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Lesson 62: Unpacking Ratios – Discussion
Posted by Jon on December 6, 2019 at 6:30 amCreate two (2) ratio relationship scenarios; one (1) that tends to promote thinking as a multiplicative comparison and another that tends to promote thinking as a composed unit.
Compare and contrast the similarities and differences of these two scenarios which promote two different ways of relative thinking.
Share your reflection below along with any wonders you still have.
Luke Albrecht replied 11 months ago 18 Members · 20 Replies 
20 Replies

Well. For some reason, the last post only put up the image, not the notes I wrote. My reflection is that the problem seems to have two parts to it. First in order to figure out the width of suitcase B, I had to think of it as a “table of values” chart where I worked out the multiplicative comparison of the the heights of A and B. Then I completed the problem by thinking of that ratio as a composed unit.
Is this the right way (one of many) to do it?
It appears that multiplicative comparisons and composed units are very closely related.
My wondering is how to draw the pictures out! I also did another problem, included in the last post which did not upload properly. It had large numbers. That can be very time consuming! Don’t I sound like a lazy kid??

Multiplicative Comparison: Amy is 8 years old; Sally is 4 years old. Amy is twice as old as Sally. Sally is 1/2 of Amy’s age.
Composed unit: Jessica’s Purple Paint. Jessica mixes 2 cups of red paint and 3 cups of blue paint to make the perfect shade of purple paint. If she needs 20 cups of paint, how many cups of red paint and how many cups of blue paint will she need to mix to keep her perfect shade of purple?
I’ve done both of these with my students, the first one to have the discussion between multiplicative and additive thinking; the second to introduce the idea of using the ratio as a unit which can be iterated to solve a problem.
They’re both similar in the sense that they use relative thinking– you have to look at the relationship between the two quantities being compared; they are not separate amounts independent of each other.
They’re different in that the ratio of ages is like looking at a snapshot of two quantities, kind of like they’re a fixed point. The paint ratio lets you take that relationship and expand it– you can scale it up or down to find different amounts for different situations. Is she painting a wall? is she painting her whole house?

<div>Multiplicative Example</div>
A building is 18 metres tall and the building is nine times Jerry’s height. How tall is Jerry?
Composed Unit Example
Last Saturday I babysat Elsie for 2 hours and was paid $11.
a. I am going to babysit her again. If the time is tripled, how much will I expect to make?
Write the composed ratio.
b. How much will I make if I only babysat for one hour?

Multiplicative Comparison:
$2 compared to $8
$8 is 4 times greater than $2. $2 is 1/4 times as large as $8
This is from my grade 8 textbook. Many of the questions where looking just for the ratio to be written in ratio form or as a fraction. No emphasis on looking at the multiplicative thinking.
Composed Unit:
A black bear eats 22 kg of berries in 2 weeks. How many berries would the bear eat in 3 weeks? in 7 days?
I’ve never thought of ratios in this manner before. I find that multiplicative comparison for ratios is what I would generally use for ratios. With composed units, I naturally progress towards unit rates as soon as I see it. I think I can make more of the connection to ratios for composed units when I build a table with the information.

I use the IM/OUR curriculum so I this is how I am thinking of these
Multiplicative Comparison:
situations involving scale factors/scaled copies
A rectangle has a width of 2 and height of 4. A scaled copy of the rectangle has a width of 6 and height of 13. What is the scale factor?
Composed Unit:
situations like recipes, shopping/costs, mixing colors, time and distance (constant of proportionality)
A turtle went 6 feet in 42 seconds? How far could it go in 3 seconds?

Multiplicative Comparison Ratio
Julia is 5 years old and Katherine is 7 years old. So Julia is 5/7th the age of Katherine and Katherine is 7/5ths the age of Julia. I am having trouble totally grasping this because I am comparing it to the Truck problem and in the truck problem they broke the lengths down into 1/2 parts. Can I do the same with this?
Composed Unit
I guess would be when I am baking. So a batch of 35 puffed pancakes (ebelskivers) calls for 2 cups of flour. How many ebelskievers can I make with 1 cup or a half cup of flour?

You’re bang on with your multiplicative comparisons!
When it is an easier ratio, like 6 and 12, you can quickly / more intuitively see the double and half relationship… but 5 and 7 is trickier because now you are multiplying by 7 and dividing by 5 or “sevenfifthing” and the inverse which is just hard on the brain (but so important to become flexible with!).


John’s 2003 Saturn wagon goes an average of 16 miles per gallon in the city streets and 24 miles per gallon on the highway.
(This is a multiplicative comparison – the highway mileage is 1.5 times greater than the city mileage. The city mileage, in average, is twothirds of the highway mileage)
It costs John $30.00 to fill his car’s 12 gallon gas tank. Typically, he has to fill up every 240 miles. What is John’s cost per gallon of gas?
(This is a composed unit. I am going to use partitive division to reach the rate per gallon of gas: $30 divided by 12 is $2.50 per gallon. For each $1 John buys twofifths of a gallon of gas.)
One thing I noticed about the tables for these is that the multiplicative comparison table has values that can only be read vertically; but the table for gallons and cost can be read in both horizontal and vertical directions. (Not sure what this means though, or if it would usually or always be true!)

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.

Love your examples.
Something I found really interesting for your multiplicative comparison example was the fact that you compared one part of two composed units! You shared two composed units (cost to weight ratios) and then chose to multiplicatively compare the costs! Super slick 🙂


Multiplicative comparison:
Running: The number y of miles you run after x weeks is represented by the equation y=8x. Graph the equation and interpret the slope. Using the understanding of composed unit in order to understanding the meaning of the slope and the growth in a table or graph.
Using the understanding of multiplicative comparison. Chris is making a drawing of his school and the grounds around it. The basketball court is 75 feet long and 40 feet wide. If Chris uses a scale factor in which 1 inch equals 10 feet, what should the dimensions of the basketball court be in his drawing?

Composed unit: You are going on a pack trip with several people. Their gear weight 300 pounds. If one llama can carry 75 pounds, how many llamas will you need?
Multiplicative comparison: One pannier has a volume of 3 square feet while another has a volume of 4 square feet. the smaller is 3/4 the volume of the larger or the larger is 4/3 (1 1/3) the volume of the smaller.

Love your flexibility! It is already coming along 😉


Multiplicative comparison: A zipper snack bag measures 6.5 in x 3.25 in and a sandwich bag measures 6 1/2 x 5 7/8 in. How much taller is the sandwich bag to the snack bag? Honestly I was at my kitchen table looking for a comparison and saw these boxes. I find it interesting that the same company wrote one set of dimensions as decimals and the other as fractions. I wondered why? What a great question to ask students as we go to actually do the comparison. Even the real world switches between fractions and decimals not just school math problems. I think I would use the fractional amounts first as it can be modeled more easily as I have seen you do with the relational rods.
Snack bag length is 3 1/4 or 13/4 or 26/8
Sandwich bag is 5 7/8 or 47/8
Sandwich bag is 47/26 times bigger than the snack bag or 1 21/26 times bigger.
Snack bag is 26/47 times smaller than the sandwich bag
Wow! I think I got there and I see multiplying by the reciprocal but this was a complex problem because of the numbers. The models that you have been doing through the lessons helped me visualize what I was doing with the algorithm. That’s a powerful testimony about why we need visual for learners.
Composed unit: A snack bag hold 3 Oreos. How many bags would I need to bag 15 Oreos?
This seems so much easier if 1 snack bag holds 3 Oreos, 2 snack bags hold 6 Oreos, and I can jump in thinking that 5 snack bags hold 15 Oreos.
So I didn’t get this from a textbook. Am I on the right track? This real world problem seemed great and interesting until it got way complex. However, I could see my students engaging like I did, after a moment of wanting to give up. But I could see some students dealing out because we haven’t done enough visual and concrete work in class. I need to work on this.

Bob has $60 to fill his truck with gas. How many gallons of gas can he get for $60.00 if the gas costs $4.25 a gallon?
If rectangle A is 6cm by 9cm and rectangle B is 18cm by 27cm. How many times bigger is rectangle B?

Next week my students will work with dilations. This feels like multiplicative comparison to me.
Here’s one problem: You can use a flashlight and your hands to make a shadow puppet of a rabbit. The length of the ears made from your hand is 3 inches. The length of the ears on the shadow is 4 inches. What is the scale factor? (And I will also be sure to talk about scaling down from the shadow to the hands.)
In January we study slope. As we determine if a a relationship is linear, that feels like a composed unit to me.
For example: You can rent a kayak for 3 hours for $27 or for 5 hours for $45. How much does it cost to rent the kayak for 8 hours.

I’m still working on fully understanding this idea, but from what I have gathered, the following would provide examples of the two types of ratios:
Consolidated unit:
I make a lot of tea. If I make 3/2 cups of tea (to fill my big mug) with 1 teaspoon of loose tea leaves, how many teaspoons of loose tea would I use to make a pot containing 6 cups of tea? This would be assuming I was making it for people who like tea with the strength that I do!
Multiplicative comparison:
One potato from my garden measured 10 cm in diameter. Another potato from the same plant measured only 2 cm in diameter. The first potato is 5 times larger than the second potato and the second potato is 1/5 the size of the first potato.
If I’m not on the right track, please leave a comment to show me back to the path! And yes, I really had this kind of diversity in my potatoes. All the resources were poured into the firstborn apparently!

Multiplicative comparison–scale factor problem where we say the base of a triangle is 4 units and the base of a similar triangle is 8 units.
Composed Unit–2 hotdogs are $5. How much for 15 hotdogs?
The how much ice cream cone per dollar was great though it felt a little like going down the rabbit hole. I think it is good for teachers to see though.