Big Idea:

Quantity: Using base ten materials and mass sets, students explore the “howmuchness” of decimal numbers to thousandths.

Operational sense: Students explore a variety of strategies for adding decimal numbers.

Relationships: Representing decimal numbers using base ten materials allows students to understand the base ten relationships in our number system (e.g., 10 thousandths is 1 hundredth).

Representation: Students represent decimal numbers visually by using concrete materials, andsymbolically by using decimal notation.

Expectations:

B2.4 represent and solve problems involving the addition and subtraction of whole numbers and decimal numbers, using estimation and algorithms

teacher background

Provide each group with a collection of base ten blocks (large cubes, flats, rods, small cubes).

Record “1.3, 1.42, 2.09” on the board, and have students read the numbers orally (“one and three tenths”, “one and forty-two hundredths”, “two and nine hundredths”).

Tell students that the flat represents one whole. Ask them to work together as a group to represent the numbers recorded on the board using base ten blocks and their place-value mats. Ask students to demonstrate and explain how they used the materials to represent each number.

Next, explain that the large cube now represents one whole, and again have students represent the numbers recorded on the board. Discuss students’ representations with the base ten blocks, and how the change in the representation of one whole (i.e., the large cube instead of the flat) affected the concrete representation of the decimal numbers.

Remind students that the large cube still represents one whole. Show the class the following collections of base ten blocks:

• 1 large cube, 2 flats, 7 rods

• 2 large cubes, 6 rods

For each collection, have students identify the decimal number orally, and ask them to record the number using decimal notation. For example, 1 large cube, 2 flats, 7 rods represents one and twenty-seven hundredths, which can be recorded as “1.27”.

Next, show a small cube, and ask students to discuss in their groups the value of the small cube if the large cube has a value of one. Discuss the idea that the small cube is one thousandth of the large cube, and that “one thousandth” can be recorded as “1/1000” and “0.001”.

Show the following collections of base ten blocks, and ask students to identify and record each decimal number:

• 3 flats, 1 small cube (“three hundred one thousandths”, 0.301)

• 1 large cube, 9 small cubes (“one and nine thousandths”, 1.009)

• 2 large cubes, 3 rods, 1 small cube (“two and thirty-one thousandths”, 2.031)

ACTIVITY 2: BACKPACK INVESTIGATION

Review the relationship between kilogram and gram. Provide each group of four students with a kilogram mass and a gram mass. Pose the following question: “If you place a kilogram mass in the ones column on the place-value mat, where would you place the gram mass?” Have students, in their groups, discuss their ideas. Then, review the concept that a gram is one thousandth of a kilogram (1/1000 or 0.001 of a kilogram).

Record the following on the board:

• 6 g = _______ kg

• 78 g = ________ kg

• 354 g = ________ kg

Have students work together in their groups to determine the missing decimal numbers in each number sentence. Discuss the solutions as a whole class. Talk to the class about the health concerns related to the mass of students’ backpacks – students who carry heavy backpacks risk back, shoulder, and neck injuries. Explain that studies have found that the maximum mass a backpack should be is 15% of the student’s mass – about 5 kg for most Grade 6 students. Explain that a recent survey found that there was a wide range in the mass of students’ backpacks, but that many students were carrying backpacks that exceeded the recommended maximum mass.

You might ask a few students to show their backpacks to the class, and have the class predict if the backpack has a mass of more than 5 kg. Use a bathroom scale to check the predictions.

Arrange students in pairs. Provide each pair with a copy of AddSub6.BLM1: Backpack Items and a few sheets of blank paper. Provide access to base ten blocks, place-value mats, and mass sets, and encourage students to use the materials, if needed.

Explain that students will work with their partner to find combinations of items that come close to, but do not exceed, the recommended maximum mass of a backpack (5 kg). Encourage students to record combinations of different items and their total mass on the paper provided. Ask students to find more than one combination of items, challenging them to get as close to 5 kg as possible without going over.

78 Number Sense and Numeration, Grades 4 to 6 – Volume 2

As students work, observe their strategies, and ask the following questions:

• “How are you solving the problem?”

• “What strategy are you using to determine the total mass of the items?”

• “Can you calculate the total using another strategy?”

• “How are you using manipulatives/mass sets/drawings/computation to help you determine the total mass?”

• “Which numbers were easy to add? Why?”

• “Which combination of items is closest to 5 kg? How do you know?”

After students have had an opportunity to find different combinations of items, have them select the combination whose mass is closest to 5 kg. Provide each pair of students with markers and a sheet of chart paper or a large sheet of newsprint. Ask them to record the combination of items and the total mass. Have students record their strategies for adding the numbers in such a way that others will understand their thinking.

Have pairs of students present their solutions and strategies to the class. Try to include two pairs who used different strategies (e.g., concrete materials, student-generated strategies, standard algorithms). Make positive comments about students’ work, being careful not to infer that some approaches are better than others. Your goal is to have students determine for themselves which strategies are meaningful and efficient, and which ones they can make sense of and use.

Post students’ work, and ask questions such as:

• “Which strategies are similar? How are they alike?”

• “Which strategy would you use if you solved a problem like this again?”

• “How would you change any of the strategies that were presented? Why?”

• “Which work clearly explains a solution? Why is the work clear and easy to understand?”

Discuss how determining the exact mass of backpack items is not practical in a real-life situation (although it provided a context for a math activity), and that estimating the mass of combinations of items might be a more appropriate approach.

Ask students to estimate the combined mass of two or three items on AddSub6.BLM1: Backpack Items (e.g., gym clothes and shoes; math textbook, binder, and agenda). Discuss students’ estimation strategies. For example, to estimate the mass of a math textbook, a binder, and an agenda, students might recognize that the combined mass of the binder and agenda would be approximately 1 kg, and that the math textbook has a mass of approximately 1.5 kg. The estimated mass of the three items would be approximately 2.5 kg.

Observe students to assess how well they:

• represent decimal numbers using materials (e.g., base ten blocks, place-value mats);

• read and record decimal numbers;

• explain concepts related to place value (e.g., 10 thousandths are 1 hundredth);

• use appropriate strategies to add decimal numbers;

• explain their strategies for adding decimal numbers;

• use appropriate estimation strategies.

SEL Self-Assessments (English) and Teacher Rubric

Pair students who might have difficulty with a partner who can help them understand the problem and different strategies, including the use of concrete materials (e.g., base ten blocks, place-value mats).