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Lesson 2 – Building A Continuum Of Measurement – Discussion
Posted by Jon on December 6, 2019 at 5:27 amSelect two (2) quantities that you can compare and/or measure.
Share how you might compare and measure at each stage along the continuum from direct comparison, to indirect comparison, to direct measurement, and finally indirect measurement.
Consider using an example that is relevant to a grade or course that you teach.
Share your reflection below along with any wonders you still have.
Luke Albrecht replied 8 months, 3 weeks ago 23 Members · 29 Replies 
29 Replies

I used the Math Learning Center apps and copied and pasted into my reply. Sadly the fraction strips I used did not show up when i submitted my reply so I edited with no pictures. (BTW it’d be really great if you could create a “redo” button for the replies. Sometimes, I accidentally delete stuff and have to retype it!)
Gr. 7 question:
5/8 – 1/2
Direct comparison: Place 2 fraction strips, of the same length, showing 1/2 and 5/8 side by side. Notice which strip’s coloured spaces is longer.
Indirect comparison: hmmm… I didn’t know what to put here. I played with pattern blocks – which don’t work at all as they do not have “eighths” – and number lines.
direct measurement : Place the strips over top of each other . Note that 5/8 is an 1/8 bigger than 1/2
Indirect measurement; we can see that 1/2 = 4/8 so…
5/8 – 1/2 = ____
5/8 – 4/8 = 1/8
This process was much more difficult than I anticipated. Thanks for taking me out of my comfort zone. I am sure the synapses in my brain are firing like crazy. Hope they are creating a pathway for all my new learning!!
But seriously, what should I use for the indirect comparison?
 This reply was modified 2 years, 2 months ago by marianne aamodt.


This is pretty challenging … a grade 6 example might be ordering numbers on a number line. I originally thought of negative numbers, but that got a bit too challenging for me —
So I’l start with just simply comparing 2 fractions, say 4/5 and 2/3
Direct comparison – looking at visual models of the fractions (bar models) to determine which is bigger
Indirect comparison – using something like a strip of paper or a string to measure the length of each fraction and then compare those strips (??? not sure about this one)
Direct measure – using unifix cubes to build equivalent models of the fractions (dividing into the same number of parts – 15) and then count the cubes to compare those quantities
Indirect measure – creating fractions that have common denominators and then comparing numerators
This does seem pretty complicated but there are some deep understandings that could be uncovered and critical connections to be made through this type of representation.

This is a great example, and such a great way to be able to see where kids are at on the continuum!

I’d say more complex than complicated. The complexity is difficult to avoid, but complication is when we add unnecessary “stuff”. Math is filled with complexity and we often add complications through tricks, memorization and otherwise.
The hard part is determining what parts are important for both the teacher and student to understand vs what is more just for the teacher to help them plan and deliver an effective lesson?


So I was thinking about the area of a circle (7th grade – and definitely building/stealing from the pizza example).
Direct Comparison – Placing circles over each other.
Indirect Comparison – Placing square tiles on the circles
Direct measurement – “Cutting up” circles to make rectangles
Indirect measurement – using the formula to find the area

Piggybacking off of Heidi’s example is this one for volume. This comes from Lessonresearch.net
Direct Comparison: Which bottle holds more?
Indirect Comparison: Pour liquid into one bottle then pour contents of that bottle into the opposite one to see whether it overflows and in that case the first bottle holds more; or if there is still room in the bottle than that bottle holds more.Indirect Measure: Use a smaller cup and fill it and count the number of cups that each container holds. Container a holds 4 cups while container b hold 3.5 cups.
Direct Measure: Volume of a liter is equal to a 10 x 10 x 10 cm cube.

Wow, I read the teachers ideas.
However, many saw a need to go many steps beyond the rectangle indirectly measured with squares. Triangles? comparing two rhombus?? Wow that is rocket science! Then liquid measures. How abstract?? As teachers, I think we needlessly complicate instruction. We need to show more empathy (get into the sneakers of our less developed students).
The most important idea I heard was: TRUST.
Students need time and experience to develop TRUST. Educators are in such a hurry to ‘cover concepts’ and get to the next topic. These lessons have impressed upon me the need to build foundational skills through play (is only). Civil engineers advocate for a sound foundation before proceeding with any construction.
Educators need to chill. My greatest thrill is watching student light bulbs lite as they grasp new concepts or express themselves in new ways.
“Be Calm and PLAY On.”

Great take away here. We all need to “chill” and really try to better understand what next step that would benefit each individual student. Easier said than done, of course!


Grade 4: Place Value
1,234 and 12,345
Direct comparison: use counters to compare each place
Indirect comparison: use tensblocks on a place value chart
Direct measurement: Placing the numbers vertically to compare
Indirect comparison: I’m not sure…can anyone help?

Christine, I think your indirect measurement might be that students are just able to identify the larger number based on their understanding they have built.


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.

I think it would be neat to play with volume but more challenging of using a rectangular prism vs a cylinder to see which one could hold more
direct comparison comparing the two forms through visual inspection (like height)
indirect comparisonfilling both forms with another object (like the popcorn challenge)
direct measurementmeasuring lengths, heights and areas
rectangular prism: measure height, width and length need for calculations of area of base times the height
cylinder: measure circumference, diameter, and height need for calculations of area of the base times the height
indirect measurementgive measurements and calculate using the formulas.

I was thinking about this example too. Working with really young learners, especially kindergartners, they just like to fill things up. Providing them with a challenge of comparing the capacity of two objects can lead them through the measurement continuum. Using candy would be even more engaging for them.
With kindergartners, even with first graders, is it even necessary to move them to the indirect measurement stage yet? The whole idea of trust really resonated with me during this lesson and I just don’t think our really young learners have a lot of trust in us yet if we were to just give them measurement attributes. I think developmentally, they would just ignore the given attributes, and find out for themselves, using whatever external measurement objects we make available for them.

Aaron,
I would agree that kindergarten student wouldn’t need to enter the indirect measurement stage at this time. I think they should be building understanding through play and inquiry which will help them when the curriculum expects them to build on that understanding.



With it being Halloween season I thought of how we could use candy for such an assignment. For example, I could get to big chocolate bars – one that is square and one that is a rectangle. Then ask which one is bigger and how could we find out. Instead of using tiles, we could use Starbursts as they are the shape of a tile. I feel using these resources would get them more excited and enjoying lesson.

Compare 60 miles in 2 hours, 45 miles in 3 hours
Direct comparison
Each bar represents an hour
60 miles in 2 hours
45 miles in 3 hours
Indirect comparison
Giving students base ten blocks and one blocks to place 60 and 45 equally on hour bars
Direct Measurement
Counting each block unit for hour in both scenarios
Indirect Measurement
Figuring out the unit value by dividing distance by time for mile/hour in each
Sorry, my bars did not show up. Please feel free to give me any feedback. Thanks

I will compare foursixths and fiveeighths. (based on same whole!)
Direct comparison – place the fraction strips or cuisenaire rods representing each on top of each other and visualize which is more/greater/longer.
Indirect comparison – I might use a onehalf strip or cuisenaire rod and compare each one to the onehalf piece.
Indirect measurement – create equivalent fractions for each one.
Thinking about where on continuum this reasoning would be: compare each mentally to benchmark of onehalf. Foursixths is onesixth more than threesixths (or onehalf). Fiveeighths is oneeighth more than foureighths (or onehalf). Therefore, foursixths is larger because the onesixth more than onehalf is larger than oneeighth more than onehalf because onesixth is longer than oneeighth (of same whole). Is this indirect comparison because I am using the benchmark onehalf to do this reasoning?

Great wonder and I think you’d be right. I can see why one might think indirect measurement, but it would seem that the 1 half “measure” is simply another unit to indirectly compare to.
Great reflection here. It is clear that you’re doing a great job making sense of these ideas with reflections like this one!


Which one is bigger? (whatever the objects are…)
direct comparison using one object of the two to measure the second
indirect comparison using other objects that we have handy to measure both and compare.
direct measurement cutting up one object to make a unit
indirect measurement using formula to find the area and compare.

Compare round hay bales to rectangular ones for volume. For direct comparison set them side by side and eyeball them. A tarp laid over each bale and marked for size might work for indirect comparison though this is surface area so I’m not sure it works. I don’t have a good thought on this one. Direct measurement might involve baling twine to compare length, height and width/diameter of the bales. indirect measurement would involve using the formulas and actual measures.

In an upcoming task with my students, I’m asking them to explore measurement of length as well as mass. I took pictures all along the harvesting of my carrots this year and I have created various explorations involving estimation of lengths of the carrots as well as ultimately calculating how many kilos of carrots we harvested.
For the measurement of the length, first students will compare relative lengths of different carrots and estimate what they believe will be the length of all the carrots lined up in a row.
Eventually they will have the measurement of one of the carrots provided to help them zero in on their estimations. I am providing linking cubes and other measurement aids to help them physically map it out.
Once they have an idea of how long the carrots would be end to end, then they will need to compare the length of that row to the length of one of the walls of the classroom. In this exploration, I am leaving it up to them to decide how to compare this as I actually am curious to see whose thinking land where on the continuum of these comparison types.
My challenge will be to not lead too much, but give just the right nudges to prompt their thinking about measurement to change. Of course the challenge with this task is that not all carrots are the same length, so this task will not have predictable/constant measurements, although they will be able to use their linking blocks as somewhat of a standard measure.
Later we will determine the average weight of a bag of carrots and based on that, as well as the number of bags (conveniently stowed away in milk crates), students should be able to calculate the approximate number of kilos of carrots. Here I am struggling a little bit with how to bring the direct and indirect comparisons into play, so I may actually bring some bags of carrots in with me, but I’m also thinking of substituting other objects that would be about the same weight to help students gain a better idea of what grams, kilograms etc are about. Or experimenting with packing a similarly filled bags into crates to simulate the situation.
If anyone has an idea of how to help the students make the leap, I would welcome any insights you’d like to offer. :).

It blew my mind the first time I saw a physical model of the Pythagorean Theorem with the squares made from each side being placed over the square made from the hypotenuse. I had spent years using the Pythagorean Theorem without really understanding what it meant. The visual model is so powerful! It’s so important not to rush over that!
So direct comparison— placing the squares from the sides over the square from the hypotenuse to compare their equality.
Indirect comparison— placing cubes from each square onto the hypotenuse square to show that they are equal.
Direct measurement— maybe cutting up the area of the 2 squares and placing these parts over the hypotenuse square.
Indirect measurement— using a ruler to measure and find area or actually giving each side of the right triangle a numeric length.

I too taught Pythagorean for years without really understanding the visual / physical relationship. How powerful of a tool to be hidden from the forefront!
Thanks for sharing your summary of the progression of measurement.


Comparing angles
Direct comparison: looking at a pair of angles and trying to decide which one is bigger/smaller; putting one angle on top of the other
Indirect comparison: I wonder if using string to overlay each angle will tell anything? I must try this.
Direct measurement: using a protractor
Indirect measurement: figuring out if they would be supplementary or complementary through measuring
I’m not sure on all of these… it’s food for thought. I was trying to come up with something different than what was already done.

I was wondering about comparing a triangle to a rectangle. In sixth grade they are to learn about the formula for the area of a triangle. I remember in school I never could remember what those formulas were, especially a trapezoid’s. But now that I have had to teach it to my students more visually than I ever learned with I can’t not remember because I see the picture in my head.
Right Isosceles Triangle and Rectangle
Direct comparison would be to place one on top of the other.
Indirect comparison would be to iterate tiles on both
Direct Measurement would be to get a count of tiles figure out how to count the cut tiles
Indirect Measurement would be to make calculations to find the missing measurement.
I would think doing this with various quadrilaterals would be beneficial before the triangle now that I have thought this example through.

In the past, I have done a similar comparison of 2 small pizzas vs 1 large as in the last lesson. I jumped right to measuring diameters and then also did a cost break down–if a square inch of pizza is about one bite, how much was it costing per bite of pizza? I also have decomposed circles to show how the area formula can be derived from a decomposed circle shaped into a parallelogram. I would really like to combine these two as we saw in the last lesson.