Lesson 7

The purpose of this warm-up is to elicit the idea that there are different ways to calculate a product, using the standard algorithm, which will be useful when students find products of a 3-digit number and a 2-digit number in a way that makes sense to them. 

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Why do you think the results of the two calculations are the same?” (The factors are the same, just in a different order. The order of the factors does not change the result of multiplication.)
• "Which way would you prefer to find the value of \(417 \times 28\)?" (I like the one with two partial products as there is less adding up to do.)
• "Today you will get to choose how to find products of a 3-digit number and a 2-digit number.

Students use their understanding of place value to generate expressions that have the greatest product. They take turns selecting number cards 0 through 9 and place them strategically to make the largest  product of a 2-digit number and a 3-digit number. Students will need to think about how the different place values influence the value of the product and choose where to put their digits accordingly. There is also a large element of chance since they do not know in advance which numbers they will select.

• Groups of 2
• Give each group a set of number cards and 2 copies of the blackline master.
• “Remove the cards that show 10 and set them aside.”
• “We’re going to play a game called Greatest Product. Let’s read through the directions and play one round together.”
• Read through the directions with the class and play a round against the class using the diagram in the student workbook:
• Display each number card.
  • Think through your choices aloud.
  • Record your move and score for all to see.

• “Now, play the game with your partner.”
• 8–10 minutes: partner work time

• Display a blank image from game board.
• “If this is your game board and you pick an 8, where would you write the 8? Why?” (I would either put it in the hundreds place of the top number or the tens place of the bottom number because it’s a big number and those are the biggest place values.)
• Write the 8 in the hundreds place of the top number.
• “What if you next select a 1? Where would you write the 1? Why?” (I would put it in the ones place of one of the numbers because I want to put a bigger number in the tens place.)
• “What did you find challenging about the game?” (Since I didn’t know what numbers I was going to get on later picks, I sometimes wasn’t sure where to put a number because I didn’t know if I would get a bigger number on a later pick.)

The purpose of this activity is for students to  explore the number of new units that can be composed when using the standard algorithm. As students experiment with the given numbers, they will find examples of 1, 2, 3, 4, 5, 6, 7, or 8 new units composed. Here they address the question of whether or not it is possible to compose 9 or more new units when using the standard algorithm. In order to make sense of and persevere in solving this problem (MP1), there are several types of arguments students may make, all of which highlight how the standard algorithm works:

• they may calculate \(999 \times 9\) and see that while they get very close to composing 9 new units, they fall 1 short
• the biggest product that can be made from multiplying a pair of 1-digit numbers is 81, which would mean that 8 units are composed
• this 81 has to be combined with whatever new units were composed before, but since that’s 8, that means the largest number you can form at each step in the standard algorithm is 89, which is 1 short of the 90 you would need to compose 9 new units

• Groups of 2

• 1–2 minutes: quiet think time
• 3–5 minutes: partner work time

If students are not fluent with their multiplication facts, offer them a multiplication table. As they use the standard algorithm to find the products in problem 1, ask them to circle each new unit. If they are not sure if it is possible to compose 9 new units, ask them to use the algorithm to find the product of \(99 \times 9\) and  \(999 \times 99\).

• Invite students to share the number of units they composed in the calculations of problem 1.
• “Was anything missing from 1 to 8?” (No, we composed everything from 1 to 8 new units.)
• “Was anyone able to compose 9 new tens in a product? Why?” (No. The new tens come when I take the product of the ones. The biggest that can be is \(9 \times 9\) which gives me 8 tens.)
• “Was anyone able to compose 9 new hundreds? Why?” (No. I could get 8 new hundreds if I have \(9 \times 90\), giving me 810. I might also have to combine that with the new tens from the product of the ones, but that would give me at most 890, not 900.)