Make Math Moments Academy › Forums › Full Workshop Reflections › Module 5: Planning Your Problem Based Lessons › Lesson 52: Where Your Students Come From › Lesson 52: Question

Lesson 52: Question
Posted by Jon on May 1, 2019 at 11:57 amShare your chosen learning goals and any comments or questions you have here:
Noel McMillin replied 1 month, 3 weeks ago 32 Members · 34 Replies 
34 Replies

Question regarding going back to reintroduce concepts and skills students will need: what do you do when the majority of the class has mastery (even unconscious mastery) of a concept but a student or two does not? Do you meet separately with those students? I have seen teachers get “hung up” on this and spend way too long on a topic that some students are not mastering. Spiraling seems like a good answer to this, since you know you’ll come back to the concept again. But if you are building towards a learning goal, such as linear relations, and someone is not understanding proportions, what do you guys do?

I usually meet individually with those students for a few minutes while the rest of the class is working on a bigger task!


Happy New Year guys! This is a unit I’ll be starting the new year with with my Year 7s (Australian, here, so we start the new year at the end of Jan). The unit is on Whole Number and I’m using as the example for this reflection the idea of index notation. There are a few outcomes relating to this broad concept in the Year 7 Australian Curriculum course but I’m focusing on representing patterns in square numbers and the connection between square numbers and index notation. Last year, with QR codes being very topical, I designed an exploration called Quirky QR codes challenging students to work out how many unique QR codes can be drawn on different sized square grids. Considering prior knowledge, like most of us, my classes tend to be very diverse with a 4 – 5 year difference in skills being typical. Some students will know what ‘square numbers’ are and have the understanding that it is x times x (eg 2 x 2, 3 x 3, 4 x 4 etc). They may know that square numbers form square arrays. There isn’t any specific prior knowledge on index notation, as it is introduced in Year 7 for the first time. My experience tells me that they confuse ‘squared’ with ‘x 2’ eg 6 squared = 12 (6 x 2 not 6 x 6) so this is a common mistake I’ll be looking out for.

My overall teaching goal is to teach my students division. The learning goal for my first lesson will be to teach them how to share equally when the number of sets is known.

This is actually how we plan our lesson for the week. I teach 2nd grade.
1. Where are our students coming from?
We spiral in. We build background knowledge. We have seen the 1st grade standards.2. Where are our students headed?
We establish the objective, teach the vocabulary (if not already known), and scaffold on to what students should already know. For instance, in 1st grade students should know the value of coins. We review that, but we move on to counting the money.3. Anticipate what students will do.
We anticipate the mistakes or misunderstandings so we can try to avoid those.
Great reflection and share for others to think on!


I am gong to be teaching how to solve equations. Here are learning goals that might be important.
1) Understand integers, negative numbers and how you can change subtraction problems to addition.
2) What is an unknown, what is a variable
3) What are the inverses of operations (+x/)
4) Be able to create fact families

Not only within the progression of our lessons, but in general, I think this is why Vertical Articulation amongst grade level teachers is important. For us to see where students are coming from (in terms of topics covered, skill level, etc). One of the topics we are looking at right now is the Pythagorean Theorem. Prior knowledge required would include: squares and square roots (in helping visualize and prove the theorem and to solve problems), types of triangles (angles and side lengths), coordinate grid system (when working on distance and midpoint). All of these concepts/skills should be reviewed in some capacity prior to proving and investigating the Pythagorean Theorem to allow for a smooth exploration.

Nice work here. And donâ€™t forget that we can also use the introduction of this topic as a means to dig deeper into those prior skills and concepts you had mentioned!



I’m introducing matrices to my 11th grade IB class. We don’t teach matrices at all earlier in the curriculum, so I thought it would be interesting to think about a learning progression for matrices.

This fits well with what we are learning next week. Volume of spheres and cones. They’re coming from 7th grade where they learned the relationship between surface area of the base of a prism or cylinder with its’ height and volume. I really like the meatball example. I think that it will be great to get students to reflect about what volume is, how it relates to the base area and depth of a cylinder. They can then speculate with regards to the volume of the spheres.

I’m currently working with a 4th grade teacher and his students on multiplying decimals by multiples of 10. Here’s my Learning Goal Progression:

The next unit I am teaching will be Graphing Systems of Equations. It is important to know what our studentsâ€™ abilities are before teaching the unit because we need to know what prior knowledge they will need or they have. Students will need to know many things before graphing systems of equations. They will need to know how to use the coordinate grid, graph lines, how to use substitution and elimination, how to find x and y intercepts manipulating the equations or using a graph. So many students in my classes have gaps in their learning, so this can be very challenging. I like the idea of spiraling lessons and want to incorporate that method more in my teaching.
 This reply was modified 9 months, 2 weeks ago by Dawn Oliver.

Our next topic is proportions, particularly looking at proportionality with equations, tables, and graphs. Students need to know fractions/ratios, how to find equivalent fractions, how to multiply, divide and simplify fractions, decimals, relationship between division and multiplication, understanding the coordinate plane, independent and dependent variable, the concept of a variable, evaluating expressions.

We are just ending the school year, so I will be using this after we come back from summer. I will probably start with solving equations, so I will need to go back and make sure they understand what a variable is; that they understand like terms and how to combine them; that they can work with positive and negative numbers; that they understand what an equal sign actually means.

This is for my Geometry class in Chapter 10 on area and at the end we do Geometric Probability.
 This reply was modified 7 months, 1 week ago by Renee Holmquist.

I will be using this planning template with solving systems of equations. It is easy to forget how many different skills they need to have mastered to be able to solve a system. It is also easy to forget how beneficial, and not that time consuming, it is to step back and reacquaint the students with the knowledge in order to catapult them forward.

I am going to tech properties of quadrilaterals this week so I will start with pairs of parallel and intersecting lines including perpendicular. Then look at the angles they make and classifying them into acute, right, obtuse and reflex. Estimate angle measurement. I will need to check if they know vocab such as polygon, sides, opposite and adjacent, as well as checking if they know how to use a ruler properly to check for side length. Without all this, the properties of the different quads will not mean anything to them!

I took a look at day 1 lesson for 8th grade students in general education setting. I will be teaching the intervention (pull out class of 8th grade students). I was actually greatly surprised at what they will be doing on day 1 Exponents! I am not confident that my students will recognize an exponent or know what a number raised to an exponent is.
After listening to yesterday’s podcast, Misconceptions About Teaching Students Who Struggle With Math I decided I will start where the general education is starting…. but I feel like I will be behind by day 2. Day 2 the general education continues with exponents by using exponents to write prime factorization of numbers.

This lesson is so timely. I started to teach graphing solutions to inequalities today but quickly found out that I needed to backup when I discovered that students did not understand how to solve equations. Doing this activity helped me realize exactly how far back building blocks could go. Time for a lesson revamp and relaunch.

Love it! Glad you found the work helpful and is giving you a reframe of how you might approach this concept!


As I was filling in the graphic organizer, I reflected on what Jon and Kyle mentioned in the beginning of the course how we don’t preteach our students before they dive into the meaningful problem and let them get curious. I wonder now how this can be achieved as the students still need prior knowledge to facilitate the discussion. It is still necessary to think about the grouping of more knowledgeable students in one group so they won’t do all the thinking work for the less knowledgeable ones. Strategic planning for each and every lesson is much needed! I really wonder where to start when I start planning for the year.

My big idea is adding and subtracting integers (planning for the fall for back to school).
Here’s what I put together. Then I’m thinking that I want to extend it to where positive and negative fractions are on a number line and adding them.

Here is my learning goal progression for Rational Number Operations. I listed the precursor skills in roughly the order they are typically taught in the U.S. (according to Common Core Standards).

My lesson is for a College Algebra Course on Quadratics.
Learning Goal:
To model and interpret real world examples with Quadratic Functions
Prior Knowledge: What is a Quadratic Function? Knowing the important parts of a quadratic function like the Vertex, line of symmetry, and zeros
Prior Knowledge: Solve Quadratic using different methods: graphing, factoring, quadratic formula, and completing the square
Prior Knowledge: Determining the independent and dependent variable and understanding what the question/situation is asking the students to solve
Learning Goal:
To model and interpret real world examples with Quadratic Functions.

I have taught 4th grade in the past and will be starting teaching math K2 this year for the first time. This planning will be incredibly important for me as I work to support students in K2. I will have the time to familiarize myself with all the standards and their relationships and how they build on each other. I also know where they are going (I have taught 4th grade and 3rd grade for the past 8 years). For each concept in the fall, I will be using this framework to help me know how to create a low floor to the tasks I present at different grade levels. I want to also remember to start concrete, then visual, then abstract. There is a lot to think about when preparing each lesson as I will also want to spark curiosity with those steps as well. I know with time and practice, this planning will become easier, but it is a lot to think about before living it.

I have done lots of work on fractions and therefore I think I would like to use this opportunity to really develop a sound trajectory of tasks to fulfill the goal of “thinking about dividing fractions as a way to figure out a unit”. I had an extra step of prior knowledge that I just couldn’t seem to be able to leave out, so I developed a flow chart with an added prior knowledge block. : )
 This reply was modified 5 months, 2 weeks ago by Kami Fevery.

For the first concept of our curriculum, the students will be working with proportional relationships as seen in graphs and tables as well as other representations. In order to be successful with this concept, students will need to: be able to find equivalent fractions and know what equivalent fractions are; simplify fractions; find unit rates; and write data as a rate/ratio showing the comparison of the two quantities.
I totally understand and see the importance of this process and agree that it is crucial that students have the foundational understanding before you can move forward successfully. However, I am concerned with the amount of time this process would take and how I would possibly be able to address ALL of the learning standards that I am expected to work through in a year, preferably before state testing which is in early to midMay. I feel a constant internal struggle with doing what is right by students and meeting the goals my administration has for me, i.e. complete ALL curriculum prior to testing.

Grade 7, Part of the Algebra: Expressions and Equations Unit

In a few weeks I will be teaching students how to write equations of a line from data points of information. To be able to do this, students need to be able to substitute in values to variables and solve, the need to understand slope and they need to be able to understand the concept of Slope Intercept Form. All of these concept have been covered in depth this year leading up to this point.

Great module today… thank you.
It really helped to consider the learning progression towards our linear equations exploration.

I am at the beginning of our linear equation unit. I started with a couple of “Notice and Wonder” activities involving pizza from two different pizza places in our town. My goal is for students to conceptually understand slope and yintercept and in this case cost per topping and the cost of cheese pizza (we even had a great discussion about whether or not cheese counts as a topping). In order to get there, students need to understand the unit rate and rate of change. We also needed to revisit collecting data, making a table, and graphing the results appropriately. Even the concept of mathematical patterns was challenging for many students. We followed up with the “Goose Egg” problems which were great for identifying the rate of change and starting value. The numbers were easy to work with and the pictures made it easily accessible for all.
Something that came up that was a little surprising was how repeated addition and multiplication are the same things. Many of the students relied on repeatedly adding the cost per topping rather than simply multiplying by the cost per topping. I did my best not to tell them to multiply but certainly asked questions to steer them in the right or better direction (how about a pizza with 37 toppings?)
Tomorrow we literally start with abstract linear equations. Something else we will need to discuss more indepth is the independent and dependent variables and just how “y” is a function of “x”. Of course, the slope will be a fraction eventually and that will add another layer of complexity for those who struggle with multiplying fractions.
Looking forward to it!