Big Idea(s):

Operational sense: Students explore a variety of strategies for adding and subtracting decimal numbers to hundredths (money amounts).

Relationships: Students compare decimal numbers expressed as money amounts. They also consider how number relationships help to determine appropriate and efficient computational strategies (e.g., when a mental strategy is more efficient than using a standard algorithm).

Representation: Students represent decimal numbers 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 100000, and of decimal numbers up to hundredths, using appropriate tools, strategies, and algorithms.

teacher background

Explain the following problem:

“The Pine Hill School community hopes to raise $500 for a charity organization. The parent group held a craft sale and raised $265.50. The students in the primary grades raised $104.25 by selling tickets to a play they performed. Next week, the students in the junior grades will be holding a Skip-a-Thon to raise the rest of the money. How much money do the students in the junior grades need to raise?”

Discuss the problem with students, and record important information about the problem on the board:

• Fundraising goal: $500

• Parent group: $265.50

• Primary classes: $104.25

• Junior classes: ?

Pose the following questions. Have students explain their thinking.

• “About how much money has been raised so far?”

• “Do the students in the junior grades have to raise more money than the parent group did?”

• “Do the students in the junior grades have to raise more money than the students in the primary grades did?”

• “About how much money will the students in the junior grades need to raise to reach the goal?”

Explain that students will work with a partner to solve the problem.

Provide each pair of students with a sheet of paper on which they can record their work. Encourage

students to think about how they might use different strategies, including algorithms (paper- and-pencil calculations) and mental computations, to help them solve the problem. Explain that students must be able to explain their strategies later, when the class discusses different ways to solve the problem.

Note: Performing mental computations does not mean that students may not use paper and pencil. Often strategies involve both mental calculations and recording numbers on page. Students may jot down numbers on paper to help them keep track of figures, but they should not per-form paper-and-pencil calculations that can be done mentally.

Have available concrete materials for representing decimal numbers to hundredths (e.g., money sets, 10 × 10 grids, base ten materials) and explain to students that representing the problem by using the materials can help them solve the problem.

STRATEGIES STUDENTS MIGHT USE

FINDING THE DIFFERENCE BETWEEN $500 AND THE AMOUNT OF MONEY RAISED SO FAR:

Students might add $265.50 and $104.25 to determine the amount of money raised so far ($369.75), and then calculate the difference between $369.75 and $500. Some students may use traditional addition and subtraction algorithms to perform these computations; however, other students may choose to use mental computations. To add $265.50 and $104.25, for example, they might use an adding-on strategy (see p. 21) by breaking $104.25 into parts, and adding on each part:

• adding $100 to $265.50 to get to $365.50;

• then adding $4 to $365.50 to get to $369.50;

• then adding $0.25 to $369.50 get to $369.75.

This strategy can be modelled by using an open number line. (In an open number line, the jumps are not to scale.) To calculate the difference between $369.75 and $500, students might use an adding-on strategy (see p. 21 in the Guide):

• add $0.25 to $369.75 to get to $370;

• then add $30 to $370 to get to $400;

• then add $100 to $400 to get to $500;

• then calculate the difference by adding $0.25 + $30 + $100. The difference is $130.25.

The following open number line illustrates this strategy:

SUBTRACTING THE AMOUNTS OF MONEY RAISED SO FAR FROM $500: Students might subtract $265.50 from $500 first, and then subtract $104.25 to determine the amount that remains to be raised. (Alternatively, they might subtract $104.25 first, and then subtract $265.50.) Students might decide to use a subtraction algorithm to perform these computations.

Note: Students may experience difficulty in using the standard algorithm, and may demonstrate little understanding of the regrouping required to perform the computations. If students are unable to explain the meaning of the algorithm, encourage them to find a method that they can understand and explain to others.

Students might also perform the subtraction computations in other ways. For example, they might combine parts of $265.50 and $104.25 according to place value, and subtract these parts from $500:

• combine the hundreds (i.e., $200 + $100 = $300), and subtract that amount from $500 (i.e., $500 – $300 = $200); then

• combine the tens (i.e., $60 + 0 = $60), and subtract that amount from $200 (i.e., $200 – $60 = $140); then

• combine the ones (i.e., $5 + $4 = $9), and subtract that amount from $140 (i.e., $140 – $9 = $131); then combine the hundredths (i.e., $0.50 + $0.25 = $0.75), and subtract that amount from $131 (i.e., $131 – $0.75 = $ 130.25).

Observe students as they are working. Ask them questions that allow them to explain and reflect on their strategies:

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

• “What did you do first? Why did you do that first? What did you do next?”

• “How are you using paper-and-pencil calculations? Mental computation?”

• “Which calculations did you do mentally? Why did you decide to do these calculations mentally?”

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

• “How could you show your strategy so that others can understand what you are thinking?"

• "How could you use a diagram, such as an open number line?”

It may be necessary to demonstrate how diagrams, such as open number lines, can be used to model strategies. (Examples of open number lines can be found on pages 17–18 in the Guide.) After pairs of students have found a solution, provide them with markers and a half sheet of chart paper or a large sheet of newsprint. Ask students to record their strategies in a way that can be clearly understood by others. Remind students that they need to be prepared to explain their strategies to the class.

Have pairs of students present their solutions to the class. Try to include a variety of computational strategies (e.g., various mental computation strategies, paper-and-pencil algorithms).

Pose questions that encourage the presenters to explain the computational strategies they used to solve the problem:

• “How does your strategy work? What steps did you take to use this strategy? Why did you do that step?”

• “Why did you choose this strategy?”

• “What worked well with this strategy? What did not work well?”

• “Was this strategy easy to use? Why or why not?”

• “How do you know that your solution is correct?”

Assist the class in understanding the strategies that are presented. For example, have students explain a strategy in their own words to a partner. As well, model strategies by drawing diagrams

(e.g., open number lines) on the board. Post students’ work in the classroom following each presentation. After several pairs have explained their strategies, ask questions that help the class to reflect on the efficiency of different strategies:

• “Which strategies worked well in solving the problem? Why?”

• “How would you explain this strategy to someone who has never used it?”

• “When is it appropriate to use this strategy? For example, with what kinds of numbers does this strategy work well?”

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

• “How would you change a strategy? Why would you change it?”

Observe students to assess how well they:

• understand the problem and formulate an approach to solving it;

• select and apply appropriate computational strategies (e.g., mental strategies, algorithms);

• demonstrate flexibility and skill in using various computational strategies;

• represent and explain computational strategies (e.g., by using an open number line);

• judge the efficiency of different computational strategies.

Provide a simpler version of the problem for students who experience difficulties. “The school community hopes to raise $500. If the school has already raised $295.50, how much more money does the school need to raise to reach its goal?” Extend the activity for students requiring a greater challenge by asking them to find different ways to solve the problem. Ask students to judge the efficiency of the different strategies.

You might also challenge students by having them determine the amount of money that needs to be raised if the goal changes (e.g., from $500 to $750).

SEL Self-Assessments (English) and Teacher Rubric