Illustrative Mathematics: Examples of Good Activities and Problems

Tags: Illustrative Mathematics , Mathematics Curriculum

(Español: Illustrative Mathematics: Ejemplos de Buenas Actividades)

The Illustrative Mathematics Curriculum/Resources are a collection of Mathematics lesson plans, homework practice problems, and family support materials for all K–12 grades. They were created during the years 2016-2021 and are licensed with the very nice Creative Commons Attribution 4.0 license (same as this blog) which allows one to freely share and adapt them as long as one keeps the same license and gives due credit. They are very high quality, promoting deep mathematical learning through problem solving, and have been rated very highly by independent non-profit third parties (see here and here). You can learn more about the resources at the official Illustrative Mathematics site.

Besides just saying “these resources are highly rated”, or “I personally think these resources are very good”, I like having at hand examples that explicitly illustrate why I think the resources are very good. The purpose of this post is to show some of these concrete examples together with the corresponding explanations.

I led and designed the translation process to Spanish for the 6–8 Curriculum in 2018 and did the same for the K–5 Curriculum in 2021. You can read about this process in the post I wrote about it. One of my “big goals” is to create an adaptation of these resources that is optimized for use in Colombia and Latin America, and I am always on the lookout for funding to be able to do this.

In any case, I think that even without an adaptation, these resources can already start to make a lasting impact in mathematics education in Colombia. I am already accompanying some institutions to implement them.

First, a note about licensing

As I mentioned above, the Illustrative Mathematics K-12 resources are licensed under the Creative Commons Attribution 4.0 license, so I can legally reproduce any parts of these resources as long as I give appropriate credit! (This is a concrete manifestation of why I think them having such a license is so nice.)

So, here is the corresponding attribution ;-)

The problem and activity statements reproduced below are Copyrighted Illustrative Mathematics and are available under a Creative Commons Attribution 4.0 license at https://curriculum.illustrativemathematics.org/k5/curriculum.html.

And we are good to go!

The Examples

Example 1

Practice Problem 6 from

  • Grade 5
  • Unit 5 (Place Value Patterns and Decimal Operations)
  • Section D (Divide Decimals)

Statement

Noah has a scale that weighs to the nearest ounce. The table shows the weights of different numbers of paper clips in ounces.

How many ounces do you think each paper clip weighs? Explain or show your reasoning.

paper clips weight
1 0
10 0
20 1
25 1
50 2
100 3

What is so good or interesting about this example?

Well, because of the rounding of the weights, one clip shows up as weighing 0 ounces! The weight, of course, is not zero, and students need to realize that it is because of the rounding. They also need to realize that they will obtain the best estimate for the weight of a clip from the measurement of the weight of 100 clips:

$$ \begin{aligned} \frac{3~\text{ounces}}{100~\text{clips}}& =\frac{3}{100}~\text{ounces}/\text{clip}\\ & =0.03~\text{ounces per clip} \end{aligned} $$

This tells us one clip weighs about 0.03 ounces (other table rows say other things, but this is the best estimate).

Note also the “Explain or show your reasoning” in the statement. Students are asked to explain how they thought.

Example 2

In-class activity 1 from

  • Grade 4
  • Unit 6 (Multiplying and Dividing Multi-digit Numbers)
  • Lesson 3 (From Visual Patterns to Numerical Patterns)

Statement

Here is a pattern of rectangles that follows a rule.

  • Priya says, “Each step increases by 1.”
  • Noah says, “Each step increases by 4.”
  • Lin says, “Each step increases by 2.”
  1. Can you think of possible reasons that all of them could be correct even though they describe the patterns differently?
  2. Revise the statement made by each student so that what they mean is clearer and more precise.
  3. Priya writes the number list 1, 2, 3, 4, 5, 6 to represent the first six steps of the pattern she sees. Write a list of numbers to represent the first six steps of the pattern that Noah and Lin see.
  4. Predict what number Priya, Noah, and Lin will write for step 20 if the pattern of rectangles continue. Explain or show your reasoning.

What is so good or interesting about this example?

Well, all three descriptions from Priya, Noah, and Lin are correct, and students are asked to justify all of them. It is a great example of students seeing that math problems can have multiple correct answers.

Also, students are then asked to justify each correct answer, remove the ambiguity from each statement, and predict what the 20th numbers in each sequence will be. Fantastic!

This happens during class, with a proposal for teachers of having students work in groups and then share their work. Some students/groups may not see reasons why all answers are correct, and they will benefit from hearing other groups when they share their work. Students will learn together. Mathematical reasoning, problem solving, and communication are involved and required throughout the activity.

Equally important, solutions and guidance for the in-class discussion are available for teachers, so in case you don’t know how Lin’s description “Each step increases by 2” could be correct, the teacher guide has you covered!

Example 3

Warm-up activity from

  • Grade 5
  • Unit 7 (Shapes on the Coordinate Plane)
  • Section C (Numerical Patterns)
  • Lesson 13 (Perimeter and Area of Rectangles)

Statement

What is the area of one window?

Record an estimate that is:

too low about right too high
 

Image attribution: Victor Garcia. CC0 license. Unsplash. Source.

What is so good or interesting about this example?

Warm-up activities are the opening activities of the lesson, designed to engage, get the neurons working, and prepare students for what they will learn that day. Even without the context of the lesson, this warm-up is certainly engaging!

The main point here is estimating an area, but students are also being asked to give reasonable and unreasonable estimates. Lots of interesting mathematical discussions can happen around this activity. Then again, the actual learning in the lesson will happen after, not here!

Example 4

Practice Problem 8 from

  • Grade 1
  • Unit 7 (Geometry and Time)
  • Section A (Flat and Solid Shapes).

Statement

Is this shape a triangle? Why or why not?

What is so good or interesting about this example?

This is a Homework Problem. Students at this point in Grade 1 are just getting acquainted with some basic geometric figures (triangle, square, rectangle, circle, …). This is a great question to spot possible misconceptions students may still have: it looks like a triangle, but it is not! A “bend” in a side makes a new side, so this shape has 4 sides, not three.

Then again, and very importantly, note that students are being asked to justify their answer. Since grade 1, and constantly throughout the resources.

Examples from grades 6–12

I purposefully only selected examples from grades 1–5 to make them accessible to a general audience. There are similar examples throughout the whole resources in all grades, and I hope to at some point make another post with examples from other grades.

But maybe even more important than these examples …

What I think is most important in the end is that these are not just a collection of good, but loose, activities and problems. Instead, they are part of a fully sequenced day-to-day proposal for math classes, structured in such a way to promote deep learning and understanding.

These are the sort of things that are lacking in Colombia, where we don’t have a national Mathematics curriculum and little in way of structure at a departamental level. The closest thing we have are our 1–3, 4–5, 6–7, 9–11 Mathematics Competencies Standards, which require something akin to a degree in math education to be understood (if at all possible in some cases). Not good, since primary teachers often don’t have math related degrees.

But I digress – this lack of year-by-year unified Math Curriculum in Colombia should be the subject for another post. Back to the Illustrative Mathematics resources!

My point is that our students and teachers can really benefit from having mathematics resources like these. A plan that teachers can follow to help them support deep learning in a coherent way through grades (even if they only play a role in one of these grades). This would be an improvement even if the activities were not as good as the examples that I showed above. So, by the fact that they are that good, this is even better!

Again, I feel I should emphasize that this is already freely available in Spanish for teachers and schools to use! (only student statements — most of the texts in the teacher guides remain in English only.)

They are, right now, the only Math learning resources for all grades which are in Spanish (currently grades K-8, but more to come!) and have an open license.

Where to access the resources

This list is up to date on the date this post was published but some links may break in the future. If so, feel free to contact me, or visit the official Illustrative Mathematics site.

Grades K-5:

Grades 6-8:

Grades 9-11(12):

More examples

With Grupo LEMA ( www.grupolema.org), a group I helped create to promote the use of openly licensed math education resources at all levels, we are building a collection of sample good activities in all grades. I will post the relevant link here once it is published!

Subscribe

Want to get an email when a new post is added? If so, subscribe here.