We believe there are 3 non-negotiable skills for fifth graders moving on to sixth grade:
Accordingly, we spend more time on these 3 units than most schools do, although we work in a handful of additional topics as well. The payoff comes later — middle school (and high school!) teachers don’t have to reteach these topics. A goal in itself! Additionally, students are able to transfer their understanding of these vital topics to word problems, problem-solving, and algebra.
Our success rate (the % of students succeeding) in Algebra I in 8th grade has gone UP since we slowed our curriculum DOWN! Our middle schoolers need the 4 whole number operations, plus fractions, decimals, percents, some geometry and some data theory. It’s not that much! Most of all, they need number sense, conceptual strength, self-knowledge and confidence.
With that in mind, let’s look at our new unit: Multiplication. Non-negotiable goals at the end:
- All students finish memorizing their times tables, or find efficient work-arounds.
- All students can build and draw an area model for 2-digit multiplication.
- Most students can multiply 2 digits mentally or algorithmically, and know the multiplicative property names.
- More advanced students can investigate factors, multiples, exponents and puzzles.
ACTIVITY ONE – Dots and Diamonds
Unfortunately, a fraction (1/5?) of our 5th graders have not yet mastered the first 9×9 multiplication facts. Not because they’re not trying; because memorization is not one of their learning style strengths. To give them some efficient work-arounds, we use the Dot Array approach.
PDF: Dots and Diamonds
Answer Key: DotsDi_Key
Dot Arrays for the 7’s:
In our experience, students usually know their ‘Fives’ multiples. If I know 5 sevens is 35, then 6 sevens is one more set of seven, or 42. The child writes “6×7 = 42” on the blank in the column on the left. They can count if that’s what they need. (Our mantra: we must take students where they are, not where they’re supposed to be!)
Students who already know their tables spend a maximum of 1 minute on the 3 dot array problems. They don’t begrudge this small time investment, and they’re actually visualizing the array relationships (perhaps for the first time).
Students who struggle with memorization are getting a visual aide to help them speed up their most efficient work-arounds (like 6 x7 is seven more than 5×7). We will repeat these arrays (in tiny doses) on a daily basis for a few weeks…. “Once a day”, like a vitamin. That visual repetition is what it might take to master the times tables.
We ask what students notice about the first two diamond examples. Then “turn and talk” to check for engagement. (Are they really looking?)
Usually, everyone notices that the numbers left and right are ADDED at the bottom and MULTIPLIED at the top of the diamond.
After a couple practice diamonds, we go on to more challenging questions. Six and WHAT add to 14?
Challenge: What adds to 15 and multiplies to 50? (5 & 10) What adds to 15 and multiplies to 56? (7 & 8), etc. We find that everyone tries these problems, especially since they have the 6, 7, and 8 tables at the top of the page.
These diamonds give enough challenge to be fun, yet help review basic facts.
Those who finish the diamond problems quickly go on to the challenge puzzles on page 2 — mixed up multiplication tables.
ACTIVITY TWO Intro to 2-digit Multiplication
Word doc: 1×2 dig mult
Answer key: 1x2digKey
As always, this classwork goes Concrete > Pictorial > Abstract (Challenge). Meeting students where they are means offering all 3 levels at all times
The second page is meant to be mental math – they do not have to shade in the rectangle. Can they see 3 x 32 as three arrays of 30, plus the 3 sets of 2 = 96?
The challenge page 3 is from the website:
ACTIVITY THREE A Game: “Close to 300”
This game (called Target 300) is taken from Marilyn Burns’ book Teaching Arithmetic: Lessons for Extending Multiplication, Grades 4–5 (Math Solutions Publications, 2001).
We start with this Power Point: Close to 300
and this Word handout with instructions and an example: CLOSE TO 300
In addition, each child will need one die, or each pair can share a die.
The Power Point asks students to look for patterns in the table. TAKE TIME! Wait while they process the possibilities. Often the first observations are trivial (Each column across increases by 10) Praise these forays into observational math. Only gradually do they see other patterns. Slides 4 – 8 in the Power Pt give a nudge for discovering further patterns. Anything they see is great.
ACTIVITY FOUR Intro to Two-digit Multiplication
This classwork uses concrete challenges (this requires base-10 blocks) to build the visualization of the 2-digit area model for multiplication.
Our students were introduced to area models with blocks in 4th grade. Otherwise, these activities might need more introduction, in the form of building more block models like the first 2 on page 1.
The 5 “What Multiplication Problem Is This” questions definitely require blocks, therefore building the conceptual understanding of multiplication rectangles. Students need to know the rules: The given blocks MUST form a rectangle, and we prefer to standardize our models by putting the 100-flat in the upper left.
Students find that they have to lay out a possible rectangle, and then slide tens rods from the right side down to the bottom side (or vice versa) until the rectangle has the correct number of ones. (see key) Once they realize that each puzzle is possible, they begin sliding tens around until each one works. The most common mistake at the beginning is not building a rectangle. Once that requirement is understood, they are much closer to conceptual understanding.
Soon someone will say “Oh… 7 rods and 12 ones. What adds to 7 and multiplies to 12?” Just a note: this exact same concept shows up in Algebra I when students learn to factor quadratics.
We will continue with the “What Multiplication Problem Is This” puzzles for a couple weeks; it’s that valuable.