Big Idea(s):

Operational sense: Students use a variety of strategies to solve a problem involving multiplicative reasoning and discuss the efficiency of various strategies.

Relationships: Students compare costs expressed as decimal numbers.

Proportional reasoning: The learning activity provides an opportunity for students to apply proportional reasoning to determine the cost of 24 cans of puppy food.

Expectations:

B2.7 represent and solve problems involving the multiplication of three-digit whole numbers by decimal tenths, using algorithms

Explain the following situation to the class:

“My friend called me last evening because he was very excited about getting a new puppy. My friend explained that the puppy needs special food to help it to grow up to be a healthy dog. He told me that there are three stores in his neighbourhood that sell the special puppy food.”

On the board, record the store names and prices for the puppy food:

• Pat’s Pet Emporium: $0.80 per can

• Pet-o-rama: $9.40 for a dozen cans

• Petmania: $2.55 for three cans

Pose the problem: “My friend wants to buy 24 cans of puppy food. How much will he pay for the puppy food at each store? At which store will he get the best price?”

Ensure that students understand the problem. Ask:

• “What do you need to find out?”

• “What information will you need to use to solve the problem?”

Organize students into groups of two or three. Encourage them to work collaboratively to solve the problem. Provide each group with a sheet of paper on which students can record their work. Observe students as they solve the problem. Ask questions that help students think about their problem-solving strategies and solutions:

• “How are you solving the problem?”

• “What part of this problem is easy for you to solve? What is difficult?”

• “How can you determine the cost of 24 cans at Pat’s Pet Emporium? Pet-o-rama? Petmania?”

• “What other strategies can you use to determine the cost of 24 cans at each store?”

• “How can you record your solution so that others will understand how you solved the problem?”

STRATEGIES STUDENTS MIGHT USE

USING DOUBLING:

Many students will double $9.40 (the cost of a dozen cans) to calculate the cost of 24 cans at Pet-o-rama ($18.80).

To determine the cost of 24 cans at Pat’s Pet Emporium and at Petmania, students might use a variety of strategies.

(In the following examples, decimal points have been included in the calculations. It is also acceptable for students to perform the calculations using whole numbers, and then add dollar signs and decimal points to the results to indicate monetary amounts.)

USING REPEATED ADDITION:

Students might repeatedly add the cost of single cans until they determine the cost of 24 cans

(e.g., adding 0.80 twenty-four times to calculate the cost of 24 cans at Pat’s Pet Emporium).

USING DOUBLING:

Students might repeatedly double the number of cans and their costs. For example, for Pat’s

Pet Emporium:

0.80 + 0.80 =1.60 (2 cans)

1.60 +1.60 = 3.20 (4 cans)

3.20 + 3.20 = 6.40 (8 cans)

6.40 + 6.40 =12.80 (16 cans)

$12.80 (the cost of 16 cans) + $6.40 (the cost of 8 cans) = $19.20 (the cost of 24 cans)

USING A RATIO TABLE:

Students might use a ratio table to generate the cost of 24 cans. For example, to determine the cost of 24 cans at Petmania, students might double the number of cans and the costs of the cans until they determine the cost of 24 cans.

USING PARTIAL PRODUCTS:

To calculate the cost of 24 cans at Pat’s Pet Emporium, students might decompose 24 into 10, 10, and 4, then multiply each number by $0.80, and then add the partial products.

Cost of 10 cans ----- 10 × $0.80 = $8.00

Cost of 4 cans ----- 4 × $0.80 = $3.20

Cost of 24 cans ----- $8.00 + $8.00 + $3.20 = $19.20

USING A MULTIPLICATION ALGORITHM:

Students might use an algorithm to calculate the cost.

USING PROPORTIONAL REASONING:

To determine the cost of 24 cans at Petmania, students might recognize that 3 is a factor of 24 (3 × 8 = 24) and use this multiplicative relationship to reason proportionally; that is, multiply $2.55 (the cost of 3 cans) by 8 to calculate the cost of 24 cans.

When students have solved the problem, provide each group with markers and a sheet of chart paper or large sheets of newsprint. Ask students to record their strategies and solutions on the paper, and to clearly demonstrate how they solved the problem. Make a note of the various strategies used by students, and consider which groups might present their strategies during Reflecting and Connecting. Aim to include a variety of strategies that range in their degree of efficiency (e.g., using repeated addition, using doubling, using partial products, using proportional reasoning).

Reconvene the class after the students have solved the problem. Begin a discussion by asking general questions about the problem-solving experience:

• “How did your group decide how to solve this problem?”

• “What was easy about solving this problem?”

• “What was difficult about solving the problem?”

Have a few groups present their strategies for determining the cost of 24 cans at the three pet stores, and for comparing the prices. As students explain their work, ask questions that probe their thinking, and encourage them to explain their strategies:

• “How did you determine the cost of 24 cans at each store?”

• “Why did you use this strategy?”

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

• “Would you use this strategy if you solved another problem like this again? Why or why not?”

• “How would you change your strategy the next time?”

• “How did you record your strategy?”

• “Which store offers the best price? How do you know?”

Following the presentations, encourage students to consider the efficiency of the various strategies that have been presented. Ask:

• “In your opinion, which strategy worked well?”

• “Why is the strategy effective in solving this kind or problem?”

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

Avoid making comments that suggest that some strategies are better than others – students need to determine for themselves which strategies are meaningful and efficient, and which ones they can make sense of and use.

Pose the following problem:

“A flyer for a pet store advertises a special – 4 cans of puppy food for $2.90. What is the cost of 24 cans?”

Have students work independently to solve the problem. Encourage them to think back to the different strategies presented by classmates, and to use an efficient strategy that makes sense to them. Have students show their strategies and solution on a sheet of paper or in their math journals.

Observe students as they solve the problem:

• How efficient are students’ strategies for determining the cost of 24 cans at each pet store?

• How well do students apply proportional reasoning?

• How accurate are students’ calculations?

• Are students able to compare prices?

• How well do students explain their strategies and solutions?

• Are students able to judge the efficiency of various strategies?

Examine students’ solutions for the problem posed at the conclusion of Reflecting and Connecting. Assess how well students selected and applied efficient strategies to solve that problem..”

Some students may benefit from solving a version of the problem that involves simpler numbers (e.g., determining the best buy given $1.00 per can, $9.50 for 10 cans, $2.20 for 2 cans). For students who require a greater challenge, extend the problem by having them determine the amount of money saved if a person buys 72 cans at Pet-o-rama rather than at Petmania.

SEL Self-Assessments (English) and Teacher Rubric