to show how strips of paper can be used to compare ratios.
The animation will walk you through a scenario and set up an example for you to view.
Notice how it shows the ratio of two to three.  This same idea could be used with cuisenaire rods.
The aim of this animation is to show how double number lines can be used to solve ratio problems like the beanies problem.
In this one, it provides you a context, so that you're able to make sense of the quantities on the double number line.
Again, these animations are for teacher use.
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