Manipulatives to Build and Model Understanding

Hello, Math Friends ⛄, so glad you are here!

I was looking over the Standards that we have coming up within the next month or so and thought it might be a great idea to include specific ways that you can use manipulatives to help your students BUILD their conceptual understanding and MODEL their understanding of one of the Standards in the Module. Remember, this is just one manipulative paired with one Standard for each of the grade levels -- there are so many ways to make learning concrete by using manipulatives to help your students build and model their understanding -- it is both CRITICAL and endless in its possibilities! (be sure to check out using Cuisenaire Rods in the November edition of MathSnack).

...and, you won't want to miss the Math in Practice suggestions given for each grade level, too!

PRE-K and KINDERGARTEN

Two-Color Counters

(P)K.CC.C.6
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.1

DROP and COMPARE

• Give pairs of students up to 20 counters (the quantity can be adjusted to meet the learning level of the students within each pair)
• Have students drop the counters into a box or other container (to keep them from getting all over the place!)
• Have students match each red with a yellow counter until they run out of one color
• Ask students to say a comparative statement that matches their counters:
• "There are more red than yellow"
• "There are fewer yellow than red"
• "I have the same number of red as yellow"
• "There are two more red than yellow"
• "There is 1 less red than yellow"

GRADE 1

Two-Color Counters

1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

DROP EQUATIONS with two-colored counters

• Give pairs of students a small bowl/cup with up to 20 counters (adjust to fit students' learning levels)
• Students take a handful of the 20 counters (they do not have to use all 20 of the counters)
• Students drop the counters in a box or other other container (less of a mess)
• Students sort the counters into red and yellow groups
• Students count the counters in each group
• Students write an equation (see attached worksheet or simply use dry erase boards) to match the counters ----  the image above could by 4 + 8 = 12
• Students pick up the counters and returns them to the cup/bowl
• Student takes a new handful and continues dropping, sorting, counting, and creating equations. 
• At the end of the activity, bring students together to share one of their equations as you chart the various equations that students created 
• For an added mathematical element have students model the shared equation using their counters -- this reverses the thinking process from the original activity -- students start with an equation to build the model rather than writing an equation from a model

🌟Get a copy of the Drop Equation workspace shown above by clicking HERE :) 

GRADE 2

Base Ten Blocks

2.NBT.A.3
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

BASE TEN MODELS
Use base ten blocks to create models of numbers

• Give pairs or small groups of students base ten blocks (lots of them!)
• Ask students to create a model to show 15 using the base ten blocks
• note that some students may count out 15 single unit cubes -- this IS a correct way to model 15 -- use questioning to help the student find a more mathematically efficient way to build the model
• Next, ask students to model the value of 42 using the base ten blocks
• Ask, "If we count each small square on all of the rods and then also count the two unit cubes, how many cubes would we have altogether?" (42)
• Ask, "How many "ones" do we have?" (2)
• Ask, "How many "ten rods" do we have?" (4) "What quantity do 4 ten rods represent?" (40)
• Write the number 42 on the board. Help students to understand that the 2 in 42 represents the 2 unit cubes and the 4 in 42 represent the 4 ten rods which is equal to 40 unit cubes if we break the rods into unit cubes. 
• Give more practice with double-digit numbers if needed. 
• Now ask students to model 245 using the base ten blocks. 
• Ask, "If we count each small square on the flats, and each small square on the rods, and then count the five unit cubes, how many cubes will we have in all?" (245)
• Follow-up this guided practice with a partner game: 
• Ask students to create a model of a 2- or 3-digit number using the base ten blocks.
• Tell students to write the value of the model on a dry erase board without letting their partner (or small group) see the number. 
• Have the partner(s) calculate the value of the model. 
• See if the values match. Discuss as needed.
• Students change roles.  

GRADE 3

Square Tiles

3.MD.D.8

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

UNDERSTANDING AREA & PERIMETER USING SQUARE TILES (Modules 5 & 8)

Building Understanding

• Give each student/pair/team12 square tiles (this may depend on the number of tiles you have - TIP: Cheez-It crackers make great tiles, too!)
• Ask students to create a rectangle using all 12 of the tiles
• Ask, "Was there only one way to use all 12 tiles to make a rectangle? Look around to see if you can find a team that created the rectangle differently than your team?"
• Ask, "How many tiles did you use to make your rectangle?" (12)
• Say, "Because we all used 12 tiles, we say that our rectangles have an AREA of 12 square tiles.  It doesn't matter if you made a long, skinny rectangle (2 by 6 or 12 by 1) or if your rectangle was almost a square (3 by 4) - all of our rectangles are 12 square units because we used 12 tiles to make the rectangle."
• Ask, "What IF we used 24 tiles? What would the area of our rectangle be?" (24 square tiles)
• Ask, "What IF we used 100 tiles? What would the area of our rectangle be?" (100 square tiles)
• Say, "So the area of the rectangle is determined by the number of same-sized tiles that we use." 
• Say, "Look at the rectangle you created using the 12 square tiles. We are going to count all of the edges that are not touching another tile." (Note: It may be helpful to use a document camera or to have an image projected on the screen to MODEL HOW to count the edges).
• Ask, "Let's make a prediction: Do you think that every rectangle we made will have the SAME number of edges?"
• Allow time for students to consider your question. 
• Ask students to discuss their ideas with their partner/team. 
• Ask students to count their edges. 
• Ask students to tell you the number of edges they counted. Record all responses.
• 14, 16, 16, 26, 14, 14, 14, 15, 16, etc.... (note: 15 is not a correct response - do not correct it at this time, simply record)
• Show all three of the possible models using 12 tiles 
• Ask students to identify the model they created and to tell how many edges they counted.
• Write the number under the model.
• Ask, "What do you notice?" 
• each model has the SAME area, but DIFFERENT perimeter
• all of the people who said "14 edges" created the 3 by 4 rectangle
• students who did not say "14", "16", or "26" miscounted and may need assistance to better understand how to count the edges

Expanding Understanding 

• Say, "Listen to this story about Claudia who is building a garden. When I am done telling you about Claudia's garden, you will use the square tiles to build what Claudia's garden might look like."
• Say, "Claudia's garden is 24 square feet. It is a rectangle-shaped garden. What could Claudia's garden look like? Build it?" (TIP: Write the mathematical information on the board)
• Give students time to contemplate, discuss, and build (and rebuild as needed)
• Have students discuss the various ways they built Claudia's garden as you model their ideas on the board or by using tiles under the document camera.
• Ask, "What is the AREA of the garden that each of you created for Claudia?" (24 sq. feet). "How do you know?"
• Say, "Claudia wants to put a fence around her garden so the animals do not eat her vegetables. Will all of these gardens need the same amount of fence?"  Give time for students to think about your question. 
• Say, "Calculate how much fence Claudia needs if she builds her garden like the one you built with your tiles."
• Allow plenty of time. Encourage students to compare and discuss with others. Help students to notice that gardens that are built the same will have the same area AND perimeter. Gardens that are built with different dimensions will have different perimeters even though the area is still 24 square feet. 

Modeling Understanding

• Say, "I want you to use the square tiles to build a garden that has an area of 20 square units and a perimeter of 42 units."
• Ask, "How many tiles will you need to use?" (20)
• Ask, "How did you know you would need to use 20 tiles?" (because the area tells you how many tiles to use)
• Say, "Okay, work with your partner/team to build a garden that has an area of 20 and a perimeter of 42." 
• Give time to trial and error. Circulate around the room observing and asking questions that help students build their understanding. 

GRADE 4

Snap Cubes

4.NF.B.3.D

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

MIXED NUMBER ADDITION WITH SNAP CUBES

• Have the following scenario posted and read it aloud to the class: "Makala is making a recipe that calls for 2-4/5 cups of apple juice and 1-2/5 cups of water. How much total liquid is in the recipe?"
• Ask, "What is this story about? Not what math are we supposed to do, just what is the story about?" (Makala is making a recipe that calls for two different amounts of liquid and we are trying to determine the total amount of liquid needed). 
• Say, "Let's use our snap cubes to help us visualize this problem."
• Ask, "What do you notice about the quantities used?" (fractions/mixed numbers)
• Ask, "When we look at the fractions, how many parts are needed to make a whole?" (5 - remind students as needed that the denominator determines how many parts are needed to make a whole).
• Say, "Go ahead and build a snap cube train that represents 1 whole using 5 cubes." 
• Give time and walk around checking and asking guiding questions as needed. 
• Say, "I notice that Makala needs 2 whole cups and a little more of apple juice. How can we represent the 2 whole cups?" (build 2 trains of 5 cubes each)
• Say, "How much more apple juice does she need?" (4/5) "How can you represent that?" (making a train of 4 instead of 5)
• Students should now have a representation of 2-4/5 (2 trains of 5 and 1 train of 4)
• Say, "Remember, in the story, Makala also needed 1-2/5 cups of water. Use more snap cubes and build a representation of the amount of water she needs." (1 train of 5 and 1 train of 2)
• Ask, "How many cubes are needed to represent 1 whole cup of liquid in this scenario?" (5) "How do we know that it is 5?" (the denominator tells how many makes the whole)
• Say, "Look at your snap cube models. How many whole cups of liquid are needed for this recipe?"
• If students say "3", ask them if they have enough cubes to combine the shorter cube trains to make a train that represents one whole. 
• If students say "4", ask them to explain how they arrived at 4 whole cups.
• Ask, "After we have made all of the whole cups that we can, how many cubes are leftover?" (1)
• Ask, "How many do we need to make another whole cup?" (students may say 5 as the total needed for a whole train or they may say 4 as the number of additional cubes needed)
• Ask "So if we have 1 remaining, what fraction does that 1 represent?" (1/5)  Ask students to explain WHY it represents 1/5 (because we need 5 to make a whole in this scenario and we only have 1 of the 5 that are needed).
• Create additional stories to reinforce the concept. There is no need to use large whole numbers which can become cumbersome to model; small whole number values and smaller denominator values will be just as effective in helping students to build their conceptual understanding.

GRADE 5

Two-Color Counters

5.OA.A.1

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

BUILDING BASIC MODELS OF COMPLEX EXPRESSIONS
For this activity, stick with small number values so the intended learning is not overshadowed by the task of counting out large quantities of counters. 

Building Understanding

• Ask students to use the two-colored counters to model 3 + 4 (students should put out 3 red and 4 yellow, or vice versa).
• Ask students to adjust their current model so it shows 1 + 4 (students should simply remove 2 of the 3 same colored counters)

Expanding Understanding

• Post this expression on the board for students to see:  2 x ( 1 + 4) 
• Say, "Do not touch your counters, yet. Think about what this expression might mean."
• Allow think time
• Say, "Think about how you could use the counters to model this new expression." Give some think time. 
• Say, "Let's talk about this new expression. What do you notice?" 
• there are parentheses
• there is a multiplication symbol
• there is an addition symbol
• 2x means "two groups of", so we need "2 groups of 1+4"
• Say, "We already have one group of 1 + 4 modeled from our last activity. How can we change it to be two groups of 1 + 4?" 
• Discuss and then ask students to show you the model for 2 x (1 + 4)  --- students should have 1 red/4 yellow and then another set of 1 red/4 yellow (remember red/yellow can be reversed as long as the 1 is represented by the same color in both sets and the 4 is represented by the same color in both sets)
• Say, "Let's try another one. Clear your counters."
• Write 3 x (2 + 4) on the board. Ask students to model this expression.
• Discuss, question, and guide as needed.

Modeling Understanding

• Show students the following model using counters under the document camera or a prepared slide on the SmartBoard. 

• Ask students to WRITE an expression that matches your image. Below are some of the expressions that students may generate. 
• Give plenty of time for students to think and discuss in small groups. 
• Encourage students to come up with multiple ways the model can be written as a mathematical expression. 
• Ask students to share their ideas as you write them on the board.
• 15 + 6
• (5 + 2) x 3
• 3 x (5 + 2)
• (3 x 5) + (3 x 2)
• 21
• 3 x 7
• ... and probably other ideas that deserve your classes time and effort to explain

The most valuable part of this activity will be the discussion as to WHY the expression represents the model.  If a student offers an incorrect/different expression than expected, do not discourage the solution; instead, ask the student to explain. In most cases, the student will self-correct if needed :)