Event: How to Provide Equal Access to High-Level Math for Students of Mixed-Ability Levels

Tomorrow we will be joined by Patricia Helmuth, our guest expert, and presenter, to discuss strategies to modify high-cognitive demand math tasks to engage a classroom of students at varying math levels.  Math and Numeracy CoP members will discuss how to uncover what students already know and then support them in developing a higher level of thinking and a new understanding of mathematics.

To get the most out of our upcoming three-day discussion, “How to Provide Equal Access to High-Level Math for Students of Mixed-Ability Levels,” you are invited to and encouraged to explore the following resources:

  1.  Watch Dr. Jo Boaler’s Video to discuss best practices in leading a Visual Dot Card Number talk:  https://www.youcubed.org/resources/jo-teaching-visual-dot-card-number-talk/
  2. Access and participate in The Apple Orchard Slide Show:  http://bit.ly/2lvsJBd
    Remember, to try each activity first as the student and then put on your teacher hat.
  3. Become familiar with the teaching approached called push and support that will be discussed by accessing the following curriculum guide: http://bit.ly/2lQVete

This discussion will focus on integrating learning through the following objectives that will guide our dialogue:

 1. Examine the instructional approach called push and support

2.  Discuss and share strategies for implementing and supporting adult math and numeracy learners.

3.  Brainstorm and come up with approach on how to assist learners who might struggle or need more challenging work to complete.

I cordially invite you to begin exploring the resources and invite you to join in our discussion with Patricia, a teacher leader, trainer, and the newsletter editor for the Adult Numeracy Network (ANN). Please be sure to become a member so you can participate in the discussion and engage with other community members.

Brooke Istas
Math and Numeracy Moderator


Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills. This is true regardless of what level a student may be assessed at when entering an adult education program or what level class they are placed in. We all want the best for our students, so this leads to questions:

  • Can we individualize instruction for our multi-level students, while at the same time, provide all students with equal access to high-level math tasks?
  • How can we level this imbalance— not going too fast for learners who need time to process a math task, while at the same time, not going too slow for learners who are done with the task and are ready for deeper exploration? 

Today we will begin with a visual number talk. Number talks are non-intimidating ways to get students talking about the way they see groups of images. Every student can participate by sharing how they see the visual model.There are no wrong ways of seeing!  If you are new to visual number talks, please watch the video at youcubed: Jo Teaching a Visual Dot Card Number Talk  to get an introduction to number talks and to see how this type of activity might play out in your classroom. We’re going to approach this task with our student hats on as we think about all the varied learners we have in our classrooms. We’ll take it one step-at-a-time, just as we would do with our students.

For Day 1 of this activity, please access The Apple Orchard slideshow  for our virtual number talk and follow the prompts for slides 1-9. The slides will advance automatically. During the slideshow you will be prompted to pause and reflect. Please take the time to do this and then respond to the following prompts: 

  1. How did you group the trees (slide 4)? Describe how you saw the visual model and share a diagram that illustrates your thinking and/or an equation that represents how you grouped the trees. (If you want to upload a picture of a diagram, see last week’s LINCS Math and Numeracy discussion: Pentominos: Introducing Algebra Through an Arithmetic Puzzle where Heidi Schuler-Jones explains how to upload images to LINCS discussions using  https://postimages.org/)
  2. Put on your teacher hat and choose one of the student-generated equations from the slideshow (slides 6-8), or use the equation that represents the way that you grouped the trees, to think about how you might use this as an opportunity to draw out students’ understanding of the order of operations, distributive property, use of parentheses, etc.
  • For example, consider Student Example 1 on slide #6 of the slideshow:     5 x 5 - 9 + 4 = 20.      Would adding parentheses to the equation change the solution sometimes, always, or never?

Please enjoy the slideshow! Many thanks to Christin Smith for the graphics design of the Apple Orchard that you see in the slideshow.


Hi Patricia,

I'm looking forward to this discussion. Looking at the apple orchard pattern after the Boaler number talk is really interesting. It definitely helped me see it in different ways. But it didn't stop me from miscounting the number of trees the first time round. :)

I initially saw the pattern like this and came up with 25 trees, which I later realized was wrong. I tried to find the number of pine trees on the border by multiplying 5 x 5 and subtracting 4, then I added the 4 apple trees in the center.

Image removed.

I should have multiplied the 5 pine trees on each side by 4 (number of sides), then subtracted the 4 corner trees that were double-counted, and then added the 4 apple trees in the center: 5 x 4 - 4 + 4


Thanks, Connie, for helping us out with a few more tips on how to upload an image. It’s not exactly intuitive, so we appreciate the help! 

That being said, I encourage those who haven’t shared their thinking yet to not hold back if you don’t have an image to share. Simply describe what you saw in terms of how you grouped the trees in the apple orchard and share what equation you came up with that represents how you grouped the trees.


Image removed.<-- I saw the trees like this

Image removed.<--Which seemed to match this expression from student example 3

If I wanted to use this activity as an opportunity to discuss the properties or order of operations, I would try to teach the controversy! 

I might choose a student-generated expression like student example 3 and ask students to work out an answer. After collecting answers and sharing ones that differ (supplying different answers if there are none: 40, 20, 84), I might challenge everyone to figure out how one might arrive at each answer (even if they think it’s wrong). 

Deciding that there is *reason* behind answers that seem *wrong* is important; then the conversation might naturally flow toward decisions we make when writing math and the purpose behind an accepted order.  

For example, this particular conversation might discuss the merits and drawbacks of writing the above expression using the distributive property: 2(5+3) + 4


A few of us have shared our thinking in terms of how we saw groups of trees in the apple orchard and I encourage all those new to this discussion to continue to share your thinking and equations with us.

In the meantime, we can start to think about the equations that have already been generated by our discussion, or the equations that were generated by students in the slideshow, in terms of how can we use those equations to draw out student thinking in regards to what they may, or may not know, about order of operations, use of parentheses, or distributive property— to name a few. 

Christin has gotten us going by sharing her thoughts on how she might use her way of seeing the orchard to teach the controversy in regards to the order of operations! Thanks for sharing that, Christin!   Similarly, let’s compare Eric’s way of seeing the Apple Orchard with slide #6.

Eric's Thinking:  5 x 4 - 4 + 4 = 20

Image removed.

Student Example #1: 5 x 5 - 9 + 4 = 20

Image removed.

How is Eric’s thinking similar to the thinking of Student Example #1?  How are they different? How could these equations be used as a jumping-off point to explore what students already know about order of operations? What questions could you ask students?  What exploring could they do?

Example: Would adding parentheses somewhere in either equation make the statement sometimes true, always true, or never true? If I switch the order in which the numbers appear in the equation, such as adding the 4 at the beginning of either of the equations above (instead of adding the 4 at the end),  is the statement still true?

Finally: Why might you use student generated equations to explore order of operations rather than examples in textbooks?


Christin and I seem to have similar thinking with Example 3 on Slide 8. Here's how I first grouped the trees.

Image removed.     Image removed.

Heidi's orchard                                                                                             Christin's orchard  and  Student Example 3


I like the idea of using these to do a notice/wonder about what's the same and different. That could lead to some great conversations about what about my groupings in the picture led to my setup of the equation and how that compared to Christin's setup.

It might also be interesting to have them rewrite the equations without parentheses or the x for the multiplication symbol. I might even open up the question to ask, "How many different ways can these equations be written?" I could have them write their equations on sticky notes or index cards, collect their responses, and post them around the room as individual stations. Beside each equation, I could have sheets of chart paper and markers available for students to illustrate how the equation might be reflected in the orchard. It would be a way of undoing the work or working backwards from their equation creations. Each design and equation could later be scanned and made into cards for a card matching activity for other classes to use or as a review throughout the term.

Makes me wish I had a class to try this on now! : )

Today we’ve seen some great examples of how a visual number talk might play out in our classrooms. We’ve demonstrated how there are many different ways of seeing a visual model and expressing our thinking as an equation. In our classrooms, when students generate their own equations, they own them!

Christin and Heidi shared some really terrific ideas about how we might use these student generated equations as a springboard to bridge to activities where students can engage with “doing and undoing”, explore the order of operations, the use of parentheses, the distributive property, the commutative property of multiplication, make their own card matching activity, and more. I recommended saving their ideas for use in your own classroom. Great stuff!

Be sure to check back tomorrow for more strategies on how to use the apple orchard activity to bridge to higher-level math topics that are accessible to students of mixed-ability levels.


For those of you who are new to this discussion, jump right in to today's exploration of the apple orchard. Let's see how far we can advance the understanding of the apple orchard pattern.

Yesterday, we approached a task in a way that all students at all levels can find their way into by using an instructional strategy called a visual number talk. Today we extend our discussion of the apple orchard with another strategy that allows equal access to a high-cognitive task by asking: What do you notice? What do you Wonder?   (Please follow the link to collectedny.org for a more detailed explanation of notice and wonder. I recommend watching the Annie Fetter video that is featured there. She tells a great story about notice and wonder.)

Every student can notice something about a visual pattern and generate a question about that pattern. This is another non-intimidating instructional routine that encourages students to talk about what they see. Again, there are no wrong answers! 

  • Today, we will put on our student hats again and think about what we notice and wonder about how the apple orchards are changing on slides 10 & 11 of  The Apple Orchard slideshow. Here is what some students noticed and wondered about the pattern.

Image removed.Image removed.

  • Please approach this task with your student hat on and post what you find interesting (notice) about the pattern as well as a question (wonder) you have about the pattern that you want to explore. Please share any conclusions you reach after exploring your question along with your solution pathway. Today we want to share and compare strategies for problem solving.
  • As we move through our discussion today I will be adding some Push and Support  cards (another strategy that lends support to students of mixed-ability levels) for the Apple Orchard Activity.  Feel free to engage with any of the cards that catch your interest. 

Hi all,

Here are a few things I notice and wonder:

Image removed.


  • Orchard #3 is 2 pine trees wider and 2 pine trees longer than Orchard #2.
  • Orchard #3 is 1 apple tree wider and 1 apple tree longer than Orchard #2.
  • The pine trees form a border (linear or a measure of length). The apple trees form a "filled-in" space (quadratic or a measure of area).


  • What would Orchard #1 look like?
  • Is there an orchard # that would have more apple trees than pine trees?
  • Is it possible to draw an Orchard #2b with 6 pine trees on a side? How many apple trees would there be?


For those of you who aren't already in the process of exploring a question, Eric has just shared some great "wonderings" that you might want to check out yourself. Here are a few others that you might want to consider:

Image removed.

After you've explored one of Eric's "wonderings", one of the above prompts (push and support cards), or one of your own questions, please come back and share:

  • ...the method you used to make sense of the prompt or question you chose.
  • ...how your exploration extended your understanding of the apple orchard pattern

Yesterday we tried out the Notice and Wonder strategy as we looked at the Apple Orchard Pattern. We also saw a few examples of Push and Support cards that enable students at all ability levels to engage with the pattern. Below is an example of how a group of participants at the Adult Numeracy Network’s Annual Meeting, at the COABE National Preconference earlier this year, engaged with one of the push cards that was featured in yesterday’s post:

Image removed.

Can you tell which card they were working with? Hint: They posted the card to the bottom right corner of their project.

If you started working with one of the push & support cards, one of Eric's wonderings, or one of your own questions about the Apple Orchard pattern, (or even if you haven't had a chance to yet), I encourage you to do so and post your problem solving methods and reflections here. Or, you might want to explore the pattern using any of the Apple Orchard  push and support cards that were created by the Adult Numeracy Network PreConference Planning Committee.


Today we will focus on why and how to use push and support cards with our students.

Why: In addition to having mixed-ability students in our adult education classrooms, many of us also have classes that are open enrollment. Thus, we need activities that new students, students who been attending class for a while, and students at all levels can engage with and walk away from the activity with new learning. With push and support cards, all students can be working with the same initial problem, but then take it as far as it is appropriate for each student. Not all students need to take the task to the same level. 

How: After you have broken the ice with a non-intimidating activity, such as a number talk or notice and wonder, have students pair/group to explore a question that interests them (or give them a guiding question or prompt). As you observe the way they are approaching the problem and listening to their conversations, you may discern that they are stuck or need something more to explore since they have already found a solution to the initial task. This is when you would give them a push or support card to keep them moving forward with their exploration of the math embedded in the problem. There is no pressure on the student or pairs/groups of students. You simply place a card in front of them and walk away.

Let’s try this out. Below are a few samples of student work for the Apple Orchard Activity.

Image removed.

(If you have some difficulty seeing the details of the samples of student work, you can access a larger view of them here.)

  • Which support or push card would you give one of those students, based on what you are seeing in the work the student has done thus far? Observe what seems to be the pathway the student is already on and use a card to help build on what he or she is already doing. For your convenience, some of the questions and prompts that are on the cards are written below.

    • Draw Orchard #5

    • Write a description of how you see the pattern changing.

    • How do you see the apple trees increasing?

    • How do you see the confer trees increasing on the top row? On the side row?

    • How many conifer trees would be in Orchard #10? How many apple trees?

    • Create a table to organize your information about how the pattern is growing.

    • Visualize your table on a graph

    • Write a description of how you would determine the number of conifer and apple trees in any Orchard #.

    • Write an explicit rule for the Orchard pattern.

  • Please choose one of the samples of student work and explain which push or support card you might give the student. Keep in mind that more than one of the cards may be helpful to the student. Additionally, a student may come to a conclusion on one of the cards and will be ready for an additional challenge. This is not a multiple choice quiz where there is only one right answer!



As we close this activity, but not the discussion, I want to thank Patricia for unpacking this high-level math activity.  Hopefully, this is a problem that you can continue to unpack and discuss here or you can take this lesson and begin using it in your classes.  Thank you for your participation in this discussion and thank you, to our expert, Patricia.  lease continue to share your thoughts on this problem and let us know if you plan to implement this into your classroom (also, how it went, what you changed, and any other feedback you want to share).   If you would like to lead a special discussion in this group please reach out to me, at brooke.istas@cowley.edu.  I would love to see more participants share their techniques and have fun in our group!

Keep thinking like mathematicians! 

Brooke Istas

Launch: Apple Orchard Visual Number Talk

  • A warm-up activity that gets students thinking about how items in a visual model are grouped. As we saw from our discussion, this grouping activity can lead to:

    •  discussions where students compare the different equations they generated 

    • Doing and undoing, order of operations, use of parentheses, distributive property, commutative property, and more.

Notice and Wonder

Push and Support

  • Included in Apple Orchard with Push and Support Cards

  • Make a copy of this document and change up the cards to reflect the needs of your students 

  • The apple orchard pattern leads to both linear and quadratic growth. Every student takes the problem to the level that is appropriate for the student when using the push and support cards

Curriculum Guide

Additional Resources

We learn so much from each other when we share instructional strategies and student stories. So, if you try out the Apple Orchard activity and the push and support cards with your students, please come back and share your reflections with us. Let’s keep the conversation going!


Thank you Patricia and Brooke for this discussion and all the activities!  I am pushing this out again to our ABE teachers because many of them as just getting back into their classroom.  I especially like your last post.  It will make it much easier for those of us who are "catching up!"