3rd Grade Math Lesson
Learning Module
- Creator(s):
This learning module will help students strengthen their pre-existing skills in multiplication and division while establishing that there is a relationship between these two functions. Then, students will move into using an unknown to represent factors in simple multiplication and division word problems.
I will use the content of this course to create functional updates which will help move the students towards greater clarity in this mathematical concept. I will use real-world examples and attempt to help students create solid mathematical knowledge, rather than emphasizing redundant, procedural learning as is typically utilized in the mathematics classroom in elementary school. Experiential Learning is a very useful and effective way to move math content from a didactic structure to a more understanding-based structure (Meyer, et. al, 1997).
I am familiar with this lesson content, as I taught a math lesson with similar content to third graders in an early-field experience context this past semester. The lesson was not entirely a success, as the curriculum I was instructed to use introduced too much foreign content into a singular lesson. By breaking the content that I covered in one lesson into six updates, and giving students actionable goals at the end of each update, I hope to help create a learning module that takes this foreign and intimidating concept and addresses it in small, easy to manage sections for children.
I have a particular interest in how mathematics is taught in elementary school. My undergraduate degree is a Bachelor of Science in Mathematics, but I am in the process of becoming an elementary school teacher. In my observation, elementary mathematics tends to be taught with a didactic approach with an emphasis on procedural fluency. I am particularly interested in how elementary teachers can work towards emphasizing understanding in math lessons, and ascertain that students will cultivate confidence as they work through math units.
For modeling in the following updates, I will use discrete models instead of continuous models, as teachers tend to use these models more (Kosko, 2019). I will also emphasize discrete models due to the fact that this lesson content will only cover multiplication and division of whole numbers.
It is important to establish meaning in multiplication and division from a young age as "Understanding of multiplication and division is needed in order to develop sound concepts of fraction, ratio and proportion" (Vale & Davies, 2007). Utilizing open-ended modeling tasks is important in order to establish meaning in multiplication and division in more than the context of solving explicit problems (Vale & Davies, 2007).
This lesson module will also utilize videos and picture books to introduce mathematical concepts, because, at the elementary level "the integration of literature and mathematics is certainly viewed as an effective instructional approach" (Cooper, et. al, 2020).
In this lesson, you will be discovering new ways to use your multiplication and division strategies!
To review multiplication and division, take a few minutes to watch The Lions Share: A Tale of Halving the Cake and Eating it Too.
(OBrein, 2019).
In order to do these lessons, you'll need a pencil and piece of paper to show your work. You can upload your work to Scholar by taking a picture of it on a cell phone or tablet and uploading it. If you are unable to take a picture of your work, you can type it.
Each update should take about 20-30 minutes in order to read the update, comment, and create your own update.
Throughout this module your classmates will engage with your work and say if they agree or disagree with your answer. If someone disagrees with your work you are expected to refute, or support your answer in a reply, or rework your answer and agree with their answer.
Good luck!
This lesson module is for third-grade students. These students should already have the ability to confidently multiply and divide single and double-digit numbers. Students should also have some knowledge of how to represent multiplication problems in an array and utilize pictorial representations of division by grouping.
The following Illinois Common Core Math standards will be addressed:
(Illinois State Board of Education, 2020)
This learning module will take a basic understanding of multiplication and division and work the students gradually towards the ability to represent word problems with an unknown through both multiplication and division. By gaining this ability, students will have a more generalized and abstract understanding of the functionality of multiplication and division. This will lead students towards an easier understanding of algebraic functions, ratios, and abstraction in mathematics.
Students need a notebook to write their work down on and something to write with. They also need the ability to capture images of their work to upload it in the course.
Each of the six updates should take 20-30 minutes of focused work by the students. As the instructor, you should check the student's progress with their updates before instructing them to move on to future updates.
Here is a list of things to consider:
5 + 3
5 + 3 = 8
2 - 5
5 - 2 = 3
4 x 8
5 x 3 = 15
We will be working in this first update to define the word equation.
What is an equation?
an equation is an expression in mathematics that shows two equal sides
An equation shows equality, which means it must have an equals sign in between two sides of the equation.
So, out of the above, which is an equation?
The three equations above would be: 5 + 3 = 8, 5 - 2 = 3 and 5 x 3 = 15
Here is a fun example of equations called Raindrop Math. Watch this video and work along with the teacher!
Now, similarly to how you balanced the equations in the video above, you are going to use a virtual balance to experiment with equations.
In the above image, the balance is illustrating that 10 = 1 + 9
http://www.didax.com/apps/math-balance/ - Click this link and experiment with the math balance. Spend a few minutes dragging the weights between different points in the scale to try to find balance.
Comment: What are three strategies you used to find balance using the Didax application?
Update: We can use a scale to illustrate equations for addition, but we can use areas to illustrate equations for multiplication. Draw two different ways to create rectangles with an area of 20. How would you write these in a multiplication equation?
Comment on Three Classmates Updates: Comment and Prove one classmates work to be true by explaining their answer. If you disagree with their answer, state why.
The first update in this lesson module is a review of prior knowledge, in order to activate this knowledge so students are able to engage with it at a higher level.
Using the scale is a great way for students to build an understanding of equality.
It is important that students build an understanding of an equals sign representing the notion of equality rather than it being an indicator of a function, so using this as a "warm-up" activity for a learning module is a good idea in many aspects!
Use the comments to assess if the students are understanding this concept, as it is integral to success in this learning module. If they need more review, provide them with class materials for further review before they move on to the following updates.
Multiplication and Division are very similar! Watch the following video to explore these similarities further:
(Math & Learning Videos 4 Kids, 2016).
Let's explore this idea further!
Here is a division problem:
24 ÷ 3 = ?
Let's write a story for this problem. There are 24 pizzas delivered to a school for their third grade classes. The school has three third grade classes. How many pizzas does each class get?
Represent this story by drawing a picture of 24 pizzas and divide the pizzas among the three classes.
You might end up with a picture of an array, like this:
So, each class gets eight pizzas.
Now, what if the question was the following:
Three third grade classes are ordering eight pizzas each, how many pizzas will be delivered for the students?
Start by representing this as an equation: 3 x 8 = ?
Now, try to create an array for this problem.
Did you notice something interesting? This array is the same dimensions as our earlier array!
So, the equations 3 x 8 = 24, 8 x 3 = 24, 24 ÷ 3 = 8 and 24 ÷ 8 = 3 all represent the same amount of pizza.
Consider the following problem: 5 people each have 2 shoes on when they arrive at the party. Each person takes off their shoes when they get to the door. How many shoes are left at the door?
Draw a picture of the shoes in an array to solve the problem.
You should solve that 5 x 2 = 10.
Now, consider that there are 10 shoes at the welcome mat. You know that each person wears 2 shoes. How many people are present at the party?
10 ÷ 2 = 5.
Comment: How are multiplication and division related?
Update: Consider the numbers 6, 4, and 24. Write two word problems, draw an array and represent these numbers with multiplication and division.
Comment on 3 Updates: Comment on 3 of your classmate's updates and check their work. Do you agree or disagree with their work? Wh
This update is about understanding the relationship between multiplication and division before moving the students into working with unknowns.
Engage with the student's updates and provide them with more scaffolding until you are certain that the students really understand this concept.
Also, when students engage with each other's work and say why they agree or disagree with each other, be sure to check those comments for accuracy. If students start to move in the wrong direction in their understanding, jump in on the comment thread to get them back on track. However, this is not necessary to do too soon, as the students may get back on the right track on their own.
Let's go back to a problem we did in the last update.
5 people each have 2 shoes on when they arrive at the party. Each person takes off their shoes when they get to the door. How many shoes are left at the door?
Whenever we have a math problem, we are trying to solve for something that we do not know. This thing that we do not know is called the Unknown because it is currently "unknown" to the mathematician trying to solve the problem! So, in the context of this problem: we do not know how many shoes are left at the door. Therefore, our unknown is total shoes.
For now, we are going to represent the unknown with a Question Mark.
Our multiplication and division problems can be broken into three parts: number of groups, group size and total. We will represent these three parts in a table:
Total, Number of Groups and Group Size are related through multiplication and division.
In this problem we have five pairs of shoes, so the group size is 2, and the number of groups is 5. The total number of shoes is the unknown, so you can represent that with a question mark.
All the following hold true and will help us solve this problem in different ways:
Your unknown does not need to be the total! In fact, the unknown can be in any position. Try this example:
There are 35 dancers at a dance studio. Each class has 7 dancers, how many dancers are in each class?
In this example we know the total number of dancers and the group size. Can you write the equations for this problem?
Comment: Write a word problem with the unknown in the "Group Size" spot. Write all 4 equations for the word problem.
Update: Use an object you can find around your house to create groups. Write 3 word problems with these groups of objects, each with the unknown in a different blank of the problem. Also, explain how total, group size and number of groups are related through multiplication and division (you can use an array diagram as in Update 2 to illustrate this).
Comment on 3 Updates: Work through the problems your classmates created. Do you agree or disagree with your classmate's work? Why?
This update is to introduce students lightly to the concept of an unknown. In the previous update we were touching on this concept, but now we are putting a word to it and filling the place of an unknown with a question mark. This idea of a "place-holder" will be expanded upon in the coming updates, so before moving students on be sure that they really grasp this.
Again, check students' engagement with their classmates to be certain they are on the way to correct understanding.
If students are struggling with this update, work through similar problems with them in the comment threads on the updates. Once it seems like students are really grasping the concepts presented in this update, they are ready to move on to the next update.
In this Update, we will be working through how to solve for an unknown. There are several strategies that you can use to do this, find the ones that make the most sense to you -- that is the most important part of learning something new in math class!
In Update 3, you set up equations with a question mark to represent word problems with an unknown. In this update, we will take this one step further and solve for the unknown in order to find the solution to our word problem!
This may sound tricky, but this is something you have done dozens of times before, just with a few new strategies to add to your tool belt.
Here is our problem:
Suppose you have 5 cakes from your favorite baker. You need 50 slices of cake. How many slices do you need to cut from each cake?
To start, let's fill out our table:
| Total (Slices of Cake) | Groups (Cakes) | # per Group (Slices per Cake) |
| 50 | 5 | ? |
Now, we will build out our four equations, and discuss some strategies for how to solve for the "?" in each equation.
Comment: Which strategy do you think is the best for solving for an unknown in this problem? Why?
Update: Create a related multiplication and division word problem with an unknown. Represent the word problem in our three-column table and draw a picture to illustrate the word problem. Then, choose 2 different ways to solve for the unknown.
Comment on Three Updates: Work through 3 of your classmate's word problems. Do you agree with their solutions and work? Why or why not
The most important part of this update is to keep in mind that it is okay and natural for students to utilize different tools in their problem solving abilities.
Encourage the students to compare their skills and tell their "solution story" to each other in order to engage each other in critical thinking conversations.
Be adamant and encouraging that students really write out their full thought process in the update and instruct students to engage with each others updates. The more ways they see these strategies written out and the more ways they engage with these strategies the better.
Again, moderate the comment sections to be sure the students are being respectful and are on the right track towards developing their understanding.
Now, we will be doing nearly the same thing that we did in the last lesson! Except, we will be using a letter in the space of where we did put the question mark.
We can use any letter in this space, but we typically choose something that could be tied to what we are trying to figure out. Let's work through an example together to make this more clear.
Take the following problem:
Each student gets 2 pieces of pizza at lunch. There are 50 pieces of pizza, how many students could be fed?
Let's fill out the table as we did in the last update:
| Total (pieces of pizza) | Groups (number of students) | # per group (pieces per student) |
| 50 | ? | 2 |
Now, a question mark does not represent something specific, just an unknown. But let's say the lunch ladies were trying to solve for the number of students but had a different problem where they were trying to solve for the number of teachers. We might need to use a letter to represent our unknown, so we can represent several different unknowns in the same project.
So, let's call our number of students "S". We are going to write "S" into our equations, but this just means that it's an unknown number we are trying to solve for!
Now to write out our expressions:
50 ÷ S = 2
50 ÷ 2 = S
2 x S = 50
S x 2 = 50
It may look kind of funny to have letters and numbers in an equation, but this will be a very helpful skill to have mastered as you move forward into your future as a strong mathematician!
Comment: Solve the above equations for S, or find out which number S is standing in place of.
Update: Create your own word problem with an unknown. Fill out the table with a question mark. Decide which letter you are using in place of the question mark. Write out the four expressions and solve for the letter.
Comment on Three Updates: Work through your classmate's problems in their update and solve for the unknown. Do you agree or disagree with your classmate's solution? Why?
This update should be pretty simple for the kids, as they aren't really learning anything new. However, the idea of putting letters into math can be quite tricky for kids. Be sure to monitor their updates and comment sections for accuracy and try to steer them back onto the right track.
This update is the last bit of new material before moving on to the project where students will create work. So, it is important to be sure that students are fluent in these concepts and are able to work through these problems with minimal hesitation.
The multimodal work for this class will be presented through FlipGrid. To start, watch a quick tutorial for FlipGrid below:
(Thomas, 2017)
You will start by thinking of a multiplication and division problem that you can create using things around your home. This could be things like cans of food stacked in even piles, shelves with equal amounts of books, arrangements of blocks, etc. Create an Update outlining your idea for a problem to get approval before starting.
Then, you will use FlipGrid to create a video to state your problem. Then, respond to your video and work through how to solve the problem using each step in our Updates. Be creative while solving it! Use pictures, your concrete representation, or other ways to explain why your problem makes sense.
Engage with 3 other classmate's sense-making projects.
Update: Create a multiplication and division word problem using things around the house.
Comment: What questions do you have about this project?
| Project Rubric | 3 | 2 | 1-0 |
| Mathematical Understanding | The student shows clear mathematical understanding. The student's skills are accurate and the student has achieved fluency in the subject area. | Students have a groundwork of understanding. Students may have made a few mistakes in their solution. | Student had little to no understanding of the concepts in the updates. |
| Sensemaking Strategies | The student had clear and proficient sensemaking strategies. The student was able to explain why multiplication and division make sense in their lives. | Students had a good grasp on sensemaking strategies, but need a little more work understanding concrete explanations of the concepts. | Student could use more work with their sensemaking strategies. |
| Clarity | Student's work and processes were clear throughout the videos. | Student's work was unclear a few times | Student's work was mostly unclear. |
In this part of the learning module the students are ready to create their own work out of things in their household. By using FlipGrid, students will be able to present their work in ways that are clear, concrete, and applicable to their lives!
By showing that math is applicable, we will keep students engaged!
Be sure to check students updates to make sure that they are creating clear and solvable problems before confirming them to start their project, so they do not do a lot of unnecessary work.
Grade students according to the rubric found in the student section: the focus is on sensemaking and mathematical understanding over procedural fluency.
Cooper, S., Rogers, R. M., Purdum-Cassidy, B., & Nesmith, S. M. (2020). Selecting Quality Picture Books for Mathematics Instruction: What Do Preservice Teachers Look for? Children’s Literature in Education, 51(1), 110–124.
Illinois State Board of Education. (2010). Common Core State Standards for Mathematics. https://www.isbe.net/Documents/math-standards.pdf.
Kids Academy. (2017, October 10). Balancing Equations | Addition | Math for 2nd Grade | Kids Academy. [Video]. YouTube. https://www.youtube.com/watch?v=KtPhbdWtw7M
Kosko, K. W. (2019). Third-Grade Teachers’ Self-Reported Use of Multiplication and Division Models. School Science and Mathematics, 119(5), 262–274.
Math & Learning Videos 4 Kids. (2016, August 4). Multiplication and Division Relationships - Fun Math Videos for Kids 3rd Grade [Video]. YouTube.https://www.youtube.com/watch?v=i31rRt5m1-4
Meyer, D.K., Turner, J.C., & Spender, C.A. (1997). Challenge in a Mathematics Classroom: Students’ Motivation and Strategies in Project-Based Learning. The Elementary School Journal, 97(5), 501.
OBrein, Denise. (2019, March 8). Lions Share [Video]. YouTube. https://www.youtube.com/watch?v=kI9kjPvKC0E
Thomas, Abbey. (2018, March 7). FlipGrid Instructions For Students. [Video]. YouTube. https://www.youtube.com/watch?v=6tfoCgQuqcw
Vale, C., & Davies, A. (2007). Dean’s Great Discovery: Multiplication, Division and Fractions. Australian Primary Mathematics Classroom, 12(3), 18–22.
2 Minute Teacher Tech. (2017, August 24). FlipGrid Tutorial. [Video]. YouTube. https://www.youtube.com/watch?v=-aZ523-HHBg