Big Idea(s):

Quantity: Students explore the “howmuchness” of decimal numbers to tenths by comparing and ordering the capacity of different containers.

Operational sense: Students apply their understanding of addition and subtraction to per-form calculations with decimal numbers.

Relationships: This learning activity allows students to see the relationship between tenths and wholes (e.g., that 10 tenths makes 1 whole).

Representation: Students represent decimal amounts visually by using concrete materials, and symbolically by using decimal notation.

Expectations:

B2.4 represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 10000 and of decimal tenths, using appropriate tools and strategies, including algorithms

teacher background

Key concepts

• Situations involving addition and subtraction may involve:

  • adding a quantity onto an existing amount or removing a quantity from an existing amount;

  • combining two or more quantities;

  • comparing quantities.

• There are a variety of tools and strategies that can be used to add and subtract numbers, including decimal tenths:

  • Acting out a situation, by representing it with objects, a drawing, or a diagram, can help support students in identifying the given quantities in a problem and the unknown quantity.

  • Set models can be used to add a quantity to an existing amount or removing a quantity from an existing amount.

  • Linear models can be used to determine the difference between two quantities by comparing them visually.

  • Part-whole models can be used to show the relationship between what is known and what is unknown and how addition and subtraction relate to the situation.

Note

• An important part of problem solving is the ability to choose the operation that matches the action in a situation. For additive situations – situations that involve addition or subtraction – there are three “problem structures”:

  • Change situations, where one quantity is changed, by having an amount either joined to it or separated from it. Sometimes the result is unknown; sometimes the starting point is unknown; sometimes the change is unknown.

  • Combine situations, where two quantities are combined. Sometimes one part is unknown; sometimes the other part is unknown; sometimes the result is unknown.

  • Compare situations, where two quantities are being compared. Sometimes the larger amount is unknown; sometimes the smaller amount is unknown; sometimes the difference between the two amounts is unknown.

• The use of drawings and models, including part-whole models, helps with recognizing the actions and quantities involved in a situation. This provides insight into which operation to use and helps in choosing the appropriate equation to represent the situation.

• A variety of strategies may be used to add or subtract, including algorithms.

• An algorithm describes a process or set of steps to carry out a procedure. A standard algorithm is one that is known and used by a community. Different cultures have different standard algorithms that they use to perform calculations.

• The most common (standard) algorithms for addition and subtraction in North America use a compact organizer to decompose and recompose numbers based on place value. They begin with the smallest unit – whether it be the ones column or decimal tenths – and use regrouping or trading strategies to carry out the computation. (See Grade 3, B2.4.)

• When carrying out an addition or subtraction algorithm, only common units can be combined or separated. This is particularly noteworthy when using the North American standard algorithms with decimals numbers because unlike with whole numbers, the smallest unit in a number is not always common (e.g., 90 − 24.7). In this case, the number 90 can be changed to 90.0 so that the units can more easily be aligned; that is, 0 is used as a placeholder.

• Making explicit the compactness and efficiency of the standard algorithm strengthens understanding of place value and the properties of addition and subtraction.

Using, BLM 1- What’s the Capacity? Draw students’ attention to the numbers on the bottom of the page and explain that these numbers represent the capacities of the different containers. Review the meaning of “capacity”(the greatest amount or volume that a container can hold).

Ask:

• “Which container do you think has the greatest capacity?”

• “Which capacity listed at the bottom of the transparency corresponds to this container?”

• “How do you know that 3.6 L is the greatest capacity?”

Have students work with a partner to match the other four capacities with containers. Have students explain their rationales for matching each capacity with a container. Record the capacity below each container (dish soap, 1.1 L; laundry soap, 2.9 L; water,0.5 L; juice, 1.4 L).

Explain that containers are sometimes reused to carry water to pour into other containers.

Ask: “How much water would fill both the dish soap container and the juice container?”

Provide access to manipulatives such as BLM 2- Paper Strips cut into tenths and invite students to use the manipulatives to help them determine a solution. Allow approximately one minute for students to work individually,and then have them explain their strategies to a partner.

Invite a few students to share their strategies with the class. Encourage them to demonstrate their thinking using manipulatives or diagrams. For example, students might use base ten blocks or fraction strips to represent the capacity of each container (1.1 L and 1.4 L), and then combine the materials to determine the total capacity. Other students might add the two capacities mentally. A possible mental computation might involve adding 1.4 L and 1 L to get 2.4 L, and then adding 0.1 to get 2.5 L. Model addition strategies by drawing a number line on the board to help students visualize addition processes.

Explain the problem situation:

“The Kindergarten teacher often needs to fill the water table basin with water. The basin is too heavy to carry when it is filled with water, so the teacher wants to use empty plastic containers to fill the basin. He doesn’t want to fill the basin to the top, because the water will spill over the sides when the children are playing in it. He has discovered that 25 L of water is the ideal capacity.”

Refer to BLM 1 -What’s the Capacity? and explain the problem: “The Kindergarten teacher found 5 empty containers that he can use to fill the water basin.The containers have capacities of 0.5 L, 1.1 L, 1.4 L, 2.9 L, and 3.6 L. He can use 1, 2, 3, 4,or all 5 containers. How can he use these containers to fill the water table basin with 25 L of water? Is there more than one way to fill the water basin?”

Organize students into pairs. Explain that students will work with their partner to solve the problem. Encourage them to use manipulatives (e.g., fraction circles, fraction strips, base ten blocks)or diagrams (e.g., open number lines) to help them determine a solution. Provide each pair of students with a sheet of paper on which they can record their work.

STRATEGIES STUDENTS MIGHT USE

• USING TRIAL AND ERROR Students might try different combinations of containers to reach a total of 25 L.

• USING FRIENDLY NUMBERS Students might find that certain combinations of containers provide whole-number capacities that are easy to work with. For example, the 3.6 L and 1.4 L containers can be added to get 5 L. Students would reason that they would need to combine five 3.6 L and five 1.4 L containers to get 25 L.

• USING REPEATED ADDITION Students might try to maximize the use of the largest container and repeatedly add 3.6 L six times until they get to 21.6 L. (Adding 3.6 seven times results in 25.2, a value that is greater than 25.) To get from 21.6 L to 25 L, students would add on containers that provide a capacity of 3.4 L (e.g., 2.9 L + 0.5 L).

Observe students as they work. Ask them questions about their strategies and solutions

:• “What strategy are you using to solve the problem?”

• “What is working well with your strategy? What is not working well?”

• “How can you prove that your solution is correct?

• “Can you find another solution?”

After pairs of students have found one or more solutions, provide them with markers and a half sheet of chart paper or a large sheet of newsprint. Ask students to show how they solved the problem in a way that can be clearly understood by the Kindergarten teacher. Encourage them to use diagrams (e.g., open number lines) to show their thinking.

Have pairs of students present their solutions to the class. Try to include a variety of strategies,solutions, and formats. Have each pair justify their solution(s) by asking them to prove that each combination of containers would provide 25 L of water. Encourage students to clarify their understanding of presented solutions and strategies by asking questions such as:

• “What strategy did the presenters use to determine the total number of litres?"

• “Why do you think the presenters used that strategy?"

• “What questions about the strategy or solution do you have for the presenters?”

Post students’ work in the classroom following each presentation. After several pairs have explained their strategies, ask:

• “Which solutions would you recommend to the Kindergarten teacher? Why?”

• “Which solutions would you not recommend? Why?”

(Students might assess the suitability of different solutions by considering the number of times containers need to be filled.)Draw students’ attention to the different formats used to record solutions. Ask questions such as:

• “In what different ways did pairs record their strategies and solutions?”

• “Which forms are easy to understand? Why is the work clear and easy to understand?”

Have students, individually, solve this problem. Ask students to record their solutions,reminding them to show their ideas in a way that can be clearly understood by others.Encourage students to use manipulatives (e.g., fraction circles, fraction strips, base ten blocks) or diagrams (e.g., open number lines) to help them determine a solution.

Observe students work to assess how well they:

• add the capacities of containers using manipulatives and/or diagrams;

• solve the problem by finding a combination of containers with a capacity of 20 L;

• demonstrate their thinking using materials, drawings, and/or written explanations;

• explain their strategy and solution.

SEL Self-Assessments (English) and Teacher Rubric

Simplify the problem for students who experience difficulties by reducing the capacity of the water table basin to 10 L and/or reducing the number of containers.

Prompt students to search for friendly numbers by asking:

• “What combination of containers would create a friendly number?”

• “How can you use this friendly number to get close to the capacity of the basin?”

Extend the activity for students requiring a greater challenge by posing the following problems:

• “What is the fewest number of containers that you could use to fill the basin?"

• "How many times would you need to fill each container?”

• “What is the greatest number of containers that you could use?"

• "How many times would you need to fill each container?”