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Lesson 52: Question
Kyle Pearce updated 1 month, 2 weeks ago 49 Members · 73 Posts 
Share your chosen learning goals and any comments or questions you have here:

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?

My learning goal is solving onestep equations. Students need to understand what a variable is. They need to understand adding, subtracting, multiplying and dividing. And they need to know that adding/subtracting and multiplying/dividing are opposites of one another.

Awesome. Have you thought about a task or unit you intend to use?


As I look at the previous grades and the expected prior knowledge coming into grade 6, students have been reading and representing numbers in increasing values and place values each year. As they enter in grade 6 and the place value has increased to one million, students are already expected to know how to compose and decompose numbers and this is no longer part of the grade level expectation.

Interesting! Now do keep in mind that just because an expectation is no longer present (ie, covered in earlier grades) doesn’t mean we don’t want to also come back and revisit in future grades. This is what makes teaching the higher grades so challenging is that there is so much prior knowledge we need to continually bring back to the forefront.

I completely agree and understand that! My question with this, is where do we draw the line to go back to for the whole class? This gets especially tricky when time is limited. I teach at a virtual school, where I only get 1 hour per class per week.

Definitely a hard question that many educators find themselves having to navigate. Unfortunately, for most, the result is drawing a line somewhere and then some (all?!) students being left behind. This is why we strive for problem based lessons that are accessible by all students to approach using their own strategies based on where they are. If we draw a hard line some where, we cut some students off. There is no easy answer to this question, but do know that all students benefit from thinking about earlier mathematics as well.



I think I am going back too far….standard is factoring algebraic expression and equivalent expressions;

Good stuff here @maryannabiedermann I don’t think you went too far back. The more we know about the progression the better prepared we’ll be in the classroom when students show understanding along that progression.

Today was complete remote, 25 minute class, we worked through Name Game slide. None of this is my idea. I am going to try to reuse/build on the attached for some of the ideas in the progresssion…any feedback is welcome. Thanks for all you do. LOVED the Mindfullness podcast



We will start Proportional Reasoning this week.

Nice work here @jody.soehner You have a clear view of the progression. What tasks are you thinking of using to see your students understanding along this progression?


I am looking at my next unit in Algebra which will be Factoring Quadratics. This was hard to narrow down to 4 previous understandings because there are many little skills to be taught to truly be successful in factoring quadratics, primarily if taught using the concreteness fading method. Do these typically look more like a spider web of previous knowledge over a linear progression? Well, here is my best attempt at this time to get a progression in 5 easy steps.
Area of Rectangle/Generic Rectangle method for multiplication
GCF–ability to look at all three terms to see if there is a common factor to factor out of the trinomial
Polynomials Identification–To be able to classify what a Quadratic trinomial is in comparison to a linear binomial
Distributive Property–To understand how product and sum form are related
Completely Factoring Quadratics–with all common factors out in product form
There are a lot of realms I can look at with this method of discovery. Like the ability to use and understand algebra tiles, multiplication of monomials, or even understanding what a square variable is. These have all been roadblocks that I have seen in the past for students understanding factoring quadratics. However that is why it is fun because it is fun to teach algebra, because I am always surprised at what previous understanding we get to reintroduce.

You’re so right about how complex ideas like factoring quadratics are.
I might start back at multiplication with partial products and area models. If students aren’t seeing that factoring is just “division with unknowns” then they will forever be stuck with memorization of a procedure as their only entry.


Learning goal: solving equations in the form p(x+r)=q. I want students to be able to see what the parenthese means and how it is a tool to understand/symbolize certain situaitons. I want them t osee how dividing by the coefficient and using distributive property both can be useful and to use both fluently depending on what is more efficient. For example, if there is a fraction as the coefficient, clearing the fraction might be better.
I started this equation unit with the shotput task. I then gave them a basketball video of differert incredible shots from long distance and they liked that but gave the same pathway with the measuring sticks and different ways to measure it. It helped us understand the variable and the construction of the px+r=q equation. They are still guessing and checking instead of using inverse operations however. So as we move towards grouping we will work to see if they can discover the inverse operations too. I find I am doing better with the intros but struggling with the bridge to sense making wihtout eventually giving them rules – especially remotely. I will keep thinking and working. This is helping. Thanks.

New learning goal: to multiply whole numbers of up to four digits by one and twodigit numbers
Prior Knowledge required:
Repeated addition
Skip counting
Multiplying one digit numbers
Multiplication using arrays
Base ten models
Multiplying by the place values
Area model
Factorisation

This all makes sense, and I have seen other teachers implement their own version of this in a pinch in the classroom. I am wondering how much time to dedicate to teaching necessary prior knowledge. I’m just trying to get a sense of how it fits into a unit plan so that this unit would take the same amount of time as a “traditional” unit would.

Our focus is on low floor high ceiling tasks. So by entering with a problem based approach, you are giving students an opportunity to access or revisit their prior knowledge.
Have you tried any of our problem based units?
learn.makemathmoments.com/tasks
Try one of these and you’ll get a better sense of what we mean!


My learning goal is to calculate an estimate of the mean from a grouped frequency table
Prior knowledge 1: Review of mean, median and mode of small discrete list using miniwhiteboards
Prior knowledge 2: Finding the mode of a large list of discrete data so we can see why a frequency table could help.
Prior knowledge 3: Finding median from a frequency table
Prior Knowledge 4: Finding the mean from a large list of discrete data and seeing how a frequency table could help get to solution faster
Then finally look at grouped frequency tables and the knowledge that online we do not know the original values if in a group but we can find the modal group and estimate mean

Super solid.
Wondering what a spark might sound like for this work? How will you get students to lean in?



That’s what I did, I had to teach about rate of change ina function. Students were supposed to know how to solve proportionality problems, how to make a table of values and a line graph, how to find the algebraic formula of the function, how to find the slope and the cutoff points on the axes … I’ve prepared an activity to see what they really know, relating diferents cards situations with som cards qüestions, card graph, cards with algebraic functions, cards with slopes and intersections on the axes. Some of them where to relate other ones where to find or calculate and another ones where to invent. So I realised that they just knew how to solve a proportional problems. Half of a class knew how to do a graph and a few where confident to give an algebraic function. I didn’t do a good work sparking curiosity but they did the posters pretty well working in groups striving for all members of the group to understand the task. We have make some other activities toi’hope ensure that they learn this concepts well. I’ve even decided to firstly have a tourné introducing quadratic functions (I’ll try to do it with the herous path) and after that they will be able to deduce the way to calculate the rate of change of a quadratic function and of any function. Lot of time, but it is not a waste of time I’m shure. It is useless to explain function analysis if they do not feel confident with the basics.
 This reply was modified 6 months ago by Laura Las Heras Ruiz.

Surface Area of a Cylinder through Investigation: Students should have prior knowledge of areas of rectangles, and circles, as well as diameter, and circumference of a circle. I will still do activities where students explore circles in pairs, and after recording diameter and circumference they will be asked what do you notice when you divide the circumference of each circle by its diameter? Students will also explore area of circles by cutting them into equal sections and looking for a relationship, and also cutting cylinders apart and being asked what they notice about the width of the rectangle and the circumference of the circle.

I’m liking this progression…I’m wondering how you could design this lesson so students “lean in” or are chomping at the bit to figure out the surface area by cutting it up. How can we spark that curiosity to start it all off?

What are you thinking here? You have sparked my curiosity.

I think your progression is great, I’m just wondering if you can use your learning from module 2 to spark curiosity around cutting up the circles…how can we create a hook to cutting up those circles so students will be eager to explore?



We are learning about angle relationships. I started with a basic reminder of the names of angles (complementary/supplementary) so that students had some vocabulary. Then I was hoping to give them a map so they could compare the intersections of streets? I don’t know…. I’m struggling to find the fun in this task!

@kathleen.bourne Have a look at the video we just released this week: https://makemathmoments.com/howtoteachanglerelationships/

ooooooh, nice timing! Thanks so much. I’ll check that out and try it. I didn’t want to just point out everything – feels so boring and for the kids, why do I need this?? This activity will help, thanks.



This was a great exercise for me to really dive deep and evaluate my students’ needs, especially since we know that this past year was so unusual. In a typical year, my students come to me underprepared with basic skills, and I suspect I will see even larger gaps next year. This helped me focus my attention on where to begin.

One of the goals we work towards in First grade is Place Value – I have narrowed down the focus to: Given a 3 digit number, students will be able to tell what “place” each digit represents using words, objects and / or pictures. The prior learning would include: be able to accurately count 10 objects 11, Be able to count by 5s / 10s, be able to represent 10 and more w/ object or pictures and tell why, be able to count and represent #s to 100.

I am planning for the Fall as school has ended. We start the year with multiplication and division in 3rd grade. We start with multiplication and division of 2,3, 4, 5,and 10. I will start with adding and subtracting 2, 3, 4; recognizing and explaining odd/even numbers and skip counting. I will have students practice making equal groups with manipulatives and explain why a group is equal or not. Then we will move into arrays. We will discuss the equal groups and arrays in terms of # of groups and # of items (4 groups of 3, 4 rows of 3) before introducing the equation 4 x 3. We will look at the connections between the 2’s and 4’s as 4 is double the 2’s, 5 is double the 10’s 3 is double the 6’s, etc. We will look for patterns in our skip counting between even and odd’s.

Prior to this video, I have grappled with how much time is required to do justice to full activation of prior knowledge of our learners, when unpacking a concept. What a relief to the realization that we keep coming down the ladder, so to speak, until we can reach our students hand. To an extent, we don’t assume that they knew the concept without checking for prior knowledge.
Wow. This clearly put into perspective some of my students peculiarities.

The first unit I teach is ordering rational and irrational numbers, which involves square roots. Eighth grade is the first time students see square roots, and the unit isn’t taught until much later in the year. The order of the curriculum is nonnegotiable, and I have only 10 days to teach the first unit. Is there a way to make the square root symbol make sense to kids in a short period of time because I was thinking of showing students how to use the calculator to get a decimal equivalent until I can get to the Pythagorean Theorem?
 This reply was modified 3 months, 2 weeks ago by Anthony Waslaske.

I find working with arrays to make rectangles and squares is important… so straight up multiplication and concrete / visuals of the array… then the question flips to what are the possible dimensions of a rectangle wjth area of… 24… 16… 18… 25… 36 etc
Then when we see some squares emerging (as well as other rectangles), we can start to “name” it…
“Find me the dimensions of a square with area of 25” is what you write down first… “sheesh… that’s a lot of writing… do you mind if I use a symbol to represent this???”
Then you’ve arrived!

Learning Goal: Place Value – grade 3
Prior Knowledge Required:
– need to know and understand numbers (to 1000)
– need to know the terms (ones, tens, hundreds, thousands)
– vocab (columns, digits, rows, place, etc.)
I am having trouble figuring out how to implement this in early elementary classrooms without examples at these ages as the examples are targeting older students.

Grade 3 here as well. I think this depends on what you mean by “Place Value” – For example, one of my standards says that students will learn how to round numbers using place value knowledge. So for rounding to the nearest ten, they would have to be able to read and write all the numbers to 100, be able to count by 10s – and be pretty familiar with the concept of which ten is before & which ten is after a number, e.g. 43 comes between 40 and 50. (That’s just to start with — there’s a lot more to unpack there.)
I completely agree with you about the example here — I skipped forward all the way to about 11:30 until it started talking about “know where your students come from,” and while I understand the value of digging deep to show all the parts, I think for me it would be more valuable to see a discussion using more examples at more levels.

If you’re looking for more examples, consider checking out the Fundamentals of Mathematics course in the Academy.

Thank you so much! Depite being at the lowest end of your target audience, I’m getting so much from this training!



Solving equations (6 & 7thgrade levels):
Prior Knowledge: rational number operations & properties; order of operations; equality; variables; inverse operations
I think I might use the Shot Put task after I have done something to remind students of prior learning, but I have a question… How do you make rational number operations engaging? I need some help here – desperately.

@andrea.cadman We’d encourage you to find a textbook question to start and then let’s see how we can apply the curiosity path to lower the floor and raise the ceiling. What can you withhold?

@andrea.cadman as rational numbers can be represented as fractions have a peek at any of the fraction activities in our catalogue. https://makemathmoments.com/tasks



I wrote my information using CCSS from achievethecore.org. This is a great resource that maps the standards.

In this response I will discuss a fourth grade learning standard which is considered a safety net standard because of its importance.
LEARNING GOAL – Compare two fractions with different denominators and different numerators and represent the comparison using the symbols < , = , and > .
The typical method of crossmultiplying is what I was told was the way we were supposed to teach this skill “because it’s so easy that way and we just need to move on.” However, it was not at all comfortable for me to show them this method without any understanding of the WHY IT WORKS. Not a single student could tell me why cross multiplying works. But I’m ashamed to say that I did fall to the pressure to teach it and move on. 🙄 However, I did take the time to use manipulatives and fraction bar representations to help them visualize the relative sizes of the pieces. What I skipped was what is actually happening when you cross multiply. All my students who had at least a basic understanding of multiplication could use the cross multiplication method and get it right, but I know it did nothing to build their understanding of why it works.
PRIOR KNOWLEDGE REQUIRED:
Ability to compare whole numbers using the symbols < , = , > to make sure they know the meanings of each symbol.
Ability to recognize benchmark fractions of half (including 2/4, 3/6, 4/8, etc. which would be learned in a lesson prior to this one.
Knowledge that a larger denominator results in smaller pieces when the whole is the same.
Ability to compare two fractions with either the same numerator or the same denominator.
SPARK CURIOSITY – Nothing is going to spark curiosity about fractions like pizza! So I would find and use pictures of samesized pizzas that are cut into various fractional pieces.

Thanks for being so honest here @terri.bond I think a lot of us here have numerous times been in your situation.
I really like how you’ve worked to learn and show why the cross multiplication “works” with fraction strips.



Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property)
Students need not use formal terms for these properties.
Prior Skills and Knowledge:
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.
Describe a context in which a total number of objects can be expressed as 5 × 7.
Be comfortable with array models, including changing the orientation of an array to show the commutative property.
Decompose numbers – within an array model, decompose the array into smaller rectangles.

Finding out where our students are when we go back in September is going to be crucial. We need to create tasks that will show not only skills they have and can retrieve but where they are in the mathematical thinking continuum.


Very cool! Wondering if you might have a context that would be helpful to emerge integers and this progression?



I am going to be starting the school year with a review of fractions and operations with fractions. We will start with the idea of what a fraction is, what the numerator and denominator stand for. I will then go to recognizing equivalent fractions, adding, multiplying and dividing fractions.

Before I introduce rates, my students need prior knowledge of multiplicative thinking, understand time: seconds, minutes, hours, days and other measurements like gallons, distance… and understand basic ratios (the prior lesson) as a comparison of 2 numbers or things.

New learning goal: 5.NF.1 – +/ fractions with unlike denominators. There is quite a bit of prior knowledge required to be successful. Here is a sample problem that I might use this coming school year: Students were challenged to see how fast they could run one mile in P.E. class yesterday. Jason ran one mile in 7 1/2 minutes. Antoine ran one mile in 7 2/3 minutes. Write a question about Jason and Antoine that would be solved by using fractions.
Prior knowledge: 4.NF.1 (equivalent fractions using visual fraction models to understand the #/size differ yet are equal.) Use # strips, number lines, model representations
Prior knowledge: 4.NF.2 (compare 2 fractions using fraction benchmarks.) This will aid in estimating
Prior knowledge: 4.NF.3 (a/b with a>1 as a sum of fractions 1/b) Students need to know that when you join/separate fractions they are being added/taken away from the same whole, how to decompose a fraction, and how to +/ mixed numbers with like denominators. Students also need to know how to convert an improper number to a mixed number. (5/3 = 3/3 + 2/3 = 1 + 2/3 = 1 2/3) It isn’t just –> divide the top number by the bottom number or some other quick “shortcut” method that students can’t explain. This goes back to the consciously incompetent phase.

One of the struggles in Sixth Grade math is converting between equivalent fractions, decimals, and percentages.
Fraction to decimal: PK understanding a fraction is division; PK know how to divide (single digit AND multidigit); PK understanding a proper fraction and a decimal amount are both parts of a whole amount; PK understanding that adding zeros in the place values to the right of the decimal does not change the value; PK rounding to a nearest specific place value; PK repeating decimals.
I would also like to add a side note; I have noticed a lot of people keep saying opposite operations and I feel that this is a very confusing way of teaching the term inverse operation. The definition of integer is all whole numbers, their opposites, and zero. When speaking of positive and negative, those two words tell us where they are on the number line, like a descriptive word. Positive five and negative five are on opposite sides of the “town” or number line. When we are speaking of operations we don’t want to use opposite, instead we could ask what “undoes” the operation. If you give someone candy how do you undo this? You take away. The operation is our verb or action and we want to continue referring to it as a verb or action rather than a description or adjective.

I am looking at my grade 8 proportions unit, I want students to understand fractions well so that learning, using and applying proportional reasoning becomes a valuable problem solving tool.

So important and it seems that I’m constantly realizing it is even more important than I thought the day / week / months before. Most of our latest problem based units are related to multiplicative thinking including division and ratios/rates. There is so much to explore there.


My learning goals: Create, interpret, and/or use the equation, graph, or table of a linear
function. Translate from one representation of a linear function to another
(i.e., graph, table, and equation). Identify, describe, and/or use constant rates of change.Before I can get to these learning goals for Algebra, I need to back all the way down to identifying arithmetic patterns, identifying missing terms in an arithmetic patter, and constructing a rule for the arithmetic pattern. We also need to determine if the relation is a function. Another important piece before getting to graphing is knowing the coordinate plane and how to plot ordered pairs.

My learning goal is to Use Angle Relationships to Solve for Unknowns.
1. Prior Knowledge 1 Angle Relationships 7th Grade Vocabulary
Angle Pairs, Complementary Angles, Supplementary Angles, Vertical Angles
2. Prior Knowledge 2 8th Grade Vocabulary Angle Relationships 8th Grade Vocabulary
Angles Created by Transversal Interior Angles, Exterior Angles, Congruent Angles, Supplementary Angles
3. Prior Knowledge 3Use Angle Relationships to Create Equivalent Equations and Solving Equations.
I am planning to use the “Sliding Shelves Unit” to Introduce and Teach this unit!

*Initial Note of Sharing:https://www.sejda.com/pdfeditor (Free PDF editor up to 3/day I think)
Filled In: File Attached Here
Comments: A lot of things lead to the new goal that build upon smaller things – reminding people about the learning and a makesense for that previous learning goal would definitely help. Especially if they didn’t see the thinking the first time.
 This reply was modified 2 months, 2 weeks ago by Velia Kearns.

I find myself struggling with the idea of this call to action, not because of the idea of going back to help the students’ access their prior knowledge but finding the spot in the curriculum to make the links matter. I like the idea of using arrays to help further their understanding of multiplication.
In the grade 7 Ontario Curriculum, students are working with decimals. I think by having them use arrays to help understand the visual with multiplication that then using a visual to understand decimals would be a logical progression. They would be able to see the relationship for pieces. I have been toying with the idea of using a 100 gird bubble popper for the students to see the size of pieces in relation to the whole. The students are to be adding and subtracting decimals. I think that using base ten blocks or the 100 girds the students will be able to see the operation movement.
I am not sure if this makes sense to others but I think the progression will be there.

My chosen standard relates to dilation, similarity … for 8th grade math.
Some of the prior knowledge required includes:
7th grade: Scale factor / scale drawings
6th grade: ratios, proportions … unit rates and percentages would also be helpful.
This is the second unit for 8th grade in my curriculum. Some of the things that are review have been historically difficult for my students when I taught those grades. So I’m thinking before this unit some hands on experiences with ratios and percentages would be helpful. Then we can review scale drawings by experimenting with digital images and what is and isn’t a scale copy. Finally, we can talk about how to know exactly how much bigger or smaller one images is than the other.
By doing something they already have experience with such as the digitial images I’m hoping they will come up with on their own some of the needed details such as the center of dilation as well as the scale factor.


As I will be in grade 2 next year, I am finding it very difficult to find a learning goal that needs much prior knowledge other than understanding basic value and order of numbers. On the other hand, this course in general is making very apparent the importance of really working through some concrete and abstract understandings of math concepts. I think back to the previous module in which we spoke of skip counting and that really being grouping. I’d like for my kiddos to see that. I want to really focus my year on using manipulatives and extending understanding so that my students are equipped with the tools to take on the upcoming challenges with a smile on their face.

Whenever we start new units we do a preassessment. This is done using the Freckle math platform, however, having this training shows how you can assess in the moment and do formative, quick assessments as well when starting a new task. I do use coherence maps to help educators select which standards lead into their unit’s focus standards. Like how standards 2.OA.1, 3, and 4 lead into 3rd grade OA. Coming back to this training though I think the low floor entry tasks are an additional and necessary way to kick off units as then teachers can observe student thinking during the task instead of just looking at preassessment data.

Grade 9
New Learning Goal: Solving an inequality and representing it on number line
Prior Knowledge: What integers look like on a numberline
Prior Knowledge: Solving equalities
Prior Knowledge: The concept of infinity
Prior Knowledge: Greater than and less than symbols and meaning
I would have to spend more time finding tasks for each of these ideas, but I think this could be fun and meaningful!



I think I will really be brining back arrays and the base foundations of multiplication and division before teaching multiplication/division with decimals! Lovely reminder to take things at the pace students need

Be sure to check out our operating with fractions problem based units before moving to operating with decimals since that is foundational for that understanding. See them here:
learn.makemathmoments.com/tasks
