Students will:

• Solve a problem about rice involving proportions. In solving the problem, they will consider ratio, rate, and scale.

The purpose of this Make Math Moments Task is to give students an opportunity to apply their understanding of ratio and rate to scale and to build fluency finding unknown values using a rate.

Some of the learning we will bring to light include?

• Estimating quantities through reasoning;
• Making connections between multiplicative situations where a quantity is iterated repeatedly and proportional relationships;
• Apply an understanding of ratio to scale and to build fluency finding unknown values using a rate; and,
• Explore and understand the difference between ratio and rate reasoning.

Watch this short clip taken from Module 8 of our course called The Concept Holding Your Student Back where we unpack the Spark portion of the lesson and set students up for a productive struggle:

Show students the video below.

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Write what students noticed and wondered on the blackboard, whiteboard, chart paper, etc.:

• I notice it is sunny outside.
• I notice a bag of rice.
• I wonder if they are going to cook rice?
• I wonder if there is enough rice for the whole family?
• I wonder how much rice is in the bag?
• I noticed that the bag appears to be opened.
• And many others…

Be sure to acknowledge each notice and wonder without over emphasizing emotion to ensure all students feel their voice is valued.

Then tell students to watch the next video to see if they have any more noticings and wonderings.

After watching this clip, students had more noticings and wonderings:

• I notice 200 mL of rice.
• I wonder how many servings he plans to make?
• I noticed that 250 mL of rice requires 500 mL of water.
• I wonder how much water he will need to make 200 mL of rice?
• And many others…

At this point, take their noticing and wonders and try to answer any of the wonders you can cross off the list right away. Things like whether it was close to dinner time or whether the person was going to use the remainder of the package to make the rice are all things you want to address and answer, if you can.

The focus question we will settle on is:

How much water will be needed to make 200 mL of rice?

Have students make estimates first by thinking about a quantity of water that they believe will be too low, followed by a quantity that they think will be too high. Challenge students to be risky with their low/high estimates. Then, have them pick their best estimate and share with their neighbours before sharing and recording on the chalk/whiteboard or chart paper as a class.

Ask students what information they might need in order to figure out the answer.

Once students ask what the rice box says about how much water they will need, show them the image below.

We will have students engage in this task without the aide of a calculator and thus we will promote the use of mathematical models as tools for thinking and representing their thinking. Encourage students to convince their peers.

Watch this short clip taken from Module 8 of our course called The Concept Holding Your Student Back where we unpack the During Moves of the lesson including unpacking student approaches and how you might consolidate this lesson:

If students are using algorithms and procedures with symbolic notation, be sure that they are able to clearly communicate why their strategy works. If they are unable to, ask them to attempt modelling either concretely or visually/pictorially.

Monitor student thinking by circulating around the room and listening to the mathematical discourse. Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:

• most accessible to least accessible solution strategies and representations;
• most common/frequent to least common/frequent strategies and representations; or,
• choose another approach to selecting and sequencing student work.

Discuss their strategies and elicit student thinking during your consolidation to build off of their current prior knowledge and understanding rather than “fixing” or “funnelling” student thinking to a strategy and/or model that does not connect to their strategy and/or approach.

I drew a model showing measuring cups to compare the amount of rice in the question with the amount of rice in the instructions.

I need to figure out how to get from 250 (the amount of rice in the instructions) to 200 (the amount of rice in the question) using a scale factor. I know that both numbers are multiples of 50.

To get from 250 to 50, I divide 250 by 5.

Then to get from 50 to 200, you can multiply 50 by 4.

I can use the same scale factor to figure out how much water I need for 200 mL of rice: divide by 5, then multiply by 4.

I would need 400 mL of water.

I used a double number line to compare the quantity of uncooked rice with the quantity of water needed to cook the rice.

The number line shows me the ratio of the quantity of rice to the quantity of water. I could tell from the number line that to make 200 mL of rice, I would need 400 mL of water.

Since 200 mL of rice is 4/5 of 250 mL of rice, then I need 4/5 of 500 mL of water to complete the receipe or 400 mL.

I can use a ratio table to scale both the quantity of uncooked rice and water at the same time.

I know that 250 and 200 are both multiples of 50.

To get from 250 to 200, you can divide by 5 and then multiply by 4.

I can do the same thing with the water. First, I divide 500 by 5. Then, I multiply 100 by 4.

Discuss and consolidate the learning to ensure that the mathematical thinking you intended to elicit is clear to all students.

Consider explicitly drawing out the idea that “dividing by 5 and multiplying by 4” is the same as “multiplying by 1 fifth and multiplying by 4” or, in other words, multiplying by 4 fifths.

You might show this in a ratio table such as this one:

I noticed that the instructions tell you to use twice as much water as rice since you use 500 mL of water for every 250 mL of rice. So, I just doubled 200 mL to get the amount of water that I needed. You need 400 mL of water if you have 200 mL of uncooked rice.

I can use the constant of proportionality or rate reasoning to utilize the fact that for every 2 mL of water, I need 1 mL of rice.

This means that I can use the rate of 2 mL of water per mL of rice to double any quantity of rice to find the quantity of water.

Inversely, I can use the inverse rate of 0.5 mL of rice per mL of water to half any quantity of water to find the quantity of rice.

During the consolidation, be sure to gather student work samples to build on both ratio reasoning (i.e.: scaling in tandem) and rate reasoning (i.e.: using a constant of proportionality or rate to solve for unknown values).

Being sure to make connections across representations especially the double number line, ratio table, proportion of equivalent ratios, and the equation of the proportional relationship is key.

Be sure not to promote tricks or to overemphasize any algorithms or procedures that are not student generated. Keep students reasoning through these types of problems rather than memorizing an approach that "should" work, given all conditions are the same.

Be sure to see the walk-through video (earlier in the online teacher guide version) to fully unpack each student solution approach.

Once students have shared their answers, show them the video below.

Revisit the student answers.

Have students discuss their thinking and what they would change if they did the task again. Return to the things students noticed and wondered from the SPARK section of the lesson, and answer any questions that have not been answered.

If you wish to pursue an extension of this activity, you could ask students extension problems such as the following:

Extend #1

How much butter would you need to add if you are cooking 200 mL of rice?

Extend #2

How much rice would you need if you plan on using 800 mL of water?

…students have difficulty figuring out how much water they need when they’re using 200 mL of uncooked rice.

• You could … Encourage students to use groupable concrete materials such as linking cubes to represent and compare the amount of rice being used and the amount of rice in the instructions. For example, each linking cube can represent 50 mL. Help students see that if there are 5 linking cubes representing the rice in the instructions and 4 linking cubes representing the rice we’re using, the rice we’re using is actually 4 fifths of the rice in the instructions. So, they need 4 fifths of the amount of water in the instructions as well.
• You could … Promote students’ use of visual models such as an area model (as in Sample Response 2) or number lines drawn to scale (as in Sample Response 3) in order to leverage their spatial reasoning as they make visual comparisons.

• Watch for students who see both types of ratios:
  • the ratio of the quantity of rice in the question to the quantity of rice in the instructions (200:250).
  • the ratio of rice to water (250:500).
• Watch for students who can articulate why the scale factor involves dividing by 5 and multiplying by 4. (E.g., The ratio 200:250 can also be written as 4:5, and this means we can use a factor of 4 fifths to figure out the amount of water.)
• Watch for students who use symbols and procedures without understanding to get to their answer. Ask these students to model their thinking concretely or visually so that they can better understand why the procedure they used actually works.