Nicole Jackson
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You used mathematical models, a ratio chart, double number lines, and a graph to illustrate various ways of solving this problem. Each was very helpful in uncovering “5/7”

Pasta is on sale for $4.23 for 3 boxes.
So that means, 3 boxes per every $4.23
6 boxes for $8.46
12 boxes for $16.92
If we fair share the $4.23 across the 3 boxes we reveal a rate of ( 4.23/3) $1.41.
Each 1 box receives $1.41.
Finding the constant of proportionality allows us to uncover proportional relationships that would be hidden if we were simply scaling up and down.
Now that I am aware of the rate, I can determine the price of 5 and 11 boxes as well.

Rate reasoning would be more helpful in the following scenario:
My car has a gas tank size of 21 gallons. On my last trip to the gas station, it cost $88 to fill up.
How much would it cost to purchase only 14 gallons of gas?
14 is about 67% of 21.
67*88 = 5,896
5,896/100 = 58.96
It would cost approximately $58.96 to purchase 14 gallons.
@jon , @kyle These numbers are not friendly, so they do not scale up and down well.
However, is my thinking correct?

I wonder when/why would it be helpful to contrive a scenario where a composed unit is expressed as a multiplicative comparison?

I would explore The Sowing Seeds activity with my students. This context requires students to divide in order to reveal the number of parts (or seeds) to each group (or cup). I could use an activity like this one to help students relate partitive division to a context. By adjusting the numbers to more friendly numbers, I could lower the floor. And I could make it a bit more challenging by altering the number of groups leaving students to grapple with leftover seeds.

You emphasized the importance of students being able to reason and justify their thinking. Then, strategically selecting representations of their thinking to reveal higher levels of understanding and skills. It’s a type of scaffolding, building upon the ideas and skills the students presently have access to.
In this context, it is so easy to see how simply having an answer–even if it is correct, just isn’t enough.
I intend to make more use of double number lines. Double number lines indicate scaling of composed units in a way that is very understandable.

To explore the relationship between multiplication and division.
I would expect to see students using:
function tables
visual representations
derived facts
number lines
double number lines.

The Hot Chocolate task could be used to solve one step problems involving multiplication by calculating the answer using concrete manipulatives or visual representations.
To lower the floor for this activity, I could use a smaller, even quantity such as the number 2 as the number of scoops needed per glass.
To extend this problem, I could ask how many glasses of hot chocolate could be prepared using [number of] scoops.

5 out of the 12 eggs in the carton were cracked.
Multiplicative Comparison:
5/12
Composed Unit:
number of cracked eggs number of eggs total
5/12 1
5 12
10 24
Rate:
1 cracked egg per 12/5 eggs total
5/12 cracked egg per 1 egg total
Part to Part Ratio:
5:7 or 5/7
Part to Whole Ratio:
5:12 or 5/12
Ratio as Quotient:
5 cracked eggs divided by 12 eggs total
Ratio as Rate:
5/12 cracked eggs per 1 egg
🙈

I see a rate as a ratio (of quantities with the same or different units) simplified by way of partitive division to reveal either of the quantities to a value of 1.

@Kyle Pearce during the video at approximately 21:41 you made a statement that you was dividing by the rate…
I watched this video several times and each time I expected you to say you were dividing by the scale factor. Can you help me better understand the difference.

Multiplicative Comparison:
The cost of 1 lb of butter at Store A is $3.60. The cost of 1 lb of butter at Store B is $4.50.
Store A butter is 4/5 the price of Store B butter..
Store B butter is 5/4 the price of Store A butter.
Composed Unit:
At my local gas station, the price of gas is $2.48 per gallon.
A customer receives 6/8 of a gallon for $1.86.
For $12.40 a customer may receive 5 gallons of gas.
Both use multiplicative thinking to determine the relationship between quantities. The multiplicative comparison example shows different quantities within the same attribute, making the quantities comparable to one another. The composed unit example shows different quantities with different attributes. These attributes are incomparable, yet the quantities increase and decrease by the same rate.

Units
One half of the screen shows quantities being related to one another and the units of both quantities are the same.
The other half of the screen shows multiplicative or fractional relationships between quantities with different units.

Math is Visual Prompt #3: How Many Are There?
(12 apples, 18 pears, and 24 lemons are arranged in rows and columns.)
Students would be given an opportunity to do a notice and wonder with this visual. They may notice: there a greater number of lemons compared to the other fruit. The apples are the least in quantity. The fruit is arranged in rows and columns which make them easier to count. They may wonder: how many more pears are there than apples? How many more lemons are there than apples? How many fewer pears are there than lemons? We would explore: How many pieces of fruit in all?
Students would be given an opportunity to make an estimation about the total number fruit. I would then provide them with some additional information to help them adjust their estimations, if necessary.
After the students are aware of the total number of fruit and each individual quantity,
we would use the following question to explore multiplicative thinking:
How would you describe the number of lemons relative to the number of apples?
and
How would you describe the number of apples relative to the number of lemons?

Using a measure of unit as a reference point to describe how one quantity is greater than or less than another is additive thinking.
Using multiplication or fractional language to describe how one quantity relates to another quantity is multiplicative thinking.
The key characteristic to keep an eye out for:
the point of reference….is it the unit or the other quantity

Tia’s plant is 12 inches tall. Mike’s plant is 4 inches tall. How would you describe Mike’s plant in relation to Tia’s plant?
Mike’s plant could be iterated 3 times over Tia’s plant. Therefore, the height of Mike’s plant is onethird the height of Tia’s plant.
When a student uses multiplicative thinking, they are making use of the multiplication/division relationships between quantities to describe how one quantity is relative to another.

The language of multiplicative thinking denotes number relationships and patterns between quantities.
Whereas, the language of additive thinking indicates greater or less than relationships between quantities, however, in absolute terms.

We want our students to be precise in translating word problems into number sentences, models, or representations that corresponds with the structuring of the problem.

Sarah has some chocolates. She gave 6 chocolates to Maya. Now she has 8 chocolates left. How many chocolates did Sarah have to start with?
Active Separation; Start Unknown
Students can access this problem by way of:
an algebraic equation using addition
a number line; counting forward
a mathematical model using “?” to represent the sum of the two sets and two smaller sets below to represent the amount Maya has (6) and the amount Sarah has now (8)

Rushing to algorithms and focusing on keywords: give students a false sense of confidence when they approach problem solving.

Jan has 7 candies. Her sister gives her 4 more. How many candies does she now have?
7 is (a quantity of) 3 and 4. I know that 4 plus 4 is 8, and if i count on 3….9, 10, 11.
I find that 11 is the total number of candies.

How much further did student A jump than student B?
Use spatial comparisons to make an estimate of the difference.
Use paperclips to iterate the distance of both jumps from the start point to the end point.
or
Use paperclips to iterate the distance of the longest jump first and then remove and use the same paperclips to iterate the distance of the shortest jump, noting how many are not used.
Provide students with the measurement of the paperclips in inches to determine the difference in the two distances.

Christine, I like this example.
If I can suggest:
using base ten blocks in place of counters for the representation of such large numbers.
using number bars for direct measurement.
and (maybe) number lines for an indirect measurement.
The use of the place value chart (the exterior object) to compare the two numbers is a great choice for indirect comparison.

With direct comparison, we use spatial reasoning skills to eye how one thing is relative to another thing. In the process, one of the “things” become the unit of measure.
With indirect comparison, an additional object is utilized as the unit of measure by which all other things will be measured. And the idea is to judge how each thing relates to this unit, or additional object.

I enjoyed the example that illustrated multiplying fractions. Making use of the commutative property of multiplication, I challenged myself to illustrate the problem as:
3 one fourth units of 1 half unit and I could see an answer of 3 one eighth units so clearly.
NICE!

So the unit of measure matters because children are wired with a intuitive sense to measure quantities by selecting units they are comfortable working with. Unless we are crystal clear about the unit of measure, we cannot be sure if our students are correct or not.

A person can be measured in more than one way.
Height, weight, body mass, temperature, and blood pressure are a few measurable attributes.
The attribute to measure is temperature.
The unit used will be degrees Fahrenheit.
(I was not sure how to answer #4) Degrees Fahrenheit is the standard unit to measure temperature in this region.

Determining the volume of a threedimensional figure in a twodimensional format requires spatial reasoning. Students have to develop the ability to envision the cubic units they cannot see on paper and count them towards the total volume of the figure.

Both scenarios offered nonthreatening ways to tap into the students’ prior knowledge and get a glimpse of their frame of thinking. If students have answers that reflect absolute thinking we guide them towards relative thinking.
I wonder how difficult of a task will this be.

In the past, teaching 3rd grade, I would teach my students to approach proportional situations with the “make a chart or table” problem solving strategy. It is only recently that I am truly understanding how abstract in nature tables and number lines are to early learners. Because students can learn to identify patterns in such tables and make predictable outcomes, I made the assumption that this was it.
So I am wondering: what concrete and representational models should my students experience before the tables.

An object or set can be measured in different aspects. So the aspect in which we want to measure (an object or set ) and the precise unit that we use to measure must cohere.
In measurement, numerical responses are too vague to be considered correct or incorrect.

Â¨A tooth,Â¨ she replied with clear understanding. I LOVE it. Absolutely adorable!
As parents and educators, our word choice is key. As we learned in 11, learning tools have to be presented with a particular timing or order to be used effectively, so does our choice of words.
As we learn to use more fractional language at home and in the classroom, our children are challenged to look at the world around them more creatively and see that there are more ways than one to describe quantity.

We unknowingly teach our children and students to memorize much of the wrong things. We short change our young learners when we teach them to count by memorizing the names of numbers without meaning. And we get in the way of them developing a true understanding of place value when we ask specific questions and train them to give a response that is incorrect.

I noticed that the number line was presented as a tool to understand the magnitude of a number <i style=””>after students have worked with the bar model.
<i style=””>So in order for a number line to prove useful for a student, they have to already have an understanding of magnitude. That makes sense!

Biggest takeaway: the emphasis on collaboration.
To not have the burden of initiating change on your own…To welcome other math professionals to assist in the planning, observation, and reflection of new, engaging techniques in the classroom to help our students develop a deep conceptual understanding of mathematics is a remarkable change.
And IÂ´m so proud of myself for completing this course.
 This reply was modified 2 years, 6 months ago by Nicole Jackson.

What an amazing way to approach math instruction. I really wish my teachers taught math this way when I was younger. Between spiraling the course content and using the curiosity path model–IÂ´m sure your students enter your classroom knowing to expect a challenge but feeling up to the task. This is what all math classrooms should be like.

Makes so much sense. And I can see how all learners can benefit from this. The strong student has the opportunity to be challenged with new topics more frequently to avoid becoming bored with a particular big idea. The notas strong student, will have multiple encounters with difficult concepts, but not in one dreadful and intimidating 2week block. And the time in between (spiraling) will help all learners have time to reflect.

I always thought the problem with memorization of multiplication facts shows up when a child’s memory fails them.
Students often resort to guessing when they lack a conceptual understanding of what multiplication means.
That was a brilliant point made in this video connecting memorization to information difficult to link to other relevant memories.
Counting to multiplying, seems to me to be a continuum. The gradual changes that occur along the way offer so many connecting points that students that simply memorize are being cheated.

Estimate to Add MultiDigit Numbers
234 + 127
Unconscious, Incompetence: “I think I am supposed to add these numbers.”
Conscious, Incompetence: “I know this has something to do with rounding, but I can’t remember the rules I learned last year.”
Consciously, Competent: “My answer will be incorrect if it is the exact sum of the numbers shown. In order to find the answer, I will round the given addends to numbers that are much easier to add. Then I will find the sum.”
Unconsciously, Competent: “Estimate means to round. I will round both numbers and find the sum.”
Consciously Masterful: “Because I am being asked to ‘Estimate to find the sum,’ I know I have options. I can either estimate using rounding or use compatible numbers. There is more than one correct answer to this problem.”
 This reply was modified 2 years, 7 months ago by Nicole Jackson.

Thanks for the feedback.
I am on my way to really understanding each ratio type. And I agree with you, some ratios fall into place quickly and easily while others are a bit more complicated. However, learning to notice and name each ratio type is fascinating.
More than likely, I will watch this video at least once more before moving on.
The introduction of quotient as ratio and rate as ratio is where things became a bit shaky for me.
Overall, I feel as though I am beginning to make sense of the information. I just need to spend a little more time unpacking it.